dip class3image enhancement

7/30/2019 Dip Class3Image Enhancement

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IMAGE ENHANCEMENT


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What Is ImageEnhancement?

Image enhancement is the process ofmaking images more useful

The reasons for doing this include: Highlighting interesting detail in images

Removing noise from images

Making images more visually appealing


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Image Enhancement

Accentuation, or sharpening of imagefeatures such as edges, boundaries, orcontrast

Does not increase the inherent information Component in the data

In general, image enhancement is used togenerate a visually desirable image.

It can be used as a preprocess or apostprocess.

Highly application dependent. A technique thatworks for one application may not work foranother.

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Image Enhancement


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Image Enhancement(Contd)

The image enhancement methods arebased on either spatial or frequencydomain techniques

spatial domain approaches : directmanipulation of pixel in an image

frequency domain approach : modify the

Fourier transform of an image

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Spatial Domain Method

Image processing function maybe expressed as

f(x, y): input image

g(x, y): processed image T: operator on fdefined oversome neighbor of (x, y)

Neighborhood shape : square orrectangular arrays are the most

predominant due to the ease ofimplementation mask processing / filtering

masks( filters, windows,templates)

e.g. Image sharpening

A 3x3 neighborhood about a

point (x, y) in an image

)],([),( yxfTyxg =

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Spatial Domain Method

Simplest form ofT: neighbor gdepends only on the value offat (x,y) T: gray-level transformation function

s = T(r)(r,s are variables denoting the gray level of f(x, y) and g(x,y) at any point(x, y) )

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Frequency DomainMethod

Convolution theoremg x y h x y f x y( , ) ( , ) ( , )=

G u v H u v F u v( , ) ( , ) ( , )=

H u v( , ): transfer function, optical transfer function

Visually, f(x, y) is given and the goal, after computation

ofF(u, v), is to select H(u, v) so that the desired image

[ ]g x y H u v F u v( , ) ( , ) ( , )= F 1.

exhibits some highlighted feature off(x, y)

where G, H, and Fare the Fourier transforms ofg, h, and f

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Image Enhancement Techniques

Point

operation

Mask

operation TransformoperationColoringoperation

Image NegativeContrast

Stretching

Compression ofdynamic range

Gray level slicing

Image

Subtraction

Image Averaging

Histogramoperations

Smoothing

operations

Median

Filtering Sharpening

operations

Derivative

operations

Histogram

operations

Low pass

filtering

High pass

Filtering Band pass

filtering

Homomorphic

filtering

Histogramoperations

False

coloring

Full colorprocessing


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Image enhancement

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Types of ImageEnhancement

There are three types of imageenhancement techniques:

Point operations: each pixel is modifiedaccording to a particular equation,independent of the other pixels.

Mask operations: each pixel is modified

according to the values of the pixelsneighbors.

Global operations: all the pixel values in theimage or subimage are taken into

consideration.05/25/13 11


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Point Operations

( )v f u=

Point operations are zero memory operations

where a given gray level u [0,L] is mapped into a

gray level v [0, L] according to a transformation

1.Contrast Stretching

2.Clipping and Thresholding

3. Digital Negative4. Intensity Level Slicing

5. Bit plane slicing

6. Log Transformation

7. Power Law Transformation05/25/13 12


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Contrast stretching

Increase the dynamicrange of the gray levels inthe image

Before the stretching can

be performed it isnecessary to specify theupper and lower pixelvalue limits over which

the image is to benormalized.

Often these limits willjust be the minimum andmaximum pixel valuesthat the image type05/25/13


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Contrast stretching

Call the lower and the upper limits a and brespectively. Scans the image to find the lowestand highest pixel values currently present in theimage. Call these c and d. Then each pixel P is

scaled using the following function:

Values below 0 are set to 0 and values about 255

are set to 255

acd

abcPP inout += ))((

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After contrast stretching, using a simple linear interpolation

between c = 79 and d = 136

This result is a significant improvement over the original, the

enhanced image itself still appears somewhat flat.

Source image

Its histogram


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Otherwise,we canachieve better resultsby contrast stretchingthe image over a more

narrow range ofgraylevel values fromthe original image

For example, by setting

the cutoff fractionparameter to 0.03, weobtain the contrast-stretched image

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Setting the cutofffraction to a highervalue, e.g.0.125,yields the contrast

stretched image

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Contrast Stretching

( )

0

a

( ) b

a

b

u u a

v u a v u b

u b v u L

= +

+

p

p

p

The gray scale intervals where pixels occur most frequentlywould be stretched to improve the overall visibility of a scene

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Contrast Stretching

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Contrast Stretching

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Clipping andThresholding

Clipping-

This is useful for noise reduction when theinput signal is known to lie in the range

[a,b]A special case of contrast stretchingwhere 0 = =

Thresholding

A special case of clipping

{ 1 u T

0 u T

v L=

p

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Thresholding

Separate out the regions of the imagecorresponding to objects in which we areinterested, from the regions of the image thatcorrespond to background

perform this segmentation on the basis of thedifferent intensities or colors in the foreground andbackground regions of an image

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A) shows a classic bi-modal intensity distribution.

