Digital Image Processing Ch3 [2] Histogram Equalization

• Monotonic Transformation of a Continuous r.v

(Output intensity) S=Random Variable=T(r)=(input intensity r) X

PDF : Probability density function == nomalized histogram

Transforming input intensity x into output y using y=T(X)

Fx(X) = CDF : Cumulative distribution funciton

 

T : monotonically increasing : if T(x1) < T(x2) for any x1<x2 <- 증가함수

T : monotonically decreasing : if T(x1) > T(x2) for any x1<x2 <- 감소함수

하지만 모두 1-1이고 invertible하다. 

 

inverse : x0 = T^(-1)(y0)이 가능하다. 

(1) for monotonically increasing T

CDF of Y(output intensity) Fy(y0) = P{Y<= x0} =P{x<= x0}= Fx(x0)

integral of PDF == CDF

fy(y) = fx(x)*(dx/dy)

(2) for monotonically decreasing T

Fu(y0)=P{y<=y0}=P{x>=x0}=1- Fx(x0)

integral of PDF = 1 - CDF

fy(y0)=-fx(x0)(dx/dy)

 

So in general

fy(y) = fx(x)*|(dx/dy)|

 

histogram equlazation을 위해 쓰이는 특징이다

 

Histogram Equalization

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