科学计算方法：经典小项目2(图像去模糊)

1、,The Curse of Dimensionality,The Blessing of Sparsity (or Structure),十阶 百阶 万阶 百万阶 亿阶,小 不大 较大 大 超大,来源 The original Harwell-Boeing collection,The University of Florida Sparse Matrix Collection (/research/sparse/matrices/),来源 3D vibro-acoustic problem, aircraft engine nacelle,来源

2、DNA electrophoresis,来源 2D FEM, Navier-Stokes, CFD,n=10000;e = ones(n,1); n2=n/2; a = spdiags(-e 3*e -e,-1:1,n,n); c=spdiags(e/2,0,n,n);c=fliplr(c);a=a+c; a(n2+1,n2) = -1; a(n2,n2+1) = -1; b=zeros(n,1); b(1)=2.5;b(n)=2.5;b(2:n-1)=1.5;b(n2:n2+1)=1; % Jacobi Method (Iterative Method) tic d=diag(a); % e

3、xtract diagonal of a r=a-diag(d); % r is the remainder x=zeros(n,1); % initialize vector x for j=1:50 % loop for Jacobi iteration x = (b-r*x)./d; end t1=toc tic,x=full(a)b,t2=toc % Back Slash (Direct Method),Demo1,f(x,y),g(x,y), Linear degradation model,Modeling Blurring Process,Given h and g, recov

4、erying f is called deblurring (or deconvolution).,一维卷积,零边界,Toeplitz矩阵,周期边界,循环矩阵,二维卷积,二维卷积,二维卷积,二维卷积,二维卷积,二维卷积,二维卷积,二维卷积,零边界,块Toeplitz Toeplitz 块矩阵,卷积,周期边界,块Circulant Circulant 块矩阵,H(模糊矩阵),f(x,y),g(x,y), Linear degradation model,Modeling Blurring Process,=,512*512,512*512,(5122)*1,(5122)*1,(5122)*(51

5、22),小项目2,1. 3*3大小模糊核, 128*128大小的图像, 请构造相应的模糊矩阵？(参考代码blur.m),2. 研究求解相应方程组的快速算法？(或者更加宽泛的问题我们可以利用那些特殊矩阵的结构设计快速算法？),计算的挑战,%图像复原 tic, H= blur(512,7,4);toc%需要存储系数矩阵 f=double(rgb2gray(imread(lena_c512.tif); figure,subplot(121),imshow(f,) tic,g=H*f(:);toc subplot(122),imshow(reshape(g,512,512),) Hg,如何生成系数矩阵（存储呢）？如何矩阵向量乘法？如何求解方程组？,Reference: 1. Deblurring Images: Matrices, Spectra, and