AVIK PAl 1001113861 Project 8AQ.1 Implement down sampling 2:1 based on DA (Dugad and Ahuja) and bilinear interpolationProgramming:clc;close all;clear all;X=double(imread('lena512.bmp'));[m n]=size(X); %Obtain the DCT kernel for 8x8 and 4x4N=8;Dct_matrix=dctmtx(N);D=dctmtx(4); % Implementing DA Method%DA_down_sample=zeros(4,4); % Get the 4x4 Matrix of zeroes to build sampling filters% Defining matrix For downsampling 2:1%downsample=zeros(4,4);% Defining matrix For usampling 1:2%upsample=zeros(8,8); for i=1:m/N for j=1:n/N Temp_1=X((i-1)*N+1:i*N,(j-1)*N+1:j*N); J=Dct_matrix*Temp_1*Dct_matrix'; %DA_down_sampling(DCT domain) is of size 256x256 as we have ignored %5:8,5:8 of J% DA_down_sample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=J(1:N/2,1:N/2); endend for i=1:m/N for j=1:n/N Temp_1=DA_down_sample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2); downsample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=D'*Temp_1*D;%256x256 lena image endendimshow(X,[]); title('original Image');figure, imshow(downsample,[]);title('2:1 downsampling by DA method');% Downsampling by bilinear interpolationbimup=imresize(X,0.5,'bilinear'); MSE_DA=sum(sum((X-upsample).^2))/numel(X);PSNR_DA=20*log10(255*255/(MSE_DA)); imshow(X,[]); title('original Image');figure, imshow(downsample,[]);title('2:1 downsampling by DA method');figure, imshow(upsample,[]);title(sprintf('upsampling by DA method MSE:%f PSNR:%f',MSE_DA,PSNR_DA));figure, 1AVIK PAl 1001113861 Project 8Aimshow(bimup,[]);title('Downsampling by Bilinear interpolation 2:1');Visualization:Downsampling:Based on Interpolation:2AVIK PAl 1001113861 Project 8AObservation:At first,I created the DCT matrix which is 4*4. Then I tried to insert values in DCT_downsampling matrix which is 256*256 in dimension. The matrix value is given below3AVIK PAl 1001113861 Project 8AQ.2 Implement Downsampling (2:1) and upsampling (1:2) as shown in last program:Programming;clc;close all;clear all;X=double(imread('lena512.bmp'));[m n]=size(X); %Obtain the DCT kernel for 8x8 and 4x4N=8;Dct_matrix=dctmtx(N);D=dctmtx(4); % Implementing DA Method%DA_down_sample=zeros(4,4); % Get the 4x4 Matrix of zeroes to build sampling filters% Defining matrix For downsampling 2:1%downsample=zeros(4,4);% Defining matrix For usampling 1:2%upsample=zeros(8,8); for i=1:m/N for j=1:n/N Temp_1=X((i-1)*N+1:i*N,(j-1)*N+1:j*N); J=Dct_matrix*Temp_1*Dct_matrix'; %DA_down_sampling(DCT domain) is of size 256x256 as we have ignored %5:8,5:8 of J% DA_down_sample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=J(1:N/2,1:N/2); endend for i=1:m/N for j=1:n/N Temp_1= DA_down_sample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2); Temp_1=[Temp_1 zeros(N/2);zeros(N/2) zeros(N/2)];%attaching zeros to increase size upsample((i-1)*N+1:i*N,(j-1)*N+1:j*N)=Dct_matrix'*Temp_1*Dct_matrix;%upsample is of size 512x512 in data domain endend for i=1:m/N for j=1:n/N Temp_1=DA_down_sample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2); downsample((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=D'*Temp_1*D;%256x256 lena image endend Visualization:Original Image;4AVIK PAl 1001113861 Project 8ADown sample:Upsample:5AVIK PAl 1001113861 Project 8AObservation:Upsample matrix which is 512*512 in dimension is given below6AVIK