image enhancement in the spatial domain_sa

7/27/2019 Image Enhancement in the Spatial Domain_SA

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IMAGE ENHANCEMENT IN THESPATIAL DOMAIN

Digital Image Processing, Chapter 3 Gonzalez Woods

Digital Image Processing In Life-Science

Sefi addadi 21-3-2012


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WHAT DID WE LEARN LAST TIME?


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WHAT WILL BE TODAY?Image enhancement in the spatial domain

Gray level transformation

Negative, Log and power law

Histogram processing

Equalization

Matching

Local enhancement

Spatial filtering

Smoothing filters (mean, Gaussian, median)

Sharpening filters

Edge detection Filter combinations

Background correction

Acquisition (a priori) correction

Retrospective (a posteriori) correction

Rolling ball background correction


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IMAGE ENHANCEMENT IN THE

SPATIAL DOMAIN

The principal objective of enhancement

Process an image so that the result is more suitable than the

originalimage for a specific application.

Spatial domain refers to the image planeitself, and approaches in

this category are based on direct manipulation of pixels in an

image.

There is no general theory of image enhancement.

When an image is processed for visual interpretation, the viewer is

the ultimate judge of how well a particular method works.

Gonzalez Woods Chapter 3


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Image g(x, y)

x

y

Target

MATHEMATICAL REPRESENTATION

g(x, y) = T{f(x, y) }General expression

Origin

x

y Image f (x, y)

(x, y)Neighbourhood


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GRAY LEVEL TRANSFORMATIONS

Among the simplest of all

image enhancement

techniques.

Generally denoted by

rvalues before

transformation

sValues after transformation

Tthe performed

transformation s = T(r)


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POWER- LAW TRANSFORMATIONS

S= cr where c and

are positive constants

Power-law curves with

fractional values ofmap a narrow range of

dark input values into a

wider range of output

values. The opposite being true

for highs.

Many devices used for image capture, printing, and display

respond according to a power lawGamma correction


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HISTOGRAM PROCESSING

Histogram in digital images is defined as a discreet

function h(rk) = nk

Where rk is the gray level of k andnk is the number of pixels of that value

Histogram normalization

Performed by dividing each value by the total number of pixels in theimage.

p(rk) = nk / n

p(rk) gives an estimate of the probability of occurrence of gray levelrk.

The sum of all components of a normalized histogram is equal to 1.


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High contrast image

Low contrast image

Bright image

Dark image


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Histogram Equalization

p(rk) = nk / n

histogram equalization

Each pixel with level

rk is mapped into a

corresponding pixel

with level sk in the

output image.

histogram linearization

Automated process

does not require

additional input


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HISTOGRAM MATCHING

The method used to generate a processed image thathas a specified histogram shape.

In contrast to the aim of histogram where we perform

automatic uniform histogram.


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HISTOGRAM MATCHING

1. Obtain the histogram of the given

image.

2. Precompute a mapped level sk for each

level rk.

3. Obtain the transformation function Gfrom the given pz(z)

4. Precompute zk for each value of sk

using the iterative scheme defined in

connection

5. For each pixel in the original image,

if the value of that pixel is rk, map this

value to its corresponding level sk;

then map then map level sk into thefinal level zk.


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Original Histogram equalizedimage

Using mappings fromcurve

ImagestakenfromGonzalez&Wood

s,DigitalImageProces

sing(2002)

HISTOGRAM MATCHING


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HISTOGRAM SUMMARY

Useful for assessment of acquisition quality.

Critical for image presentation

Used as basis for subsequent analysis steps such as

thresholding etc.


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NEIGHBORHOOD OPERATIONS

Neighborhood is defined asany area bigger than thesingle pixel.

May be of any size orshape.

Most used rectangle are

around the central pixel

Might be referred to asfilters, kernels, templates,or windows.

Origin x

y Image f (x, y)

(x, y)Neighbourhood


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NEIGHBOURHOOD OPERATIONS

For each pixel in the origin image, the outcome is

written on the same location at the target image.

Originx

y Image f (x, y)

(x, y)Neighbourhood

Target


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SIMPLE NEIGHBOURHOOD

OPERATIONS

Simple neighbourhood operations example:

Min: Set the pixel value to the minimum in the

neighbourhood

Max: Set the pixel value to the maximum in the

neighbourhood


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Max Filter

Radius = 3

Original Min Filter

Radius = 3

SIMPLE NEIGHBOURHOOD OPERATIONS

http://www.biologyimagelibrary.com/


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48 4 5 2 082 29 5 1 2115 66 9 1 0135 103 33 12 0142 132 82 43 9121 140 124 80 3086 121 135 108 6763 98 129 125 10252 70 108 125 11753 55 83 113 12372 62 66 98 12477 73 63 77 10755 69 69 62 8132 61 80 63 7214 45 71 70 742 28 53 67 732 7 35 64 697 10 25 52 644 5 7 31 554 1 0 14 43

