HOWARD UNIVERSITY

Blind Image Deconvolution Via Penalized Maximum Likelihood Estimation

A ThesisSubmitted to the Faculty of the

Graduate School

of

HOWARD UNIVERSITY

in partial fulfillment ofthe requirements for the

degree of

Master of Engineering

Department of Electrical and Computer Engineering

by

Khalid A. Abubaker

Washington, D.C.May 2007

© 2007 Khalid A. Abubaker

All Rights Reserved

HOWARD UNIVERSITYGRADUATE SCHOOL

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

THESIS COMMITTEE

Mohamed Chouikha, Ph.D.Chairman

John M.M. Anderson, Ph.D.

Charles J. Kim, Ph.D.

John M.M. Anderson, Ph.D.

Thesis Advisor

Candidate: Khalid A. Abubaker

Date of Defense: April 20, 2007

ii

Dedication

I dedicate this work to my late father, Abdurahman Gidaya who passed away very

early in his life without enjoying the fruit of his work and to my late grand mother, Fatuma

Yonis whose wisdom has contributed so much to the success of my endeavors.

iii

Acknowledgments

First of all, I would like to thank my academic advisor and mentor, Dr. John M.

M. Anderson, for his excellent guidance and support of my research work. Without Dr.

Anderson’s help, this research work could not have come to its final stage. I would also like

to thank my thesis defense committee members Dr. Mohamed Chouikha and Dr. Charles

J. Kim for their comments and corrections. I thank all the professors in the Department of

Electrical and Computer Engineering and the Department of Systems and Computer Science

for sharing their priceless knowledge with me.

Furthermore, I thank all my colleagues in the Electrical Engineering Lab for their

help and support. I would specially like to thank Aaron Jackson for providing me with his

Howard University Thesis and Dissertation writing Latex template and for all the technical

help he has given me .

I also would like to acknowledge the support provided by the U.S. Army High Perfor-

mance Computing Research Center located at the University of Minnesota.

Finally, my special thanks goes to Muna Mahdi and the rest of my family for their

continued love and support throughout all the good and bad times.

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Abstract

In this thesis, we present image restoration algorithms that were motivated by the

problem of reducing blur in hyperspectral images. We address the problem of interest,

known as the blind image deconvolution problem, by applying a penalized maximum

likelihood (PML) algorithm to each spectral plane of hyperspectral images. The quantities

to be estimated are the mean value of the true spectral plane, referred to as the reflectance

parameters, and hyperspectral sensor’s point spread function (PSF). In the PML method,

an objective function equal to the negative log likelihood function plus a penalty function

is minimized with respect to the unknown parameters. To minimize the PML objective

function, we use an iterative technique known as the iterative majorization algorithm. Given

initial estimates of the parameters, the PML objective function is decreased with respect to

the reflectance parameters while setting the PSF parameters to their current estimate. Then,

the PML objective function is decreased with respect to the PSF parameters while setting

the reflectance parameters to their current estimate. The algorithm alternates between the

two steps until some chosen stopping criterion is met. In experimental studies, the PML

algorithm was stable and produced promising results.

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Table of Contents

Committee Approval Form . . . . . . . . . . . . . . . . . . . . . . . . . ii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Chapters

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. History of Hyperspectral Imagery . . . . . . . . . . . . . . . . . . . . . 1

1.2. Hyperspectral Image Data Collection . . . . . . . . . . . . . . . . . . . 2

1.3. Source of Image Degradation and Need for Restoration . . . . . . . . . . . 3

1.4. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5. Background on a Poisson Model for Hyperspectral Image Data and MaximumLikelihood Image Deconvolution Algorithms. . . . . . . . . . . . . . . . 9

1.5.1. A Poisson Model for Hyperspectral Image Data . . . . . . . . . . . 9

1.5.2. Outline of Maximum Likelihood Image Deconvolution Algorithm . . . 12

1.5.3. Outline of Maximum Likelihood Blind Image Deconvolution Algorithm 14

1.6. Outline of Penalized Maximum Likelihood Blind Image DeconvolutionAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

vi

2. Maximum Likelihood Expectation Maximization Image DeconvolutionAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3. Penalized Maximum Likelihood Image Deconvolution Algorithm . . . . . . 28

3.1. Penalized Maximum Likelihood Algorithm Via Majorizing Functions . . . . . 29

4. Maximum Likelihood Blind Image Deconvolution Algorithm . . . . . . . . 36

5. Penalized Maximum Likelihood Blind Image Deconvolution Algorithm . . . 42

5.1. Penalized Maximum Likelihood Algorithm for Estimating the Point SpreadFunction With Unity DC Gain Penalty Function . . . . . . . . . . . . . . 43

6. Experimental Study and Simulation Results . . . . . . . . . . . . . . . . 49

6.1. Maximum Likelihood Image Deconvolution . . . . . . . . . . . . . . . . 49

6.2. Penalized Maximum Likelihood Image Deconvolution . . . . . . . . . . . 54

6.3. Maximum Likelihood Blind Image Deconvolution . . . . . . . . . . . . . 59

6.4. Penalized Maximum Likelihood Blind Image Deconvolution . . . . . . . . . 62

7. Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . 67

7.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Appendices

Matlab Source Code for PML Blind Image Deconvolution Algorithm . . . . . . 70

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

vii

List of Figures

Figure1.1. 2-Dimensional Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2. Lexicographical Representation of a 3 � 3 PSF . . . . . . . . . . . . . . . . 10

1.3. Reflect-Extending an Image . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4. A 1-D Illustration of Iterative Majorization Technique . . . . . . . . . . . . . 17

2.1. A 9 by 9 example image for visualizing the elements of Sj when the PSF maskis centered at an interior pixel j D 42 . . . . . . . . . . . . . . . . . . . . . . 24

2.2. Examples showing some of the elements of Sj when the PSF mask is notcentered at an interior pixel j D 42. . . . . . . . . . . . . . . . . . . . . . . 25

2.3. A 9 by 9 example image for visualizing the elements of Sj when the PSF maskis centered at a border pixel j D 1. . . . . . . . . . . . . . . . . . . . . . . . 26

2.4. Examples showing some of the elements of Sj when the PSF mask is notcentered at border pixel j D 1. . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.1. MLEM image deconvolution of an image with Poisson Noise . . . . . . . . . 50

6.2. MLEM image deconvolution of an image with Poisson noise and plot of theobjective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3. MLEM image deconvolution of an image without Poisson Noise and plot of theobjective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.4. MLEM image deconvolution of an image without Poisson Noise . . . . . . . 53

6.5. PML image deconvolution of an image with Poisson noise and plot of theobjective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.6. PML image deconvolution of an image without Poisson noise and plot of theobjective function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

6.7. PML deblurred images with different ˇ values and plot of the objective functionfor noisy observed image . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

viii

6.8. PML deblurred images with different ˇ values and plot of the objective functionfor noiseless observed image . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6.9. ML BID results for an image blurred with Poisson noise . . . . . . . . . . . 60

6.10. ML BID results for landmine field image . . . . . . . . . . . . . . . . . . . . . 61

6.11. PML BID results for an image blurred with Poisson noise . . . . . . . . . . . 63

6.12. PML BID results for landmine field image with ˛ D 0:1 . . . . . . . . . . . 65

6.13. PML BID results for the bright spiral galaxy Messier 61 (NGC 4303) in theVirgo cluster with unity DC gain penalty function when ˛ D 0:0001 . . . . . 66

ix

List of Abbreviations

PSF – Point Spread Function

HYDICE – Hyperspectral Digital Image Collection Experiment

AVIRIS – Airborne Visible/Infrared Imaging Spectrometer

ER – Earth Resources

ERS – Earth Resources Survey

MSS – Multispectral Scanner

TM – Thermic Mapper

CCD – Charge-Coupled Device

PET – Positron Emission Tomography

MLEM – Maximum Likelihood Expectation Maximization

PML – Penalized Maximum Likelihood

OSL – One-Step-Late

ENR – Edge-to-Noise Ratio

MRP – Median Root-Prior

SNR – Signal to Noise Ratio

BID – Blind Image Deconvolution

GMRF – Gauss-Markov Random Field

RHS – Right Hand Side

LHS – Left Hand Side

x

List of Symbols

F – Poisson random variable for which the observations are pixels of true image of a scene

G – Poisson random variable for which the observations are pixels of the observed image

f – True image of a scene

g – Observed/Blurred image

� – Mean value of the true image of a scene

– Mean value of observed/blurred image

h – Point spread function

N – Negative of the log likelihood function.

Nm – Majorizing function for N when PSF is known

P – PML objective function for estimating �

Pm – Majorizing function for P

L – Edge preserving penalty function

ˇ – Penalty parameter controlling the edge preserving penalty function L

NLm – Huber’s majorizing function for L

Lm – Chang and Anderson’s majorizing function for L

– A general neighbor smoothing function

N m – Huber’s majorizing function for

m – Proposed majorizing function for

� – Neighbor smoothing function used in L

N�m – Huber’s majorizing function for �

�m – Proposed majorizing function for �

P� – First derivative of �

e – A non-increasing function derived from P�r – Gradient of a function

xi

Bi – Set of pixels in the neighborhood of the i th pixel

Q – Majorizing function for N when PSF is unknown

R – PML objective function for estimating h

Rm – Majorizing function for R

K – Unity DC Gain penalty function

˛ – Penalty parameter controlling the function K

Km – Majorizing function for K

� – Standard deviation used for generating Gaussian PSF

Convention: All bold symbols represent a vector whereas all non-bold symbols

represent either a scalar value, a matrix or a function name.

xii

Chapter 1. Introduction

Our motivation for investigating blind image deconvolution algorithms is to restore

hyperspectral images on a plane by plane basis. It is our belief that the performance

of anomaly detection algorithms for hyperspectral images will improve when a suitable

restoration is applied.

1.1. History of Hyperspectral Imagery

In hyperspectral imaging, an airborne or spaceborne sensor captures images that provide

spectral information on the terrain under investigation. Every physical object, based on its

shape, molecular composition, and surrounding terrain, absorbs and reflects electromagnetic

radiation from the sun. The reflected electromagnetic energy, measured by a hyperspectral

sensor, is divided into hundreds of contiguous narrow frequency bands known as channels.

For each channel, the hyperspectral sensor generates an image, referred to as a spectral

plane, with pixel values that are proportional to the reflected electromagnetic energy that

lies within the channel’s frequency band. Therefore, in hyperspectral imaging, a "stack" of

spectral planes are generated for each scene captured by the sensor. Hyperspectral imaging

is used in civil, commercial, and military applications where the problem is to detect and

identify objects of interest, geological features, and topographical features. For example, an

important military and humanitarian de-mining application is landmine detection.

Space technology developments in the late 1950s [3] allowed artificial satellites to

make observations of the earth from space and acquire some useful information helping

many science and engineering developments in civil, commercial and military remote

sensing applications such as civil engineering, Petroleum exploration, mining, forestry,

water management, landmine detection, and military information gathering. Starting with

the Earth Resources Survey (ERS) program launched by the United States in the mid-1960s,

1

NASA has been the pioneer launching the first ERS satellite in July of 1972, later renamed

Landsat I. It was on this Landsat I, and the later launched Landsat II satellites that NASA

has used its successive generations of multispectral scanner instruments, each having few

spectral bands, to collect images of the earth remotely. Electrooptical remote sensing is a

process of gathering information about an object or scene by a sensor array without coming

to physical proximity. When the reflected electromagnetic radiation from an object or scene

arrives at the airborne or spaceborne image sensor array, it is measured at sufficiently high

number of spectral channels for every pixel to form multispectral image of the same object

or scene. These multispectral images are different from images captured by conventional

devices in that they not only have spatial information but also wealth of information from

the different spectral dimensions which are studied in the field of Spectroscopy 1. Remote

sensing has included multispectral imaging devices (i.e., devices which image over a small

number of broad spectral bands) since Landsat I was launched in 1972. The two main

Landsat multispectral imaging instruments were the Multispectral Scanner (MSS) launched

aboard Landsat I, and the Thematic Mapper (TM), a second generation Landsat sensor. Both

instruments used a system of filters and discrete detectors to split image data into multiple

spectral bands, four filter/detector pairs in the case of the MSS, and seven for the TM.

1.2. Hyperspectral Image Data Collection

Hyperspectral image sensors, as the name indicates differ from their predecessors (i.e.,

multispectral Image sensors) in that the number of channels is much higher. The images most

often obtained by space imagery applications are hyperspectral images which are images

acquired by the hyperspectral image sensors. All the spectral sensors used onboard NASA-

developed Landsat satellites as well as all the others developed and used before late 1980s,

only had few spectral bands on the order of ten. The first sensor collecting data in several

hundreds of contiguous narrow spectral bands (hyperspectral sensor) emerged in the late

1Spectroscopy is the study of matter and its properties by investigating light, sound, or particles that areemitted, absorbed or scattered by the matter under investigation.

2

1980s. The airborne visible/infrared imaging spectrometer (AVIRIS), developed by NASA,

was tested in 1987, and then launched two years later in 1989. AVIRIS is mounted and flown

on the NASA ER-2 airplane at an altitude of 12.5 miles, and speed of 454 mi/h [4]. Among

the many hyperspectral sensors developed following the development of AVIRIS, was the

Hyperspectral Digital Imagery Collection Experiment (HYDICE), owned and operated by

the U.S. Navy Research Laboratory and used for surveillance missions[4]. HYDICE uses an

airborne camera to acquire 210 band images whose electromagnetic spectrum ranges from

visible light to the near Infrared in the wavelength of 0:4 � 2:5�m . Real data experiments

in this paper are performed using HYDICE images obtained from the Countermine Division

of the Night Vision and Electronic Sensors Directorate in Fort Belvoir, Virginia.

1.3. Source of Image Degradation and Need for Restoration

Hyperspectral image processing is becoming increasingly popular due to the wealth

of information hyperspectral images can provide to civilian, commercial, and military

applications. By exploiting fine spectral differences between various natural and man-made

materials of interest, it can support improved detection and classification capabilities of

hyperspectral remote sensors. One of the important military and humanitarian de-mining

applications is landmine detection. According to the landmine facts published by United

Nation [5], it is estimated that there are more than 110 million active mines scattered in 70

countries and an equal number stockpiled around the world waiting to be planted. Over

2,000 people are killed or maimed by mine explosions every month. Most of the casualties

of mine explosions are civilians who are killed or injured after hostilities of war have ended.

