ece542-4

# ece542-4 - ECE542-4 Digital Image Processing Image...

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1 Digital Image Processing ECE542-4 3- Image Enhancement in the Spatial Domain - II Dr. Z. Aliyazicioglu Cal Poly Pomona Electrical & Computer Engineering 1 Cal Poly Pomona Electrical & Computer Engineering Office 9-143 Histogram Equalization As the low-contrast image’s histogram is narrow and centered toward the middle of the gray scale, if we distribute the histogram to a wider range the quality of the image will be improved the quality of the image will be improved. We can do it by adjusting the probability density function of the original histogram of the image so that the probability spread equally where 0 r 1 T(r) satisfies T(r) is single valued and s () ,0 1 sT r r  1 ECE542-4 2 Cal Poly Pomona T(r) is single-valued and monotonically increasingly in the interval 0 r 1 0 T(r) 1 for 0 r 1 01 r k s k = T(r k ) r T(r)

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2 Histogram Equalization Single-valued (one-to-one relationship) guarantees that the inverse transformation will exist Monotonically condition preserves the increasing order from black to white in the output image thus it won’t cause a negative image 0 T ( r ) 1 for 0 r 1 guarantees that the output gray levels will be in the same range as the input levels . The inverse transformation from s back to r is 1 () , 0 1  rT s s ECE542-4 3 Cal Poly Pomona If T ( r ) satisfy conditions (a) and (b), it is possible that corresponding inverse T -1 ( s ) may fail to be single valued Histogram Equalization The gray levels in an image may be viewed as random variables in the interval [0,1] PDF is one of the fundamental descriptors of a random variable Let p r (r) denote the PDF of random variable r p s (s) denote the PDF of random variable s If p r (r) and T(r) are known and T -1 (s) satisfies condition (a) then p s (s) can be obtained using a formula : dr ECE542-4 4 Cal Poly Pomona The PDF of the transformed variable is determined by the gray-level PDF of the input image and by the chosen transformation function sr p(s ) p (r) ds
3 Histogram Equalization A transformation function is a cumulative distribution function (CDF) of random variable r : r where w is a dummy variable of integration Note: T ( r ) depends on p r r CDF is an integral of a probability function (always positive) is the area under the function 0 () ( ) r sT r pw d w ECE542-4 5 Cal Poly Pomona Thus, CDF is always single valued and monotonically increasing Thus, CDF satisfies the condition (a) We can use CDF as a transformation function Histogram Equalization Finding p s (s) from given T(r) ds dT r 0 r r r dr dr d p wdw dr pr    1 1 where 0 1 sr r r dr ps pr ds s  Substitute and yield ECE542-4 6 Cal Poly Pomona Substitute and yield

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4 Histogram Equalization p s (s) As p s (s) is a probability function, it must be zero outside the interval [0 1] in this case because its integral over all values of
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ece542-4 - ECE542-4 Digital Image Processing Image...

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