This image can be successfully segmented usinga single threshold T1. B) is slightly morecomplicated. Here we suppose the central peakrepresents the objects we are interested in and sothreshold segmentation requires two thresholds:

T1 and T2. In C), the two peaks of a bi-modaldistribution have run together and so it is almostcertainly not possible to successfully segmentthis image using a single global threshold

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using a single threshold at

a pixel intensity value of 120

Input

Output


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Thresholding

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Adaptive Thresholding

Whereas the conventionalthresholding operator uses a globalthreshold for all pixels, adaptive

thresholding changes the thresholddynamically over the image

This more sophisticated version of

thresholding can accommodatechanging lighting conditions in theimage, e.g. those occurring as a

result of a strong illumination05/25/13


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For each pixel in the image, a thresholdhas to be calculated. If the pixel valueis below the threshold it is set to the

background value, otherwise itassumes the foreground value

Two main approaches to finding the

threshold: the Chow and Kaneko approach

local thresholding

Assumptions: smaller image regions are morelikely to have approximately uniform illumination,05/25/13


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For each pixel in the image, a thresholdhas to be calculated. If the pixel valueis below the threshold it is set to the

background value, otherwise itassumes the foreground value

Two main approaches to finding the

threshold: the Chow and Kaneko approach

local thresholding

Assumptions: smaller image regions are morelikely to have approximately uniform illumination,05/25/13


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Chow and Kanekoapproach

Divide an image into an array ofoverlapping subimages and then findthe optimum threshold for each

subimage by investigating itshistogram

The threshold for each single pixel is

found by interpolating the results ofthe subimages

The drawback of this method is that

it is computational expensive and,05/25/13


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Local Thresholding

Finding the local threshold is to statisticallyexamine the intensity values of the localneighborhood of each pixel

The statistic which is most appropriate depends

largely on the input image. Simple and fastfunctions include the meanof the local intensitydistribution,

the median value,

or the mean of the minimum and maximum

values,

meanT=

medianT

=

2

maxmin+=T

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Image contains a

strong illuminationgradient, global

thresholding produces

a very poor result

Source Image

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Using the mean of a 77neighborhood, adaptive

thresholding

However, the mean of thelocal area is not suitable

as a threshold,because the

range of intensity values

within a local neighborhood

is very small and theirmean

is close to the value of the

center pixel

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The larger window yields the poorer result,

because it is more adversely affected by theillumination gradient

Mean, a 77

neighborhood

and C=7

Mean, a 7575

neighborhood

and C=10

Median, a 77

neighborhood

and C = 405/25/13


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How to choose a thresholdvalue?

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Image negatives

Display medicalimages

and photographing ascreen with

monochrome positivefilm

Reverse the orderfrom black to white so

that the intensity ofthe output imagedecreases as theintensity of the input

increases

L is the number of gray levels,r and N denote the input and

output gray levels

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Digital Negative

Applications:

display of medical images

produce negative prints of images

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an image its negative


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Digital Negative

Original Image Negative Image

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G /I i l l


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Gray/Intensity-levelslicing

Highlighting a specific range of graylevels is often desired

Various way to accomplish this Highlight some range and reduce all others

to a constant level

Highlight some range but preserve all otherlevels

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Intensity Level Slicing

, a0 otherwiseL u bv =

Without Background

, a

u otherwise

L u bv

=

With Background

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Intensity-level slicing:

(a) a transformation function that highlights a range [A, B] of intensities whilediminishing all others to a constant ,low level

(b) a transformation that highlights a range [A, B] of intensities but preserves all others

(c) Original image

(d) result of using the transformation in (a)

These transformations permits segmentation of certain gray level regions from

the rest of the image

a)

d)c)

b)


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Examples of display transfer functions

Manipulation of the grey scale transfer function:

a) an original, moderately low-contrast transmission light microscope image

(prepared slide of a head louse)

b) expanded linear transfer function adjusted to the minimum and maximumbrightness values

c) positive gamma (log) function

d) negative gamma (log) function

e) negative linear transferfunction

f) nonlinear transfer function (high slope linear contrast over central portion of

brightness range, with negative slope or solarization for dark and bright portions)


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Bit-plane slicing

Highlighting the contribution madeto the total image appearance byspecific bits

Higher-order :The majority of the visually significant data

Lower-order :

Subtle details

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Bit Extraction

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Bit-plane slicing

3 2

1 0

7 6

5 4

Original image


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Bit-plane slicing

Plane 7 containsthe mostsignificant bits,

and plane 0contains theleast significant

bits of the pixelsin the originalimage 012

345

67

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Log Transformation

Fourier Spectrum Log Transformed image

( )10log 1v c u= +

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Gamma Correction

A variety of devices used for imagecapture, printing, and displayrespond according to a power law.

