數字信號處理Day2－小波基與規範正交化

我們有這麼一張灰度圖64*64

我們可以定義出4096個基，分別是某一位是0其他是1，在這種情況下，如果我們傳輸圖片，那麼就相當於傳輸原始數據

假設傳到一半，網絡壞了。

於是，我們得到

我們可以計算原圖像和這圖像的差距

error = I - I_approx; 
distance = sqrt(sum(sum(error.*error))) 
distance = 
713.5559

假設我們使用Haar基

function h = haar(N)

    % Assume N is power of 2
    h = zeros(N,N); 
    h(1,1:N) = ones(1,N)/sqrt(N);

    for k = 1:N-1 
        p = fix(log(k)/log(2)); 
        q = k-(2^p); 
        k1 = 2^p;
        t1 = N/k1; 
        k2 = 2^(p+1);
        t2 = N/k2; 
        for i=1:t2 
            h(k+1,i+q*t1) = (2^(p/2))/sqrt(N); 
            h(k+1,i+q*t1+t2) = -(2^(p/2))/sqrt(N); 
        end 
    end

然後，我們看一下只接受一半的圖片可以恢復成什麼樣子，這個東西可以用到圖片壓縮（PCA降維）

clear all
close all
clc

% Load image
I = double(imread('camera.jpg'));

% Arrange image in column vector
I = I(:);

% Generate Haar basis vector (rows of H)
H = haar(4096);

% Project image on the new basis
I_haar = H*I;

% Remove the second half of the coefficient
I_haar(2049:4096) = 0;

% Recover the image by inverting change of basis
I_haar = H'*I_haar;

% Rearrange pixels of the image
I_haar = reshape(I_haar, 64, 64);

% Display image
figure
imshow(I_haar,[]);

效果還可以

error = I - I_haar; 
distance = sqrt(sum(sum(error.*error))) 
distance = 
350.6765

Haar基是正交基

如何將普通基變成正交基呢

線性代數中的拉格慕－施密特正交化就行

function E=gs_orthonormalization(V)
% V is a matrix where each column contains the vectors spanning the space of which we want to compute the orthonormal base
% E is a matrix where each column contains an ortho-normal vector of the base of the space

[numberLines,numberColumns]=size(V);

% U is a matrix containing the orthogonal vectors (non normalized)
U=zeros(numberLines,numberColumns);

for indexColumn=1:numberColumns
    U(:,indexColumn)=V(:,indexColumn);

    for index=1:(indexColumn-1)
        R(index,indexColumn) =U(:,index)'*V(:,indexColumn);
        U(:,indexColumn)=U(:,indexColumn)-R(index,indexColumn)*U(:,index);
    end

    R(indexColumn,indexColumn)=norm(U(:,indexColumn));
    U(:,indexColumn)=U(:,indexColumn)./R(indexColumn,indexColumn);

end

E=U;

return