Although the use of hyperspectral imagery for automatic target detection and recogni-

tion is a fairly new area of research, it has a high potential in detection and identification

of man-made anomalies from their natural background. Literature reveals that spectral

characteristics of man-made objects differ greatly from spectral characteristics of natural

background [6],[7],[8] . However, due to the hyperspectral sensor’s optical system, atmo-

3

spheric conditions, and relative motion between the sensor and the scene, hyperspectral

images may suffer varying degrees of blur. If these images can be restored from the blur, then

landmines can be detected easily. Although some of the blur factors such as aircraft motion

and non-uniformity of the photodetector can be measured and corrected beforehand, the

biggest factor that makes the hyperspectral image restoration problem extremely challenging

is the continuously varying nature of atmospheric condition which is the media for the

reflected signal to be detected by the airborne camera. Because of this changing nature of

the atmosphere, atmospheric blur compensation has been a very challenging area of research

for the remote-sensing society. Mathematically speaking, image blurring happens when

clean image is convolved with system’s point spread function. The point spread function h,

in our case, models the blur due to motion, atmospheric distortion caused by scattering of

photons in the reflected electromagnetic radiation, and the sensor’s optical system. A typical

convolution process is shown in Figure 1.1.

Figure 1.1. 2-Dimensional Convolution : (a) blurred Image (b) clean Image of a scene (c)point spread function (d) additive Noise.

Recovering the original scene image from a degraded or observed image using knowl-

edge about its nature is called image Restoration. Image restoration is a useful technique

relevant in many fields: medicine and astronomy among others. Because the image data

recorded by an airborne camera, miles away from the scene, has to pass through different

blurring factors as mentioned above, the acquired image data may be highly blurred and

noisy. Hence, image restoration plays a major role in hyperspectral imagery. HYDICE

4

images of land-mine fields can have highly blurred mine signatures which are very hard to

detect without restoring the images; Specially small mine signatures. Therefore, it is highly

important to restore hyperspectral images before performing any image analysis or landmine

detection. Our objective in this thesis is to develop an algorithm to restore the original scene

image and, at the same time, estimate the blurring PSF merely using the observed image

and no prior information about the image or the PSF. Once the hyperspectral images are

restored, anomaly detector algorithms such as the RX algorithm [9] and the Gauss-Markov

Random Field (GMRF) algorithm [10] can be applied.

1.4. Literature Review

Image Restoration is the process of recovering the original scene image from a degraded

or observed image using knowledge about its nature. Many image processing applications

critically depend on image restoration; it is a useful technique relevant in many fields

including medicine and astronomy. We classify image restoration into two major categories.

The first category, also called image deconvolution, is a linear image restoration problem

where the parameters of the true image are estimated using the observed or degraded image

and a known point spread function (PSF) [11], [12]. In reality, the degrading PSF of a

system is sometimes hard or even impossible to identify due to physical, economical, or

strategic limitations. The second category, which is called blind image deconvolution (BID),

is a more difficult image restoration problem where image recovery is performed with little

or no prior knowledge of the degrading PSF. Since our goal in this paper is to find a BID

algorithm that works well for the hyperspectral image restoration problem, we focus on the

later category problem.

Although the BID problem has been a focus of study for many researchers in the last

few decades, this notorious but very important problem still remains to be open-ended.

Many of the previously proposed BID methods are outlined in [13]. There are several

BID algorithms that estimate the blurring PSF from the observed image and then solve the

5

linear image restoration problem to estimate the true image using the estimated PSF and

the degraded or observed image. This approach, which was used in [14] and [15], is called

the "a priori blur identification" method because the blurring kernel is first estimated using

prior knowledge. Several other BID algorithms have been proposed for simultaneously

estimating the clean image and the blurring kernel [16], [17] (see [13] for a comprehensive

review). This later method usually employs maximum likelihood estimators sometimes

with incorporated priors. The proposed method uses PML algorithms for the restoration

of spectral planes from blurred hyperspectral images. In the PML method, an objective

function equal to the negative log likelihood function plus a penalty function is minimized

with respect to the unknown parameters. The penalty function accounts for certain a priori

information about the unknown parameters. To solve the optimization problem, we use

an iterative technique commonly known as the alternating minimization technique initially

developed by Holmes [18]. Given initial estimates of the parameters, the PML objective

function is minimized with respect to the reflectance parameters (spectral plane parameters)

while setting the PSF parameters to their current estimate. Then, the PML objective function

is minimized with respect to the PSF parameters while setting the reflectance parameters to

their current estimate. The algorithm alternates between the two steps until some chosen

stopping criterion is met.

In a general sense, the a priori blur identification method is used only when enough

prior information is known about the blurring PSF. However, the proposed PML method can

be applied to any BID problem with little or no prior knowledge of either the original scene

image or the PSF. Moreover, the PML approach takes into account the statistical nature of

quantum photon emission and a basic rule of probability theory to enforce a nonnegativity

constraint on the reflectance means and the PSF respectively. A major advantage of the

approach is the ability to incorporate multiple regularizing penalty functions among which is

the very important noise suppression function. The problems associated with the alternating

PML method include the burden of computational complexity and determination of the

6

stopping criteria for the alternating state.

The most widely used BID approach is a parametric PSF estimation as opposed to

non-parametric estimation. In parametric PSF estimation, the degrading kernel is modeled

with a known statistical distribution so that the parameters of the model are estimated.

Review of BID algorithms [13], [19], [20], reveals that almost all BID algorithms assume

irreducible PSF. An irreducible signal is a signal that can not be exactly expressed as the

convolution of two or more component signals of the same family on the assumption that a

two-dimensional delta function is not a component signal. However, a Gaussian PSF, which

is the most natural image blurring kernel, is reducible which means that G� D G�1�G�2

for �2 D �12 C �2

2. It is this reducible nature of the Gaussian PSF that rendered the BID

of hyperspectral images most difficult. This problem, even with the noiseless case is well

known as the ill-posed inverse problem of heat transfer and still remains to be a challenge

[21].

In 1982, Shepp and Vardi [22] proposed MLEM algorithm for estimating emission

means in PET. The problem with the ML based method for PET was that the images

produced were very noisy because, as discussed in Section 1.5.1, the data obeys a Poisson

distribution which made the problem ill-posed. Different methods have been tried to address

this issue. One of the methods used to obtain ML images with reduced noise was to stop

the MLEM algorithm before the objective function is completely minimized. But the

major problem associated with this method is that it is impossible to exactly know where

to stop the algorithm. Although, there is ambiguity on how to choose the filter, people

have tried to obtain a noisy image from the MLEM algorithm and then post-process the

image with a lowpass filter. In 1990, [23], Silverman suggested filtering the image after

each iteration. An alternative way researchers have used to try to remove noise from ML

images is pre-processing the observed data. In [24], and [25], Lu and Anderson proposed

the pre-processing technique to denoise a PET emission data before using the ML algorithm.

In [20] and [21], Jian and Wang, assuming an isotropic Gaussian PSF, proposed ML based

7

alternating optimization technique used to approximate an oblique computed tomography

image by iteratively maximizing edge-to-noise ratio (ENR) to characterize the image quality

due to image deconvolution.

The most widely used regularization method is through the introduction of penalty

functions in order to force estimates of neighboring pixels to have similar values. In PML

algorithm, which is a maximum a posteriori (MAP) method, an objective function equal

to the negative log likelihood function plus a penalty function is minimized with respect

to the unknown parameters. The penalty function accounts for certain a priori information

about the unknown parameters. The Median Root-Prior (MRP) algorithm was introduced by

Alenius et al. [26] that suggests Gaussian-like prior depending on the median of the pixels

within local neighborhoods. Although the MRP algorithm generates "good" images in terms

of the Signal to Noise Ratio (SNR) of the reconstructed images, it can not be considered a

PML algorithm because of the fact that it is based on iteration dependant objective function.

In 1990, Green [27] proposed the one-step-late (OSL) PML algorithm derived from the

Kuhn-Tucker equations [28] for the PET PML optimization problem. Implementation of the

OSL algorithm is easy but it neither guaranties non-negative estimates nor does it guarantee

convergence just like most algorithms. Shepp and Vardi showed that the MLEM algorithm

could be derived from the same Kuhn-Tucker condition by minimizing the Kullback Leibler

distance between the expected emission mean values and the observed data. The PML

algorithm for PET proposed by Levitan and Herman [29] was based on the assumption that

the prior distribution of the true emission means was a multivariate Gaussian distribution.

The penalty function that was followed by this assumption was in the form of a weighted

least-squares distance between the true emission means and a reference image. The problem

is that the authors did not provide information on how to choose the reference image.

De Pierro [30, 31] came up with an algorithm that minimizes majorizing functions

constructed for certain penalty functions. He used the fact that the negative log likelihood

function and penalty functions such as quadratic functions are convex [32, pp. 860 – 862], in

8

order to construct his majorizing functions. Closed form expressions for the minimizers of

the majorizing functions do not exist except for the quadratic penalty functions. Therefore,

Newton’s method or similar optimization method is required to find the minimizers of the

majorizing functions. A quadratic majorizing function was developed by Huber [1, pp. 184

– 186] which was later used by many researches including Chang and Anderson [2], and

Erdogan and Fessler [33].

1.5. Background on a Poisson Model for Hyperspectral Image Data and MaximumLikelihood Image Deconvolution Algorithms

In this section, we first introduce a Poisson model for a single spectral plane from a

hyperspectral image data set. Then, we outline approaches for deriving image deconvolution

algorithms for the cases where the PSF is known and unknown.

1.5.1. A Poisson Model for Hyperspectral Image Data

Shepp and Vardi [22] proposed a Poisson model for positron emission tomography

(PET) data. Later, Snyder et al. extended Shepp and Vardi’s model to hyperspectral images

[34]. Now, we will describe the model we use for hyperspectral images. The difference

between our model and the one put forth by Snyder et al. is the way border pixels are

modeled and their model includes Gaussian readout noise 2. Let f denote the true M1�M2

image. Further, let g denote the observed image (i.e., blurry image) obtained from an

airborne sensor. The pixel values f .m1;m2/ and g.m1;m2/ are assumed to be observations

of Poisson random variables F.m1;m2/ and G.m1;m2/ respectively. Moreover, the random

variables fF.m1;m2/g and fG.m1;m2/g are assumed to be independent. Let �.m1;m2/ and

.m1;m2/ denote the mean values of F.m1;m2/ and G.m1;m2/, respectively. Additionally,

let the point spread function h model the blur due to motion, atmospheric distortion, and

the sensor’s optical system. In the Poisson model, the key relationship is that equals the

2The additive Gaussian readout noise is often left behind in the model not only to keep the model simplerbut also by considering the fact that the effect of the additive noise is mostly minimal compared to theconvolution of the input signal and the system’s PSF.

9

convolution of � and h. This relationship is what connects the blurry image to the true

image.

.m1;m2/ DXiD1

XjD1

h .i; j / � .m1 � i;m2 � j /: (1.1)

The problem of estimating the mean values of the true image f�.m1;m2/g given the

observed blurry image and the PSF is known as a deconvolution problem. When the PSF is

unknown, the problem is known as a blind deconvolution problem. It will be convenient to

use vector notation and standard lexicographical ordering to convert the two-dimensional

notation to one-dimensional notation. For example, an M1 �M2 image x.m1;m2/ can be

viewed as an M1M2 � 1 vector x by stacking the rows of x.m1;m2/. In other words, the

first M2 elements of x are the elements of the first row of x.m1;m2/. The next set of M2

elements of x are the elements of the second row of x.m1;m2/ and so on. Thus, with the

lexicographical ordering just described, the images g, f , , and � can be represented by

the I � 1 vectors g, f , , and � respectively, where I DM1 �M2, g D Œg1;g2; :::;gI �,

f D Œf1; f2; :::; fI �, D Œ 1; 2; :::; I � , and � D Œ�1; �2; :::; �I �. Let the size of the

PSF be L1 by L2 , where L1 and L2 are both odd. Also, let L , .L1�L2�1/

2. Then,

after rearranging it in lexicographical order, the PSF can be represented as the vector

h D Œh�L; :::; h�1; h0; h1; :::; hL�.

We now depart from Snyder’s model [34] and express the mean value of the observed

image i as

i D< h; �i >; i D 1; 2; :::; I; (1.2)

where < h; �i >,LP

lD�L

hl�iland �i ,

��i�L

; �i�LC1; :::; �iL�1

; �iL

�: The key question at

this point is how should the indices filg be defined for each i ? We will define the indices filgso that each mean value i is expressed as a linear combination of all the PSF coefficients

and certain mean values of the true image f�ig. The reason we express filg in this way is

that when i is a border pixel, the standard two-dimensional deconvolution is ill-posed. We

10

Figure 1.2. Lexicographical Representation of a 3 � 3 PSF.

will now consider some examples.

Case1: When pixel i does not lie on the border, we choose the indices filg such that

< h; �i >DL1�1

2PiD1

L2�12P

jD1

h .i; j / � .m1 � i;m2 � j /. To make this statement clearer,

we consider an example where M1 D M2 D 9 and L1 D L2 D 3, and explicitly

determine the indices filg for j D 42 (note: pixel 42 is not on the border of the

image). Figure 1.3 (a) illustrates the lexicographical ordering for the 9 � 9 image �.

Figure 1.3 (b) shows the location of the PSF mask that corresponds to the convolution

expression in (1.1) for i D 42.

42 D h�4�42�4C h�3�42�3

C h�2�42�2C h�1�42�1

C h0�420C h1�421

C h2�422C h3�423

C h4�424

D h�4�32 C h�3�33 C h�2�34 C h�1�41 C h0�42

C h1�43 C h2�50 C h3�51 C h4�52: (1.3)

As seen from Figure 1.3 (b), �42lD Œ�32; �33; �34; �41; �42; �43; �50; �51; �52�.

Case2: When i is a border pixel, we modify the image space so that the indices filg can

be obtained. To do that, we symmetrically reflect-extend the blurred image before

11

performing the deconvolution process. By reflect-extend, we mean reflecting some of

the border rows and columns of the image and extending the image by the reflected

data so that all the L1 � L2 elements of �ilexist when i is at the border and h0

is located at �i . The number of top and bottom rows that should be reflected is

equal to .L1�1/

2, whereas the number of left and right columns to reflect equals .L2�1/

2.

Using the example image used in Case1, where M1 D M2 D 9 and L1 D L2 D 3,

Figure 1.3 (c) illustrates the reflect-extended example image. Figure 1.3 (d) shows the

location of the PSF mask that corresponds to the convolution expression in (1.1) for

i D 1.

1 D h�4�1�4C h�3�1�3

C h�2�1�2C h�1�1�1

C h0�10C h1�11

C h2�12C h3�13

C h4�14

D h�4�1 C h�3�1 C h�2�2 C h�1�1 C h0�1

C h1�2 C h2�10 C h3�10 C h4�11: (1.4)

As seen from Figure 1.3 (d), �1lD Œ�1; �1; �2; �1; �1; �2; �10; �10; �11�. At this point, we

have established a technique by reflect-extending the blurred image, to allow us to use (1.2)

for all i regardless of its location in the image.

