Digital Image Analysis and Processing CPE 0907544 Image Enhancement - - PowerPoint PPT Presentation

• Dr. Iyad Jafar

Digital Image Analysis and Processing CPE 0907544

Image Enhancement – Part I Intensity Transformation

Chapter 3 Sections: 3.1 – 3.3

Outline

 What is Image Enhancement ?  Background  Intensity Transformation Functions

 Negatives  Log and Inverse Log  Power-Law  Piecewise Transformation  Graylevel and Bit Slicing

 Histogram Processing

 Equalization  Specification  Local Processing

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What is Image Enhancement?

 The purpose of image enhancement is to process an

image such that it is more suitable than the original image for a specific application

 The word specific is important, because algorithms

developed for some images may not work for others

 There is no general theory for enhancement and the

evaluation of its outcome is highly subjective

 Enhancement can be performed in

 Spatial domain: direct operation on the pixel values  Frequency

domain: modify the image frequency components (Ch. 4)

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Background

 Spatial domain of the image is the set of

pixels composing the image

 Enhancement in the spatial domain involves direct

• peration on the pixel intensities

 This can be expressed mathematically as

g(x,y) = T[f(x,y)]

f(x,y) is the input image g(x,y) is the output image T[ ] is an operator defined over some neighborhood of (x,y)

 Important Keep in mind that g(x,y) may take any value from the set of available gray levels only. Thus, when mapping we should assign the mapped value to the closest level

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Background

 Defining the neighborhood around (x,y)

 Use a square/rectangular subimage that is centered at (x,y)

 Operation

 Move the center of the subimage from pixel to pixel and apply the

• peratorT at each location (x,y) to compute the output g(x,y)

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Background

 The simplest form of the operator T is when the

neighborhood size is 1x1 pixel.

 Accordingly, g(x,y) is only dependent on the value of f

at (x,y)

 In this case, T is called the gray-level or intensity

transformation function that can be represented as s = T(r)

s is a variable denoting g(x,y) r is a variable denoting f(x,y)

 This is kind of processing is referred as point

processing

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Background

 Intensity transformation function examples

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T(r) performs contrast stretching by mapping levels less than k to narrow range while those above k are mapped to wider range T(r) reduces the number of levels in the image to two

Point Processing Example Thresholding

 Thresholding transformations are particularly useful

for segmentation in which we want to isolate an

• bject of interest from a background

s = 1.0 0.0 r <= threshold r > threshold

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Basic Gray Level Transformations

 Mapping

can be performed by mathematical substitution or lookup tables

 Some common functions are

 Linear (negative/identity)  Logarithmic (log/inverse log)  Power law (nth power/nth root)

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Basic Gray Level Transformations

 Image Negatives

 Can be performed by using

s = L – 1 – r

where L-1 is the maximum intensity value

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Basic Gray Level Transformations

 Log and inverse Log Transformations

 The general form of the log transformation  b is the base  Maps narrow range of low intensity levels to wider range

and wide range of high intensity levels to narrower range

 Usually used to expand the values of dark pixels and

compress the higher level values

 The general form of the inverse log  Its operation is the opposite of the log transformation

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b

s = clog (1+r) 1

cr

s = b 

Basic Gray Level Transformations

 Log Transformation Example

 It is very important in mapping wide dynamic ranges into

narrow ones

 Fourier spectrum values in the range [0,1.5x106] transformed

to [0,255] using log transformation

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s = log(1 + r)

Basic Gray Level Transformations

 Inverse Log Transformation Example

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ecr -1

Basic Gray Level Transformations

 Power-Law transformations

 The general form

 Power law is similar to log

when gamma < 1 and similar to inverse log when gamma > 1

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γ

s = cr

Basic Gray Level Transformations

 Power-Law transformations

 Gamma-correction



Display devices have intensity-to-voltage response that is a power functions. Thus, images tend to be darker when displayed.



