《数字图像处理》课程教学课件（Digital Image Processing）灰度变换与空间滤波 3.2 直方图 Histogram processing

3 Image Enhancement 3.Intensity transformations spatial filtering Background Some basic intensity transformation Histogram-based image processing Fundamentals of spatial filtering Smoothing spatial filters Sharpen spatial filters 第3章第1页

3 Image Enhancement 第3章 第1页 3. Intensity transformations & spatial filtering  Background  Some basic intensity transformation  Histogram-based image processing  Fundamentals of spatial filtering  Smoothing spatial filters  Sharpen spatial filters

3 Image Enhancement 3.3 Histogram processing Motivation Intensity transform can enhance image because it properly changes the image histogram.So we can directly design an intensity transform function based on histogram 第3章第2页

3 Image Enhancement 第3章 第2页 3.3 Histogram processing  Motivation Intensity transform can enhance image because it properly changes the image histogram. So we can directly design an intensity transform function based on histogram

3 Image Enhancement 3.3 Histogram processing > Definition of histogram (hist=bar,gram=) If the intensity levels is in the range [0,L-1],the histogram is a discrete function h(rg)=ng,(K=0,1,.,L-1). where rg is the kth intensity value,and ny is the number of pixels in the image with intensity rk. 6 Na.of pixels 2 3 2 5 4 3 4 3 2 histogram 3 5 3 2 4 2 2 1 4x4 image Gray scale [0,9] 0 1 2 3 4 5 6 7 8 9 Gray level 第3章第3页

3 Image Enhancement 第3章 第3页 3.3 Histogram processing  Definition of histogram (hist=bar,gram=图) If the intensity levels is in the range [0,L-1], the histogram is a discrete function h(rk )=nk , (k=0,1,...,L-1). where rk is the kth intensity value, and nk is the number of pixels in the image with intensity rk . 2 3 3 2 4 2 4 3 3 2 3 5 2 4 2 4 4x4 image Gray scale = [0,9] histogram 1 2 3 4 5 6 No. of pixels 0 1 2 3 4 5 6 7 8 9 Gray level

3 Image Enhancement 3.3 Histogram processing Motivation Dark image Low-oontrast image Bright image High-contrast image 第3章第4页

3 Image Enhancement 第3章 第4页 3.3 Histogram processing  Motivation

3 Image Enhancement 3.3 Histogram processing histogram processing histogram equalization (HE) histogram specification (HS) Local HE 第3章第5页

3 Image Enhancement 第3章 第5页 3.3 Histogram processing histogram equalization (HE) histogram processing histogram specification (HS) Local HE

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 0rL-1. 2) 0T(r) L-1 for 0rL-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 ,0s 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 ,0s 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 ,0s 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