1.5.2. Outline of Maximum Likelihood Image Deconvolution Algorithm

Based on the model assumptions, the likelihood function of the observed data is given

by

Pr ŒG D gj�� D Pr ŒG1 D g1;G2 D g2; :::;GI D gI j�� (1.5)

DIY

iD1

Pr ŒGi D gij�� (1.6)

DIY

iD1

e�

LP

lD�L

hl�il

! �LP

lD�L

hl�il

�gi

gi !; (1.7)

12

(a) (b)

(c) (d)

Figure 1.3. Reflect-Extending an Image: (a) original Image, (b) when i is not a border pixel,(c) reflect-extended Image , and (d) when i is a border pixel.

13

where G D ŒG1;G2; :::;GI �. Therefore, maximum likelihood (ML) estimates for the

unknown mean values f�ig, which we now refer to as reflectance means, can be obtained by

maximizing the likelihood function in (1.7) with respect to � under the constraint � � 0.

Equivalently, ML estimates can be determined by minimizing the negative log likelihood

function N , as defined below, for � � 0:

N .�/ , � log Pr ŒG D gj�� (1.8)

D �IX

iD1

gi log

LX

lD�L

hl�il

!C

IXiD1

LXlD�L

hl�ilC

IXiD1

log gi ! (1.9)

Stated succinctly, ML estimates for the reflectance parameters are obtained by solving the

following optimization problem

O� D arg min�>0

N .�/ : (1.10)

Shepp and Vardi developed the Maximum Likelihood Expectation Maximization (MLEM)

algorithm for solving (1.10). Their derivation of the MLEM algorithm will be provided in

Chapter 2.

1.5.3. Outline of Maximum Likelihood Blind Image Deconvolution Algorithm

This section outlines a BID algorithm when the PSF is unknown. Keeping in mind

the discussion in Section 1.5.2, ML estimates for the reflectance means f�ig and the PSF

coefficients fhlg can be found by maximizing the log likelihood function Pr ŒG D gj�;h�or minimizing the negative log likelihood function under the constraints � � 0 and h � 0.

N .�;h/ , � log Pr ŒG D gj�;h� (1.11)

D �IX

iD1

gi log

LX

lD�L

hl�il

!C

IXiD1

LXlD�L

hl�ilC

IXiD1

log gi ! : (1.12)

14

In other words, the ML estimates for the reflectance means and the PSF coefficients can be

expressed as

� O�; Oh� D arg min�>0;h>0

N .�;h/ : (1.13)

Since solving (1.13) involves the challenging BID problem that tries to estimate both the

reflectance means and the underlying PSF from the blurred image (i.e., the observed image),

it is impossible to find a straightforward solution for the problem. Therefore, we use an

alternating minimization approach initially suggested by Holmes [18] and later used by other

researchers as well (see [21] for example). Given the current estimates for the reflectance

means and PSF coefficients, we obtain an improved estimate for the reflectance means by

minimizing N��;h.n/

�with respect to �, where h.n/ is the current estimate of the PSF

coefficients. The resulting estimate, �.nC1/, is then used to obtain the next estimate for the

PSF coefficients, h.nC1/.

Summarizing, a modified alternating minimization for BID consists of the following steps:

Given �.0/ > 0 and h.0/ > 0,

Step 1 Find �.nC1/ � 0 such that N��.nC1/;h.n/

�� N

��.n/;h.n/

�(1.14)

Step 2 Find h.nC1/ � 0 such that N��.nC1/;h.nC1/

�� N

��.nC1/;h.n/

�(1.15)

Step 3 Iterate between Step 1 and Step 2 until chosen convergence criterion is met.

1.6. Outline of Penalized Maximum Likelihood Blind Image DeconvolutionAlgorithm

The optimization problem in (1.13) is ill-posed in nature, meaning that a unique optimal

solution does not exist for the convex objective function. The most widely used method

for addressing this ill-posed nature of the optimization problem is by introducing certain

regularizing penalty functions. Therefore, based on some a priori information, we add

characteristic penalty functions for both the reflectance parameters and PSF coefficients.

Therefore, the Penalized Maximum Likelihood (PML) estimates for the reflectance means

15

and the PSF coefficients can be expressed as

� O�; Oh� D arg min�>0;h>0

N .�;h/C ˇL .�/C ˛K .h/ : (1.16)

L and K are the penalty functions for � and h respectively Penalty parameters ˇ and ˛.

In Chapter 3, under the assumption that the PSF is known, we consider the problem of

minimizing the negative log likelihood function plus a penalty function for the reflectance

parameters. The penalty function is used to address the ill-possed nature of the problem of

simply minimizing the negative log likelihood function.

Unfortunately, closed form solutions for the PML optimization problem does not exist.

Therefore, employing iterative optimization techniques is a must. The proposed PML

algorithm is based on iterative majorizing technique [30, 31, 35–37] where at each iteration

of the optimization problem, a function called a majorizing function is constructed for the

objective function that is easier to solve and satisfies certain conditions. Then, the minimizer

of the majorizing function is used as the next iterate of the objective function.

For demonstration purposes, consider a one-dimensional minimization problem that

does not have a closed form solution

Ot D arg mint>0

f .t/: (1.17)

Suppose a function fm can be determined such that

(C1.1) fm

�t; t .n/

�> f .t/ , for t > 0:

(C1.2) fm

�t .n/; t .n/

� D f �t .n/� .

Then, in the iterative majorizing technique, the next iterate is defined to be

t .nC1/ D arg mint>0

fm

�t; t .n/

�: (1.18)

In this example, and more generally, the primary motivation for the iterative majorizing

technique is that it produces a sequence of iterates that monotonically decreases the objective

16

function.

In summary, a modified alternating minimization for BID using PML based algorithms

consists of the following steps:

Given �.0/ > 0 and h.0/ > 0,

Step 1 Find �.nC1/ � 0 such that N��.nC1/;h.n/

�C ˇL

��.nC1/

�� N

��.n/;h.n/

�C ˇL

��.n/

�(1.19)

Step 2 Find h.nC1/ � 0 such that N��.nC1/;h.nC1/

�C ˛K

�h.nC1/

�� N

��.nC1/;h.n/

�C ˛K

�h.n/

�: (1.20)

Step 3 Iterate between Step 1 and Step 2 until chosen convergence criterion is met.

In Chapter 2, we derive the MLEM algorithm and in Chapter 3, a PML algorithm with a

class of edge-preserving penalty functions, both for estimating the reflectance parameters by

assuming that we know the underlying PSF responsible for the degradation of the true image.

In Chapter 4, we derive a BID algorithm based on the a modified alternating minimization

technique using the MLEM algorithm. In chapter 5, we derive the proposed alternating BID

algorithm that is based on a PML algorithm. In the proposed method, we bring together

the ideas discussed and/or used in previous chapters and also derive a PML algorithm and

associated penalty functions for estimating the PSF.

Summarizing the proposed BID algorithm, given a strictly positive initial estimates

�.0/ > 0, and h.0/ > 0 the steps for n D 1; 2; :::; are as follows:

Step 1 Let �.0/ > 0 and h.0/ > 0, be the initial estimates.

Step 2 Estimate �.n/ using h.n�1/.

Step 3 Estimate h.n/ using �.n/.

17

Figure 1.4. A 1-D illustration of iterative majorization technique. At each iteration, amajorizing function is obtained and minimized. The minimizer of the majorizingfunction is used as the next iterate of the algorithm.

18

Step 4 Iterate between Steps 2 and 3 until chosen convergence criterion is met.

At this point, it is important to note that Step 4 is the main iteration of the algorithm that

includes Steps 2 and 3 within a single iteration. Although a person can choose how many

iterations to run for each sub-iterations (i.e., Steps 2 and 3), for practical reasons, a single

sub-iteration usually suffices within the bigger iteration.

19

Chapter 2. Maximum Likelihood Expectation Maximization Image DeconvolutionAlgorithm

In this chapter, we present an ML algorithm for estimating the reflectance means˚�j

under the assumption that the PSF coefficients fhlg are known.

Recall from Section 1.5.2 that the negative log likelihood function is given by

N .�/ , � log Pr ŒG D gj�� (2.1)

D �IX

iD1

gi log

LX

lD�L

hl�il

!C

IXiD1

LXlD�L

hl�ilC

IXiD1

log gi ! : (2.2)

D �IX

iD1

gi log < h; �i > CIX

iD1

< h; �i > CIX

iD1

log gi ! : (2.3)

where < h; �i > ,�

LPlD�L

hl�il

�. Therefore, ML estimates of the reflectance parameters

are obtained by determining a solution to the following optimization problem

(P1) O�ML D arg min�>0

N .�/ . (2.4)

Unfortunately, it is impossible to find a closed form solution to the problem (P1). Conse-

quently, iterative optimization techniques must be employed. In 1982, Shepp and Vardi used

the expectation maximization algorithm [38] to develop an iterative algorithm for solving an

optimization problem in positron emission tomography that is essentially identical to the

problem (P1)[22]. Their algorithm is known as the Maximum Likelihood Expectation Max-

imization (MLEM) algorithm. 3. Later, De Pierro demonstrated that the MLEM algorithm

could be derived using certain functions known as majorizing functions. We use De Pierro’s

approach to derive an algorithm for solving (P1).

Consider an arbitrary I �1 vector �0 > 0 and a function Nm that satisfies the following

3The Richardson-Lucy algorithm, also known as the Richardson-Lucy deconvolution algorithm, is aniterative procedure for recovering an image that has been blurred by a known point spread function [39], [40]

20

conditions:

(C2.1) Nm.�;�0/ > N .�/ for all � > 0

(C2.2) Nm.�0;�0/ D N .�0/ for �0 > 0.

The function Nm . � ;�0/ is said to be a majorizing function for the function N at the point

�0. Given an initial estimate for � and a majorizing function Nm, an iterative algorithm for

solving (P1) is

(P2) �.nC1/ D arg min�>0

Nm.�;�.n//; (2.5)

where �.n/ > 0 is the current estimate for �. The algorithm described above is known by

several names including optimization transfer, iterative majorization, and minorize-maximize

algorithms.

We will now use properties (C2.1) and (C2.2) to show that the iterative majorization

algorithm (2.5) monotonically decreases the negative log likelihood function with increasing

iterations. The inequality N��.nC1/

� � Nm.�.nC1/;�.n// follows from (C2.1) and by

(C2.2), the equality Nm.�.n/;�.n// D N

��.n/

�holds true. Now, from (2.5) it follows that

Nm.�.nC1/;�.n// � Nm.�

.n/;�.n//. Therefore, we get the desired result

N��.nC1/

�� Nm.�

.nC1/;�.n// � Nm.�.n/;�.n// D N

��.n/

�. (2.6)

We will now explicitly determine a majorizing function for the negative log likelihood

function N by exploiting the concavity of the log function. It will be convenient to express

the negative log likelihood function as follows

N .�/ D �IX

iD1

gi log

LX

lD�L

hl�il

�.n/il

�.n/il

< h; �.n/i >

< h; �.n/i >

!

CIX

iD1

< h; �i >CIX

iD1

log gi ! (2.7)

where < h; �.n/i >,

�LP

kD�L

hk�.n/ik

�and �.n/ � 0: Continuing, we express the negative

21

log likelihood function as

N .�/ D �IX

iD1

gi log

LX

lD�L

d.n/

il< h; �

.n/i >

�il

�.n/il

!

CIX

iD1

< h; �i >CIX

iD1

log gi ! ; (2.8)

where d.n/

il,

hl�.n/

il

<h;�.n/

i>

. Using the concavity of the log likelihood function [32, pp. 860 –

862] and the fact thatLP

lD�L

d.n/

ilD 1, we obtain the following inequality

log

LX

lDL

d.n/

il< h; �

.n/i >

�il

�.n/il

!�

LXlD�L

d.n/

illog

< h; �

.n/i >

�il

�.n/il

!: (2.9)

Therefore, it follows that

N .�/ 6 �IX

iD1

gi

LXlD�L

d.n/

illog

< h; �

.n/i >

�il

�.n/il

!

CIX

iD1

< h; �i >CIX

iD1

log gi ! : (2.10)

We will now show that the function on the RHS of (2.10) is a majorizing function for the

negative log likelihood function N .

Let Nm

� � ;�.n/� denote the function on the RHS of (2.10)

Nm.�;�.n// D �

IXiD1

gi

LXlD�L

d.n/

illog

< h; �

.n/i >

�il

�.n/il

!

CIX

iD1

< h; �i >CIX

iD1

log gi ! . (2.11)

Clearly, by construction, the function Nm

� � ;�.n/� satisfies (C2.1) for �.n/ > 0. We will

now show that condition (C2.2) is also satisfied. Using straightforward calculations, it can

be seen that

22

Nm.�.n/;�.n// D �

IXiD1

gi log < h; �.n/i >

LXlD�L

d.n/

ilC

IXiD1

< h; �.n/i >

CIX

iD1

log gi ! . (2.12)

SinceLP

lD�L

d.n/

ilD 1, it follows that

Nm.�.n/;�.n// D �

IXiD1

gi log < h; �.n/i >C

IXiD1

< h; �.n/i >C

IXiD1

log gi !

D N��.n/

�. (2.13)

Thus, the function Nm

� � ;�.n/� satisfies (C2.2).

At this point, we have constructed a majorizing function for the negative log likelihood

function N . To obtain the next iterate �.nC1/, the majorizing function N at a point �.n/ > 0

must be minimized (see (2.5)). In the remainder of the chapter, we drive an explicit

expression for �.nC1/ by initially ignoring the constraint � � 0 and solving the system of

equations

@

@�j

Nm.�;�.n// D 0; j D 1; 2; :::;J: (2.14)

Straightforward calculations lead to the following expression for the derivative of Nm. � ; �.n//with respect to �j

@

@�j

Nm.�;�.n// D � @

@�j

IXiD1

gi

LXlD�L

d .n/il

�log < h; �

.n/i >C log�il

� log�.n/il

�C @

@�j

IXiD1

< h; �i >C @

@�j

IXiD1

log gi ! (2.15)

D � @

@�j

IXiD1

gi

LXlD�L

d.n/

illog�il

C @

@�j

IXiD1

LXlD�L

hl�il: (2.16)

To complete the calculation, we must consider the values for i and l such that il D j .

23

For j D 1; 2; :::;J , let Sj , f.i; l/ W il D j g. It will be convenient to denote the elements

of Sj by the ordered pairs�.aj1; bj1/; .aj2; bj2/; :::; .ajT ; bjT /

�, where T , 2L C 1 is

the number of PSF coefficients. Using this notation, we can express the derivative of

Nm. � ; �.n// with respect to �j as follows

@

@�j

Nm.�;�.n// D � @

@�j

TX

tD1

gaj td.n/

aj t bj tlog�j

!C @

@�j

TXtD1

hbj t�j : (2.17)

Note that ajt represents an index of a pixel whereas bjt represents the index for a PSF

coefficient.

