伽马变换

	close all;clc;clear all;
	I=imread('MizunoAi.jpg');
	subplot(1,3,1);imshow(I);title('原始图像');
	c=cat(3,gamma(I(:,:,1),2),gamma(I(:,:,2),2),gamma(I(:,:,3),2));
	subplot(1,3,2);imshow(c);title('伽马变换，gamma=2');
	c=cat(3,gamma(I(:,:,1),0.5),gamma(I(:,:,2),0.5),gamma(I(:,:,3),0.5));
	subplot(1,3,3);imshow(c);title('伽马变换，gamma=0.5');
function J=gamma(I,g)
	c = 255^(1-g);
	J = c * double(I).^g;
	J = mat2gray(J);
end

PROJECT 03-02 [Multiple Uses] Histogram Equalization

1. Write a computer program for computing the histogram of an image.
2. Implement the histogram equalization technique discussed in Section 3.3.1.
3. Download Fig. 3.8(a) and perform histogram equalization on it.

As a minimum, your report should include the original image, a plot of its histogram, a plot of the histogram-equalization transformation function, the enhanced image, and a plot of its histogram. Use this information to explain why the resulting image was enhanced as it was.

变换函数是：$s_k=T(r_k)=(L-1)\sum_{j=0}^kp_r(r_j)=(L-1)\sum_{j=0}^k\frac{n_j}{n}$，其中$k=0,1,\ldots,L-1$

我想做一些加强，针对彩色图片做直方图均衡化。查了一些网上资料，就是对三维 RGB 图像的三种颜色分别做一次灰度均衡。

	I=imread('MizunoAi.jpg');
	subplot(2,3,1);imshow(I);title('原始图像');subplot(2,3,4);imhist(I);
	c=histeq(I);
	subplot(2,3,2);imshow(c);title('直方均衡化·调库');subplot(2,3,5);imhist(c);
	c=cat(3,histogram(I(:,:,1)),histogram(I(:,:,2)),histogram(I(:,:,3)));
	subplot(2,3,3);imshow(c);title('直方均衡化');subplot(2,3,6);imhist(c);
function J=histogram(I)
	J=I;
	[n,m]=size(I);
	a=zeros(1,256);
	b=zeros(1,256);
	for i=1:n
		for j=1:m
			a(1,I(i,j)+1)=a(1,I(i,j)+1)+1;
		end
	end
	sum=0;
	for i=1:256
		sum=sum+a(1,i);
		b(1,i)=255*sum/(m*n);
	end
	for i=1:n
		for j=1:m
			d=J(i,j)+1;
			J(i,j)=b(1,d);
		end
	end
end

运行结果如下，同时输出调用 matlab 自己的直方图均衡函数的结果作为对比，发现自己写的图像均衡偏绿，不知道彩色图像做均衡化要做什么黑科技…不过右侧阴影处的细节确实变得清晰了。

PROJECT 03-05 Enhancement Using the Laplacian

1. Use the programs developed in Projects 03-03 and 03-04 to implement the Laplacian enhancement technique described in connection with Eq. (3.7-5). Use the mask shown in Fig. 3.39(d).
2. Duplicate the results in Fig. 3.40. You will need to download Fig. 3.40(a).

原理

对连续函数情形，最简单且各向同性的二阶微分算子是拉普拉斯（Laplacian）算子$\triangledown^2f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$。

离散情况下，$\frac{\partial^2f}{\partial x^2}=f(x-1,y)+f(x+1,y)-2f(x,y)$，$\frac{\partial^2f}{\partial y^2}=f(x,y-1)+f(x,y+1)-2f(x,y)$，于是$\triangledown^2f=f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)-4f(x,y)$

1. 当拉普拉斯算子中心系数为正的时候，有$g(x,y)=f(x,y)+\triangledown^2f$
2. 当拉普拉斯算子中心系数为正的时候，有$g(x,y)=f(x,y)-\triangledown^2f$

源代码

close all;clc;
I=imread('liftingbody.png');
I=im2double(I);
J=zeros(size(I));
[M,N]=size(I);
for x=2:M-1
	for y=2:N-1
		J(x,y)=9*I(x,y);
		for dx=-1:1
			for dy=-1:1
				J(x,y)=J(x,y)-I(x+dx,y+dy);
			end
		end
	end
end
subplot(1,3,1);imshow(im2uint8(I));
subplot(1,3,2);imshow(im2uint8(J));
subplot(1,3,3);imshow(im2uint8(I+J));

运行结果

左一是原图，左二是模板，左三是拉普拉斯变换得到的最终图像，可以看到，增强后的第三张图中阴影边缘对比变强了。

PROJECT 03-06 Unsharp Masking

1. Use the programs developed in Projects 03-03 and 03-04 to implement highboost filtering, as given in Eq. (3.7-8). The averaging part of the process should be done using the mask in Fig. 3.34(a).
2. Download Fig. 3.43(a) and enhance it using the program you developed in (a). Your objective is to choose constant A so that your result visually approximates Fig. 3.43(d).

原理

1. 当拉普拉斯算子中心系数为正的时候，有$f_{hb}(x,y)=Af(x,y)+\triangledown^2f$
2. 当拉普拉斯算子中心系数为正的时候，有$f_{hb}(x,y)=Af(x,y)-\triangledown^2f$

可以看出，基于拉普拉斯算子的高提升滤波，当$A=1$时就是拉普拉斯图像增强方法，当$A$足够大时，锐化效果将变得不明显

源代码

因为拉普拉斯变换就是特定条件的基于拉普拉斯算子的高提升滤波，所以和上一个代码几乎相同。

close all;clc;
I=imread('liftingbody.png');
I=im2double(I);
J=zeros(size(I));
[M,N]=size(I);
for x=2:M-1
	for y=2:N-1
		J(x,y)=9*I(x,y);
		for dx=-1:1
			for dy=-1:1
				J(x,y)=J(x,y)-I(x+dx,y+dy);
			end
		end
	end
end
subplot(1,3,1);imshow(im2uint8(I+J));
subplot(1,3,2);imshow(im2uint8(1.5*I+J));
subplot(1,3,3);imshow(im2uint8(2*I+J));

运行结果

如图，上面三张图自左向右是分别赋值A=1、A=1.5、A=2时的非锐化掩模，其中A=1时就是标准拉普拉斯锐化。自左向右图像的边缘突出效果不断增加。