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Dr. Yoram Tal � � � � � � � � � � � � � �� � � � � � � � � � � � � !

"minmax

min255

−

−⋅=II

IIIout

brighten darken

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Dr. Yoram Tal# $ % & ' ( ) * + , - . * / $ 0 * & $ ' 1 2 3 4Adam

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Dr. Yoram Tal

Histograms

A histogram of a discrete (integer) data set is a frequency table which counts the number of appearances of each member

First introduced by Abu Yusuf Yaqub ibnIshaq al-Sabbah Al-Kindi as a tool for deciphering the substitution code – a breakthrough at that time, 850 AD. (see: simon singh “the code book” 1999)

Who invented it and why?

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Dr. Yoram Tal

Which Histogram ?

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

0 50 100 150 200 250

0

0

0

0

0

0

0 5 0 1 0 0 1 5 0 2 0 0 2 5 0

Red Histogram: imhist(im(:,:,1))

Green Histogram: imhist(im(:,:,2))

Blue Histogram: imhist(im(:,:,3))

0 50 100 150 200 250

Gray Histogram: (rgb2gray)

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Dr. Yoram Tal � � � � � � � �Given an array of numbers: �� �� ��� ��������� �������� !� �""��# $#�"�����

A = { }Find the frequency of each number:

%&'()*+,-. /01223/202Number:

Count:

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Dr. Yoram Tal4 5 6 7 8 9 : ; < = > 8 : < ; ? @x image matr ix (intensity)

imhist (x) display histogram[count, bin] = imhist (x) ;

Gray level

Count# pixels

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Dr. Yoram Tal A B C D E F G H I C J K L M N O PThe histogram of an image contains valuable informationconcerning the distr ibution of gray levels

I t does not contain any spatial information

All the following images have exactly the same histograms!

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Dr. Yoram Tal

Q R S T U V W X Y S X S Z U T R [ VA histogram is a result of voting it counts the number of supporters (i.,e., pixels) of each candidate (graylevel)

In the simple case of a binary image there are only two candidates: Mr zero and Ms one.

Voting has many useful applications in image processing (as well as in democracy).

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Dr. Yoram Tal

\ ] ^ _ ` _ a b c ^ a d ` e b f ] ^ g h b i j k l m

Too dark

Too bright

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Dr. Yoram Tal

n o p q r q s t u p s v r w t x o p y z t { | } ~ �

Low contrast

High contrast

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Dr. Yoram Tal� � � � � � � � � � � � � � � � � � �I

I = imread('pout.tif');a = min(I(:));b = max(I(:));J = 255*(I-a)/(b-a);J = uint8(J);

a = 74 b = 224

Histogram of I

255minmax

min ⋅−

−=II

IIJ

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Dr. Yoram Tal

� � � � � � � � � � � � � � � � � � � � � � � � � �J Histogram of J

Histogram of I

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Dr. Yoram Tal� � � � � � � � � � � � � � � � � �   � ¡¢ £ ¤ ¥ £ ¤ ¦ ¦ § ¨ § © ª« ¬ ­ ® ¯ ° ± ² ­ ³ ± °´ µ ¶ ® ¯ ±· ° ¸ ¬ ¹ ± ¬ º ¶ ° ­ ® ¯

The minimum is 0 and the maximum is 255

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Dr. Yoram Tal» ¼ ½ ¾ ¿ À Á Â Ã Â Ä Å Æ ½ ¾ Ã Ç È ¾

a = 74

Histogram of I

255 if I i >= b

255*(I-a)/(b-a) if a <= I i < b

0 if I i < a

b = 170

K =

K = imadjust(I,[0.3 0.67],[]);

This is a non-linearoperation

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Dr. Yoram TalÉ Ê Ë Ì Í Î Ï Ð Ñ Ð Ò Ó Ô Ë Ì Ñ Õ Ö Ì × Ø Ù Ú Û Ü ÝI J K

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Dr. Yoram TalÞ ß à á â ã ä å æ ç è é å ê ß ë å á ß â ìy = histeq(x,256);

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Dr. Yoram Tal í î ï ð ñ ò ó î ô õ ð ö ÷ ø ù ðOriginal Histogram