Now, we will discuss the members of the set Sj in more detail. The key idea is that the

model represented by (1.2) is equivalent to the two-dimensional convolution model (1.1)

when the two-dimensional PSF mask (i.e., hŒ�m;�n�) centered at a pixel i lies within the

spatial extent of �.m; n/. Pixels that satisfy this condition are said to be interior pixels. For

pixels that do not meet this condition, an alternative model is used and they are referred to

as border pixels. The alternative model used for border pixels helps address the well known

"border problem" in image deconvolution. Finding the members of Sj requires moving

the PSF mask around the image and centering it at each pixel location i . If there is an i

value PSF coefficient index l such that il D j , then .i; l/ 2 Sj . We will now explicitly

demonstrate this point by considering examples for the two cases.

Interior Pixel Case:

When the pixel j does not lie on the border of the image, the PSF mask will totally lie

within the boundary of the image and all the elements of Sj do exist in the neighborhood of

j . We will now demonstrate this case by finding the members of Sj for j D 42 which can

serve as an example of a pixel that does not lie on the border. Consider a 9 by 9 image and a

3 � 3 PSF . Finding Sj requires moving the PSF around the image centering the mask at

each i value. If there is an i and an associated l value for which il D 42, then .i; l/ will be

an element of S42 .

From Figure 2.1, we see that when the mask is centered at i D 42, then i D 42 and

24

Figure 2.1. A 9 by 9 example image for visualizing the elements of Sj when the PSF maskis centered at an interior pixel j D 42.

l D 0 corresponds j D 42 such that il D j ; Which means that .42; 0/ 2 S42. Note that for

all i and j , if i D j , then i0 D j and .i; 0/ is always an element of Sj . In Figure 2.2 below,

we elaborate more on the example above and show in more detail, the elements of S42 when

the mask is not centered at the pixel in question. When the mask is centered at i D 41 as

shown in Figure 2.2 (a), j D 42 corresponds to i D 41 and l D 1 such that il D j . Similarly,

by centering the mask at i D 43, Figure 2.2 (b) shows that i D 43 and l D �1 such that

il D j . Also, Figure 2.2 (c), and (d) tells us that .32; 4/ 2 S42, and .52;�4/ 2 S42

respectively. If we keep on centering the mask at each i value, we will finally arrive

at4 S42 D f.32; 4/; .33; 3/; .34; 2/; .41; 1/; .42; 0/; .43;�1/; .50;�2/; .51;�3/; .52;�4/g .

Therefore, the ordered pairs representation of S42 corresponds to a.42; 1/ D 32, b.42; 1/ D4, a.42; 2/ D 33, b.42; 2/ D 3,..., a.42;T / D 52, b.42;T / D �4:

Border Pixel Case:

When j is at the border of the image, then portion of the PSF mask lies outside of the range

4It should be noticed that the PSF index vector is in reverse order when the l values of S42 viewedseparately. This reverse ordering is the basis for solving the inverse problem.

25

Figure 2.2. Examples showing some of the elements of S42 when the PSF mask is notcentered at an interior pixel j D 42.: (a) i D 41 and l D 1, (b) i D 43 andl D �1, (c) i D 32 and l D 4, and (d) i D 52 and l D �4 .

of the image and some members of Sj do not exist within the boundary of the image. To

deal with this border problem we symmetrically reflect-extend the image as discussed in

Section 1.5.1. We will now illustrate the usage of the reflect-extend method by using a 9� 9

example image and a 3 � 3 PSF to find Sj when j D 1.

From Figure 2.3, we see that when the mask is centered at i D 1, then i D 1 and l D 0

corresponds j D 1 such that il D j ; Which means that .1; 0/ 2 S1. In Figure 2.4 we give

examples of i and l elements of S1 when the mask is not centered at j D 1. When the mask

is centered to the left of j D 1 as shown in Figure 2.4 (a), j D 1 corresponds to i D 1 and

l D 1 from the extended section of the image such that il D j . Similarly, by centering the

mask one pixel to the right at i D 2, Figure 2.4 (b) shows that i D 2 and l D �1 correspond

to j such that il D j . Also, Figure 2.4 (c), and (d) tells us that .1; 4/ 2 S1, and .11;�4/ 2S1 respectively. If we keep on centering the mask at each i value, we will finally arrive at

S1 D f.1; 4/; .1; 3/; .2; 2/; .1; 1/; .1; 0/; .2;�1/; .10;�2/; .10;�3/; .11;�4/g . Therefore,

for when j D 1, the ordered pairs representation of S1 is given by a.1; 1/ D 1, b.1; 1/ D 4,

26

Figure 2.3. A 9 by 9 example image for visualizing the elements of Sj when the PSF maskis centered at a border pixel j D 1 .

Figure 2.4. Examples showing some of the elements of Sj when the PSF mask is notcentered at border pixel j D 1: (a) i D 1 and l D 1, (b) i D 2 and l D �1, (c)i D 1 and l D 4, and (d) i D 11 and l D �4 .

27

a.1; 2/ D 1, b.1; 2/ D 3,..., a.1;T / D 11, b.1;T / D �4.

From (2.17) and the discussion on Sj above, it follows that the derivative of the

majorizing function can be reduced to the following expression

@

@�j

Nm.�;�.n// D � @

@�j

TX

tD1

gaj td.n/

aj t bj tlog�j

!C @

@�j

TXtD1

hbj t�j (2.18)

D �TX

tD1

gaj td.n/

aj t bj t

1

�j

CTX

tD1

hbj t(2.19)

A necessary condition for the minimum of Nm. � ;�.n// is

@

@�j

Nm.�;�.n//j

�jD�.nC1/

j

D 0; j D 1; 2; :::;J: (2.20)

Or, equivalently

�TP

tD1

gaj td.n/

aj t bj t

�.nC1/j

CTX

tD1

hbj tD 0: (2.21)

Therefore, since d.n/

il,

h;�.n/

il

<h;�.n/

i>; the next iterate �.nC1/

j can be expressed as

�.nC1/j D

TPtD1

gaj td.n/

aj t bj t

TPtD1

hbj t

(2.22)

D

TPtD1

gaj t

hbj t�

.n/

j

<h;�.n/aj t>

TPtD1

hbj t

: (2.23)

Using the fact thatTP

tD1

hbj tD

LPlD�L

hl , the iterative majorizing algorithm for obtaining ML

reflectance parameter estimate is

�.nC1/j D �.n/j

1

LPlD�L

hl

TXtD1

gaj thbj t

< h; �.n/aj t>; j D 1; 2; :::;J: (2.24)

28

Chapter 3. Penalized Maximum Likelihood Image Deconvolution Algorithm

As discussed in Section 1.4, the MLEM algorithm developed in Chapter 2 can be

used to restore / deblur hyperspectral images from the degradation they suffer due to many

degradation factors such as atmospheric blur, motion blur and sensor noise (see [22] for

usage in PET data). However, the problem of restoring hyperspectral images is ill-posed

in nature due to the fact that the data obeys Poisson statistics. Therefore, a unique optimal

solution does not exist for the convex objective function in (1.10). As a result, the MLEM

algorithm fails to recover the true image and after certain number of iterations, the image

becomes more and more noisy with increasing iterations. The most widely used method

for addressing this ill-posed nature of the minimization problem is by introducing a penalty

function. In the area of medical imaging, specially in Positron Emission Tomography (PET)

image reconstruction algorithms, there are certain types of penalty functions used for noise

reduction. These penalty functions are designed in such a way that estimates of neighboring

pixels are forced to be similar in value unless there is an edge within neighbors. An edge

occurs in an image, whenever there is distinctive activity in group of connected pixels that

is different from the neighboring pixels. In a monochrome image, like the hyperspectral

image used in this thesis, an edge can be identified because of the difference in pixel values

(i.e., contrast). The hyperspectral image restoration problem is similar to the PET image

reconstruction problem in that the data obeys Poisson distribution in both problems. Hence,

the use of edge-preserving penalty functions developed for PET yields good results in the

hyperspectral image restoration problem. Since in the context of our research, restoring

hyperspectral images is important mainly for the identification of anomalies in the image, it

is extremely important to use a penalty function that can reduce the noise while keeping the

edges of the anomalies (landmines).

In Section 3.1, using majorizing functions, we will introduce a penalized maximum

29

likelihood (PML) algorithm for image deconvolution.

3.1. Penalized Maximum Likelihood Algorithm Via Majorizing Functions

We now present a modification of an algorithm developed by Chang and Anderson [2]

for PET that provides PML estimates of the reflectance means˚�j

under the assumption

that the PSF coefficients are known. Our derivation closely follows their own, except we

use the model in (1.2). Recall from Section 1.5.2 that the negative log likelihood function is

given by

N .�/ , � log Pr ŒG D gj�� (3.1)

D �IX

iD1

gi log

LX

lD�L

hl�il

!C

IXiD1

LXlD�L

hl�ilC

IXiD1

log gi ! . (3.2)

For a user specified penalty function L, PML estimates of the reflectance parameters are

obtained by solving the following optimization problem

(P3) O�PML D arg min�>0

P .�/; (3.3)

where the PML objective function P is given by

P .�/ , N .�/C ˇL .�/ : (3.4)

The constant ˇ, which is called the penalty parameter, controls the degree of influence of

the penalty function. Ideally, the penalty function L is an edge-preserving penalty function

in the sense that it forces neighboring pixel values to have similar values, unless they occur

at an edge. Like a number of researchers, the penalty functions we consider are of the form

L .�/ DIX

iD1

Xk2Bi

wik .�i; �k/; (3.5)

where is a cost function and Bi is a set of pixels in the neighborhood of the i th pixel that

excludes the i th pixel itself. Further, the neighborhoods fBig are defined in such a way that

for a pixel k in Bi , it is always true that the pixel i is in Bk . In our formulation, Bi is the

30

eight nearest neighbors of pixel i . The weight wik is chosen to be inversely proportional to

the distance between pixels i and k.

Let .u; v/ , � .u � v/, where the function � satisfies the following assumptions:

(AS1) � .t/ is symmetric

(AS2) � .t/ is differentiable everywhere

(AS3):

� .t/ , ddt� .t/ is increasing for all t

(AS4) e .t/ ,:

�.t/

tis non-increasing function for all t > 0

(AS5) � .0/ D limt!0

z .t/ is finite and nonzero

(AS6) � .t/ is bounded from below.

Note that assumption (AS3) implies that � is strictly convex function and assumption (AS6)

implies that L .�/ is bounded from below. Some of the functions that satisfy (AS1)-(AS6)

include Green’s log-cosh function [27] � .t/ D log .cosh .t// and the quadratic function

� .t/ D t2.

Since it is impossible to find a closed form solution for the minimization problem (P3),

we use the iterative majorization method to minimize the PML objective function P . The

PML objective function in (3.4) has two components: the negative log likelihood function

N and the penalty function L To determine a majorizing function for P , we use De Pierro’s

majorizing function for N and the majorizing function for L developed in [2].

Now, we introduce a majorizing function for the penalty function for L. In [1], Huber

developed a majorizing function for � under the assumptions (AS1)-(AS6). Given an

arbitrary point t .n/, Huber’s majorizing function for � is defined by

N�m

�t; t .n/

�, �

�t .n/�C :

�

�t .n/� �

t � t .n/�C 1

2e�t .n/� �

t � t .n/�2

; (3.6)

where e .t/ ,:

�.t/

t: The majorizing function has the following three properties:

31

(C3.1) N�m

�t; t .n/

�> � .t/ for all t

(C3.2) N�m

�t .n/; t .n/

� D � �t .n/�(C3.3) PN�m

�t .n/; t .n/

� D :

��t .n/�

.

Note, the dot over the function represents the first derivative of the function. Replacing t by

�i � �k and t .n/ by �.n/i � �.n/k, a majorizing function for L is given by

NLm

��;�.n/

�D

IXiD1

Xk2Bi

wikN m

��i; �k ; �

.n/i ; �

.n/

k

�; (3.7)

where N m

�u; v;u.n/; v.n/

�, N�m

�u � v;u.n/ � v.n/�. By the convexity of the square func-

tion, it is clear that the following inequality holds true

h�i � �k �

��.n/i � �.n/k

�i2 D�

1

2

�2�i � 2�

.n/i

�C 1

2

�2�

.n/

k� 2�k

��2

6�

1

2

�2�i � 2�

.n/i

�2 C 1

2

�2�

.n/

k� 2�k

�2�

. (3.8)

Using Huber’s[1] majorizing function N�m as a starting point, Chang and Anderson [2] used

the above inequality to construct a separable majorizing function for L defined as follows

m .�i; �k/ , ���.n/i � �.n/k

�C :

�

��.n/i � �.n/k

� h��i � �.n/i

����k � �.n/k

�iC 1

4e��.n/i � �.n/k

� ��2�i � 2�

.n/i

�2 C�2�k � 2�

.n/

k

�2�

. (3.9)

By construction, it is clear that the following statements hold true: (1) m .�i; �k/ >

.�i; �k/ for all �i > 0 and �k 2 Bi , (2) m

��

.n/

i ; �.n/

k

�D

��

.n/

i ; �.n/

k

�. The differ-

ence between m and N m is that m is de-coupled or separable in the sense that m does

not have coupled terms of the form �i�k . This de-coupled nature of m will later help us

find a closed form expression for the next iterate of the PML iteration. Given the advantages

of m as compared to N m, the majorizing function for L at the point �.n/ we use is

Lm

��;�.n/

�D

IXiD1

Xk2Bi

wik m

��i; �k ; �

.n/i ; �

.n/

k

�(3.10)

32

Using the majorizing function Nm for the negative log likelihood function N (see

(2.11)) and the majorizing function for the penalty function Lm, the majorizing function for

the PML objective function P at the point �.n/ is given by

Pm.�;�.n// D Nm.�;�

.n//C ˇLm

��;�.n/

�. (3.11)

From the properties of Nm and Lm, it follows that the majorizing function Pm satisfies:

(C3.4) Pm.�;�.n// > P .�/ for all � for all � > 0

(C3.5) Pm

��.n/;�.n/

� D P��.n/

�for �.n/ > 0

Now that we have developed a majorizing function for our PML objective function P , given

the previous iterate �.n/, the next iterate �.nC1/ can be found by minimizing the majorizing

function Pm at the point �.n/. Stated mathematically, an iterative algorithm for solving (P3)

is

(P4) �.nC1/ D arg min�>0

Pm

��;�.n/

�. (3.12)

We will now solve the minimization problem (P4). In [2], it is shown that Lm can be written

as

Lm

��;�.n/

�D 2

JXjD1

Xk2Bj

�.n/

jk

��j

�CC.n/

2 (3.13)

where

�.n/

jk

��j

�, wjke

��.n/j � �.n/k

� ��j � �jk

.n/�2

(3.14)

�ik.n/ ,

�.n/j C �.n/k

2(3.15)

33

C.n/

2 ,JX

jD1

Xk2Bj

wjk

����.n/j � �.n/k

�� 1

2

:

�

��.n/j � �.n/k

� ��.n/j � �.n/k

��. (3.16)

We can also rewrite the majorizing function for the negative log likelihood function in (2.11)

as

Nm

��;�.n/

�D �

IXiD1

gi

LXlD�L

d.n/

illog

��il

�C IXiD1

< h; �i >CC.n/

1 ; (3.17)

where

C.n/

1 , �IX

iD1

gi< h; �.n/i >

LXlD�L

d.n/

illog

1

�.n/il

!C

IXiD1

log gi ! . (3.18)

From (3.13) and (3.17), Pm can be written as

Pm

��;�.n/

�D Nm

��;�.n/;h.n/

�C ˇLm

��;�.n/

�(3.19)

DIX

iD1

< h; �i > � gi

LXlD�L

d.n/

illog

��il

�!

