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BIM472 Image Processing

Processing in Spatial Domain

Part 1. Intensity Transformations

BIM472 Image Processing 2Processing in Spatial Domain - Intensity Transformations

Outline

• Spatial Domain vs. Transform Domain

• Spatial Domain Processing

• Intensity Transformations

– Image Negative

– Log Transformation

– Power-Law (Gamma) Transformation

• Piecewise-Linear Transformations

– Contrast Streching

– Intensity Level Slicing

– Bit-plane Slicing

• Histogram Processing

– Histogram Equalization

– Histogram Matching

• Matlab Functions

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Spatial domain:

• Spatial domain refers to image plane itself

• Processing in spatial domain indicates direct manipulation

of pixels (intensity values ) in an image.

Transform domain:

• Image is first transformed into another domain (using FFT,

DCT, etc.).

• Processing in transform domain indicates manipulation of

transform coefficients, not the intensity values, of the image.

Spatial Domain vs. Transform Domain


Spatial domain processing is divided into two categories:

• Intensity transformation (point processing) : operates on

single pixels of an image, principally for the purpose of

contrast manipulation and image thresholding

• Spatial filtering: operates in a neigborhood of every pixel in

an image for performing image enhancement (i.e.,

sharpening, etc.).

Spatial Domain Processing

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( , ) [ ( , )])

( , ) : input image

( , ) : output image

: an operator on defined over

a neighborhood of point ( , )

g x y T f x y

f x y

g x y

T f

x y

=




1x1 neighborhood = Intensity Transformation

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Intensity transformation function

( )s T r=

Intensity Transformation

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Intensity Transformation


Image negatives

1s L r= − −

Image Negative

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Small lesion

Image Negative


Log Transformations

log(1 )s c r= +

Log Transformation

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Log Transformation


Power-Law (Gamma)

Transformations

s crγ=

Power-Law (Gamma) Transformation

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Example: Gamma Transformations



Example: Gamma Transformations


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• Contrast Stretching

– Contrast is the difference in visual properties that makes an object

distinguishable from other objects and the background.

– Contrast stretching expands the range of intensity levels in an image

so that it spans the full intensity range of the recording medium or

display device.

• Intensity-level Slicing

– Highlighting a specific range of intensities in an image often is of

interest.

Piecewise-Linear Transformations


Contrast Stretching

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Highlight the major blood vessels and study the shape of flow of the contrast medium (to detect blockages, etc.) Measuring the actual

flow of the contrast medium as a function of time in a series of images

Intensity Level Slicing


Bit Plane Slicing

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Bit Plane Slicing


• The values of the 8th bit are 0 for byte values [0,127] and 1

for [128,255].

• The 7th bit plane outputs a 0 for byte values in the range [0,

63], a 1 for byte values in the range [64, 127], a 0 for byte

values in the range [128, 191], and a 1 for byte values in the

range [192, 255].

• and so on...

Bit Plane Slicing

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Bit Plane Slicing

Original image is reconstructed using formula below:

Image = 2*(2*(2*(2*(2*(2*(2*b7+b6)+b5)+b4)+b3)+b2)+b1)+b0;


• Histogram Equalization

• Histogram Matching

Histogram Processing

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• Histogram is a graph indicating the number of times each

grey level occurs in the image.

• p(rk) is an estimate of the probability of occurence of

intensity level rk in an image. Sum of all components is

equal to 1.


Histogram ( )

is the intensity value

is the number of pixels in the image with intensity

k k

th

k

k k

h r n

r k

n r

=

Normalized histogram ( )

: the number of pixels in the image of

size M N with intensity

kk

k

k

np r

MN

n

r

=

×


We can gather a great deal about the appearance of an image

from its histogram, as the following examples indicate.

• In a dark image, the grey levels (and hence the histogram)

would be clustered at the lower end.

• In a uniformly bright image, the grey levels would be

clustered at the upper end.

• In a well contrasted image, the grey levels would be well

spread out over much of the range.


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Histogram Equalization

• Histogram equalization is a method for contrast adjustment using

the histogram of the image.

• In other words, it produces a uniform histogram

• Transformation function (continuous) that produce an output intensity

level s for every pixel in the input image having intensity r :

w is a dummy variable of integration

p(w) represents the probability density function (PDF)

L is the maximum intensity level

0( ) ( 1) ( )

r

rs T r L p w dw= = − ∫

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( ) 0 1s T r r L= ≤ ≤ −. T(r) is a strictly monotonically increasing function

in the interval 0 -1;

. 0 ( ) -1 for 0 -1.

a

r L

b T r L r L

≤ ≤

≤ ≤ ≤ ≤


• Assumptions on transformation function:


Example

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Suppose that the (continuous) intensity values

in an image have the PDF

2, for 0 r L-1

( 1)( )

0, otherwise

Find the transformation function for equalizing

the image histogra

r

r

Lp r

≤ ≤ −=

m.