135 135 115 82 29142 142 135 115 66142 142 142 135 103142 142 142 142 132142 142 142 142 140142 142 142 142 140142 142 142 140 140142 142 140 140 135140 140 140 135 135129 135 135 135 129113 129 129 129 12898 124 126 127 14180 107 124 127 14480 81 110 127 14580 80 102 127 14580 80 96 126 14571 80 86 119 14464 71 77 115 14452 64 75 111 14431 55 74 108 145

1 0 0 0 01 0 0 0 01 0 0 0 01 0 0 0 09 0 0 0 033 9 0 0 052 30 7 0 052 52 30 7 152 52 52 30 752 52 52 52 3532 32 52 53 5514 14 32 55 622 2 14 32 612 2 2 14 452 2 2 2 282 2 2 2 71 0 0 0 70 0 0 0 00 0 0 0 00 0 0 0 0R=3


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48 4 5 2 082 29 5 1 2115 66 9 1 0135 103 33 12 0142 132 82 43 9121 140 124 80 3086 121 135 108 6763 98 129 125 10252 70 108 125 11753 55 83 113 12372 62 66 98 12477 73 63 77 10755 69 69 62 8132 61 80 63 7214 45 71 70 742 28 53 67 732 7 35 64 697 10 25 52 644 5 7 31 554 1 0 14 43

3 2 0 0 03 1 1 0 09 1 0 0 033 9 0 0 082 33 9 0 063 80 30 7 052 63 67 30 752 52 63 67 3052 52 52 70 6752 52 53 55 8352 53 55 62 6632 55 62 62 6214 32 55 62 622 14 32 61 622 2 14 45 632 2 2 28 532 2 2 7 352 1 0 7 250 0 0 0 70 0 0 0 0

115 82 48 6 6135 115 82 29 5142 135 115 65 12142 142 135 103 43142 142 142 132 82142 142 140 140 124142 140 140 134 134129 140 134 134 129108 129 134 129 12583 113 129 125 12677 98 124 126 12677 77 107 124 12677 80 81 110 12780 80 80 102 12771 80 80 96 12653 71 80 86 11935 64 71 77 11525 52 64 75 11110 31 55 74 1087 14 42 69 100 R=1.5


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62 82 91 87 5680 91 105 105 80

106 115 123 119 100133 154 151 137 115147 181 181 159 130125 163 186 174 14889 125 164 170 15855 89 116 133 13542 71 80 102 12026 47 69 84 10214 24 59 74 929 11 27 50 797 7 7 27 496 2 7 13 203 0 2 4 80 0 13 7 10 0 6 3 00 1 0 0 00 1 0 0 01 0 4 0 0

62 62 56 19 562 62 56 19 562 62 80 33 580 80 91 55 1489 89 115 79 3255 55 89 98 5142 42 55 80 8226 26 42 69 8014 14 24 47 699 9 11 24 507 7 7 7 272 2 2 7 70 0 0 2 40 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

114 123 123 122 119154 154 154 151 137181 180 181 181 159181 186 186 186 181186 186 185 186 186186 186 186 186 186186 186 186 186 186164 186 186 186 174124 164 170 171 17089 116 134 136 15072 83 102 120 13359 74 92 102 10927 59 79 92 9211 27 50 79 8213 13 27 49 5513 13 13 30 5213 12 13 22 506 13 13 14 394 6 6 8 294 4 4 5 16

R=2


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THE SPATIAL FILTERING PROCESS

j k l

m 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*3Neighbourhood

e 3*3 Filter

a b c

d e f

g h i

OriginalImage Pixels

*

Performed for each pixel in the original image

to generate the filtered image


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SMOOTHING SPATIAL FILTERS

Simple spatial filter

Average all of the pixels in a neighbourhood around a

central value according to selected mask.

Especially useful

in removing noise

from images

Also useful for

highlighting gross

detail

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SMOOTHING SPATIAL FILTERING

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9

1/9

1/9

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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*3

Neighbourhood106

104

99

95

100 108

98

90 85

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1/91/9

1/9

1/91/9

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3*3 Smoothing

Filter

104 100 108

99 106 98

95 90 85

Original

Image Pixels

*


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Original Mean R=1

Mean R=2 Mean R=4

MEAN FILTER

SMOOTHING

As R grows

Fine details and

noise vanish

AVERAGING

FILTER


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WEIGHTED AVERAGE

Created by defining different weight for different

pixels around the center pixel.

Pixels closer to the center pixel

are contribute more to the average

Size of objects in the image should

taken into account.

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a

as

b

bt

a

as

b

bt

tsw

tysxftsw

yxg

),(

),(),(

),(General annotation for

weighted average filtering

an MxN image


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Comparison ofMean and

Median

R=2

Mean Filter Median Filter


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GENERAL FILTERING REMARKS

There is no such thing as GOOD FILTER

but a SUITABLE filter for a specific need / task.