C 2ˇ

JXjD1

Xk2Bj

�.n/

jk

��j

�CC.n/

1 CC.n/

2 . (3.20)

In order to simplify (3.20), we have to find i and l values such that il D j . From the

discussion in Chapter 2 about the set Sj , f.i; l/ W il D j g (see discussion and (2.18)), the

majorizing function for the PML objective function can be written as follows

Pm

��;�.n/

�D

JXjD1

(TX

tD1

hbj t�j �

TXtD1

gaj td.n/

aj t bj tlog�j

)

C 2ˇ

JXjD1

8<:Xk2Bj

wjke��.n/j � �.n/k

���2

j � 2�jk.n/�j C

��jk

.n/�2�9=;

CC.n/

1 CC.n/

2 (3.21)

DJX

jD1

nX.n/

j log�j C Y.n/

j �2j CZ

.n/j �j

oCC

.n/

3 ; (3.22)

34

where

X.n/

j , �TX

tD1

gaj td.n/

aj t bj t(3.23)

Y.n/

j , 2ˇ

JXjD1

Xk2Bj

wjke��.n/j � �.n/k

�(3.24)

Z.n/j ,

TXtD1

hbj t� 4ˇ

JXjD1

8<:Xk2Bj

wike��.n/j � �.n/k

��jk

.n/

9=; (3.25)

C.n/

3 , C.n/

1 CC.n/

2 C 2ˇ

IXiD1

Xk2Bi

wike��.n/i � �.n/k

� ��ik

.n/�2

. (3.26)

Since Pm

��;�.n/

�is de-coupled, as can be seen from (3.22), the solution to (3.12) is given

by

(P5) �.nC1/j D arg min

�>0Pm;j

��j

�; j D 1; 2; :::;J: (3.27)

where Pm;j is defined to be

Pm;j .t/ , X.n/

j log .t/C Y.n/

j t2 CZ.n/j t . (3.28)

Before we try to solve (P5), lets first show that the function Pm;j is strictly convex for all

j and n under the assumption that �.n/j > 0 for all j and n. In [2], a similar function was

proven to be strictly convex by showing that the second derivative of a function similar to

Pm;j is positive when �.n/i > 0 for all j and n. For completeness and reader’s convenience,

we will now provide our proof by mimicking their own. First note that X.n/

j is negative and

Y.n/

j is positive for all j and n. The fact that the function e, weights˚wjk

, and ˇ are all

positive is the reason for Y.n/

j > 0. Recall by (AS1) and (AS3) that � .t/ is symmetric and

strictly convex function. It follows that:

� .t/ > 0 over .0;1/ and:

� .t/ < 0 over .�1; 0/.

35

Using the fact that e .0/ is finite and non zero (see (AS5)), we have that e .t/ > 0 for

�1 < t <1. Using easy calculations, the second derivative of Pm;j can now be shown

to be RPm;j .t/ D��X

.n/

j

t2 C 2Y.n/

j

�, where the double dot over a function represents the

second derivative of a function. Since Y.n/

j > 0 and X.n/

j < 0, the second derivative of Pm;j

is positive for all j and n. Therefore, Pm;j is strictly convex for all j and t > 0. It can then

be concluded from (3.22), that Pm is strictly convex over the set f� W � > 0g. Since Pm .�/

is de-coupled, Pm;j .t/ is strictly convex, and Pm;j .t/!1 as t ! 0C, it follows that

�.nC1/j > 0 and PPm;j

��.nC1/j

�D 0. (3.29)

Note that (3.29) satisfies the assumption that �.n/ > 0 for all j .

Now that we have shown strict convexity of the majorizing function for the PML

objective function, the minimization problem (P5) can easily be solved by computing the

first derivative of Pm;j and setting it to zero. Since X.n/

j < 0 and Y.n/

j > 0, the root of the

quadratic equation in (3.28) that satisfies the non-negativity constraint is

�.nC1/j D

�Z.n/j C

r�Z.n/j

�2 � 8X.n/

j Y.n/

j

4Y.n/

j

; j D 1; 2; :::;J: (3.30)

Given a strictly positive initial estimate �.0/ > 0, the PML algorithm can be summarized as

follows:

For n D 1; 2; :::;

Step 1 Construct majorizing function for P from the current iterate �.n/

using.3:22/ � .3:25/:

Step 2 Estimate �.nC1/using.3:30/:

Until stopping criterion is satisfied

36

Chapter 4. Maximum Likelihood Blind Image Deconvolution Algorithm

In this chapter, we use the ML method to develop a BID algorithm for jointly estimating

the reflectance means and PSF coefficients.

From discussions in Section 1.5.3, it follows that the negative log likelihood function

under the assumption that the PSF coefficients are unknown is given by

N .�;h/ , � log Pr ŒG D gj�;h� (4.1)

D �IX

iD1

gi log

LX

lD�L

hl�il

!C

IXiD1

LXlD�L

hl�ilC

IXiD1

log gi ! (4.2)

D �IX

iD1

gi log< h;�i >CIX

iD1

< h;�i >CIX

iD1

log gi ! : (4.3)

where < h;�i >,LP

lD�L

hl�il. The optimization problem for obtaining ML estimates of

the reflectance means and the PSF coefficients follows

(P6)� O�; Oh� D arg min

�>0;h>0N .�;h/ : (4.4)

To solve (P6), we use an alternating minimization approach that has been used by Holmes

[18] and other researchers (see [21] for example). Given the current estimates for the

reflectance means �.n/ and PSF coefficients h.n/, we obtain an improved estimate for the

reflectance means by minimizing N��;h.n/

�with respect to �, where h.n/ is the current

estimate of PSF coefficients. The resulting estimate �.nC1/ is then used to obtain the next

estimate for the PSF coefficients h.nC1/ by minimizing N��.nC1/;h

�. Summarizing, the

ML BID consists of the following steps:

37

Get initial estimates �.0/ > 0, and h.0/ > 0

Step 1. �.nC1/ D arg min�>0

N��;h.n/

�(4.5)

Step 2. h.nC1/ D arg minh>0

N��.nC1/;h

�(4.6)

Step 3. Iterate between Step 1 and Step 2 until chosen convergence criterion is met.

The optimization problems in Steps 1 and 2 are difficult to solve. Therefore, for practical

reasons, we replace Step 1 and Step 2 above with Step 1a and Step 2a respectively.

Step 1a: Find �.nC1/ � 0 such that N��.nC1/;h.n/

�� N

��.n/;h.n/

�(4.7)

Step 2a: Find h.nC1/ � 0 such that N��.nC1/;h.nC1/

�� N

��.nC1/;h.n/

�(4.8)

In Chapter 2, where the PSF coefficients were assumed to be known, we constructed

a majorizing function Nm that enabled us to determine iterates such that N��.nC1/

� �N��.n/

�: Thus, by viewing h.n/ as the known PSF coefficient in Step 1a, it follows from

(2.24) that a solution to Step 1a is

�.nC1/j D �.n/j

1

LPlD�L

hl

TXtD1

gaj thbj t

< h; �.n/aj t>; j D 1; 2; :::;J: (4.9)

The focus of the remainder of this chapter is to determine a solution to Step 2a .

Towards this end, we construct a majorizing function for the function N��.nC1/; � � at an

arbitrary point h0, where h0 is T � 1, that satisfies the following conditions:

(C4.1) Q��.nC1/;h;h0

�> N

��.nC1/;h

�, for all h > 0

(C4.2) Q��.nC1/;h0;h0

� D N��.nC1/;h0

�for h0 > 0 :

From (4.1), N��.nC1/;h

�can be expressed as

38

N��.nC1/;h

�D �

IXiD1

gi log< h;�.nC1/i >C

IXiD1

< h;�.nC1/i >

CIX

iD1

log gi ! (4.10)

D �IX

iD1

gi log

LX

lD�L

hl�.nC1/il

!C

IXiD1

< h;�.nC1/i >

CIX

iD1

log gi ! (4.11)

D �IX

iD1

gi log

LX

lD�L

hl�.nC1/il

h.n/

l< h.n/;�

.nC1/i >

h.n/

l< h.n/;�

.nC1/i >

!

CIX

iD1

< h;�.nC1/i >C

IXiD1

log gi ! ; (4.12)

where< h;�.nC1/i > ,

LPlD�L

hl�.nC1/il

, < h.n/;�.nC1/i > ,

�LP

kD�L

h.n/

k�.nC1/ik

�; h.n/ � 0;

and �.nC1/ � 0. Continuing, we write N��.nC1/;h

�as

N��.nC1/;h

�D �

IXiD1

gi log

LX

lD�L

d.n/

ilhl

< h.n/;�.nC1/i >

hl.n/

!

CIX

iD1

< h;�.nC1/i >C

IXiD1

log gi ! ; (4.13)

where d.n/

il,

h.n/

l�

.nC1/

il LP

kD�L

h.n/

k�

.nC1/

ik

! . Using the concavity of the log likelihood function [32,

pp. 860 – 862] and the fact thatLP

lD�L

d.n/

ilD 1, we obtain the following inequality

log

LX

lD�L

d.n/

ilhl

< h.n/;�.nC1/i >

h.n/

l

!�

LXlD�L

d.n/

illog

hl

< h.n/;�.nC1/i >

h.n/

l

!:

(4.14)

Applying the inequality in (4.14) to (4.13) yields the desired result

39

N��.nC1/;h

�6 �

IXiD1

gi

LXlD�L

d.n/

illog

hl

< h.n/;�.nC1/i >

h.n/

l

!

CIX

iD1

< h;�.nC1/i >C

IXiD1

log gi ! . (4.15)

We will now show that the function on the RHS of (4.15) ,which we denote by Q��.nC1/; � ;h.n/�,

is a majorizing function for the function N��.nC1/; � � at the point h.n/.

Q��.nC1/;h;h.n/

�, �

IXiD1

gi

LXlD�L

d.n/

illog

hl

< h.n/;�.nC1/i >

h.n/

l

!

CIX

iD1

< h;�.nC1/i >C

IXiD1

log gi ! : (4.16)

By construction, it is clear that the function Q��.nC1/; � ;h.n/� satisfies (C4.1) for�.nC1/ �

0 and h.n/ � 0. Next, we show that (C4.2) is also satisfied. Straightforward calculations

lead to the result below

Q��.nC1/;h.n/;h.n/

�D �

IXiD1

gi

LXlD�L

d.n/

illog < h.n/;�

.nC1/i >

CIX

iD1

LXlD�L

hl.n/�

.nC1/il

CIX

iD1

log gi ! (4.17)

D �IX

iD1

gi log < h.n/;�.nC1/i >

LXlD�L

d.n/

il

CIX

iD1

< h.n/;�.nC1/i >C

IXiD1

log gi ! . (4.18)

Using the fact thatLP

lD�L

d.n/

ilD 1, it follows that

40

Q��.nC1/;h.n/;h.n/

�D �

IXiD1

gi log < h.n/;�.nC1/i >

CIX

iD1

< h.n/;�.nC1/i >C

IXiD1

log gi ! (4.19)

D N.�.nC1/;h.n//. (4.20)

Thus, the function Q��.nC1/; � ;h.n/� satisfies (C4.2).

At this point, we have developed a majorizing function for N��.nC1/; � � using ideas

from De Pierro’s work [30, 31]. To obtain the next iterate h.nC1/ defined in Step 2a, we

must solve the following optimization problem

(P7) h.nC1/ D arg minh>0

Q��.nC1/;h;h.n/

�: (4.21)

For s D �L; .�L C 1/; :::; .L � 1/; L ; the derivative of the majorizing function

Q��.nC1/;h;h.n/

�with respect to hs is given by

@

@hs

Q��.nC1/;h;h.n/

�D @

@hs

�IX

iD1

gi

LXlD�L

d.n/

il

log hl C log

< h.n/;�.nC1/i >

h.n/

l

!

C @

@hs

IXiD1

< h;�.nC1/i >C @

@hs

IXiD1

log gi ! (4.22)

D �IX

iD1

gid.n/is

�1

hs

�C

IXiD1

�.nC1/is

. (4.23)

Ignoring the constraint h � 0 momentarily, a necessary condition for h.nC1/ would be

@

@hs

Q��.nC1/;h;h.n/

�ˇ̌̌̌hsDh

.nC1/s

D 0; (4.24)

or equivalently

�IX

iD1

gid.n/is

�1

h.nC1/s

�C

IXiD1

�.nC1/is

D 0: (4.25)

41

After replacing d.n/is by

h.n/s �

.nC1/

is

<h.n/;�.nC1/

i>

in (4.25) and solving the resulting equation, a solution

to Step 2a is given by

h.nC1/s D h.n/s

1

IPiD1

�.nC1/is

IXiD1

gi�.nC1/is

< h.n/;�.nC1/i >

; s D �L; .�LC 1/; :::; .L� 1/;L:

(4.26)

Since gi � 0 for all i , it can be seen from (4.26) that h.nC1/s � 0 for all s provided that

�.0/ � 0 and h.0/ � 0.

The ML BID algorithm can be summarized as follows:

Get initial estimates �.0/ > 0, and h.0/ > 0

For n D 0; 1; 2; :::;

Step 1 Estimate �.nC1/ from �.n/ and h.n/ using .4:9/:

Step 2 Estimate h.nC1/ from �.nC1/ and h.n/using .4:26/:

Iterate between Steps 2 and 3 until chosen convergence criterion is met.

42

Chapter 5. Penalized Maximum Likelihood Blind Image Deconvolution Algorithm

In this chapter, we present a BID algorithm where both the reflectance parameters and

PSF coefficients are estimated using the penalized maximum likelihood method.

As discussed in Chapter 4, the negative log likelihood function when the PSF coeffi-

cients are unknown is given by

N .�;h/ , � log Pr ŒG D gj�;h� (5.1)

D �IX

iD1

gi log< h;�i >CIX

iD1

< h;�i >CIX

iD1

log gi ! : (5.2)

Unfortunately, the problem of minimizing N .�;h/ with respect to � and h is ill-posed.