By convention, the exponent in thepower-law equation is referred to asgamma

The process used to correct thispower-law response phenomena iscalled gamma correction.

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Example

cathode ray tube (CRT) devices havean intensity-to-voltage response thatis a power function, with exponents

varying from approximately 1.8 to2.5

With reference to the curve for g=2.5

wesee that such display systems would

tend to produce images that are

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Gamma Correction

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What is a histogram?

A graph indicating the number of times each graylevel occurs in the image, i.e. frequency of thebrightness value in the image

The histogram of an image with L gray levels isrepresented by a one-dimensional array with L

elements Algorithm:

Assign zero values to all

elements of the array hf;

For all pixels (x,y) of the

image f, increment

hf[f(x,y)] by 1.

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Histogram Processing

Histogram of a digital image withgray levels in the range [0,L-1] is adiscrete function

h(rh(rkk) = n) = nkkwherer

k: the kth gray level

nk : the number of pixels in the imagehaving gray level r

k

h(rk) : histogram of a digital image with

gray levels rk

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Normalized Histogram

dividing each of histogram at graylevel rr

kkby the total number of

pixels in the image, nn

p(rp(rkk) = n) = nkk/ n/ nfor k = 0,1,,L-1

p(rp(rkk)) gives an estimate of the

probability of occurrence of graylevel rr

kk

The sum of all components of a

normalized histogram is equal to 105/25/13 57


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Histogram

An image histogram is a plot of the gray-level frequencies.

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Image Histogram

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Properties of Image


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Properties of ImageHistogram

Histograms with small spread correspond to lowcontrast images (i.e., mostly dark, mostly bright,or mostly gray).

Histograms with wide spread correspond to high

contrast images.

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Properties of Image


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Properties of ImageHistogram

Histograms clustered at the low end correspond to

dark images.Histograms clustered at the high end correspond to

bright images.

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Example

Dark Image

Components of

histogram are

concentrated on

the low side of thegray scale.

Bright Image

Components of

histogram areconcentrated on the

high side of the

gray scale.

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Example

Low contrast image

histogram is narrow and

centered toward the

middle of the gray scale

High contrast image

histogram covers broad

range of the gray scale

and the distribution of

pixels is not too far from

uniform, with very fewvertical lines being much

higher than the others

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Histogram Processing

Histogramscorresponding to fourbasic image types

Dark image

Bright image

Low-contrast image

High-contrast image05/25/13


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Histogram equalization

Goal: to produce an image withequally distributed brightness levelsover the whole brightness scale

Effect: enhancing contrast forbrightness values close to histogrammaxima, and decreasing contrast near

minima. Result is better than just stretching,

and method is fully automatic

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Histogram Equalization

Let us assume for the moment that the input image to beenhanced has continuous gray values, with r= 0 representing

black and r= 1 representing white.

We need to design a gray value transformation s = T(r), basedon the histogram of the input image, which will enhance the

image.

he histogram equalization is an approach to enhance a given image.

The approach is to design a transformation T(.) such that the gray

values in the output is uniformly distributed in [0, 1].

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As before we assume that:


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As before, we assume that:

(1) T(r) is a monotonically increasing function for 0 r

1 (preserves order from black to white).

(2) T(r) maps [0,1] into [0,1] (preserves the range of allowed

Gray values).

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f ( )


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Let us denote the inverse transformation by r= T-1(s) . We

assume that the inverse transformation also satisfies the above

two conditions.

We consider the gray values in the input image and output

image as random variables in the interval [0, 1].

Letpin(r) and pout(s)denote the probability density of the

Gray values in the input and output images.

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If ( ) d T( ) k d T 1( ) ti fi diti 1


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If pin(r) and T(r) are known, and r= T-1(s) satisfies condition 1, we can

write (result from probability theory):

)(1

)()(sTr

inoutdsdrrpsp

=

=

One way to enhance the image is to design a transformation

T(.) such that the gray values in the output is uniformly

distributed in [0, 1], i.e.pout(s) = 1, 0 s 1

In terms of histograms, the output image will have all

gray values in equal proportion .

This technique is called histogram equalization.

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Equalization

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N t d i th l i th t t i if l


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Consider the transformation

10)()( ,0

== rdwwprTsr

in

Note that this is the cumulative distribution function (CDF) ofpin

(r)

and satisfies the previous two conditions.

From the previous equation and using the fundamental

theorem of calculus,

)(rpdr

dsin=

Next we derive the gray values in the output is uniformly

distributed in [0, 1].

1)()( =

=

ds

drrpsp inout

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Th f th t t hi t i i b


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Therefore, the output histogram is given by

[ ] 10,11)(

1)()(

)()(

1

1

==

=

==

srp

rpspsTr

sTrininout

The output probability density function is uniform, regardless of

the input.