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Example

0( ) ( 1) ( )

r

rs T r L p w dw= = − ∫20

2( 1)

( 1)

r wL dw

L= −

−∫2

1

r

L=

−



0

Continuous case:

( ) ( 1) ( )r

rs T r L p w dw= = − ∫

0

Discrete values:

( ) ( 1) ( )k

k k r j

j

s T r L p r=

= = − ∑

0 0

1( 1) k=0,1,..., L-1

k kj

j

j j

n LL n

MN MN= =

−= − =∑ ∑


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Example:

Suppose that a 3-bit image (L=8) of size 64 × 64 pixels (MN = 4096) has the intensity distribution shown in following table. Get the histogram equalization transformation function and give the ps(sk) for each sk.



0

0 0

0

( ) 7 ( ) 7 0.19 1.33r j

j

s T r p r=

= = = × =∑ 1→1

1 1

0

( ) 7 ( ) 7 (0.19 0.25) 3.08r j

j

s T r p r=

= = = × + =∑ 3→

2 3

4 5

6 7

4.55 5 5.67 6

6.23 6 6.65 7

6.86 7 7.00 7

s s

s s

s s

= → = →

= → = →

= → = →


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• New histogram values:

– s0 : 790 / 4096

– s1 : 1023 / 4096

– s2 : 850 / 4096

– s3, s4 : (656 + 329) / 4096

– s5, s6, s7 : (245+122+81) / 4096




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• Histogram matching (histogram specification) generate a processed image that has a specified histogram

• Obtain pr(rj) from the input image and then obtain the values of sk, round the value to the integer range [0, L-1].

• Use the specified PDF and obtain the transformation function G(zq), round the value to the integer range [0, L-1].

• Mapping from sk to zq

0 0

( 1)( ) ( 1) ( )

k k

k k r j j

j j

Ls T r L p r n

MN= =

−= = − =∑ ∑

0

( ) ( 1) ( )q

q z i k

i

G z L p z s=

= − =∑1( )q kz G s−=

Histogram Matching

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Example: Histogram Matching

Suppose that a 3-bit image (L=8) of size 64 × 64 pixels (MN = 4096) has the intensity distribution shown in the following table (on the left). Get the histogram transformation function and make the output image with the specified histogram, listed in the table on the right.

Histogram Matching



Obtain the scaled histogram-equalized values,

Compute all the values of the transformation function G,

0 1 2 3 4

5 6 7

1, 3, 5, 6, 7,

7, 7, 7.

s s s s s

s s s

= = = = =

= = =

0

0

0

( ) 7 ( ) 0.00z j

j

G z p z=

= =∑

1 2

3 4

5 6

7

( ) 0.00 ( ) 0.00

( ) 1.05 ( ) 2.45

( ) 4.55 ( ) 5.95

( ) 7.00

G z G z

G z G z

G z G z

G z

= =

= =

= =

=

0→

0→ 0→1→ 2→

6→5→

7→

Histogram Matching

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Histogram Matching



Obtain the scaled histogram-equalized values,

Compute all the values of the transformation function G,

0 1 2 3 4

5 6 7

1, 3, 5, 6, 7,

7, 7, 7.

s s s s s

s s s

= = = = =

= = =

0

0

0

( ) 7 ( ) 0.00z j

j

G z p z=

= =∑

1 2

3 4

5 6

7

( ) 0.00 ( ) 0.00

( ) 1.05 ( ) 2.45

( ) 4.55 ( ) 5.95

( ) 7.00

G z G z

G z G z

G z G z

G z

= =

= =

= =

=

0→

0→ 0→1→ 2→

6→5→

7→

s0

s2 s3

s5 s6 s7

s1

s4

Histogram Matching

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0 1 2 3 4

5 6 7

1, 3, 5, 6, 7,

7, 7, 7.

s s s s s

s s s

= = = = =

= = =

0

1

2

3

4

5

6

7

kr

Histogram Matching



0 3

1 4

2 5

3 6

4 7

5 7

6 7

7 7

k qr z→

→

→

→

→

→

→

→

→

Histogram Matching

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Histogram Matching



Histogram Matching

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Histogram Matching


Matlab Functions

• Image arithmetic.– imadd - Add two images or add constant to image.

– imsubtract - Subtract two images or subtract constant from image.

– imlincomb - Linear combination of images.

– imcomplement - Complement image (negative)

• Spatial transformations.– imresize - Resize image.

– imrotate - Rotate image.

– imadjust - Adjust image intensity values

• Histogram operations– imhist - Display histogram of image data.

– histeq - Enhance contrast using histogram equalization.

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Summary

• Spatial Domain vs. Transform Domain

• Spatial Domain Processing

• Intensity Transformations

– Image Negative

– Log Transformation

– Power-Law (Gamma) Transformation

• Piecewise-Linear Transformations

– Contrast Streching

– Intensity Level Slicing

– Bit-plane Slicing

• Histogram Processing

– Histogram Equalization

– Histogram Matching

• Matlab Functions


References

• “Digital Image Processing, 3/E”, R. C. Gonzalez & R. E.

Woods, Prentice Hall.

• “Lecture Notes”, Frank (Qingzhong) Liu, University of New

Mexico Tech.

• “Lecture Notes”, Richard Alan Peters, Vanderbilt University.

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