stopping criteria image restoration alfonso limon claremont graduate university

Stopping Criteria Image Restoration

Alfonso Limon

Claremont Graduate University

Outline

• Image Restoration

• Noise and Image Details

• Observations using Wavelets

• Some Preliminary Results

• Future Work

CGU IPAM 2003: Inverse Problems

Image Restoration

The goal of image restoration is to improve a degrade image in some predefined sense. Schematically this process can be visualized as

where f is the original image, g is a degraded/noisy version of the original image and f ̃ is a restored version.


Image Degradation / Restoration

Image restoration removes a known degradation. If the degradation is linear and spatially-invariant

where F - original image, H - degradation, N - additive noise and G - recorded image. Given H, an estimate of the original image is

*( , ) ( , ) ( , ) ( , )G u v H u v F u v N u v

( , ) ( , )ˆ ( , ) ( , ) .( , ) ( , )

G u v N u vF u v F u v

H u v H u v

Notice that if H ~ 0, the noise will be amplified.


Restoration Model

Many restoration filters can be applied to recover the degraded image: Rudin-Osher-Fatemi, Wiener, Inverse FFT with threshold, etc. But all require a stopping criteria based on a noise measure.

2

2

( ) ( ) ( )

2

2

ˆ ˆ ˆ( , ) inf ( , ) ( , ) ( , ) ( , )

( , ) ( , )ˆ ( , )( , ) ( , ) ( , ) / ( , )

( , ) ( , )if 1/

( , )ˆ ( , )( , )

otherwise( , )

g BV BV L

F

f x y f x y h x y f x y g x y

G u v H u vF u v

H u v H u v S u v S u v

H u v G u vH

H u vF u v

G u v

H u v


Stopping the Restoration

Noise in the data is amplified as the number of iterations increase, while the blur decreases as the number of iteration increase.

This is the prototypical behavior which was illustrated by Professor Heinz Engl during the inverse problem tutorials. CGU IPAM 2003: Inverse Problems

Discrepancy Principle

The discrepancy principle gives a natural stopping criteria, namely to stop the restoration process when the residual is of the same order as the noise.

The drawback to this approach for image processing is that the discrepancy principle tends to recover an image that is still too blurred.

Question: How many more times can we iterate passed the threshold given by the discrepancy principle and still get an image that is not predominately noisy.


Objective

The objective is to highlight fine details in the image which were suppressed by the blur.

The problem is that enhancement of fine detail (or edges) is equivalent to enhancement of noise.

The challenge is to enhance details as much as possible before the noise overtakes the image details.


Wavelets and Details

Wavelets provide a nature way to separate signal scales, especially for 1D signals.

Taking the fine scale wavelet coefficients of the diagonal of the image provides a measure of the noise and image details.

The next set of sides show the fine scale coefficients for a reconstructed version of the quarter image (to the right) at various iterations steps.


Fine Wavelet Coefficients (1)


Observations

The discrepancy principle gives a good starting guess for the image details as the noise level is low.

Noise will eventually overtakes details, therefore tracking changes in the fine wavelet coefficient gives a possible way to differential between image details and noise.

A natural stopping criteria is to stop iterating when the fine wavelet coefficients that have been identified as details are of the same order as the fine wavelet coefficients that have been identified as noise, i.e., the discrepancy principle applied to fine wavelet coefficients.


Stopping Algorithm

1. Choose your initial image according to the discrepancy principle.

2. Calculate fine wavelet coefficients corresponding to the image’s diagonal.

3. Calculate the mean of the fine wavelet coefficients.

4. Assign the fine wavelet coefficients which are larger than the mean to the set: details.

5. If there is a previous details set, take the intersection of it with the new details set.

6. Repeat steps 2 through 5 until the mean of the details set is equal to the mean of the noise set; noise is the set of wavelet coefficients not in details.

Stopping Criteria Results: eight

Stopping Criteria Results: autumn

Stopping Criteria Results: rice

Stopping Criteria Results: alumgrns

Stopping Criteria Results: bacteria

Stopping Criteria Results: cameraman

Stopping Criteria Results: mri

Stopping Criteria Results: ic

Stopping Criteria Results: moon

Preliminary Results


Future Work

• Define an energy measure to compare wavelet coefficients instead of using the mean value.

• Apply a moving class to the wavelet coefficients to take into account the difference in behavior between details and noise.

• Apply different deblurring/noise models to test that the stopping criteria is not sensitive to the deblurring/noise process.

• Expend the method from a global to local stopping criteria.


Special Thanks to the Organizing Committee and

IPAM Staff.

ReferencesM. Bertero, Image Deconvolution, Inverse Problems: Computational Methods and Emerging Applications, IPAM 2003. (http://www.ipam.ucla.edu/publications/invws1/invws1_3804.pdf)

P. Blomgren and T. F. Chan, Modular Solver for Constraint Image Restoration Problems Using the Discrepancy Principle, Numerical Linear Algebra with Applications,Vol 9, issue 5, pp 347-358.

David Donoho, De-Noising by Soft-Thresholding (http://www-stat.stanford.edu/~donoho/reports.html)

H. W. Engl, Inverse Problems 1, Inverse Problems: Computational Methods and Emerging Applications Tutorials, IPAM 2003. (http://www.ipam.ucla.edu/publications/invtut/invtut_hengl1.pdf)

R. Gonzalez and R. Woods, Digital Image Processing, second edition, Prentice Hall, Inc., 2002.

K. Lee, J. Nagy and L. Perrone, Iterative Methods for Image Restoration: A Matlab Object Oriented Approach. Iterative Methods for Image Restoration, May 15, 2002.

R. Ramlau, Morozov’s Discrepancy Principle for Tikhonov regularization of nonlinear operators, Report 01-08, Berichte aus der Technomathematik, July 2001.

stopping criteria image restoration alfonso limon claremont graduate university

Documents

image processing

degraded image

f original image

image details observations

g recorded image

inverse problemswavelets

inverse fft

enhancement of noise