3 Image Enhancement 3.3.1 Histogram Equalization (HE) >What's histogram equalization Histogram of dark image A process that seek an intensity transform function s=T(r)so thati Histogram of light image the histogram of transformed image becomes flat. The advantages of HE Histogram of ky-contrast image Automatically and adaptly determine an optimal Histogram of high-contrast image transform function 第3章第6页
3 Image Enhancement 第3章 第6页 3.3.1 Histogram Equalization (HE) What’s histogram equalization ？ A process that seek an intensity transform function s=T(r) so that the histogram of transformed image becomes flat. The advantages of HE Automatically and adaptly determine an optimal transform function
3 Image Enhancement 3.3.1 Histogram Equalization (HE) >The constraints of HE Let re[0,L-1]be the input intensity,s=T(r)is the transformed intensity.It is required that the function T satisfies: 1) T(r)is a monotonically increasing function in the interval 0sr<L-1. 2) 0sTr≤L-1for0srs4-l, 第3章第7页
3 Image Enhancement 第3章 第7页 3.3.1 Histogram Equalization (HE) The constraints of HE Let r[0,L-1] be the input intensity, s=T(r) is the transformed intensity. It is required that the function T satisfies: 1) T(r) is a monotonically increasing function in the interval 0rL-1. 2) 0T(r) L-1 for 0rL-1, 0 1 0 1 1 r r s r s s
3 Image Enhancement 3.3.1 Histogram Equalization (HE) How to HE based on the relationship The formula of HE If's =T(r)=(L-1)fo Pr(w)dwthen p.(s)= 10ss≤L-1 Cumulative distribution function(CDF)is used as the HE transform function Proof:Because s=T(r)=(L-1)p,(w)dw So aT(r)d[(L-1)SPr(w)dw] dr =(L-1)p(r) dr substitute p.(s)=p.((r) 1 =p(r)*亿-10p) L-1 随机变量s具有均匀PDF表征 第3章第8页
3 Image Enhancement 第3章 第8页 3.3.1 Histogram Equalization (HE) How to HE based on the relationship The formula of HE Proof: ,then ,0s L-1 Because so substitute 1 ( ) ( ) ( ) / s r p s p r dT r dr Cumulative distribution function (CDF) is used as the HE transform function 随机变量s具有均匀PDF表征
3 Image Enhancement 3.3.1 Histogram Equalization (HE) he formula of HE 1 If s =T(r)=(L-1)fo Pr(w)dw,then p.()=1-1,05s sL-1 Cumulative distribution function(CDF)is used as the HE transform function P,(r) P,(s) →Eq.(3.3-4)→ L-1 L-1 L-1 ab FIGURE 3.18(a)An arbitrary PDF.(b)Result of applying the transformation in Eq.(3.3-4)to all intensity levels,r.The resulting intensities,s,have a uniform PDF, independently of the form of the PDF of the r's. 第3章第9页
3 Image Enhancement 第3章 第9页 3.3.1 Histogram Equalization (HE) If Cumulative distribution function (CDF) is used as the HE transform function The formula of HE ,then ,0s L-1
3 Image Enhancement 3.3.1 Histogram Equalization (HE) The formula of HE s=Tm)=u-。Pr(w)dw,,then.=1,0ss≤L-1 p.g)=%k=0,1,L-1 5.=)-(亿-0空P)=(L-2gk=0,lL-1 第3章第10页
3 Image Enhancement 第3章 第10页 3.3.1 Histogram Equalization (HE) The formula of HE If ,then ,0s L-1 ( ) 0,1,..., 1 k k k n p r k L n 0 0 ( ) 1 ( ) 1 0,1,..., 1 k k j k k r j j j n s T r L p r L k L n discrete