PAl 1001113861 Project 8AQ 3. See MM transformation and implement using SB and TR using MM transformationProgramming;Downsampling;%Using MM and SB%Du=dctmtx(2*N);%DCT kernel of size 16%Downsampling and Upsampling by SB methodfor k=1:N for l=1:N c(k,l)=(4*cos(pi*k/(2*N))*cos(pi*l/(2*N))); endend for i=1:m/N for j=1:n/N Temp_2=X((i-1)*N+1:i*N,(j-1)*N+1:j*N); J=Dct_matrix*Temp_2*Dct_matrix'; DA_down_sampling((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=J(1:N/2,1:N/2)./c(1:N/2,1:N/2); %modifying DA method by dividing it with cos endend for i=1:m/N for j=1:n/N Temp_2=DA_down_sampling((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2); imdownSB((i-1)*N/2+1:i*N/2,(j-1)*N/2+1:j*N/2)=D'*Temp_2*D; %256x256 endend for i=1:m/N/2 for j=1:n/N/2 Temp_2=imdownSB((i-1)*N+1:i*N,(j-1)*N+1:j*N); Temp_2=[Dct_matrix*Temp_2*Dct_matrix'.*c zeros(N);zeros(N) zeros(N)];%increasing size after .*c imup((i-1)*2*N+1:i*2*N,(j-1)*2*N+1:j*2*N)=Du'*Temp_2*Du; endend MSE_SB=sum(sum((X-upsample).^2))/numel(X); PSNR_SB=20*log10(255*255/(MSE_SB)); figure;imshow(imup,[]);title(sprintf('upsampling by SB MSE:%f PSNR:%f',MSE_SB,PSNR_SB));figure;imshow(imdownSB,[]);title('2:1 downsampling by SB method');Upsampling by SB method:%-------------------------------------------------------------------------%Upsampling by TR methodfor i=1:m/N/2 for j=1:n/N/2 Temp_3=downsample((i-1)*N+1:i*N,(j-1)*N+1:j*N); %starting with DA downsamp7AVIK PAl 1001113861 Project 8A Temp_3=[Dct_matrix*Temp_3*Dct_matrix' zeros(N);zeros(N) zeros(N)];%size is 16x16 imup((i-1)*2*N+1:i*2*N,(j-1)*2*N+1:j*2*N)=Du'*Temp_3*Du; endend MSE_TR=sum(sum((X-imup).^2))/numel(X); PSNR_TR=20*log10(255/(MSE_TR)); figure; imshow(imup,[]);title('1:2 upsampling by TR');title(sprintf('upsampling by TR MSE:%f PSNR:%f',MSE_TR,PSNR_TR));Visualization:Downsample by SB method:Upsample by SB method:8AVIK PAl 1001113861 Project 8AUpsample by TR methodObservation:DA _Downsampling matrix is given below which is 256*256 in dimension.9AVIK PAl 1001113861 Project 8AIn case of Upsampling method, the matrix is given as which is 16*16 in dimensionMSE for Upsampling in SB method is;10AVIK PAl 1001113861 Project 8A>> disp (MSE_SB) 21.9303And the PSNR in SB method is>> disp(PSNR_SB) 69.4407In case of TR methodUpsampling matrix which is 16*16 is given belowAnd MSE for this method is>> disp (MSE_TR) 20.3709PSNR for this TR method>> disp (PSNR_TR) 21.9506Q.5 Calculate using CR method described in CLProgramming:%CR method upsampling11AVIK PAl 1001113861 Project 8Axll=zeros(N*N,N*N); %64x64xhl=zeros(4*N*N-N*N,N*N);%192x64%Dd16=zeros(N/2,N/2);%4x4 for i=1:m/N/2 for j=1:n/N/2 Temp_4=Du*X((i-1)*N*2+1:i*N*2,(j-1)*N*2+1:j*N*2)*Du';%16x16 [r c]=size(Temp_4); Dd16((i-1)*N+1:i*N,(j-1)*N+1:j*N)= Temp_4(1:r/2,1:c/2); L=reshape(Temp_4(1:r/2,1:c/2),r*c/4,1); H=[reshape(Temp_4(1:r/2,1+c/2:c),r*c/4,1);reshape(Temp_4(1+r/2:r,1:c/2),r*c/4,1);reshape(Temp_4(1+r/2:r,1+c/2:c),r*c/4,1)]; xll=xll+L*L'; xhl=xhl+H*L'; endendxll=xll/(m/N/2)^2;xhl=xhl/(m/N/2)^2;A=xhl*xll^-1; A=A/max(max(A)); for i=1:m/N/2 for j=1:n/N/2 Temp_4=Dd16((i-1)*N+1:i*N,(j-1)*N+1:j*N); Temp_4=reshape(Temp_4,N*N,1); Temp4=A*Temp_4; Temp5=[Dd16((i-1)*N+1:i*N,(j-1)*N+1:j*N) reshape(Temp4(1:N*N),N,N); reshape(Temp4(N*N+1:2*N*N),N,N) reshape(Temp4(2*N*N+1:3*N*N),N,N)];