Thus, using a transformation function equal to the CDF of input

gray values r, we can obtain an image with uniform gray values.

This usually results in an enhanced image, with an increase inthe dynamic range of pixel values.

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H t i l t hi t


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Step 1:For images with discrete gray values, compute:

n

nrp kkin =)( 10 kr 10 Lk

L: Total number of gray levels

nk: Number of pixels with gray value rk

n: Total number of pixels in the image

Step 2: Based on CDF, compute the discrete version of the previous

transformation :

=

==k

j

jinkk rprTs0

)()( 10 Lk

How to implement histogramequalization?

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Example: ==k

jinkk rprTs )()( 10 Lk


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Consider an 8-level 64 x 64 image with gray values (0, 1, ,7). The

normalized gray values are (0, 1/7, 2/7, , 1). The normalized

histogram is given below:

NB: The gray values in output are also (0, 1/7, 2/7, , 1).

=j 0

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Gray value

# pixels

Normalized gray value

Fraction of

# pixels

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Applying the transformation, ==k

jinkk rprTs )()( we have


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=j

jinkk p0

)()(

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Notice that there are only five distinct gray levels (1/7


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Notice that there are only five distinct gray levels --- (1/7,

3/7,5/7, 6/7, 1) in the output image. We will relabel them

as (s0,s1, , s4 ).

With this transformation, the output image will have

histogram

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Histogram of output


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Histogram of output

image

# pixels

Gray values

Note that the histogram of output image is only approximately, and not exactly,

uniform. This should not be surprising, since there is no result that claims

uniformity in the discrete case.

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Example Original image and its histogram


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Histogram equalized image and its histogram


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Histogram equalized image and its histogram

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Comments:

Histogram equalization may not always produce desirable


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Histogram equalization may not always produce desirable

results, particularly if the given histogram is very narrow. It

can produce false edges and regions. It can also increase

image graininess and patchiness.

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Hi t E li ti


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Histogram Equalization

Form the cumulative histogram

Normalize the value by dividing it bythe total number of pixels

Multiply these values by themaximum gray level value and roundoff the value

Map the original value to the result ofstep 3 by a one-one correspondence

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Histogram (Matching)


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Histogram (Matching)Specification

Histogram equalization has adisadvantage which is that it cangenerates only one type of outputimage.

With Histogram Specification, we canspecify the shape of the histogram thatwe wish the output image to have.

It doesnt have to be a uniformhistogram05/25/13 86

Histogram specification /


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Histogram specification /matching

Motivation: Sometimes, the ability to specify particular

histogram shapes capable of highlightingcertain gray-level ranges in an image is

desirable

The aim is to produce an image withdesired distributed brightness levels

over the whole brightness scale, asopposed to uniform distribution

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Histogram specification)(rp

r :The original probability density function: The desired probability density function)(zpz


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Example

Illustration of the histogram specification method:

(a)original image; (b)image after histogram equalization;

(c)image enhanced by histogram specification; (d)histograms

(a) (b)

(c) (d)

original

equalized

specified

resulting

Histogram Specification


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Histogram Specification

Find the mapping table of thehistogram equalization

Specify the desired histogram.

Equalize the desired histogram Perform the mapping process so that

the values of step 1 can be mapped

to the results of step 2.

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Local enhancement


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Local enhancement Motivation:

To enhance details over small areas in an image

The computation of a global transformation does notguarantee the desired local enhancement

Solution:

Define a square / rectangular neighborhood Move the center of this area from pixel to pixel

Histogram equalization in each neighborhood region

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Image Subtraction


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Image SubtractionThe difference between two imagesf(x, y) and h(x, y), expressed as

h(x, y) : mask - an x-ray image of a region of a

patients body

f(x, y) : image of the same anatomical region butacquired after injection of a dye into thebloodstream

),(),(),( yxhyxfyxg =

(a) mask image

(b) image (after injectionof dye into the bloodstream)

with mask subtracted out

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Example

Showing image differences by subtraction:

a) original image; b) image after moving one coin; c) difference image after pixel-by pixel subtraction

Different images for quality control. A master image is subtracted from

images of each subsequent part. In this example, the missing chip in a printed

circuit board is evident in the difference image

Applications of ImageSubtraction and change


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Subtraction and changedetection

Medical imaging application : displayblood-flow paths

Automated inspection of printed

circuits Security monitoring

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Two frames from a videotape sequence of free-swimming

single-celled animals in a drop of pond water , and the

difference image. The length of the white region divided bythe time interval gives the velocity

