Digital Image Processing: Image Enhancement - Duong Anh Duc

Digital Image Processing: Image Enhancement - Duong Anh Duc presents about Image Enhancement; Point Operations; Image Negative; Contrast Stretching; Compression of Dynamic Range; Image Averaging for noise reduction; Some Averaging Filters; Some Typical Histograms.

Digital Image Processing

Image Enhancement

– Spatial domain: operate on the original image

g ( m , n ) = T [ f ( m , n )]

– Frequency domain: operate on the DFT of the original image

Image Enhancement Techniques

Point Operations Mask Operations Transform Operations Coloring Operations

• Median Filtering

• Sharpening operations

• Derivative operations

• Histogram operations

• Low pass Filtering

• Hi pass Filtering

• Band pass Filtering

• Homomorphic Filtering

• Histogram operations

• False Coloring

• Full color Processing

Point Operations

 Output pixel value g ( m , n ) at pixel ( m , n ) depends only on the input

pixel value at f ( m , n ) at ( m , n ) (and not on the neighboring pixel values)

 We normally write s = T ( r ), where s is the output pixel value and r is the input pixel value

Image Negative

T(r) = s = L-1-r, L: max grayvalue

Negative Image

Contrast Stretching

 Increase the dynamic range of grayvalues in the input image

 Suppose you are interested in stretching the input intensity values in the interval [ r1, r2]:

 Note that ( r - r ) < ( s - s ) The grayvalues in the range [ r , r ] is stretched into

only interested in the

shape of the objects

and on on their actual

grayvalues

Contrast Stretching

1

1

, 1 ,

, 0

r r

r r

r r

r

r r

r r

r T

g

Contrast Stretching

Gamma correction

Compression of Dynamic

Range

 When the dynamic range of the input

grayvalues is large compared to that of the display, we need to “compress” the grayvalue range - example: Fourier

transform magnitude

 Typically we use a log scale

Compression of Dynamic

Range

in log scale

Compression of Dynamic

Range

 Graylevel Slicing : Highlight a specific

range of grayvalues

Compression of Dynamic

Range

 Bitplane Slicing : Display the different

bits as individual binary images

Compression of Dynamic

Range

Image Subtraction

 In this case, the difference between two

“similar” images is computed to highlight

or enhance the differences between

them:

g ( m,n ) = f 1 ( m,n ) -f 2 ( m,n )

 It has applications in image segmentation

and enhancement

Example: Mask mode radiography

f1(m, n): Image before dye injection

f2(m, n): Image after dye injection

g(m, n): Image after dye injection,

followed by subtraction

Image Averaging for noise

reduction

 Noise is any random (unpredictable)

phenomenon that contaminates an image

Noise

Image Averaging for noise

reduction

 The noise h ( m , n ) at each pixel ( m , n ) is modeled

as a random variable

 Usually, h ( m , n ) has zero-mean and the noise values

at different pixels are uncorrelated

 Suppose we have M observations { gi( m , n )}, i= 1, 2, …,

M , we can (partially) mitigate the effect of noise by

“averaging”

M i

g M

n m

g

1

, 1

,

Image Averaging for noise

reduction

 In this case, we can show that:

 Therefore, as the number of observations

increases ( M ), the effect of noise

tends to zero

n

m M

n m g

n m f

n m g

E

, Var

1 ,

Var

, ,

Image Averaging Example

Noise Variance = 0.05

Image Averaging Example

Image Averaging Example

Image Averaging Example

Some Averaging Filters

 The histogram of a digital image with

grayvalues r 0 , r 1 , …, r L-1 is the discrete function

 The function p(r k ) represents the fraction of

the total number of pixels with grayvalue r k

appearance of the image

image

in    pixels

#   total

  e  with valu pixels

#  

n r

k

 If we consider the grayvalues in the image

as realizations of a random variable R , with some probability density, histogram

provides an approximation to this probability density In other words,

Pr[ R=r k ] p(r k )

Some Typical Histograms

information for contrast enhancement

Some Typical Histograms (cont.)

information for contrast enhancement

Example: Histogram Stretching

The original image displayed

The stretched image

0 5000

10000

The histogram of the original image

0 5000 10000

The stretched histogram

Histogram Equalization

 Idea: find a non-linear transformation

g = T(f )

to be applied to each pixel of the input

image f(x,y), such that a uniform distribution

of gray levels in the entire range results for

the output image g(x,y).

