4
\$\begingroup\$

I am trying to implement 3D Discrete Cosine Transformation calculation in Matlab with parallel computing parfor. The formula of 3D Discrete Cosine Transformation is as follows.

The experimental implementation

The experimental implementation of 3D Discrete Cosine Transformation function is DCT3D.

function X=DCT3D(x)
[N1,N2,N3] = size(x);
X=zeros(N1,N2,N3);
for k1=0:N1-1
 for k2=0:N2-1
 for k3=0:N3-1
 sumResult=0; 
 parfor n1=0:N1-1
 for n2=0:N2-1
 for n3=0:N3-1
 sumResult=sumResult+...
 x(n1+1,n2+1,n3+1)*...
 cos(pi/(2*N1)*(2*n1+1)*k1)*...
 cos(pi/(2*N2)*(2*n2+1)*k2)*...
 cos(pi/(2*N3)*(2*n3+1)*k3);
 end
 end
 end
 X(k1+1,k2+1,k3+1)=8*sumResult*CalculateK(k1)*CalculateK(k2)*CalculateK(k3)/(N1*N2*N3); 
 end
 end
end

Moreover, the used function CalculateK:

function output = CalculateK(x)
 output = ones(size(x));
 output(x==0) = 1 / sqrt(2);

Test cases

%% Create test cells
testCellsSize = 10;
test = zeros(testCellsSize, testCellsSize, testCellsSize);
for x = 1:size(test, 1)
 for y = 1:size(test, 2)
 for z = 1:size(test, 3)
 test(x, y, z) = x * 100 + y * 10 + z;
 end
 end
end
%% Perform test
result = DCT3D(test);
%% Print output
for z = 1:size(result, 3)
 fprintf('3D DCT result: %d plane\n' , z);
 for x = 1:size(result, 1)
 for y = 1:size(result, 2)
 fprintf('%f\t' , result(x, y, z));
 end
 fprintf('\n');
 end
 fprintf('\n');
end
%% Visualize result
for z = 1:size(result, 3)
 figure;
 mesh(result(:, :, z));
end

The output of the test code above:

3D DCT result: 1 plane
1726.754760 -80.720722 -0.000000 -8.646042 -0.000000 -2.828427 0.000000 -1.143708 -0.000000 -0.320717 
-807.207224 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
-86.460422 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-28.284271 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
-11.437076 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
-3.207174 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
3D DCT result: 2 plane
-8.072072 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
3D DCT result: 3 plane
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
-0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
3D DCT result: 4 plane
-0.864604 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
3D DCT result: 5 plane
-0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
3D DCT result: 6 plane
-0.282843 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
3D DCT result: 7 plane
0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
3D DCT result: 8 plane
-0.114371 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
3D DCT result: 9 plane
-0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 
3D DCT result: 10 plane
-0.032072 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
-0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

Another test pattern:

%% Create test cells
testCellsSize = 10;
test = zeros(testCellsSize, testCellsSize, testCellsSize);
for z = 1:size(test, 3)
 test(:, :, z) = mod(z, 2) * 255;
end
%% Perform test
result = DCT3D(test);
%% Print output
for z = 1:size(result, 3)
 fprintf('3D DCT result: %d plane\n' , z);
 for x = 1:size(result, 1)
 for y = 1:size(result, 2)
 fprintf('%f\t' , result(x, y, z));
 end
 fprintf('\n');
 end
 fprintf('\n');
end
%% Visualize result
for z = 1:size(result, 3)
 fig = figure;
 mesh(result(:, :, z));
 saveas(fig, [num2str(z) '.bmp'], 'bmp');
end

The output of the test code above:

3D DCT result: 1 plane
360.624458 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 
3D DCT result: 2 plane
51.635721 -0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
3D DCT result: 3 plane
-0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 
-0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 
3D DCT result: 4 plane
57.238638 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
3D DCT result: 5 plane
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 
3D DCT result: 6 plane
72.124892 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
3D DCT result: 7 plane
-0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 
0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
-0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
3D DCT result: 8 plane
112.337152 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 
0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 
3D DCT result: 9 plane
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 
-0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 
3D DCT result: 10 plane
326.015114 0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 
-0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 0.000000 -0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 
0.000000 -0.000000 -0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 -0.000000 
0.000000 -0.000000 0.000000 0.000000 0.000000 0.000000 -0.000000 0.000000 -0.000000 -0.000000 
-0.000000 0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 -0.000000 0.000000 -0.000000 
0.000000 0.000000 0.000000 0.000000 -0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.000000

The output figures:

Plane1

Plane2

Plane3

Plane4

Plane5

Plane6

Plane7

Plane8

Plane9

Plane10

Test Platform Information

Matlab version: '9.10.0.1629609 (R2021a)'

Reference

  • Malavika Bhaskaranand and Jerry D. Gibson, "Distributions of 3D DCT coefficients for video," in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing, 2009.

