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Histogram Equalization Histogram equalization is a technique for adjusting image i ntensities to enhance contrast. Let be a given image represented as a by matrix of integer pixel intensities ranging from 0 to 1. is the number of possible intensity values, often 256. Let denote the normalized histogram of with a bin for each possible intensity. So number of pixels with intensity total number of pixels = 0 , ..., L The histogram equalized image will be deﬁned by i,j = ﬂoor(( 1) i,j =0 (1) where ﬂoor() rounds down to the nearest integer. This is equi valent to transforming the pixel intensities, , of by the function ) = ﬂoor(( 1) =0 The motivation for this transformation comes from thinking of the intensities of and as continuous random variables on [0 , L 1] with deﬁned by ) = ( 1) dx, (2) where is the probability density function of is the cumulative distributive function of multiplied by ( 1). Assume for simplicity that is diﬀerentiable and invertible. It can then be shown that deﬁned by ) is uniformly distributed on [0 , L 1], namely that ) = dz = probability that 0 = probability that 0 dw dy dz ) = )) dy ))

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original image 100 200 0.1 0.2 0.3 original histogram transformed image 100 200 0.1 0.2 0.3 transformed histogram Figure 1: Histogram equalization applied to low contrast im age Note that dy )) = dy = 1, so dT dx dy )) = ( 1) )) dy )) = 1 which means ) = Our discrete histogram is an approximation of ) and the transformation in Equation 1 approximates the one in Equation 2. While the discrete vers ion won’t result in exactly ﬂat histograms, it will ﬂatten them and in doing so enhance th e contrast in the image. The result of applying Equation 1 to the elvis low contrast.bmp test image is shown in Figure 1. MATLAB: To test the accompanying code, hist eq.m, type g = hist_eq(’elvis_low_contrast.bmp’); Histogram equalization is also built into MATLAB. Type

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help histeq to see how it works. Question: What happens if Equation 1 is applied twice? Reference: R. C. Gonzalez and R. E. Woods Digital Image Processing , Third Edition, 2008.

Let be a given image represented as a by matrix of integer pixel intensities ranging from 0 to 1 is the number of possible intensity values often 256 Let denote the normalized histogram of with a bin for each possible intensity So number of pixels w ID: 22043

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Page 1

Histogram Equalization Histogram equalization is a technique for adjusting image i ntensities to enhance contrast. Let be a given image represented as a by matrix of integer pixel intensities ranging from 0 to 1. is the number of possible intensity values, often 256. Let denote the normalized histogram of with a bin for each possible intensity. So number of pixels with intensity total number of pixels = 0 , ..., L The histogram equalized image will be deﬁned by i,j = ﬂoor(( 1) i,j =0 (1) where ﬂoor() rounds down to the nearest integer. This is equi valent to transforming the pixel intensities, , of by the function ) = ﬂoor(( 1) =0 The motivation for this transformation comes from thinking of the intensities of and as continuous random variables on [0 , L 1] with deﬁned by ) = ( 1) dx, (2) where is the probability density function of is the cumulative distributive function of multiplied by ( 1). Assume for simplicity that is diﬀerentiable and invertible. It can then be shown that deﬁned by ) is uniformly distributed on [0 , L 1], namely that ) = dz = probability that 0 = probability that 0 dw dy dz ) = )) dy ))

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original image 100 200 0.1 0.2 0.3 original histogram transformed image 100 200 0.1 0.2 0.3 transformed histogram Figure 1: Histogram equalization applied to low contrast im age Note that dy )) = dy = 1, so dT dx dy )) = ( 1) )) dy )) = 1 which means ) = Our discrete histogram is an approximation of ) and the transformation in Equation 1 approximates the one in Equation 2. While the discrete vers ion won’t result in exactly ﬂat histograms, it will ﬂatten them and in doing so enhance th e contrast in the image. The result of applying Equation 1 to the elvis low contrast.bmp test image is shown in Figure 1. MATLAB: To test the accompanying code, hist eq.m, type g = hist_eq(’elvis_low_contrast.bmp’); Histogram equalization is also built into MATLAB. Type

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help histeq to see how it works. Question: What happens if Equation 1 is applied twice? Reference: R. C. Gonzalez and R. E. Woods Digital Image Processing , Third Edition, 2008.

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