Therefore, we propose a penalized maximum likelihood method that incorporates penalty

functions K and L for the PSF coefficients and reflectance parameters, respectively

(P8)� O�; Oh� D arg min

�>0;h>0N .�;h/C ˇL .�/C ˛K .h/ : (5.3)

The strictly positive constants ˛ and ˇ are penalty parameters that control the influence of

the penalty functions K and L, respectively.

Given initial estimates �.0/ > 0 and h.0/ > 0, an alternating minimization algorithm

for solving the optimization problem (P8) is

Step 1. �.nC1/ D arg min�>0

N��;h.n/

�C ˇL .�/ (5.4)

Step 2. h.nC1/ D arg minh>0

N��.nC1/;h

�C ˛K .h/ (5.5)

Step 3. Iterate between Step 1 and Step 2 until chosen convergence criterion is met.

For convenience, we define the objective function in the optimization problem of Step 2 as

43

follows

R��.nC1/;h

�, N

��.nC1/;h

�C ˛K .h/ : (5.6)

Closed form expressions for the solutions of the optimization problems in Steps 1 and 2

are unavailable. Consequently, we replace Step 1 and Step 2 with Step 1b and Step 2b,

respectively:

Step 1b: Find �.nC1/ such that

N��.nC1/;h.n/

�C ˇL

�h.n/

�� N

��.n/;h.n/

�C ˇL

�h.n/

�Step 2b: Find h.nC1/ such that R

��.nC1/;h.nC1/

�� R

��.nC1/;h.n/

�: (5.7)

Exploiting the results in Chapter 3 and viewing h.n/ as the known PSF coefficients in Step

1b, a solution to Step 1 is given by (3.30) with h replaced by h.n/.

5.1. Penalized Maximum Likelihood Algorithm for Estimating the Point SpreadFunction With Unity DC Gain Penalty Function

In this section, we will determine a solution to Step 2b when the penalty function K is

chosen so that the sum of the PSF coefficients approximately equals one. We choose K in

this way so that the sum of the observed and restored images are approximately equal. The

implication is that the overall brightness of the observed and restored images are the same.

We refer to the penalty function as the unity DC Gain penalty function becauseLP

lD�L

hl D 1

implies that the Fourier transform at � D 0 equals one. Given, the above discussion, the

unity DC gain penalty function is defined as follows

K.h/ ,

LXlD�L

hl � 1

!2

: (5.8)

We will now find a majorizing function Rm for the objective function R��.nC1/; � �

at the point h.n/. Since R��.nC1/; � � is a sum of two functions N

��.nC1/; � � and K , we

have to find majorizing functions for each of these functions. Using the results in Chapter 4 ,

44

a majorizing function for N��.nC1/; � � at a point h.n/ is given by

Q��.nC1/;h;h.n/

�, �

IXiD1

gi

LXlD�L

d.n/

illog

hl

< h.n/;�.nC1/i >

h.n/

l

!

CIX

iD1

< h;�.nC1/i >C

IXiD1

log gi ! : (5.9)

where d.n/

il,

h.n/

l�

.nC1/

il LP

kD�L

h.n/

k�

.nC1/

ik

! . Once we find a majorizing function Km for the penalty

term K , a solution to Step 2b is the minimizer of the function Rm

��.nC1/; � ;h.n/�

(P10) h.nC1/ D arg minh>0

Rm

��.nC1/;h;h.n/

�; (5.10)

where the majorizing function for R��.nC1/; � � at a point h.n/ is defined as

Rm

��.nC1/; � ;h.n/

�, Q

��.nC1/; � ;h.n/

�C ˛Km

�� ;h.n/

�: (5.11)

We will now start with the definition of K and construct a majorizing function Km for

the penalty function. We first express K in the following manner

K.h/ ,

LXlD�L

hl � 1

!2

(5.12)

D

LXlD�L

hl

!2

� 2 NhC 1; (5.13)

where Nh ,LP

lD�L

hl : Now, it will be convenient to express the penalty function as

K.h/ D

LXlD�L

hl

h.n/

l

h.n/

l

Nh.n/Nh.n/

!!2

� 2 NhC 1 (5.14)

D

LXlD�L

a.n/

l

Nh.n/h.n/

l

hl

!!2

� 2 NhC 1 ; (5.15)

where Nh.n/ ,LP

kD�L

h.n/

kand a

.n/

l, h

.n/

l

Nh.n/. Using the convexity of the square function and

45

the fact thatLP

lD�L

a.n/

lD 1, the following inequality holds true

LX

lD�L

a.n/

l

Nh.n/h.n/

l

hl

!2

6LX

lD�L

a.n/

l

Nh.n/h.n/

l

hl

!2

: (5.16)

Therefore, it follows that

K.h/ 6� Nh.n/�2

LXlD�L

0@a.n/

l

hl

h.n/

l

!21A � 2 NhC 1. (5.17)

We will now show that the function on the RHS of (5.17) is a majorizing function for the

penalty function K .

Let Km. � ;h.n// denote the function on the RHS of (5.17)

Km.h;h.n// ,

� Nh.n/�2LX

lD�L

a.n/

l

hl

h.n/

l

!2

� 2 NhC 1. (5.18)

In order for Km to be the majorizing function for the penalty function, it must satisfy the

following two conditions:

(C5.1) Km

�h;h.n/

�> K .h/ , for all h > 0

(C5.2) Km

�h.n/;h.n/

� DK�h.n/

�, for h.n/ > 0.

Clearly, by construction, (C5.1) is satisfied for h.n/ � 0. All that remains is to show that

(C5.2) is satisfied as well. SinceLP

lD�L

a.n/

lD 1 , evaluating Km

�h;h.n/

�at h.n/ yields the

following expression

Km

�h.n/;h.n/

�D� Nh.n/�2

LXlD�L

a.n/

l

h.n/

l

h.n/

l

!2

� 2 NhC 1 (5.19)

D� Nh.n/�2 � 2 Nh.n/ C 1 (5.20)

DK�h.n/

�. (5.21)

Since both Q and Km satisfy the requisite conditions for majorizing functions, it

46

follows by construction that a majorizing function for R is

Rm

��.nC1/;h;h.n/

�, Q

��.nC1/;h;h.n/

�C ˇKm

�h;h.n/

�: (5.22)

The reason is because Rm

��.nC1/; � ;h.n/� satisfies the following conditions:

(C5.3) Rm

��.nC1/;h;h.n/

�> R .h/ , for all h > 0

(C5.4) Rm

��.nC1/;h.n/;h.n/

� D R�h.n/

�, for h.n/ > 0.

All that remains now is to solve the minimization problem in (5.10). We will find the

minimizer of Rm

��.nC1/; � ;h.n/� by initially ignoring the constraint h � 0; and com-

puting its derivative with respect to hs and setting the result to zero. The derivative of

Rm

��.nC1/; � ;h.n/� with respect to hs can be written as

@

@hs

Rm

��.nC1/;h;h.n/

�D @

@hs

Q��.nC1/;h;h.n/

�C ˛ @

@hs

Km

�h;h.n/

�. (5.23)

In Chapter 4, we showed that derivative of the majorizing function Q��.nC1/; � ;h.n/�

with respect to hs is given by (see (4.23))

@

@hs

Q��.nC1/;h;h.n/

�D �

IXiD1

gid.n/is

�1

hs

�C

IXiD1

�.nC1/is

. (5.24)

Straightforward calculations lead to the following expression for the derivative of the

majorizing function Km with respect to hs

@

@hs

Km

�h;h.n/

�D @

@hs

24� Nh.n/�2LX

lD�L

a.n/

l

hl

h.n/

l

!2

� 2 NhC 1

35 (5.25)

D� Nh.n/�2

a.n/s

�1

h.n/s

�2

2hs � 2. (5.26)

After replacing a.n/s by h

.n/s

Nh.n/, we arrive at the following

@

@hs

Km

�h;h.n/

�D Nh.n/ 2hs

h.n/s

� 2 : (5.27)

47

Substituting (5.24) and (5.27) into (5.23), yields the derivative of the majorizing function

Rm

��.nC1/; � ;h.n/� with respect to hs

@

@hs

Rm

��.nC1/;h;h.n/

�D �

IXiD1

gid.n/is

�1

hs

�C

IXiD1

�.nC1/is

(5.28)

C

LXkD�L

h.n/

k

!˛2hs

h.n/s

� 2˛. (5.29)

Ignoring the constraint h � 0 momentarily, the necessary condition for the minimizer in

(P10) is

@

@hs

Rm

��.nC1/;h;h.n/

�jhsDh

.nC1/sD 0 (5.30)

Or, equivalently,

�IX

iD1

gid.n/is

�1

h.nC1/s

�C

IXiD1

�.nC1/is

C

LXkD�L

h.n/

k

!2˛h

.nC1/s

h.n/s

� 2˛ D 0: (5.31)

After replacing d.n/is in (5.31) by

0@ h.n/s �

.nC1/

is

LPlD�L

h.n/

l�

.nC1/

il

1A, we arrive at the following equivalent

expression0BBB@2˛

LPkD�L

h.n/

k

h.n/s

1CCCA�h.nC1/s

�2 C

IXiD1

�.nC1/is

� 2˛

!h.nC1/

s

�IX

iD1

gi

0BBB@ h.n/s �

.nC1/is

LPlD�L

h.n/

l�.nC1/il

1CCCA D 0. (5.32)

For convenience, we introduce the following simplified notation

a ,2˛

LPkD�L

h.n/

k

h.n/s

: (5.33)

48

b ,IX

iD1

�.nC1/is

� 2˛ : (5.34)

c , h.n/s

IXiD1

0BBB@ �gi�.nC1/is

LPlD�L

hl.n/�

.nC1/il

1CCCA : (5.35)

Considering the non-negativity constraint h � 0, we are only interested in the positive root

of the quadratic equation in 5.32. For h.0/ > 0 and �.0/ > 0, it follows that a > 0 and c � 0.

Therefore, the next iterate for hs is given by

h.nC1/s D �b Cpb2 � 4ac

2a; s D 1; 2; :::; .2LC 1/: (5.36)

Summarizing, the proposed PML BID algorithm consists of the steps below:

Get initial estimates �.0/ � 0, and h.0/ � 0

For n D 0; 1; 2; :::;

Step 1 Estimate �.nC1/ from �.n/ and h.n/ using .3:30/:

Step 2 Estimate h.nC1/ from �.nC1/ and h.n/using .5:33/ � .5:36/:

Until stopping criterion is satisfied

49

Chapter 6. Experimental Study and Simulation Results

The algorithms developed in Chapters 2, 3, 4, and 5 were implemented using Matlab

and their performances were assessed using images with real and synthetic blur. To test the

image deconvolution algorithms of Chapters 2 and 3, we convolved the well known Lena

image with a lowpass filter. A landmine field image as well as an image of the spiral galaxy,

Messier 61 (NGC 4303), were used in the experiments to test the BID algorithms in Chapters

4 and 5. We now present the simulation results for each of the algorithms developed.

6.1. Maximum Likelihood Image Deconvolution

The MLEM algorithm developed in Chapter 2 was applied to Lena, an image widely

used to test image processing algorithms. In this experiment, it is assumed that the true PSF

coefficients are known where the problem is to estimate the reflectance means f�ig. For

the observed image, Lena was blurred with a known 5 � 5 gaussian PSF with � D 3 and

then Poisson noise was applied to it. The MLEM algorithm produces iterates that become

noisier and noisier with increasing iterations. Figure 6.1 shows the result of the MLEM

algorithm after applying it to the blurred Lena image. Figures 6.1 (a) and (b) show the true,

and observed images respectively. Observing Figures 6.1 (c)-(h), it can be seen that the best

quality reconstruction occures at the 10th iteration. After the 10th iteration, the ML estimates

become noisier and noisier.

50

PSfrag replacements

(a) True Image (b) Blurred Image with Poisson Noise

(c) MLEM: 10 (d) MLEM: 20

(e) MLEM: 30 (f) MLEM: 40

(g) MLEM: 50 (h) MLEM: 60

5010015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

100150200250

Figure 6.1. MLEM results: (a) true image, (b) blurred image with Poisson noise, (c) MLEMresult at 10th iteration, (d) MLEM result at 20th iteration, (e) MLEM resultat 30th iteration, (f) MLEM result at 40th iteration, (g) MLEM result at 50th

iteration, and (h) MLEM result at 60th iteration.

51

In Figures 6.2 (d), we show the plot of the ML objective function, which is the negative

log likelihood function. The true, observed, and restored images are shown in Figures 6.2

(a), (b), and (c) respectively. As mathematically proven in Chapter 2, the objective function

monotonically decreases with increasing iterations.

PSfrag replacements

(a) True Image (b) Blurred Image with Poisson Noise

(c) MLEM: 10 (d) ML Objective Function (�107)

2 4 6 8 10

5010015020025050

10015020025050

100150200250

-2.4078

-2.4076

-2.4074

-2.4072

-2.407

5010015020025050

10015020025050

100150200250

Figure 6.2. MLEM results: (a) true image, (b) blurred image with Poisson noise, (c) MLEMresult at the 10th iteration, and (d) objective function plot.

We now investigate the performance of the MLEM algorithm when it is applied to

simply the blurred Lena image (i.e., no noise). The true image, noiseless blurred image, and

20th MLEM iterate are shown in Figures 6.3 (a), (b), and (c), respectively. As expected, the

MLEM algorithm is more efficient when it is applied to blurred images without Poisson

noise. As in the noisy case, the plot of the negative log likelihood function shown in

Figure 6.3 (d) is monotonically decreasing by the MLEM algorithm.

52

We see from Figure 6.4 that the MLEM algorithm did not produce noisy iterates when

Poisson noise was absent. Figures 6.4 (a) and(b) show the true and observed Lena images

respectively whereas Figures 6.4 (c)-(h) show the result of the MLEM algorithm at different

iterations.

PSfrag replacements

(a) True Image (b) Blurred Image w/o Poisson Noise

(c) MLEM: 20 (d) ML Objective Function (�107)

5 10 15 20

5010015020025050

10015020025050

100150200250

-2.4054

-2.4052

-2.405

-2.4048

5010015020025050

10015020025050

100150200250

Figure 6.3. MLEM result: (a) true image, (b) blurred image without Poisson noise, (c)MLEM result at 20th iteration, and (d) objective function plot.

53

PSfrag replacements

(a) True Image (b) Blurred Image w/o Poisson Noise

(c) MLEM: 10 (d) MLEM: 20

(e) MLEM: 30 (f) MLEM: 40

(g) MLEM: 50 (h) MLEM: 60

5010015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

10015020025050

100150200250

Figure 6.4. MLEM results: (a) true image, (b) blurred image without Poisson noise, (c)MLEM result at 10th iteration, (d) MLEM result at 20th iteration, (e) MLEMresult at 30th iteration, (f) MLEM result at 40th iteration, (g) MLEM result at50th iteration, and (h) MLEM result at 60th iteration.