T ( r ), based on the histogram of the input image, which will enhance the image.

– T ( r ) is a monotonically increasing function for 0 r 1

(preserves order from black to white)

– T ( r ) maps [0,1] into [0,1] (preserves the range of

allowed grayvalues)

Histogram Equalization

Histogram Equalization

 Let us denote the inverse transformation by r =

T - 1 ( s ) We assume that the inverse transformation also satisfies the above two conditions

and output image as random variables in the

interval [0, 1]

 Let p in ( r ) and p out ( s ) denote the probability density

of the grayvalues in the input and output images

Histogram Equalization

 If p in ( r ) and T ( r ) are known, and T - 1 ( s ) satisfies

condition 1, we can write (result from probability theory):

transformation T (.) such that the grayvalues in the output is uniformly distributed in [0, 1], i.e

p ( s )=1, 0 s 1

s T r

in out

ds

dr r

p s

p

1

r dw

w p

r T

Histogram Equalization

 Note that this is the cumulative distribution function (CDF) of p in ( r ) and satisfies the

previous two conditions

 From the previous equation and using the fundamental theorem of calculus,

r

p dr

ds

in

Histogram Equalization

 Therefore, the output histogram is given by

 The output probability density function is

uniform, regardless of the input

1 0

for   , 1 1

r p s

p

s T r

s T r in

in out

Histogram Equalization

 Thus, using a transformation function

equal to the CDF of input grayvalues r ,

we can obtain an image with uniform

grayvalues.

 This usually results in an enhanced image, with an increase in the dynamic range of

pixel values

Example: Histogram Equalization

Original image Pout after histogram equalization

Example: Histogram Equalization

Histogram Equalization

 For images with discrete grayvalues, we

have

– L : Total number of graylevels

– n k : Number of pixels with grayvalue rk

– n : Total number of pixels in the image

1 0

  and  ,

1 0

k

Histogram Equalization

 The discrete version of the previous

transformation based on CDF is given by:

1 0

for     

,

0

L k

r

p n

n r

i k

k

Example(cont.)

 Notice that there are

only five distinct

195.0

7

689.0

7

681.0

7

565.0

7

344.0

7

119.0

7 in 1

in 0

in

7 in 7

7

6 in 1

in 0

in 6

6 6

5 in 1

in 0

in 5

5 5

4 in 1

in 0

in 4

4 4

3 in 1

in 0

in 3

3 3

2 in 1

in 0

in 2

2 2

1 in 0

in 1

1 1

0 in 0

0 0

r p r

p r p r p r

T s

r p r

p r p r p r

T s

r p r

p r p r p r

T s

r p r

p r p r p r

T s

r p r

p r p r p r

T s

r p r p r p r p r

T s

r p r p r p r

T s

r p r p r

T s

 Note that the histogram of output image is only

approximately, and not exactly, uniform This should not be surprising, since there is no result that claims uniformity in the discrete case.

Example(cont.)

Histogram of original image Histogram of equalized image

Example(cont.)

Histogram of original image Histogram of equalized image

Example(cont.)

Histogram of original image Histogram of equalized image

Example(cont.)

Histogram of original image Histogram of equalized image

Histogram Equalization

desirable results, particularly if the given histogram

is very narrow It can produce false edges and

regions It can also increase image “graininess”

and “patchiness.”

