Histogram Processing - National Institute of Technology ... · Histogram Processing Histogram...

Histogram Processing

Dr. Praveen Sankaran

Department of ECE

NIT Calicut

December 31, 2012

Dr. Praveen Sankaran (Department of ECE NIT Calicut )DIP Winter 2013 December 31, 2012 1 / 20

Outline

1 Review

2 Histogram ProcessingHistogram EqualizationHistogram Matching (Speci�cation)


Review

Contrast

De�ned as the di�erence in intensity between the highest and thelowest intensity levels in an image.

Also can be explained as - the di�erence in luminance and/or colorthat makes an object (or its representation in an image or display)distinguishable.1

1http://en.wikipedia.org/wiki/Contrast_%28vision%29Dr. Praveen Sankaran (Department of ECE NIT Calicut )DIP Winter 2013 December 31, 2012 3 / 20

Review

Some Calculations - Contrast

Let g be an M×N digital image with l = 0,1, ...,L−1 possible graylevels.

Dispersion → Image pixel value variance.

‖g −g‖2 = 1

MN∑M−1m=0 ∑

N−1n=0

(g [m,n]−g)2

g = 1

MN∑M−1m=0 ∑

N−1n=0

g [m,n]

Units are squared here.

Contrast = ‖g −g‖=√‖g −g‖2 → standard deviation.

Note that it would take an order O (MN) algorithm to �nd this.


Review

Histogram

Let g be an M×N digital image with l = 0,1, ...,L−1 possible graylevels.

c [l ] =the number of pixels with gray level l .

De�ne relative frequency

p [l ] = c[l ]MN

, ∑L−1l=0

p [l ] = 1

→ digital image gray level distribution.

The probability that a randomly selected pixel has value l .


Review

Contrast Stretching

Idea → expand the range of intensity levels in an image so that it spans thefull intensity range of the recording medium or display device.


Histogram Processing Histogram Equalization

Outline

1 Review




Variables so far

1 g = M×N digital image with l = 0,1, ...,L−1 possible gray levels.

2 r = intensities of image g to be processed, [0 · · ·L−1].

3 s = output intensity level of g after transformation, T (r).

4 c [l ] =the number of pixels with gray level l .

5 p [l ] = c[l ]MN

, gives gray level distribution of image g .

6 And more importantly, ∑L−1l=0

p [l ] = 1.



Monotonic and Strictly Monotonic Functions



Cumulative Distribution Function

p [l ] = The probability that a randomly selected pixel has value l .

pr (r) =probability density function of r .

ps (s) =probability density function of s.

Figure: Slope of the r -s curve gives the transform function, ps (s) = pr (r)∣∣ drds

∣∣CDF of r → s = (L−1)

∫ r0 pr (w)dw = area under the probability

density function curve of r .→ What type of curve is this going to be?



What about ps (s)?

ds

dr=

dT (r)

dr= (L−1)

d

dr

[∫ r

0pr (w)dw

]=(L−1)pr (r)

ps (s) = pr (r)∣∣drds

∣∣ps (s) = pr (r)

∣∣∣ 1(L−1)pr (r)

∣∣∣= 1L−1 → Uniform distribution.



Discrete Equivalent

From earlier we have a general form, p [l ] = c[l ]MN

.

So modifying for current speci�cs, pr [lr ] =cr [lr ]MN

.

ls = T (lr ) = (L−1)∑lj=0 pr [jr ], l = 0,1, · · ·L−1.

Histogram equalization or histogram linearization.



Process Example

Objective → intensity values that cover the entire gray scale.Note → see the error creeping in?



Real Image Output Example


Histogram Processing Histogram Matching (Speci�cation)

Outline

1 Review




Some More Continuous Function Theory!

s = T (r) = (L−1)∫ r0 pr (w)dw → Histogram equalization.

Let z be another random variable like r ,s, such that,

G (z) = (L−1)∫z

0pz (t)dt = s

G (z) = T (r)!→ z = G−1 (T (r)) = G−1 (s)

What does all this mean anyway?



Explanation

To start with we have the image values we have to modify → r .

Idea → modify the image based on the histogram of some other image(with values z)!

Why? →because we like this second image for some reason!

1 Find CDF of Image 1, to perform histogram equalization (r → s).

2 Find CDF of Image 2.

3 Now assuming that values s is on the y-axis of image 2 CDF, �nd thecorresponding nearest z value.



Illustration

2

2http://en.wikipedia.org/wiki/File:Histogram_matching.PNGDr. Praveen Sankaran (Department of ECE NIT Calicut )DIP Winter 2013 December 31, 2012 18 / 20


Summary

Histograms.

Monotonic/Strictly Monotonic.

Cumulative distribution/density function of an image.

Equalization.

Speci�cation/Matching.



Questions

3.1, 3.2, 3.3, 3.4, 3.5

3.6, 3.7, 3.11


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