Correction is needed using nth root before feeding the image to the monitor

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Basic Gray Level Transformations

 Power-Law Transformation

 The images to the

right shows a magnetic resonance (MR) image of a fractured human spine

 Different curves

highlight different detail

s = r 0.6 s = r 0.4 s = r 0.3

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Piecewise-Linear Transformations

 Can represent arbitrarily complex functions to achieve

different results

 Contrast stretching

 r1 ≤ r2 and s1 ≤ s2

to preserve the order

• f gray levels

 The result depends

• n the values of r1, r2,

s1, and s2

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Piecewise-Linear Transformations

 Gray-level Slicing

 Used to highlight specific range of gray levels  Two approaches

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Piecewise-Linear Transformations

 Bit-plane Slicing

 Highlight

the contribution

• f

specific bits to the appearance of the image

 Each pixel value is represented by a set of bits  Lower bits correspond to fine details while higher bits

correspond to the global visual content

 Useful in image compression !

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Piecewise-Linear Transformations

 Bit-plane Slicing - example

20 Plane 0 Plane 1 Plane 2 Plane 3 Plane 4 Plane 5 Plane 6 Plane 7

 Bit-plane Slicing - example

Piecewise-Linear Transformations

21 Planes 7 & 6 Planes 7,6,5 Planes 7,6,5,4

 Bit-plane Slicing - example

Piecewise-Linear Transformations

22 Plane 7 Plane 6 Plane 5 Plane 4 Plane 3 Plane 2 Plane 1 Plane 0

Histogram Processing

 For an image with gray levels in [0,L-1] and MxN

pixels, the histogram is a discrete function given by where rk is the kth intensity value and nk is the number

• f pixels in the image with intensity rk

 It is a common practice to normalize the histogram

function by the number of pixels in the image by

 The normalized histogram can be used as an estimate

• f the probability density function of the image

 Histograms are widely used in image processing:

enhancement, compression, segmentation …

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k k

h( r ) n 

k k

n p( r ) MN 

Histogram Processing

 For enhancement, histograms can be used to infer the

type of image quality: dark, bright, low or high contrast

24 Dark Image Bright Image Low Contrast High Contrast

Histogram Equalization

 It is quite acceptable that high contrast images have

flat histograms (uniform distribution)

 Histogram equalization attempts to transform the

• riginal histogram into a flat one for the goal of better

contrast

 In the following, we derive the transformation function

that achieves this task

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Histogram Equalization

 Let r be a continuous variable that represents the

intensity values in the range [0,L-1]

 A valid transformation function for enhancement

purposes s = T(r) should satisfy

 T(r) is monotonically increasing in the interval 0 ≤ r ≤

L-1

 T(r) is bounded by [0,L-1] for all values of r

 The inverse transformation function that maps s back

to r requires that T(r) to be strictly monotonically increasing

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1

r T ( s )





Histogram Equalization

 Examples of transformation functions

Monotonically Increasing Strictly monotonically increasing

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Histogram Equalization

 Consider the gray level intensity represented by r as a

random variable in the interval [0,L-1]

 We can use the normalized histogram pr(r) as the

probability density function for r

 If a transformation function s = T(r) is used to map

the pixels into s in the range [0,L-1], then the following relation holds where ps(s) is the normalized histogram of the output image

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1 s r r T ( s )

dr p ( s ) p ( r ) ds

 

      

Histogram Equalization

 Now, if we know that the output image has a flat

histogram, i.e. we can substitute in the equation in the previous slide and solve for s by integrating both sides

 This gives the desired transformation function s=T(r) that

performs histogram equalization

 For digital images, the transformation function is simply

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1 1

s

p ( s ) , s [0,L-1] L     1

r r

s T( r ) ( L ) p ( w )dw   



 

  

 

r r k k k r w w w w

(L-1) s T( r ) (L-1) p ( r ) n MN

Histogram Equalization

 The function given before maps the image histogram

into a flat histogram regardless of its original shape

 Example 3.1:

consider an image with intensity distribution given by show that applying the histogram equalization function transforms this histogram into a flat one.

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2

2 1

r

r p ( r ) , r [0,L-1] ( L )   

Histogram Equalization

 Example 3.1:

31 2

2 1

r

r p ( r ) , r [0,L-1] ( L )   

1 s r r T ( s )

dr p ( s ) p ( r ) ds

 

      

Histogram Equalization

 Example 3.2. A 10x10 3-bit image has the following

intensity counts. Determine the histogram of the image after applying histogram equalization.

