What are the cosine functions in JPEG's DCTII table?

Question

As I understand it from this video, in a JPEG image an 8x8-pixel block is made up of weighted cosine waves, calculated using DCTII. There are lots of visualizations of these waves, such as this one from the Wikipedia article on JPEG compression:

I'd like to be able to visualize each of the waves for a given 8x8-pixel block in an image, weighted with values calculated by DCTII.

Where can I find (or how can I calculate) the 64 cosine functions used in JPEG compression?

Answer 1

Those 2D cosine functions are independent of your input image. They are just "cosine waves", or the 64 basis functions that yield 64 coefficients when transforming "$8\times 8$" blocks. Given a $D$ $8\times 8$ matrix for the DCT-II, and a $8\times 8$ image patch $I$, you'll get $64$ coefficients $C$ by:

$$C=DID^T$$

[EDIT] If you want to display the 2D DCT-II functions only, you have to create 2D arrays $D_{h,v}(m,n)$ ($0\le m< M$ and $0\le n< N$) for each couple of horizontal/vertical integer indices $(h,v)$, with $0\le h< H$ and $0\le v< V$, with $H=M$ and $V=N$. Classically for JPEG, $M=N=8$. A formula is (up to a transposition, and possibly a scale factor):

$$D_{h,v}(m,n) = 2\sum_{m=0}^{M-1}\sum_{n=0}^{N-1} \frac{1}{\sqrt{M}}\frac{1}{\sqrt{N}}\eta_h \eta_v \cos\left(\frac{\pi h}{2M}(2m+1)\right)\cos\left(\frac{\pi v}{2N}(2n+1)\right)$$

with $\eta_x = \frac{1}{\sqrt{2}}$ if $x=0$, $\eta_x = 1$ if $x\neq0$. There are a lot of clever implementations, I'll provide the most straightforward. The goal is to get a picture like the following:

A pseudo-code version (from Matlab) could be (with the issue of 0-based or 1-based indices):

nRow = 8;nCol = 8;
nFreqHoriz = 8;
nFreqVerti = 8;
for iFreqH = 1:nFreqHoriz
    for iFreqV = 1:nFreqVerti
        iFreqHoriz = iFreqH-1;
        iFreqVerti = iFreqV-1;
        normFreqDC = 1/sqrt(2);
        matFreq2D = zeros(nRow,nCol);
        for iRow = 1:nRow
            for iCol = 1:nCol
                iRow0 = iRow-1;
                iCol0 = iCol-1;
                matFreq2D(iRow,iCol) = 2/(sqrt(nFreqHoriz)*sqrt(nFreqVerti))*cos(pi*iFreqHoriz*(2*iRow0+1)/(2*nRow))*cos(pi*iFreqVerti*(2*iCol0+1)/(2*nCol));
                if ~iFreqHoriz ,  matFreq2D(iRow,iCol) =  matFreq2D(iRow,iCol)*normFreqDC;end
                if ~iFreqVerti ,  matFreq2D(iRow,iCol) =  matFreq2D(iRow,iCol)*normFreqDC;end
            end
        end
        sum(matFreq2D(:).^2)
        subplot(nFreqHoriz,nFreqVerti,8*(iFreqV-1)+iFreqH)
        imagesc(matFreq2D);axis off;axis square;colormap gray
    end
end