UNIT-II: Image Enhancement in Spatial Domain

UNIT-II: Image Enhancement in Spatial Domain

Presented by:

Shahnawaz Uddin

DIGITAL IMAGE

PROCESSING (WLE-306)



Spatial domain process

 where is the input image, is the processed image, and T is an operator on f,

defined over some neighborhood of

)]

, ( [ )

,

( x y T f x y

g 

) , ( x y

f

) , (x y

Intensity Transformations and Spatial Filtering

• Intensity transformations operate on single pixel of an image, principally for the purpose of contrast manipulation

& image thresholding

• Spatial Filtering deals with performing operations, such as image sharpening by working in a neighborhood of every pixel in an image



Neighborhood about a point

Intensity Transformations and

Spatial Filtering



Gray-level transformation function

 where r is the gray level of f(x, y) and s is the gray level of g(x, y) at any point (x, y)

)

(r

T

s 



Contrast enhancement

 For example, a thresholding function



Masks (filters, kernels, templates, windows)

 A small 2-D array in which the values of the mask coefficients determine the nature of the process

Some Basic Gray Level

Transformations



Image negatives

 Enhance white or gray details

r L

s   1 

 Log transformations:

 where, c is a constant, and r≥0

 This transformation maps a narrow range of low intensity values into a wider range of output levels and opposite is true of higher values of input levels, (i.e., to expand the values of dark pixels while

compressing the higher values in an image)

 Compress the dynamic range of images with large variations in pixel values

 The inverse logarithm transformation does the inverse of the log transformation

) 1

log( r c

s  

 From the range 0- to the range 0 to 6.2

10

5 .

1 

Power-law transformations:

or

 maps a narrow range of dark input values into a wider range of output values, while maps a

narrow range of bright input values into a wider range of output values

 : gamma, gamma correction

cr

s  s  c ( r   )

 1



 1







Monitor, ^{  2 . 5}



Piecewise-linear transformation functions

 The form of piecewise functions can be arbitrarily complex

 Contrast stretching

 Gray-level slicing

 Bit-plane slicing

Histogram Processing



Histogram

where r_k is the k^th gray level and n_k is the number of pixels in the image having gray level r_k

 Normalized histogram

k

k

n

r

h ( ) 

n n

r

p (

) 

/

Histogram Equalization

1 0

),

(   

T r r L s

1 0

),

1(   

 T ^ s s L r

• Here, we focus our attention on transformations of the from

that produce an output intensity level s for every pixel in the input image having intensity r. We assume that:

(a)T(r) is monotonically increasing function in 0<= r <= L-1 and (b) 0<= T(r) <= L-1 for 0<= r <= L-1

In some formulations, we use the inverse

in which case, we change condition (a) to (a’)

(a’) T(r) is strictly monotonically

increasing function in 0<= r <= L-1

 Probability density functions (PDF)

ds r dr p

s

p

( ) 

( )







 T r L

p

w dw s ( ) ( 1 )

( )

) ( )

1 (

) ( )

1 ) (

(

0

p w dw L p r

dr L d

dr r dT dr

ds

r r

r

   

 





 

1 ) 1

(  

s L

p

1 ,...,

2 , 1 , 0

, )

1 (

) ( )

1 (

) (

0 0













  





L n k

L n r

p L

r T s

k

j k j

j

j r k

k

• For the discrete form of transformation, the histogram equalization transformation is given by:

• Inverse Transformation from s back to r is denoted by

 The method used to generate a processed image that has a specified histogram is called histogram matching/specification.







 T r L

p

w dw s ( ) ( 1 )

( )

 ^



 L

p

t dt s z

G ( ) ( 1 )

( )

)]

( [ )

(

1

s G T r

G

z 



) ( z

p

Histogram Matching (Specification)

1 ,...,

2 , 1 , 0

, )

1 (

) ( )

1 (

) (

0 0













  





L n k

L n r

p L

r T s

k

j k j

j

j r k

k

1 ,...,

2 , 1 , 0

, )

( )

1 (

) (

0











 



L k

s z

p L

z G

v

k

i

i z k

k

1 ,...,

2 , 1 , 0

)], (

1

[  

 G

T r k L

z



Histogram matching

 Obtain the histogram of the given image, T(r)

 Precompute a mapped level for each level

 Obtain the transformation function G from the given

 Precompute for each value of

 Map to its corresponding level ; then map level into the final level

) ( z p

s

r

z

s

r

s

s

z

Example 3.8: A simple example of

histogram matching/specification

Histogram Matching (Specification)



Local enhancement

 Histogram using a local neighborhood, for example 7*7 neighborhood

 Histogram using a local 3*3 neighborhood

Fundamentals of Spatial Filtering



The Mechanics of Spatial Filtering

 Image size:

 Mask size:

 and

 and

N M 

n m 

 







a s

b

b t

t y

s x

f t s w y

x

g ( , ) ( , ) ( , )

2 / ) 1

( 

 m

a b  ( n  1 ) / 2 1

,..., 2

, 1 ,

0 

 M

x y  0 , 1 , 2 ,..., N  1



Spatial Correlation and Convolution

Vector Representation of Linear Filtering

Generating Spatial Filter Masks

Smoothing Spatial Filters



Smoothing Linear Filters

 Noise reduction

 Smoothing of false contours

 Reduction of irrelevant detail that is smaller than the filter mask

 Blurring the edges in an image









1 9

1

16

1 9

1

i

i i

i

and R z

z R

To reduce the blurring effect during smoothing filtering

 

 



 



 







a s

b

b t a

a s

b

b t

t s w

t y

s x

f t s w y

x g

) , (

) ,

( ) , ( )

,

(



Order-statistic (Non-linear) spatial filters

 median filter: Replace the value of a pixel by the median of the gray levels in the

neighborhood of that pixel

 Noise-reduction

Note:

Sharpening Spatial Filters



Foundation

 The first-order derivative

 The second-order derivative

) ( )

1

( x f x

x f

f   





) ( 2 )

1 (

) 1

2

(

2

x f x

f x

x f

f     







Use of second derivatives for enhancement-The Laplacian

 Development of the method

) , ( 2 )

, 1 (

) , 1

2

(

2

y x f y

x f y

x x f

f     





2 2 2

2 2

y f x

f f



 



 



) , ( 2 )

1 ,

( )

1 ,

2

(

2

y x f y

x f y

x y f

f     





) , ( 4 )]

1 ,

(

) 1 ,

( )

, 1 (

) , 1 (

2

[

y x f y

x f

y x f y

x f y

x f f





















 





 

















positive is

mask Laplacian

the of

t coefficien center

the if

) , ( )

, (

negative is

mask Laplacian

the of

t coefficien center

the if

) , ( )

, ( )

, (

2 2

y x f y

x f

y x f y

x f y

x

g

Note:

 Simplifications

)]

1 ,

(

) 1 ,

( )

, 1 (

) , 1 (

[ )

, ( 5

) , ( 4 )]

1 ,

(

) 1 ,

( )

, 1 (

) , 1 (

[ )

, ( )

, (







































y x f

y x f y

x f y

x f y

x f

y x f y

x f

y x f y

x f y

x f y

x f y

x

g



Unsharp masking and highboost filtering

 Unsharp masking

 Substract a blurred version of an image from the image itself

 : The image, : The blurred image

) , ( )

, ( )

,

( x y f x y f x y

g

 

) , ( x y

f f ( x , y ) ) , (

* )

, ( )

,

( x y f x y k g x y

g  

, k  1

 High-boost filtering

) , (

* )

, ( )

,

( x y f x y k g x y

g  

, k  1

Salt-and-pepper noise in an image