32  

  

 

r r k k k r w w w w

(L-1) s T( r ) (L-1) p ( r ) n MN

rk 1 2 3 4 5 6 7 h(rk) 10 20 20 35 10 5 p(rk) 0.1 0.2 0.2 0.35 0.1 0.05 ∑p(rk) 0.1 0.3 0.5 0.85 0.95 1.0 1.0 1.0 T(rk) 0.7 2.1 3.5 5.95 6.65 7 7 7 s 1 2 4 6 7 7 7 7

Histogram Equalization

 Example 3.3

33 Original Equalized Output Histogram Transformation for the processed images 1 2 3 4

r s

Histogram Equalization

 Notes

 The number of output levels in the equalized image is

usually less than the original image due to the finite number of available levels

 Histogram equalization always produces an image whose

mean intensity value is in the middle

• f

[0,L-1], irrespective of the original mean. This might not be acceptable in some situations

 If the inverse mapping for histogram equalization is

needed, it is required that there are no empty bins in the

• riginal histogram

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Histogram Equalization

 The Constrained

Variational Histogram Equalization (CVHE)

 Performs equalization under the constraint that the

• utput image mean is as close as possible to the original

mean

35 Original CVHE Image HE Image

Histogram Specification

 Histogram equalization might not be useful in all

cases

 Another

approach is to specify the desired histogram of the output image (based on some knowledge) and then perform the transformation

 We can use the method used in deriving the

transformation function of histogram equalization to find transformation function for the desired histogram, however

 This requires the availability of ps(s) in mathematical

form and the ability to express s in terms of r

 Example !

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Histogram Specification

 Alternatively

 Let pr(r) and pz(z) denote the original and desired

histograms and assume that there exist two transformation functions This implies that we can find the mapping from r to z by knowing the inverse of G(z)

 This is a simple operation if G-1(s) can be obtained !!!!

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s T( r ) and s = G(z) 

1 1

z G ( s ) G (T( r ))

 

 

rk sk r T(r) zq sq z G(z) sk = sq G-1(s)

Histogram Specification

 In discrete images, the situation is simpler since we

• nly have finite set of values, i.e. for each sq we know

the corresponding zq

 The algorithm

 Compute sk = T(rk) using  Compute sq = G(zq) using  For each rk find zq such that minimizes

for all i in [0,L-1].

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zk q q z j j

s G( z ) (L-1) p ( z )



 



rk k k r i i

s T( r ) = (L-1) p ( r )







k i

T( r ) G( z ) 

 Illustration

Histogram Specification

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rk sk r T(r) zq z G(z) Which value in G(zq) is closer to sk ? From G(z), we know which zq maps to this sq

Histogram Specification

 Example 3.4. Transform the histogram of to pz(z) =

[0.5 0.2 0.1 0.05 0.05 0.1 0 0]

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rk 1 2 3 4 5 6 7 p(rk) 0.1 0.2 0.2 0.35 0.1 0.05 sk 1 2 4 6 7 7 7 7 zq 1 2 3 4 5 6 7 p(zq) 0.5 0.2 0.1 0.05 0.05 0.1 sq 4 5 6 6 6 7 7 7 zq rk 1 2 3 4 5 6 7 3 7 7 7 7

Histogram Specification

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Histogram Equalization Histogram Specification Desired Histogram Original Image

Local Histogram Processing

 Both histogram equalization and specification methods

discussed earlier are considered global

 In

• ther

words, pixels are modified using a transformation function that is defined using all pixels in the image

 Such methods are suitable for overall enhancement

and may not be suitable in situations where we want to enhance small areas in the image whose pixel count contributes less to the global transformation function

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Local Histogram Processing

 Solution  Consider local processing such that for each pixel

(x,y) in the image

 Define a small neighborhood of size mxn that is centered

around the pixel

 Use

pixels inside the neighborhood to construct the transformation function

 Use the computed function map the pixel at (x,y)  Repeat for all pixels

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Local Histogram Processing

 Example 3.5

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Local histogram processing using 3x3 neighborhood Global Histogram Equalization

Related Matlab Functions

Check Matlab help for the following

functions

 imadjust  imhist  hist  bar  histeq

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Readings

Read subsection 3.3.4 titled “Using

Histogram Statistics for Image Enhancement”

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