54

6.2. Penalized Maximum Likelihood Image Deconvolution

In Chapter 3, a PML algorithm with a class of separable majorizing function was

presented for estimating the reflectance means f�ig, with the assumption that the PSF

coefficients are known. We will now follow with the simulation results of the PML algorithm

applied to the same synthetic image used in Section 6.1. The true and the blurred images

are shown in Figures 6.5 (a) and (b) respectively whereas Figure 6.5 (c) shows the restored

image at convergence. Note that the convergence criterion is defined by the user. In this

and all PML based algorithms to follow, we let the algorithm stop once the difference

between the mean square error of two successive iterates fall below � D 0:00001. As

seen from Figure 6.5 (d), the PML objective function is monotonically decreasing with

increasing iterations. The synthetic image in Figure 6.6 (a) was blurred without Poisson

PSfrag replacements

(a) True Image (b) Blurred image with Poisson noise

(c) PML: 129, beta = 500 (d) PML Objective Function (�107)

0 50 100 150

5010015020025050

10015020025050

100150200250

-2.408

-2.4075

-2.407

-2.4065

5010015020025050

10015020025050

100150200250

Figure 6.5. PML results: (a) true image, (b) blurred image with Poisson noise, (c) PMLdeblurred at 129th iteration, ˇ D 500, and (d) objective function plot.

noise (see Figure 6.6 (b)). The blurred image was then restored using the PML algorithm.

55

The optimal ˇ value for the noiseless case was found to be ˇ D 20. Figure 6.6 (c) shows the

reconstructed image at convergence. As expected, the objective function shown in Figure 6.6

(d) decreases monotonically.

PSfrag replacements

(a) True Image (b) Blurred Image w/o Poisson Noise

(c) PML: 60, beta=20 (d) PML Objective Function (�107)

0 20 40 60

5010015020025050

10015020025050

100150200250

-2.4056

-2.4054

-2.4052

-2.405

-2.4048

-2.4046

5010015020025050

10015020025050

100150200250

Figure 6.6. PML results: (a) true image, (b) blurred image without Poisson noise, (c) PMLresult at 60th iteration, ˇ D 20, and (d) objective function plot.

The choice of ˇ dictates the outcome of the PML algorithm. Since ˇ controls the

influence of the penalty function L on the PML objective function P , using a ˇ value too

large makes the algorithm give heavier weight to the neighbor smoothing penalty function

L, rather than minimizing the negative log likelihood function. The result is an image

smoothed too much such that it looses some contrast information. Such an image is shown

in Figure 6.7 (e). Figures 6.7 (a) and (b) show the true and observed images respectively.

Also, by using too small ˇ value, we can make the algorithm give little weight to the penalty

term and focus on minimizing the negative log likelihood function. Note that the smaller

the ˇ value we use, the closer the PML algorithm gets to the MLEM algorithm where we

56

had noise build up problem. Figure 6.7 (f) shows a noisy image reconstructed using too

small ˇ value. The most important question is then how to determine the penalty parameter

value ˇ � 0, that balances between image deconvolution and noise reduction. The answer

is no one came up with a theoretical method to determine the optimal ˇ value as of now.

Therefore, ˇ must be determined by experiment. After trying several ˇ values, the optimal

value that balances between image deconvolution and noise reduction was found to be

ˇ D 500 for the synthetic image with Poisson noise and ˇ D 20 for the noiseless case.

The reconstructed image with the optimal ˇ value, and the plot of its objective function

are shown in Figures 6.7 (c) and (d) respectively for the noisy image. Also, note that the

number of iterations required for convergence is different for different ˇ values. In the case

where we used too large value for the noisy image, ˇ D 1000, 87 iterations were enough to

converge. The required number of iterations was determined to be 129 for the optimal value

of ˇ D 500. 322 iterations were required for the noisy image in Figure 6.7 (f) to converge

when ˇ D 5 was used, which was too small for the algorithm to give enough weight so that

the noise is suppressed in the reconstructed image. Comparison of different ˇ value was

also done for the case where the blurred image has no Poisson noise. The reconstructed

image with the optimal ˇ value, ˇ D 20 and the plot of its objective function are shown in

Figures 6.8 (c) and (d) respectively for the noiseless blurred image. Figure 6.8 (b) shows the

blurred image without Poisson noise whereas (e) and (f) of Figure6.8 show PML images

reconstructed with too large and too small ˇ values respectively. The true image is shown

in Figure6.8 (a). Note that for the case where ˇ was too small, the reconstructed image

resembles that of the MLEM algorithm. Recall from Section 6.1 that MLEM reconstruction

was not as bad for noiseless images as it was for blurred images with Poisson noise.

57

PSfrag replacements

(a) True Image (b) Blurred Image with Poisson Noise

(c) PML: 129, beta = 500 (d) PML Objective Function (�107)

(e) PML: 87, beta = 1000 (f) PML: 322, beta = 5

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Figure 6.7. PML deblurred images with different ˇ values and plot of the objective functionfor noisy observed image: (a) true image, (b) blurred image with Poisson noise,(c) PML result at 129th iteration with ˇ D 500, (d) objective function plot,(e) PML result at 87th iteration with ˇ D 1000 , and (f) PML result at 322th

iteration with ˇ D 5 .

58

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(a) True Image (b) Blurred Image w/o Poisson Noise

(c) PML: 60, beta=20 (d) PML Objective Function (�107)

(e) PML: 47, beta=200 (f) PML: 55, beta=0.002

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Figure 6.8. PML deblurred images with different ˇ values and plot of the objective functionfor noiseless observed image: (a) true image, (b) blurred image without Poissonnoise, (c) PML result at 60th iteration with ˇ D 60, (d) objective functionplot, (e) PML result at 47th iteration with ˇ D 200, and (f) PML result at 55th

iteration with ˇ D 0:002 .

59

6.3. Maximum Likelihood Blind Image Deconvolution

In Chapter 4, an alternating BID algorithm was developed to jointly estimate the

reflectance means f�ig and the PSF coefficients fhlg based on the ML estimation method

without incorporating any kind of penalty function. Although, the alternating BID algorithm

estimates both f�ig and fhlg at the same time, the noise build up problem of ML estimation

method still exists. Figure 6.9 shows the different iterates of the ML BID algorithm.

Figures 6.9 (a), (b), (g), and(h) show the true image, observed image, the initial PSF and the

estimated PSF.

It is obvious from the images of Figures 6.9 (c)-(f) that the noise build up is not any

better than the MLEM algorithm with a known PSF. However, the one advantage that the

ML BID algorithm has is that we don’t need to know the actual PSF that is responsible for

the blur. Instead, we can just start the algorithm with a uniform PSF as the initial guess.

However, the more the initial guess is closer to the true PSF, the better the final ML estimate

will be. Figure 6.10 is the result of the ML image deconvolution algorithm applied to

a real landmine field image (HYDICE image) obtained from the Countermine Division

of the Night Vision and Electronic Sensors Directorate in Fort Belvoir, Virginia. Visual

comparison of the observed image with the reconstructed image at the 50th iteration shows

that the reconstructed image in Figure 6.10 (f) reveals less mine signatures than the observed

image in Figure 6.10 (a). The reason for the disappearance of the small mine signatures is

that the noise build up is covering up the contrast of mine signatures. Observing the images

Figures 6.10 (b) through (f), it is obvious that the mine signatures are disappearing with

increasing iterations. Figures 6.10 (g) and (h) depict the initial guess and the final estimate

of the PSF.

60

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(a) True Image (b) Blurred image with Poisson noise

(g) Starting PSF

(c) ML-based blind deblurred: 10 (d) ML-based blind deblurred: 20

(e) ML-based blind deblurred: 30 (f) ML-based blind deblurred: 40

(h) Estimated PSF

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Figure 6.9. ML BID results for an image blurred with Poisson noise: (a) true image, (b)blurred image with Poisson noise, (c) blind deblurred at 10th iteration, (d)blind deblurred at 20th iteration, (e) blind deblurred at 30th iteration, (f) blinddeblurred at 40th iteration, (g) starting PSF, and (h) estimated PSF (sum=1.3) .

61

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(a) Observed image

(g) Starting PSF

(b) ML-based blind deblurred: 10

(c) ML-based blind deblurred: 20 (d) ML-based blind deblurred: 30

(e) ML-based blind deblurred: 40 (f) ML-based blind deblurred: 50

(h) Estimated PSF

1 2 3 4 5

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Figure 6.10. ML BID results for landmine field image: (a) observed image, (b) blind de-blurred at 10th iteration, (c) blind deblurred at 20th iteration, (d) blind deblurredat 30th iteration, (e) blind deblurred at 40th iteration, (f) blind deblurred at 50th

iteration, (g) starting uniform PSF, and (h) estimated PSF (Sum = 1.002) .

62

6.4. Penalized Maximum Likelihood Blind Image Deconvolution

A PML based alternating BID algorithm was developed in Chapter 5 for joint estimation

of the reflectance means f�ig and the PSF coefficients fhlg. In Section 6.2, we have identified

ˇ D 500 to be the optimal penalty parameter value for the neighbor smoothing penalty

function L with the noiseless blurred synthetic image. In this section, we use ˇ D 500,

for the PML BID experiment with the synthetic data. For the landmine and astronomical

images, the optimal ˇ values were experimentally determined to be ˇ D 100 and ˇ D 400

respectively. Together with ˇ, the penalty parameter ˛ D 1000, ˛ D 0:1, and ˛ D 0:0001

for the synthetic, land mind, and astronomical images respectively, were used for controlling

the influence of the unity DC gain penalty function K on the PML minimization problem.

Again, just like ˇ, ˛ was determined by experiment. Figures 6.11 (a) and (b) show the true

and observed images. Also, the plot of the initial flat PSF and the final estimate of the PSF

are shown Figures 6.11 (e) and (f) respectively. It is obvious that the image in Figure 6.11 (c),

which is restored blindly out knowing the blurring PSF, is almost identical to the image in

Figure 6.5 (c), that is deblurred with the true PSF. The plot of the monotonically decreasing

objective function is shown in Figure 6.11 (d).

The PML based alternating BID algorithm was also tested using a real landmine image

(HYDICE image) shown in Figure 6.12 (a). As shown in Figure 6.12 (c), the algorithm

did not lead to improved contrast as hoped and we believe this is due to the fact that the

observed image did not have significant blur and also the intensity of the pixels with small

mine signature are very close to the pixel intensity of their background. Furthermore, the

unknown blurring PSF might be a reducible Gaussian PSF. Figure 6.12 (b) shows the result

of using too small ˇ value in which case, the deblurred image is much noisier with lost

contrast compared to the blurred/observed image in (a). Also, when too large penalty

parameter, ˇ D 5000, was used to deblur the landmine image, Figure 6.12 (b) reveals a

highly smoothed out image where most of the contrast is lost because of the effect of too

much smoothing due to the penalty function. The image in Figure 6.12 (c) is deblurred with

63

PSfrag replacements

(b) Blurred Image with Poisson noise

(d) PML Objective Function (�107)

(f) Estimated PSF

(a) True Image

(c) Blind PML: 110 iterations, beta=500

(e) Starting PSF

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Figure 6.11. PML BID results for an image blurred with Poisson noise: (a) true image,(b) blurred image with Poisson noise, (c) PML result at 110th iteration withˇ D 500 and ˛ D 1000, (d) objective function, (e) starting uniform PSF, and(f) estimated PSF (Sum = 1.002 ).

64

the optimal ˇ value which was found to be 100. The plot of the objective function using the

optimal ˇ value is shown in Figure 6.12 (d) whereas (e) of the same figure shows another

restored image with too large ˇ value.

We claim that the PML BID algorithm developed in Chapter 5 works as equally, if not

more, for astronomical images captured with CCD cameras as it works for a spectral plane of

hyperspectral images. To demonstrate this, we present, in Figure 6.13 (a), a blurred/observed

image of the bright spiral galaxy Messier 61 (NGC 4303) taken from NASA’s Hubble space

telescope and the restored image using optimal ˇ value in Figure 6.13 (c) along with the

plot of the monotonically decreasing objective function shown Figure 6.13 (d). It is obvious

from the images that some of the small or hazy stars, which are not showing in Figure 6.13

(a) seem to show up in Figure 6.13 (c). Moreover, the restored image appears to have lesser

noise and interstellar dust than the observed image. The smallest ˛ value that will keep the

sum of the PSF at unity was by experiment found to be ˛ D 0:0001 for the astronomical

image. Also, the optimal ˇ value for the same data was found to be ˇ D 400. In Figure 6.13

(f), we show the galaxy image restored with too small ˇ value (ˇ D 4) demonstrating that

the smaller the ˇ value we use that is far from the optimal value, the noisier the image

gets. On the contrary, Figure 6.13 (e) illustrates the image deblurred with too large ˇ value

(ˇ D 4000). The restored image in Figure 6.13 (e) and even worst in Figure 6.13 (b) shows

that the image is smoothed out too much to the point where it lost its contrast information.

65

PSfrag replacements

(a) Observed Image (b) Blind PML: 97 iterations, beta=5000

(c) Blind PML: 119 iterations, beta=100 (d) PML Objective Function (�1010)

(e) Blind PML: 88 iterations, beta=1000 (f) Blind PML: 100 iterations, beta=5

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-1.7964

-1.7964

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Figure 6.12. PML BID results for landmine field image with ˛ D 0:1: (a) observed image,(b) blind PML result with ˇ D 50000 after 97 iterations, (c) blind PML resultwith ˇ D 100 after 119 iterations, (d) objective function plot when ˇ D 100

(optimal ˇ value), (e) blind PML result with ˇ D 1000 after 88 iterations, (f)blind PML result with ˇ D 5 after 100 iterations.

66

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(a) Observed Image (b) Blind PML: 80 iterations, beta=400000

(c) Blind PML: 99 iterations, beta=400 (d) PML Objective Function (�107)

(e) Blind PML: 67 iterations, beta=4000 (f) Blind PML: 101 iterations, beta=4

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-7.6

-7.55

-7.5

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Figure 6.13. PML BID results for the bright spiral galaxy Messier 61 (NGC 4303) in theVirgo cluster with unity DC gain penalty function when ˛ D 0:0001: (a)observed image, (b) blind PML result with ˇ D 400000 after 80 iterations, (c)blind PML result with ˇ D 400 after 99 iterations, (d) objective function plotwhen ˇ D 400 (optimal ˇ value), (e) blind PML result with ˇ D 4000 after67 iterations, and (f) blind PML result with ˇ D 4 after 101 iterations.