Spatial filters can and should be applied in

combination for optimal results.

Spatial filters changethe numbers of signal

intensities and might change the area of an objectshould be remembered in quantification

GO BACK TO THE ORIGINAL


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SHARPENING SPATIAL FILTERS

The principal objective of sharpening is to highlight fine

detail in an image or to enhance detail that has been

blurred, either in error or as a natural effect of a

particular method of image acquisition.

Sharpening filters are based on spatial differentiation

Averaging Integration

= Blurring

Sharpening

spatial differentiation

Gonzalez Woods Chapter 3


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SPATIAL DIFFERENTIATION

Differentiation measures the rate of change of a function.For a simplified explanation we will use a 1 dimensional

example

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)


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SPATIAL DIFFERENTIATION

ImagestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

A B


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1ST DERIVATIVE

The formula for the 1st derivative of a function is as

follows:

Calculates difference between subsequent values and

measures the rate of change of the function

)()1( xfxfxf


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1STDERIVATIVE (CONT)

0

1

2

3

4

5

6

7

8

-8

-6

-4

-2

0

2

4

6

8

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)


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2ND DERIVATIVE

The formula for the 2nd derivative of a function is as

follows:

Takes into account the values both before and after the

current value

)(2)1()1(2

2xfxfxf

xf


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1ST AND 2ND DERIVATIVE

-15

-10

-5

0

5

10

f(x)

f(x)

f(x)

0

2

4

6

8

-8

-6

-4

-2

0

2

4

6

8


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2ND DERIVATIVES FOR IMAGE

ENHANCEMENTThe 2nd derivative is more useful for image enhancement

Stronger response to fine detail Simpler implementation

The approach

Define a discrete formulation of the second-order derivative. Construct a filter mask based on that formulation.

We are interested in isotropic filters

Independent of the direction of the discontinuities in the image

Rotation invariant

Laplacian - the first sharpening filter we will look at

One of the simplest sharpening filters Isotropic

Linear


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THE LAPLACIANThe Laplacian is defined as follows:

Partial 1st order derivative in thex :

In they direction as follows:

y

f

x

ff

2

2

2

22

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

2

yxfyxfyxfx

f

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

2

yxfyxfyxfy

f


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THE LAPLACIAN (CONT)

The sum is given by:

A filter built based on this

),1(),1([2 yxfyxff

)]1,()1,( yxfyxf ),(4 yxf

0 1 0

1 -4 1

0 1 0


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THE LAPLACIAN (CONT)

Applying the Laplacian to an image we get a new

image that highlights edges and other discontinuities

Ima

gestakenfromGonzalez&Woods,DigitalIm

ageProcessing(2002)

Original

Image

Laplacian

Filtered Image

Laplacian

Filtered Image

Scaled for Display


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LAPLACIAN IMAGE ENHANCEMENT

In the final sharpened image edges and fine detail aremuch more obvious

Ima


ageProcessing(2002)

- =

Original

Image

Laplacian

Filtered Image

Sharpened

Image

fyxfyxg 2),(),(


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LAPLACIAN IMAGE ENHANCEMENT

Ima


ageProcessing(2002)


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Original channel Laplace filter Subtraction result

http://www.biologyimagelibrary.com/Image - 21481_0_Miller_28_BIL27090


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SHARPENING SUMMARY

A derivative operator is proportional to the degree of

discontinuity of the image at the point at which the

operator is applied.

Thus, image differentiation enhances edges and other

discontinuities (such as noise) and deemphasizes

areas with slowly varying gray-level values

Gonzalez Woods 2nd eddition Chapter 3

EDGE DETECTION AND FILTER


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Original Gaussian blur

Laplace Gradient

magnitude

EDGE DETECTION AND FILTER

COMBINATIONS

X


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COMBINING SPATIAL

ENHANCEMENT METHODS

Applying a single spatial

operation is usually not enough

Acombination of a range of

techniques in will usually result

in a better final result.

This example will focus on

enhancing the bone scan to the

right

Im

agestakenfromGonzalez&Woods,DigitalImageProcessing(2002)

COMBINING SPATIAL


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COMBINING SPATIAL

ENHANCEMENT METHODS (CONT)

Ima


ageProcessing(2002)

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)

COMBINING SPATIAL


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COMBINING SPATIAL


Ima


ageProcessing(2002)

The product of (c)

and (e) which will

be used as a mask

Sharpened image

which is sum of (a)

and (f)

Result of applying

a power-law trans.to (g)

(e)

(f)

(g)

(h)

Image (d) smoothed with a5*5 averaging filter


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COMBINING SPATIAL


Compare the original and final images


ageProcessing(2002)

image enhancement in the spatial domain_sa

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