How to generate a 2D periodic pattern like below image from pseudorandom binary sequence? - image-processing

Question

I want to generate a 2 dimensional periodic pattern from a pseudorandom binary sequence like this one with the following context:
A periodic pattern satisfies the equations (1) and (2)
W(x + q0N0, y) = W(x, y); q0, N0 >1 (1)
W(x, y + q1N1) = W(x, y); q1, N1 > 1, (2)
where N0 and N1 determine the periodicity of repetitions and q0 and q1 a repetition number on the horizontal and vertical directions. Generation from pseudorandom values {−1, 1} produces a rectangular, binary valued pattern.

Answer 1

One way to achieve that would be to take a small pseudo-random 2D pattern (the sequence) and repeat it periodically so that neighboring tiles are always mirrored giving a sense of smooth continuity.
After you specified your requirements with W(x + q0N0, y) = W(x, y) and W(x, y + q1N1) = W(x, y); it becomes clear, that this is exactly (without the mirroring part) what you want.
You simply have to repeat a random pattern a certain number of times in both directions.
Example (similar to your image where the period length in the vertical direction is longer than in the horizontal direction)
Code (in Matlab)
% base pattern
N0 = 20;
N1 = 5;
base = rand([N0, N1]) > 0.5; % pseudo-random
% periodically repeating the pattern
Q0 = 5;
Q1 = 20;
pattern = zeros([N0*Q0,N1*Q1]);
for q0 = 1 : Q0
for q1 = 1 : Q1
pattern((q0-1)*N0+1:q0*N0, (q1-1)*N1+1:q1*N1) = base;
end
end
% save
imwrite(pattern, 'test.jpg');
% display
imagesc(pattern);
axis image;
colormap(gray);
The first lines just compute a random, binary pattern of a certain size N0 x N1
% base pattern
N0 = 20;
N1 = 5;
base = rand([N0, N1]) > 0.5; % pseudo-random
Then comes the definition of the number of repetitions of the pattern in each direction
Q0 = 5;
Q1 = 20;
Finally, in two nested but rather simple for loops, the base pattern is repeated
pattern = zeros([N0*Q0,N1*Q1]);
for q0 = 1 : Q0
for q1 = 1 : Q1
pattern((q0-1)*N0+1:q0*N0, (q1-1)*N1+1:q1*N1) = base;
end
end
The computation of the indices (where to place the base patterns) fulfills your requirement equations
(q0-1)*N0+1:q0*N0, (q1-1)*N1+1:q1*N1
Old Example (with mirroring)
Code (in Matlab)
% base pattern
N = 20;
base = rand(N) > 0.5; % pseudo-random
% multiplying the pattern
M = 4;
pattern = zeros(N*M);
for i = 1 : M
for j = 1 : M
b = base;
% mirroring the base
if mod(i, 2) == 1
flip(b, 2);
end
if mod(j, 2) == 1
flip(b, 1);
end
pattern((i-1)*N+1:i*N, (j-1)*N+1:j*N) = b;
end
end
% save
imwrite(pattern, 'test.jpg');
% display
imagesc(pattern);
axis image;
colormap(gray);
The patterns are flipped (mirrored) in one or two direction sometimes to simulates some kind of smoothness (symmetry) of the pattern.