Histogram Specification

pixels are (in theory) uniformly distributed among all graylevels.

may want a transformation that yields an output

image with a prespecified histogram This

grayvalues

Histogram Specification

 Suppose, the input image has probability density pin( r ) We want to find a transformation z = H ( r ), such that the

probability density of the new image obtained by this

transformation is pout( z ), which is not necessarily uniform.

 First apply the transformation

This gives an image with a uniform probability density.

* 1

0

,

in

r o

r dw

w p

r T s

Histogram Specification

 If the desired output image were available, then the following transformation would

generate an image with uniform density:

 From the grayvalues we can obtain the

grayvalues z by using the inverse

transformation, z = G -1 ( ).

*

* 1

w p

z G

will generate an image with the specified density

will generate an image with the specified density pout( z ),

from an input image with density pin( r )

 For discrete graylevels, we have

1 0

for      ,

and  

1 0

for      ,

out

0

L k

z p

z G

L

k n

n r

T s

k

i k

k

k i

i k

k

Histogram Specification

 If the transformation z k G ( z k ) is

one-to-one, the inverse transformation s k G -1 ( s k ), can be easily determined, since we are

dealing with a small set of discrete

grayvalues.

 In practice, this is not usually the case (i.e.,

z k G ( z k ) is not one-to-one) and we assign grayvalues to match the given histogram, as closely as possible.

Ex.: Histogram Specification

Ex.: Histogram Specification

 It is desired to transform this image into a new image,

using a transformation z=H ( r ) = G-1[ T(r )], with histogram as

Ex.: Histogram Specification

 The transformation T ( r ) was obtained earlier (reproduced below):

Ex.: Histogram Specification

the transformation

G as before

100.1

7

685.0

7

565.0

7

235.0

7

115.0

000.0

000.0

000.0

7

6 out 1

out 0

out 6

6 6

5 out 1

out 0

out 5

5 5

4 out 1

out 0

out 4

4 4

3 out 1

out 0

out 3

3 3

2 out 1

out 0

out 2

2 2

1 out 0

out 1

1 1

0 out 0

0 0

z p z

p z p z p z

G

z p z

p z p z p z

G

z p z

p z p z p z

G

z p z

p z p z p z

G

z p z

p z p z p z

G

z p z p z p z p z

G

z p z p z p z

G

z p z p z

Ex.: Histogram Specification

 Notice that G is not invertible But we will do the best possible by setting

G-1(0) = ? (This does not matter since s 0)

G-1(1/7) = 3/7

G-1(2/7) = 4/7 (This does not matter since s 2/7)

G-1(3/7) = 4/7 (This is not defined, but we use a close match)

G-1(4/7) = ? (This does not matter since s 4/7)

G-1(5/7) = 5/7

G-1(6/7) = 6/7

G-1(1) = 1

Ex.: Histogram Specification

 Combining the two transformation T and G-1, we get our

Ex.: Histogram Specification

 Applying the transformation H to the original image yields

an image with histogram as below:

Ex.: Histogram Specification

Ex.: Histogram Specification

 Again, the actual histogram of the output

image does not exactly but only

approximately matches with the specified

histogram This is because we are dealing with discrete histograms.

Original image and its histogram

Histogram equalized image Actual histogram of output

Histogram specified image, Actual Histogram, and Specified Histogram

Enhancement Using

Local Histogram

 Used to enhance details over small portions of the image

 Define a square or rectangular neighborhood, whose center moves from pixel to pixel

neighborhood for each point and apply a histogram

equalization or histogram specification transformation to the center pixel

 Non-overlapping neighborhoods can also be used to reduce computations But this usually results in some artifacts (checkerboard like pattern)

Enhancement Using

Local Histogram

enhancement is the statistical moments associated with the histogram (recall that the histogram can be thought of as a probability density function)

variance to determine the local

brightness/contrast of a pixel This information can then be used to determine what, if any

transformation to apply to that pixel

non-uniform in the sense that a different

transformation is applied to each pixel