Analysis of motion in a more complex

situation than shown in fig. Where the

paths of the swimming microorganisms

cross, they are sorted out by assuming

that the path continues in a nearly

straight direction. (Gualtieri &Coltelli,

1992)05/25/13

Image Averaging


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Image Averaging

Motivation: Imaging with very low light levels is

routine, causing sensor noise frequently

to render single images virtually uselessfor analysis

Solution: image averaging

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Image Averaging


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Image Averaging

Noisy image g(x, y)=noise image

f(x, y)=original image

n(x, y)=noise

Assumption: at every pair of coordinates (x, y), thenoise is uncorrelated & has zero average value

uncorrelated: covariance

If an image is formed by averaging K

different noisy images

),(),(),( yxnyxfyxg +=

0)])([( = jjii mxmxE),( yxg

{ }

2

),(

2

),(

1

1

),(),(

),(1),(

yxnyxg

K

i

i

K

yxfyxgE

yxgK

yxg

=

=

= =

As K increases: the variability

(noise) of the pixel values at

each location (x,y) decreases

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Example


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Example

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Spatial Filtering


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p g-Contents

What is spatial filtering?

Smoothing Spatial filters.

Sharpening Spatial Filters.

Combining Spatial EnhancementMethods

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Mask Operation

Linear systems and linear filtering

Smoothing operations

Median Filtering

Sharpening operations

Derivative operations

Correlation

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Neighbourhood


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Operations

Neighbourhood operations simplyoperate on a larger neighbourhood ofpixels than point operations

Neighbourhoods aremostly a rectanglearound a central pixel

Any size rectangleand any shape filterare possible

Origin x

y Image f (x, y)

(x, y)Neighbourhood

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Neighbourhood


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Operations

For each pixel in the origin image, theoutcome is written on the samelocation at the target image.

Origin x

y Image f (x, y)

(x, y)

Neighbourhood

TargetOrigin

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Simple Neighbourhood


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Operations

Simple neighbourhood operationsexample:

Min: Set the pixel value to the minimumin the neighbourhood

Max: Set the pixel value to the

maximum in the neighbourhood

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Image Enhancement:

S ti l Filt i


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Spatial Filtering Image enhancement in the spatial domain can be represented

as:

[ ] ),(),( nmfTnmg =

Enhanced Image TransformationGiven Image

The transformation Tmaybe linear or nonlinear. We will mainly study linear

operators Tbut will see one important nonlinear operation.

There are two closely related concepts that must be understood when

performing linear spatial filtering. One is correlation, the other is convolution.

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How to specify T


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If the operatorTis linear and shift invariant (LSI), characterized by

the point-spread sequence (PSS) h(m, n), then (recall convolution)

=

=

=

=

=

=

=

l k

l k

lkhlnkmf

lkflnkmh

nmfnmhnmg

),(),(

),(),(

),(),(),(

In practice, to reduce computations, h( n , m )is of finite extent:

where is a small set (called neighborhood). is also called as the support

of h.

h( n , m ) =0, for (k,l)

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(2)The image is extended by appending extra rows/columns at the boundaries. The

extension is done by repeating the first/last row/column or by setting them to some

t t (fi d b d )


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constant (fixed boundary).

(3)The boundaries wrap around (periodic boundary).

For one dimension

=

)1(

)(

)0(

)(~

Nf

xf

f

xf

1)2/( +


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image f(m, n).

The mask operation can be implemented in matlab using the filter2command, which is based on the conv2command.

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The Spatial Filtering


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Processj k lm n o

p q r

Origin x

y Image f (x, y)

eprocessed= n*e +

j*a + k*b + l*c +

m*d + o*f +p*g + q*h + r*i

Filter (w)Simple 3*3

Neighbourhoode 3*3 Filter

a b cd e f

g h i

Original Image

Pixels

*

The above is repeated for every pixel in the

original image to generate the filtered image05/25/13 111

Spatial Filtering:


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Equation Form

= =

++= aas

b

bt

tysxftswyxg ),(),(),(

Filtering can be given

in equation form asshown above

Notations are based

on the image shownto the left

05/25/13 112

Smoothing Filters


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Image smoothing refers to any image-to-image transformation designed to

smooth or flatten the image by reducing the rapid pixel-to-pixel variation in

gray values.

Smoothing filters are used for:

(1) Blurring: This is usually a preprocessing step for removing small

(unwanted) details before extracting the relevant (large) object, bridging gapsin lines/curves,

(2)Noise reduction: Mitigate the effect of noise by linear or nonlinear

operations.

Image smoothing by averaging (lowpass spatial filtering)

05/25/13 113

Smoothing is accomplished by applying an averaging mask.


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An averaging mask is a mask with positive weights, which sum to 1. It

computes a weighted average of the pixel values in a neighborhood. This

operation is sometimes called neighborhood averaging.

Some 3 x 3 averaging masks:

010

111

010

5

1

010

141

010

8

1

111

111

111

9

1

131

3163

131

32

1

This operation is equivalent to lowpass filtering.

