DIGITAL IMAGE PROCESSING

Spatial Domain &

Spatial FilteringMay 1, 2023 1

INTRODUCTION TO IMAGE ENHANCEMENT

Enhancement is to process an image so that the result is more suitable than the original image for a specific application

Two approaches use for image enhancement:Spatial domain: Refer to the image plane itself. Direct manipulation of pixels in

an imageFrequency Domain: Based on modifying the Fourier Transform of an image

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GOOD IMAGES• For human visual

– The visual evaluation of image quality is a highly subjective process.

– It is hard to standardize the definition of a good image.• For machine perception

– The evaluation task is easier.– A good image is one which gives the best machine

recognition results.• A certain amount of trial and error usually is

required before a particular image enhancement approach is selected.

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SPATIAL DOMAIN• Procedures that operate

directly on pixels.g(x,y) = T[f(x,y)]g(x,y) = T[f(x,y)]

where – f(x,y) f(x,y) is the input image– g(x,y) g(x,y) is the processed

image– T T is an operator on f f

defined over some neighborhood of (x,y)(x,y)

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MASK/FILTER

• Neighborhood of a point (x,y) can be defined by using a square/rectangular (common used) or circular subimage area centered at (x,y)

• The center of the subimage is moved from pixel to pixel starting at the top of the corner

•(x,y)

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POINT PROCESSING

• Neighborhood = 1x1 pixel• gg depends on only the value of ff at (x,y)(x,y)

• TT = gray level (or intensity or mapping) transformation function

s = T(r)s = T(r)• Where

– rr = gray level of f(x,y)f(x,y)– ss = gray level of g(x,y)g(x,y)

GRAY LEVEL IMAGE TRANSFORMATION

The value of pixels before and after processing are denoted by r and s,s = T(r), Where T is Transformation

Basic types of function used for image enhancement:

Linear: Negative and identity transformationsLogarithm: Log and inverse log transformationsPower-low: n-th power and n-th root transformations

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LOG AND POWER TRANSFORMATIONS

Log transformations:s = clog(1+r) where, c is a constant, r>=0

Power-low transformations:s=cr, where c and are positive constants

Gamma Corrections:

The exponent of the power law equation is referred to as gamma. The process used to correct this power law response phenomena is called gamma correction.

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IMAGE ENHANCEMENT IN THE SPATIAL DOMAIN(CONTRAST STRETCHING)

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Contrast stretching is piecewise-linear transformation function.

to increase the dynamic range of the gray levels in the image being processed

T(r)

(r2, s2)

L/2

L/4

3L/4

(r1, s1)L-1L/2 3L/4

L-1

L/40

Input gray level, r

outp

ut g

ray

leve

l, s

IMAGE ENHANCEMENT IN THE SPATIAL DOMAIN(HISTOGRAM PROCESSING)

Intensity distribution in the imageThe histogram functions count the number of elements within a range and display each range

as a rectangular bin. The height (or length when using rose) of the bins represents the number of values that fall within each range.

for (int i=0;i<width;i++){ for (int j=0;j<height;j++){ int x = pixels[i][j]; histData[x] =histData[x]+1; } }

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IMAGE WITH HISTOGRAM

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HISTOGRAM EQUALIZATION

The probability of occurrence of gray level rk in an image is approximated by,

Where k =0,1,2,---L-1; n is the total number of pixels, nk is the number of pixels that have gray level rk.

A plot pr(rk) vs. rk is called histogramThe transformation (mapping) sk is

called histogram equalization

k

j

k

j

jjrkk n

nrPrTs

0 0

)()(

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( ) kk

nP r

n

HISTOGRAM EQUALIZATION ALGORITHM

1. Calculate Histogramloop over i ROWS of input image

loop over j COLS of input imagek = input_image[i][j]hist[k] = hist[k] + 1end loop over j

end loop over i2. calculate the sum of hist

loop over i gray levelssum = sum + hist[i]sum_of_hist[i] = sumend loop over i

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3. Transform input image to output image

area = area of image (ROWS x COLS)Dm = number of gray levels in output

imageloop over i ROWSloop over j COLSk = input_image[i][j]out_image[i][j] = (Dm/area) x sum_of_hist[k]

end loop over jend loop over i

COMPARISON OF IMAGE:BEFORE AND AFTER HISTOGRAM EQUALIZATION

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Make low bright region to higher bright

Reduce the number of gray level

IMAGE ENHANCEMENT IN THE SPATIAL DOMAIN:

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Local Enhancement:Enhancement of image in a particular area to reduce noise.

Image ArithmaticImage ArithmaticImage subtraction, addition, multiplication, divisiong (x, y) = f (x, y) – h (x, y)g (x, y) = f (x, y) + h (x, y)

BASICS OF SPATIAL FILTERINGSome neighborhood operations work with the values of the

image pixels in the neighborhood and the corresponding values of a sub-image that has a same dimensions as the neighborhood.

This sub-image is called filter, mask, kernel, template or window.

The values in a filter sub-image are referred to as coefficients, rather than pixels

Filtering operations that are performed directly on the pixels of an image is called spatial Filtering.

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BASICS OF SPATIAL FILTERING

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(0,0)

Mask

Image f(x,y)

y

x

W(-1,-1) W(-1,0) W(-1,1)

W(1,0)W(1,-1)

W(0,-1) W(0,0) W(0,1)

W(1,1)

f(x+1,y+1)f(x+1,y)f(x+1,y-1)

f(x,y-1) f(x,y+1)

f(x-1,y+1)f(x-1,y)f(x-1,y-1)

f(x,y)

Pixels of image section under mask

Mask coefficients

BASICS OF SPATIAL FILTERING

Moving the filter mask from point to point in an image At each point (x,y), the response of the filter at that point is calculated using a predefined relationship

The response is given by a sum of products of the filter coefficients and the corresponding image pixels in the area spanned by the filter mask.

Limitation near boundary: to limit the excursions of the center of the mask to be at a

distance no less than (n-1)/2padding by 0 or constant gray level, then stripped

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SMOOTHING SPATIAL FILTERS

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Smoothing Linear Filter:The output (response) of a smoothing, linear spatial filter is simply the average of the pixels contained in the neighborhood of the filter mask. These filters sometimes are called averaging filters or low pass filters.

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1111

11

1

11

116

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421

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3x3 smoothing filter masks

SMOOTHING SPATIAL FILTERS

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Non-Linear Spatial Filter:The nonlinear spatial filter whose response is based on ordering the pixels contained in the image area encompassed by the filter and then replacing the value of the center pixel with the value determined by the ranking result. The best known filter is median filters.

Max Filters: use to find the brightest point

Min Filters:use to find the darkest point

SHARPENING SPATIAL FILTERS

Sharpening is to highlight fine detail in an imageFirst-order derivative: of an one dimensional function, f(x)

Second-Order derivative

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( 1) ( )f f x f xx

2

2 ( 1) ( 1) 2 ( )f f x f x f xx

EXAMPLE OF FIRST AND SECOND ORDER DERIVATIVES

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1st derivative

2nd derivative

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f(x-1) f(x) f(x+1)

0 6 0 0 0

-1, -1, -1, -1, -1, 0, 0, 6, -6, 0

7 7 ..

-1, 0, 0, 0, 0, 1, 0, 6, -12, 6First order derivative produce thick edge

2nd order derivative produce much finer one and strong response at isolated point

Thank You