  • J. Augustin Jacob and N. Senthil Kumar, "Determining Optimal Cube for 3D-DCT Based Video Compression for Different Motion Levels," ICTACT Journal on Image and Video Processing, Vol. 03, November 2012.

  • Related Code Golf coding challenge.

If there is any possible improvement, please let me know.

asked Apr 30, 2021 at 22:43
\$\endgroup\$
3
  • 2
    \$\begingroup\$ I don’t have time to write a full answer, but I’ll leave three short comments: (1) parfor is always best used as the outer loop. (2) Vectorize your code. (3) The DCT is separable. \$\endgroup\$ Commented May 21, 2021 at 1:43
  • \$\begingroup\$ @Cris Luengo Please check the vectorized part (Sep. 9, 2021 Update). \$\endgroup\$ Commented Oct 26, 2024 at 7:49
  • 2
    \$\begingroup\$ This question is very similar to the IDCT one. Everything above the update would have very similar comments as in my IDCT answer. The update is my proposed IDCT code, modified to compute the DCT. There is nothing for me to comment on it. IMO, you could have posted the update as an answer instead. \$\endgroup\$ Commented Oct 26, 2024 at 14:25

1 Answer 1

4
\$\begingroup\$

Considering Cris Luengo's answer, I realized that not only 3D IDCT, but also 3D DCT can be vectorized as the following code.

function out = DCT3D2(x)
 [N1,N2,N3] = size(x);
 out = DCT1D(x, 1);
 out = DCT1D(out, 2);
 out = DCT1D(out, 3);
 out = 8 .* out ./ (N1 * N2 * N3);
end
function out = DCT1D(x, dim)
 N = size(x, dim);
 sz = ones(1,3);
 sz(dim) = N;
 index = {':',':',':'};
 out = zeros(size(x));
 for n = 0:N-1
 alpha = cos(pi/(2*N)*(2*(0:N-1)+1)*n);
 alpha = reshape(alpha, sz);
 index{dim} = n + 1;
 sum_result = sum(x .* alpha, dim);
 if n == 0
 sum_result = sum_result / sqrt(2);
 end
 out(index{:}) = sum_result;
 end
end

Performance Comparison

%% Create test cells
testCellsSize = 10;
test = zeros(testCellsSize, testCellsSize, testCellsSize);
for x = 1:size(test, 1)
 for y = 1:size(test, 2)
 for z = 1:size(test, 3)
 test(x, y, z) = x * 100 + y * 10 + z;
 end
 end
end
%% Performance test
timeElapseds = zeros(1, 10);
for index = 1:10
 tic
 result = DCT3D(test);
 timeElapseds(1, index) = toc;
end
timeElapseds
mean(timeElapseds)

The code snippet above is for testing the performance of the origin DCT3D function and the output is as follows.

timeElapseds =
 7.3854 7.3691 7.2833 7.2259 7.2568 7.2323 7.2916 6.9258 6.9233 6.8979
ans =
 7.1792

The average time of calculating 10x10x10 size 3D DCT is 7.1792s. Let's change DCT3D to DCT3D2 (run the vectorized code). The output is as follows.

timeElapseds =
 0.0267 0.0014 0.0004 0.0002 0.0006 0.0001 0.0001 0.0001 0.0001 0.0001
ans =
 0.0030

The average execution time comes to 0.0030s.

Notes: Tests run on the machine with the following spec.

CPU: 12th Gen Intel(R) Core(TM) i9-12900K 3.20 GHz

RAM: 128 GB (128 GB usable)

OS: Windows 11 Pro 22H2, OS build: 22621.4317

answered Oct 28, 2024 at 3:10
\$\endgroup\$
1
  • 1
    \$\begingroup\$ You might want to edit the question before you answer it next time. \$\endgroup\$ Commented Oct 28, 2024 at 12:03

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