67

Chapter 7. Conclusions and Future Work

7.1. Conclusions

In Chapter 2, we derived a modification of Shepp and Vardi’s MLEM algorithm for

estimating the reflectance means˚�j

under the assumption that the PSF coefficients fhlg

are known. The optimization problem that must be solved to obtain the desired ML estimates

is ill-posed and the negative log likelihood function is convex but not strictly convex. As

a result, after certain number of iterations, the reconstructed images become noisier and

noisier with increasing iterations.

The second algorithm presented was the PML algorithm developed in Chapter 3 for

estimating the reflectance means˚�j

under the assumption that the PSF coefficients, fhlg

are known. The PML algorithm can incorporate a class of penalty function,

L .�/ DIX

iD1

Xk2Bi

wik� .�i � �k/; (7.1)

where the function � satisfies the assumptions (AS1)-(AS6). The penalty functions are

designed in such a way that estimates of neighboring pixels are forced to be similar in value

unless there is an edge within the neighborhood. Our choice for � was � .t/ D log cosh .t/.

As shown in Section 6.2, the PML algorithm, unlike the MLEM algorithm, can produce

reconstructed images without excessive noise. However, the biggest challenge lies in

choosing the penalty parameter ˇ. To date, no one has developed theoretical way to

choose the penalty parameter. Thus, in practice, the penalty parameter must be obtained

experimentally through trial and error.

In Chapter 4, we derived an ML BID algorithm using an alternating minimization

technique to jointly estimate the reflectance means f�jg and the PSF coefficients fhlg. Like

the ML image deconvolution algorithm discussed in Chapter 2, the ML BID algorithm

68

produces images that become noisier with increasing iterations. Moreover, the reducible

nature of the Gaussian PSF implies that the ML BID algorithm can not be used for Gaussian

PSFs.

Finally, a PML BID algorithm was developed in Chapter 5. In Section 6.4, the algorithm

was shown to work as good as the PML algorithm with known PSF, and at the same time,

estimate the blurring PSF using the unity DC gain penalty function together with its penalty

parameter ˛. The limitation due to the reducible nature of a Gaussian PSF still applies to

the PML BID algorithm. Moreover, the challenge of obtaining optimal ˇ and ˛ values is

still unsolvable. In simulation studies, the PML BID algorithm was shown to address the

noise build up problem of the ML BID algorithm. In summary,

1. Blind PML algorithm produces nonnegative estimates.

2. Blind PML algorithm monotonically decreases the PML objective function with

increasing iterations.

3. Blind PML algorithm can incorporate a class of penalty functions for noise suppres-

sion.

4. Performance of PSF estimation indicates that additional a prior information is needed.

5. Preliminary experiments suggest that blind PML algorithm is promising.

7.2. Future Work

A weakness of the blind PML algorithm is that it produces poor estimates for the

PSF coefficients. Recently, Bioucas-Dias et al. [41] developed a non separable majorizing

function for the total variation penalty function, which has been used successfully in a

number of estimation problems (see [42] for example). Therefore, in the future, we intend

to incorporate the total variation penalty function into the blind PML algorithm. A separable

majorizing function for the total variation can be constructed by combining Bioucas-Dias’s

69

result with Huber’s [1] majorizing function for quadratic functions.

Additionally, we plan to:

1. Perform additional experiments to assess the performance of the deconvolution algo-

rithms.

2. Investigate approaches for automatically determining penalty parameters ˇ and ˛

from the data.

3. Asses if the algorithms developed in this thesis could improve anomaly detection

algorithms that identity landmines in hyperspectral images.

4. Determine more accurate initial estimate for the PSF coefficients.

5. Investigate approaches for estimating the spatial extent of unknown PSF.

70

Appendix

Matlab Source Code for PML Blind Image Deconvolution Algorithm

1 clear all % Clear all variables2 load lamd.mat; %load blurred image3 psfsize=5; %Size of deblurring PSF4 sigvalue=5;5 psf=ones(psfsize,psfsize)./(psfsize.^2); % Initial guess for the deblurring PSF6 Bigiterations=500; %Max number of iteration to run if convergence criteria is not

met early.7 smallNumber= 0.0000000001;8 alpha=1; %Parameter for unity DC gain penalty function.9 weight=sqrt(0.5);

10 beta=500; %Parameter for log(cosh) neighbor smoothing penalty function.11 converge=0; %Convergence flag12 input=im(129:640,1:512); % Take 512x512 part of the blurred image.13 [M N]=size(input);14 Npixels=M�N; %Number of pixels in the lexicographically ordered image.15 [n1 n2]=size(psf); %Size of PSF16 inputimg=input;17 blrd=inputimg;18 origi=reshape(input,Npixels,1);1920 %Reflect�Extend the input image21 for i=1:M,22 for j=1:N,23 locmatrix(i,j)= (i�1)�N+j;24 end,25 end;26 extndlocmatrix=zeros(M+(n1�1),N+(n2�1));27 extndlocmatrix((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2)=locmatrix;2829 % Vertical extension30 extndlocmatrix(1:(n1�1)/2,:)=flipdim(extndlocmatrix((n1�1)/2+1:n1�1,:),1);31 extndlocmatrix(M+(n1�1)/2+1:M+(n1�1),:)=flipdim(extndlocmatrix(M+1:M+(n1

�1)�(n1�1)/2,:),1);3233 %Horizontal Extension34 extndlocmatrix(:,1:(n2�1)/2)=flipdim(extndlocmatrix(:,(n2�1)/2+1:n2�1),2);35 extndlocmatrix(:,N+(n2�1)/2+1:N+(n2�1))=flipdim(extndlocmatrix(:,N+1:N+(n2

�1)�(n2�1)/2,:),2);36 reference= reshape(extndlocmatrix’,size(extndlocmatrix,1)�size(extndlocmatrix,2)

,1);37 data=reshape(inputimg,Npixels,1); % data is in format needed by program

71

38 originaldata=data;39 mlOutput=inputimg; % Deblurred image is the initial image at the start4041 [m n]=size(extndlocmatrix); % size of the extended image42 numcoefficients=n1�n2; % number of PSF coefficients43 dumm=(n1�1)/2;44 dumn=(n2�1)/2;45 numpixels=m�n; % number of pixels in the extended image46 repPSF=zeros(numpixels,numcoefficients); % matrix of probabilities47 repIndex=zeros(numpixels,numcoefficients); % matrix of pixels48 outputimg=zeros(m,n);49 flatImg=ones(m,n);50 onesPSF=ones(n1,n2);51 weights=conv2(flatImg,onesPSF,’same’);52 for i=0:m�1,53 for j=0:n�1,54 SUM=0;55 counter=0;56 outpixelnum=(i�n+j)+1; % pixel number of current output image57 if (weights(i+1,j+1)==numcoefficients),58 for k1=�dumm:dumm,59 for k2=�dumn:dumn,60 val1=i�k1;61 val2=j�k2;62 if val1>=0 & val1<=m�1 & val2>=0 & val2<=n�1,63 counter=counter+1;64 index_k1=k1+dumm+1; % +1 terms are due to

matlab’s array convention65 index_k2=k2+dumn+1;66 index_val1=val1+1;67 index_val2=val2+1;68 inpixelnum=(val1�n+val2)+1;69 repPSF(outpixelnum,counter)=psf(index_k1,

index_k2);70 repIndex(outpixelnum,counter)=reference(

inpixelnum);71 end72 end73 end74 end75 end76 end77 Pn=reshape(repPSF,m,n,numcoefficients);78 PT=Pn((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2,1:numcoefficients);79 P=reshape(PT,M�N,numcoefficients);

72

80 Vn=reshape(repIndex,m,n,numcoefficients);81 VT=Vn((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2,1:numcoefficients);82 V=reshape(VT,M�N,numcoefficients);8384 for outerloop=1:Bigiterations, % Start the main iteration85 if(converge==1), % Don’t do anything if PML algorith has converged86 continue,87 end;88 inputimg=mlOutput;89 gausspsf=customgaussian(psfsize,sigvalue); %Blurring90 sigmavalue(outerloop)=sigvalue;91 Zs=find(inputimg==0);92 inputimg(Zs)=smallNumber; % If there s zero value in the starting image,

replace it with smallNumber93 sm(outerloop)=sum(sum(psf));94 smx(outerloop)=sum(sum(mlOutput));95 niterations=1; %Number of sub iterations to run for estimating reflectance

parameters96 blurryimg=inputimg;97 [m n]=size(blurryimg);98 Npixels=m�n;99 innerProduct=zeros(Npixels,1); % Calculate the inner product of PSF and

certain reflectance values100101 %get initial estimate102 x=reshape(blurryimg,Npixels,1); % x is in format needed by program103 meanval=mean(blurryimg(:));104105 % get scale factor so that sum =1106 scale=sum(psf); % update term of ml algorithm107108 %Reflect�Extend the current image109 TMPextended=zeros(M+(n1�1),N+(n2�1));110 TMPextended((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2)=blurryimg;111112 % Vertical extension113 TMPextended(1:(n1�1)/2,:)=flipdim(TMPextended((n1�1)/2+1:n1�1,:),1);114 TMPextended(M+(n1�1)/2+1:M+(n1�1),:)=flipdim(TMPextended(M+1:M+(

n1�1)�(n1�1)/2,:),1);115116 %Horizontal Extension117 TMPextended(:,1:(n2�1)/2)=flipdim(TMPextended(:,(n2�1)/2+1:n2�1),2);118 TMPextended(:,N+(n2�1)/2+1:N+(n2�1))=flipdim(TMPextended(:,N+1:N+(

n2�1)�(n2�1)/2,:),2);

73

119 input=TMPextended((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2); %Extract the non�extended part

120 innerWide=conv2(TMPextended,psf,’same’);121 innerProduct=reshape(innerWide((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1

�1)/2),Npixels,1);122 cost(outerloop)=sum(innerProduct�(originaldata.�log(innerProduct))); %Cost

function123 for i=1:niterations, %Estimate reflectance parameters124 update=zeros(Npixels,1); % update term of ml algorithm125 % COMPUTE UPDATE TERM126 for tnum=1:Npixels,127 dum=length(find(V(tnum,:)));128 Vvalues=V(tnum,1:dum)’; % transponse => column vectors129 Pvalues=P(tnum,1:dum)’; % transponse => column vectors130 update(Vvalues)=update(Vvalues)+PSFvalues�(data(tnum)/

innerProduct(tnum));131 end132 Tmpx=TMPextended’;133 [g1 g2]=size(Tmpx);134 Ftemp=zeros(g1,g2);135 Gtemp=zeros(g1,g2);136 E=zeros(Npixels,1); % update term of PML algorithm137 tnum=0;138 for mt=2:g1�1,139 for nt=2:g2�1,140 Gt=0;141 gam=0;142 sumgammaFn=0;143 tnum=tnum+1;144 for k1=�1:1,145 index_k1=mt+k1; % +1 terms are due to matlab’s array

convention146 for k2=�1:1,147 index_k2=nt+k2;148 if (~((k1==0)&(k2==0)))149 dif=Tmpx(mt,nt)�Tmpx(index_k1,index_k2);150 if(abs(dif) >=smallNumber)151 gam=(tanh(dif/delta))/(delta�dif);152 else gam= 1/(delta�delta);153 end;154 sumgammaFn=sumgammaFn+gam;155 Gt=Gt+gam.�((Tmpx(mt,nt)+Tmpx(index_k1,

index_k2))/2);156 end; %if (~((k1==0)&(k2==0)))157 end;

74

158 end;159 Ftemp(mt,nt)=sumgammaFn;160 Gtemp(mt,nt)=Gt;161 end;162 end;163 E=�x.�update;164 F=2�beta.�reshape((Ftemp((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1

�1)/2))’,Npixels,1);165 G=(scale�4�beta.�reshape((Gtemp((n1�1)/2+1:M+(n1�1)/2,(n2�1)

/2+1:N+(n1�1)/2))’,Npixels,1));166 Deriv=(Ftemp((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2))’;167 Deriv= reshape(Deriv,Npixels,1);168 for tnum=1:Npixels,169 dum=length(find(V(tnum,:)));170 pixellocations=V(tnum,1:dum)’; % transponse => column vectors171 PSFvalues=P(tnum,1:dum)’; % transponse => column vectors172 end173 xnew = ((�G+ sqrt((G.^2)�(8.�E.�F)))./(4.�F)); %PML estimate of

reflectance parameter174 MSE(outerloop)=sqrt(sum( (abs(xnew.^2 � x.^2))./((abs(xnew)).^2)));175 if (outerloop>=2),176 if ((MSE(outerloop�1)�MSE(outerloop))<=0.000000001),177 converge=1; ’PML solution found’,178 end;179 end;180 x=xnew; update current iterate181 x=x�(meanval/mean(x(:))); % scale iterate so that perceived brightness is

retained182 for tnum=1:Npixels,183 dum=length(find(V(tnum,:)));184 pixellocations=V(tnum,1:dum)’; % transponse => column vectors185 PSFvalues=P(tnum,1:dum)’; % transponse => column vectors186 innerProduct(tnum,1)=x(pixellocations)’�PSFvalues;187 end188189 end % End of iterations for estimating reflectance parameters.190191 %Next, we will estimate the PSF192 Inp=conv2(extended,psf,’same’);193 convd=Inp((n1�1)/2+1:M+(n1�1)/2,(n2�1)/2+1:N+(n1�1)/2);194 diff(outerloop)=sum(sum((blrd�convd).^2));195 psf=reshape(psf,numcoefficients,1);196 for tnum=1:numcoefficients,197 dum=length(find(V(:,tnum)));198 pixellocations=V(1:dum,tnum)’; % transponse => column vectors

75

199 PSFvalues=P(1:dum,tnum)’; % transponse => column vectors200 pro(pixellocations)=x(pixellocations)’.�PSFvalues;201 sumX(tnum)=sum(x(pixellocations));202 updt=0;203 upd=0;204 for len=1:dum,205 updt=updt�data(pixellocations(len)).�x(pixellocations(len))./innerProduct

(pixellocations(len));206 upd=upd+data(pixellocations(len)).�x(pixellocations(len))./innerProduct(

pixellocations(len));207 end; %len208 C=psf(tnum)�updt;209 B=sumX(tnum)�2�alpha;210 A=2�alpha�sum(psf)/psf(tnum);211 r1=(�B+sqrt(B.^2�4�A�C))/(2�A);212 r2=(�B�sqrt(B.^2�4�A�C))/(2�A);213 if r1>0,214 psf(tnum)=r1; %’Positive root’215 else, psf(tnum)=r2; %’Negative root’216 end;217 updateValue(tnum)=upd/sumX(tnum);218 end; % End of for loop, tnum=1:numcoefficients,219220 P=repmat(psf’,Npixels,1); %Replicate lexicographical PSF vector221 psf=reshape(psf,n1,n2); %Reshape PSF back to 2�Dimention222 psf=psf’;223 mlOutput=reshape(x,m,n); %Current iterate of the PML algorithm (updated image

)224 end; % End of main iteration

76

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