05/25/13 114

Smoothing Spatial Filters


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Smoothing Spatial Filters

One of the simplest spatial filteringoperations we can perform is asmoothing operation

Simply average all of the pixels in aneighbourhood around a central value

Especially usefulin removing noisefrom images

Also useful forhighlighting gross

detail

1/91/9

1/9

1/91/9

1/9

1/91/9

1/9

Simpleaveraging

filter

05/25/13 115

Smoothing Spatialil i


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Filtering1

/91

/91

/91/9

1/91/9

1/91/9

1/9

Origin x

y Image f (x, y)

e = 1/9*106 +1/9*104 +

1/9*100 +1/9*108 +

1/9*99 +

1/9*98 +1/9*95 +

1/9*90 +1/9*85

= 98.3333

Filter

Simple 3*3Neighbourhood

106

104

99

95

100 108

98

90 85

1/91/9

1/9

1/91/9

1/9

1/91/9

1/9

3*3 SmoothingFilter

104 100 108

99 106 98

95 90 85

Original Image

Pixels

*

The above is repeated for every pixel in the

original image to generate the smoothed image05/25/13 116

Image Smoothingl


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Example

The image at the top leftis an original image ofsize 500*500 pixels

The subsequent imagesshow the image afterfiltering with an averaging

filter of increasing sizes 3, 5, 9, 15 and 35

Notice how detail begins

to disappear05/25/13 117

Image Smoothing Example


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Weighted SmoothingFilt


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Filters

More effective smoothing filters can begenerated by allowing different pixelsin the neighbourhood different weights

in the averaging function Pixels closer to the

central pixel are moreimportant

Often referred to as aweighted averaging

1/16 2/16 1/16

2/164/16

2/16

1/162/16

1/16

Weighted

averaging filter05/25/13 124

Another Smoothing Example


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Another Smoothing Example

By smoothing the original image weget rid of lots of the finer detail whichleaves only the gross features forthresholding

Original Image Smoothed Image Thresholded Image

* Image taken from Hubble Space Telescope

05/25/13 125

Averaging Filter Vs. MedianFilter Example


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Filter Example

Filtering is often used to remove noise

from images

Sometimes a median filter worksbetter than an averaging filter

Original Image

With Noise

Image After

Averaging Filter

Image After

Median Filter

05/25/13 126

Averaging Filter Vs. Median FilterExample


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Example

Original

05/25/13 127



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Example

Averaging

Filter

05/25/13 128



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Example

Median

Filter

05/25/13 129

Strange Things Happen At TheEdges!


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Edges!

Origin x

y Image f (x, y)

e

e

e

e

At the edges of an image we are missingpixels to form a neighbourhood

e e

e

05/25/13 130

Strange Things Happen At TheEdges! (cont )


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Edges! (cont)

There are a few approaches to dealingwith missing edge pixels:

Omit missing pixels

Only works with some filters Can add extra code and slow down

processing

Pad the image

Typically with either all white or all blackpixels

Replicate border pixels

Truncate the image05/25/13 131

Correlation &Convolution


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Convolution

The filtering we have been talkingabout so far is referred to ascorrelation with the filter itself referredto as the correlation kernel

Convolution is a similar operation, withjust one subtle differenceeprocessed= v*e +

z*a + y*b + x*c +

w*d + u*e +t*f +s*g + r*h

r s t

u v w

x y z

Filter

a b c

d e e

f g h

Original Image

Pixels

*

05/25/13 132

Sharpening SpatialFilters


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Filters

Previously we have looked atsmoothing filters which remove finedetail

Sharpening spatial filters seek tohighlight fine detail

Remove blurring from images

Highlight edgesSharpening filters are based on spatialdifferentiation05/25/13 133

Image Sharpening


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This involves highlighting fine details or

enhancing details that have been blurred.Basic highpass spatial filtering

This can be accomplished by a linear shift-

invariant operator, implemented by meansof a mask, with positive and negativecoefficients.This is called a sharpening mask, since ittends to enhance abrupt gray levelchanges in the image.

05/25/13 134

The mask should have a positive coefficient at thecenter and negative coefficients at the periphery.


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The coefficients should sum to zero. Example:

111

181111

9

1

This is equivalent to highpass filtering.

A highpass filtered image g can be thought of asthe difference between the original image fand a

lowpass filtered version off:g(m,n) = f(m,n) lowpass(f(m,n))

05/25/13 135

Example:


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05/25/13 136

High-boost filtering


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This is a filter whose output g is produced bysubtracting a lowpass (blurred) version offfroman amplified version offg(m,n) = A f(m,n) lowpass(f(m,n))

This is also referred to as unsharp masking.

Observe that

g(m,n) = A f(m,n) lowpass(f(m,n))

= (A-1) f(m,n) + f(m,n) lowpass(f(m,n))

= (A-1) f(m,n) + highpass(f(m,n))

ForA> 1, part of the original image is added back to the highpass filteredversion of f.

05/25/13 137

The result is the original image with the edges enhanced relative to the original

image.