Equalized Histogram

n = prod(size(x))tf = cumsum(255*h/n )

h

tf

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Dr. Yoram Talú û û ü ý þ ÿ � � � � � ú ý ÿ �

X (in [0 255])

Linear approx. of tf

L(1:150) = 0;L(151:240) = linspace (0,255 , 90) ;L(241:255) = 255;

y = L(x+1);

y

tf

L

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Dr. Yoram Tal5 6 7 8 9 : ; < = > ? @ < A 6 B < 8 6 9 C D E FPout

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Dr. Yoram Tal

G H I J K L M N O P Q R N S H T N J H K U V W XY Z [ Z \ ] ^Y Z [ Z \ ] ^ _ _ ^ _ ` [ a Z ` \ b c d^ _ ` [ a Z ` \ b c d e e b f Z g \ Z e ^ d ^ h _ h ib f Z g \ Z e ^ d ^ h _ h i b b d e b _ _ Z \d e b _ _ Z \

paper Histogram of paper

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Dr. Yoram Tal

Histogram Manipulation

• Correct illumination problems • Correct contrast problems

• Use histogram information to create a Look-Up Table (LUT)

• Apply LUT to the image

Image

Histogram LUT

New_imge = LUT(image)

Why?

How?

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Dr. Yoram Tal

Gamma Correction(point operation)

γ

minmax

min255 ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

⋅=II

IIIout

brighten darken

Input graylevelOut

put

gray

leve

l

[H, b] = imhist(I);p = H/sum(H); (p is in [0,1])LUT = p.^gamma

Iout = LUT(I+1); (if Imin is 0)(gamma + stretching)

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Option Ioperate on each channel separately

R = brighten(I(:,:,1),0.4);G = brighten(I(:,:,2),0.4);B = brighten(I(:,:,3),0.4);

RGB = cat(3,R,G,B);

Warning: distortion of colors!

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Dr. Yoram Tal

Option IIoperate on the intensity channel

RGB Image HSV Image V = HSV(:,:,3);

Correct V => Vc

rgb2hsv

HSV(:,:,3) = Vc;RGB Imagehsv2rgb

brighten

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Dr. Yoram Tal

Histogram Equalization

Assumption: A “good” image has a “flat” histogram, i.e., all graylevels have the same probability.

How ?See histeq

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Dr. Yoram Tal

Histogram Equalization - Lemma

• Let I be an image having N pixels and and G+1 graylevels such that g is a graylevel of I iff

g ε {0, 1, 2, …, G}

• Let h be the histogram of I where h(g) is the number of pixels in I having graylevel g.

• Let H be the cumulative histogram of I:

• Then L(x) = (G/N) H(x) is a lookup table which transforms I into a histogram equalized image.

( )0

H ( )g x

gx h g

=

=

= ∑

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Dr. Yoram Tal

Histogram Equalization - how

[H, b] = imhist(I);p = cumsum(H); p = p/p(end);LUT = p*255;

LUT

graylevel

1 Make LUT 2 Apply LUT

Iout = LUT(I+1); (if I is uint8)

In this case, a similar effect could be obtained with imadjust(use stretchlim to find thresholds)

The same result can be achieved with histeq

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Dr. Yoram Tal

Histogram Equalization problem: a narrow histogram

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Dr. Yoram Tal

Histogram Equalizationresult: too mach contrast

LUT

Resulting histogram

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Dr. Yoram Tal

CLHEContrast Limited Histogram Equalization

Source histogram Cut at some threshold(the histogram is very high & therefore the LUT is steep)

Redistribute the “left over”Now integrate

Original LUT“softer” LUT

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CLHE - Result

Source HE

CLHE

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Dr. Yoram Tal

CLAHE Contrast Limited Adaptive Histogram equalization

Problem: a big illumination difference between the two sides of the image (High Dynamic Range).

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Try Histogram Equalization

It certainly helps, but does not solve the main problem.

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Dr. Yoram Tal

Adaptive Histogram Equalizationrgb => hsv;v1 = adapthisteq(v,{parameters});hsv1 => rgb1

• The image is partitioned into blocks.

• Each block is enhanced with CLHE.

• The results are stitched back smoothly.

“Parameters” are very important!

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CLAHE – another exampleSource image Histogram Eq. adapthisteq

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Dr. Yoram Tal