Intensity Transformations and Spatial Filtering

Basics of Intensity Transformation and Spatial Filtering

Spatial Domain Process

Neighborhood is rectangle, centered on (x,y), and much smaller in size than image.

Neighborhood size is 1x1, 3x3, 5x5, etc.

, ,g x y T f x y Origin (0,0)

(x,y)

(M-1,0)

(0,N-1)

3x3 Neighborhood of (x,y)

Intensity TransformationT[f(x,y)] is Intensity

Transformation, if the neighborhood size is 1x1.

Intensity Transformation can be written as follows

s = T[r],

where s = g(x,y), and r = f(x,y)

Image Negatives s = L-1 – r

where intensity level is in the range[0, L-1]

Log Transformations s = c Log(1+r)

Log Transformation is used to expand the value of the dark pixels while compressing the higher-level value.

It is used to compress the intensity of an image which has very large dynamic range.

Log Transformations of Fourier Spectrum

Original Image

Fourier Spectrum

Log Transform of

Fourier SpectrumWe cannot see the Fourier spectrum,

because its dynamic range is very large.

Power-Law (Gamma) Transformations

If <1, expand dark pixels, compress bright pixels.

If >1, compress dark pixels, expand bright pixels.

0.04 0.10

0.20 0.40

0.64

1.0

1.5

2.5 5.0

10.0

s cr

Examples of Gamma Transformations

3.0

4.0 5.0

Contrast StretchingIf r<r1 then

s = r*s1/r1If r1<= r<=r2 then

s = (r-r1)*(s2-s1)/(r2-r1)+s1If r>r2 then

s = (r-r2)*(255-s2)/(255-r2)+s2If r1=r2 and s1=0,s2=255, the

transform is called “Threshold Function”.

Examples of Contrast Stretching

Contrast Stretching in Medical Image

Window Width/Level(Center) s1=0,s2=255

width (w)=r2-r1, level (c)=(r1+r2)/2

Histogram & PDF

h(r) = nr

where nr is the number of pixels whose intensity is r.

The Probability Density Function (PDF) h r

p rM N

Cumulative Distribution Function (CDF)

PDF CDF

Transfer Function

r

s

0

rCDF r p r dr

Example of Histogram and Cumulative Distribution Function (CDF)

Low Contrast Image

The image is highly concentrated on low intensity values.

The low contrast image is the image which is highly concentrated on a narrow histogram.

HighConcentra

te

LowConcentra

te

Histogram Equalization

The Histogram Equalization is a method which makes the histogram of the image as smooth as possible

The PDF of the Transformed Variable

s = Transformed Variable.

= The PDF of r = The PDF of s

s T r

rp r

sp s

1

/

s r

r

drp s p r

ds

p rdT r dr

Transformation Function of Histogram Equalization

The PDF of s

0255

r

rs T r p r dr

0255

255

1

255

r

r

r

s r

dT rds

dr drd

p r drdrp r

drp s p r

ds

Histogram Equalization Example

Intensity # pixels

0 20

1 5

2 25

3 10

4 15

5 5

6 10

7 10

Total 100

CDF of Pr

20/100 = 0.2

(20+5)/100 = 0.25

(20+5+25)/100 = 0.5

(20+5+25+10)/100 = 0.6

(20+5+25+10+15)/100 = 0.75

(20+5+25+10+15+5)/100 = 0.8

(20+5+25+10+15+5+10)/100 = 0.9

(20+5+25+10+15+5+10+10)/100 = 1.0

1.0

Histogram Equalization Example (cont.)

Intensity (r)

No. of Pixels(nj)

Acc Sum of Pr

Output value Quantized Output (s)

0 20 0.2 0.2x7 = 1.4 1

1 5 0.25 0.25*7 = 1.75 2

2 25 0.5 0.5*7 = 3.5 3

3 10 0.6 0.6*7 = 4.2 4

4 15 0.75 0.75*7 = 5.25 5

5 5 0.8 0.8*7 = 5.6 6

6 10 0.9 0.9*7 = 6.3 6

7 10 1.0 1.0x7 = 7 7

Total 100

Histogram MatchingHow to transform the variable r

whose PDF is to the variable t whose PDF is .

0

0

1

255

255

r

r

t

t

s T r p r dr

G t p t dt s

t G t

rp r

tp t

r T( ) s G-1( ) t