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Example:

05/25/13 138

Spatial Differentiation


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p

Differentiation measures the rate ofchange of a function

Lets consider a simple 1 dimensionalexample

05/25/13 139

Spatial Differentiation


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A B

05/25/13 140

1st Derivative


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The formula for the 1st derivative of afunction is as follows:

Its just the difference betweensubsequent values and measures the

rate of change of the function

)()1( xfxf

x

f+=

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1st Derivative (cont)


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5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

0 -1 -1 -1 -1 0 0 6 -6 0 0 0 1 2 -2 -1 0 0 0 7 0 0 0

f(x)

f(x)

05/25/13 142

2nd Derivative


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The formula for the 2nd derivative of afunction is as follows:

Simply takes into account the valuesboth before and after the current value

)(2)1()1(2

2

xfxfxfx

f

++=

05/25/13 143

2nd Derivative (cont)


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5 5 4 3 2 1 0 0 0 6 0 0 0 0 1 3 1 0 0 0 0 7 7 7 7

-1 0 0 0 0 1 0 6 -12 6 0 0 1 1 -4 1 1 0 0 7 -7 0 0

f(x)

f(x)

05/25/13 144


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05/25/13 145

1st and 2nd Derivative


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f(x)

f(x)

f(x)

146

Using Second Derivatives ForImage Enhancement


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Image Enhancement

The 2nd

derivative is more useful forimage enhancement than the 1stderivative

Stronger response to fine detail

Simpler implementation

We will come back to the 1st orderderivative later on

The first sharpening filter we will lookat is the Laplacian

Isotropic

One of the simplest sharpening filters05/25/13 147

The Laplacian


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The Laplacian is defined as follows:

where the partial 1

st

order derivative inthex direction is defined as follows:

and in the y direction as follows:

y

f

x

ff

2

2

2

22

+

=

),(2),1(),1(2

2

yxfyxfyxf

x

f++=

),(2)1,()1,(2

2

yxfyxfyxf

y

f++=

05/25/13 148

The Laplacian (cont)


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So, the Laplacian can be given asfollows:

We can easily build a filter based on

this

),1(),1([2 yxfyxff ++=)]1,()1,( +++ yxfyxf

),(4 yxf

0 1 0

1 -4 1

0 1 005/25/13 149

Laplacian Mask


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05/25/13 150

The Laplacian (cont)


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Applying the Laplacian to an image weget a new image that highlights edgesand other discontinuities

Original

Image

Laplacian

Filtered Image

Laplacian

Filtered Image

Scaled for Display05/25/13 151

But That Is Not VeryEnhanced!


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Enhanced!The result of a Laplacianfiltering is not an enhancedimage

We have to do more work inorder to get our final image

Subtract the Laplacian resultfrom the original image to

generate our final sharpenedenhanced image

Laplacian

Filtered Image

Scaled for Display

fyxfyxg 2),(),( =05/25/13 152


153/194

Background features can be recovered whilestill preserving the sharpening effect of theLaplacian operation simply by adding the originaland Laplacian images

05/25/13 153

Laplacian ImageEnhancement


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Enhancement

In the final sharpened image edgesand fine detail are much more obvious

- =

Original

Image

Laplacian

Filtered Image

Sharpened

Image

05/25/13 154


155/194

Simplified ImageEnhancement


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Enhancement

The entire enhancement can becombined into a single filteringoperation

),1(),1([),( yxfyxfyxf ++= )1,()1,( +++ yxfyxf)],(4 yxf

fyxfyxg 2),(),( =

),1(),1(),(5 yxfyxfyxf +=)1,()1,( + yxfyxf

05/25/13 156

Simplified Image Enhancement(cont)


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( )

This gives us a new filter which doesthe whole job for us in one step

0 -1 0

-1 5 -1

0 -1 0

05/25/13 157

Simplified Image Enhancement(cont)


158/194

( )

05/25/13 158


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Unsharp Masking andHighboost Filtering


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g g

Unsharp maskingSharpen images consists of subtracting anunsharp (smoothed) version of an image from theoriginal image

e.g., printing and publishing industrySteps

1. Blur the original image

2. Subtract the blurred image from the original

3. Add the mask to the original

05/25/13 160

Unsharp masking


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( , ) ( , ) ( , )sf x y f x y f x y=

( , )sf x y - sharpened image obtained by unsharp masking

( , )f x y - Blurred version of f(x,y)

161

High-Boost Filtering


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Generalization of unsharp masking iscalled high-boost filtering

( )

( , ) ( , ) ( , )

( , ) ( , ) ( , ) ( , ) ( , )

( , ) 1 ( , ) ( , )

hb

hb

hb s

f x y Af x y f x y

f x y Af x y f x y f x y f x y

f x y A f x y f x y

=

= + = +

( ) 2

2

( , ) 1 ( , ) ( , ) ( , )

( , ) ( , ) ( , )

hb

hb

f x y A f x y f x y f x y

f x y Af x y f x y

= + =

05/25/13 162

High-Boost Filtering


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05/25/13 163

High-Boost Filteredimage


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image

05/25/13 164

Highboost Filtering


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05/25/13 165

1st Derivative Filtering- TheGradient


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Implementing 1st derivative filters isdifficult in practice

For a function f(x, y) the gradient offatcoordinates (x, y) is given as the column

vector:

=

=yf

x

f

G

G

y

xf

05/25/13 166

1st Derivative Filtering(cont)


167/194

( )The magnitude of this vector is given by:

For practical reasons this can besimplified as:

)f(= magf

[ ] 21

22

yx GG +=

21

22

+

=

y

f

x

f

yx GGf +

05/25/13 167

1st Derivative Filtering(cont)


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( )There is some debate as to how best tocalculate these gradients but we will use:

which is based on these coordinates

( ) ( )321987 22 zzzzzzf ++++

( ) ( )741963

22 zzzzzz +++++

z1 z2 z3

z4 z5 z6

z7 z8 z9

05/25/13 168


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Sobel Example


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Sobel filters are typically used for edge

detection

An image of acontact lens which

is enhanced in

order to make

defects (at four

and five oclock in

the image) moreobvious

05/25/13 170


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Combining SpatialEnhancement Methods


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Successful imageenhancement is typicallynot achieved using asingle operation

Rather we combine arange of techniques inorder to achieve a final

resultThis example will focuson enhancing the bone05/25/13 172


(cont )


173/194

(cont)

Laplacian filter of

bone scan (a)

Sharpened version of

bone scan achieved

by subtracting (a)

and (b) Sobel filter of bone

scan (a)

(a)

(b)

(c)

(d)05/25/13 173


174/194


(cont)


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(cont)

Compare the original and final images

05/25/13 175

Filtering in FrequencyDomain


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05/25/13 176

Notch Filter


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( , ) 0 ( , ) ( / 2, / 2)1 otherwise

H u v if u v M N= =

05/25/13 177

Transfer Function of IdealLowpass Filter


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0

0

0

12 2 2

( , ) 1 ( , )

0 ( , )

is the cutoff frequency

( , )2 2

H u v if D u v D

if D u v D

D

M ND u v u v

=

= +

f

05/25/13 178


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05/25/13 182

Butterworth Low pass Filter

1


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[ ]2

0

1

( , ) 1 ( , ) /nH u v D u v D= +

05/25/13 183

Resu ts o F ter ng w tBLPF


184/194

05/25/13 184


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05/25/13 185

Gaussian Lowpass Filter


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2 2

0( , )/2

( , )D u v D

H u v e

=

05/25/13 186

High Pass Filter


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05/25/13 187

Transfer function of HPF

HPFId l


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[ ]

2 20

0

0

0

2

0

( , )/2

HPF

( , ) 0 ( , )

1 ( , )

is the cutoff frequency

HPF

1( , )

1 ( , ) /

HPF

( , ) 1

n

D u v D

Ideal

H u v if D u v D

if D u v D

D

Butterworth

H u vD u v D

Gaussian

H u v e

=

=

+

=

f

05/25/13 188

Results of HPF usingButterworth HPF


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05/25/13 189

Homomorphic FilteringApproach


190/194

05/25/13 190

Homomorphic Filtering

( , ) ( , ) ( , )f x y i x y r x y=


191/194

{ } { } { }

{ } { }

{ } { }

( , ) ( , ) ( , )

( , ) ( , ) ( , )

( , ) ln ( , )

= ln ( , ) ln ( , )( , ) ln ( , )

= ln ( , ) ln ( , )

( , ) ( , ) ( , )i r

f x y i x y r x y

F f x y F i x y F r x y

z x y f x y

i x y r x yF z x y F f x y

F i x y F r x y

Z u v F u v F u v

=

+=

+

= +

05/25/13 191

we process ( , ) by means of a filter function H(u,v)

( , ) ( , ) ( , )

( ) ( ) ( ) ( )

If Z u v

S u v H u v Z u v

H F H F

=

+


192/194

{ }

{ }

1

1 1

= ( , ) ( , ) ( , ) ( , )

the spatial domain

( , ) ( , )

( , ) ( , ) ( , ) (

i r

i r

H u v F u v H u v F u v

In

s x y F S u v

F H u v F u v F H u v F u

+

=

+ { }, )

v

{ }

{ }

1

1

( , )

'( , ) '( , )

0 0

'( , ) ( , ) ( , )

'( , ) ( , ) ( , )

( , ) '( , ) '( , )( , )

= .

( ) ( )

i

r

s x y

i x y r x y

i x y F H u v F u v

r x y F H u v F u v

s x y i x y r x yg x y e

e e

i

=

=

= +=

05/25/13 192

Filter Function H(u,v)


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The filter function tends to decrease the contribution made by the low

frequencies (illumination) and amplify the contributions made by the

high frequencies (reflectance)

05/25/13 193


194/194

Thank You

dip class3image enhancement

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