Vectorising 5 nested FOR loops - matlab

Question

I am writing a program in MATLAB as a part of my project based on DFT.
Let the N x N data matrix be X and the corresponding DFT matrix be Y, then the DFT coefficients can be expressed as
Y(k1,k2) = ∑(n1=0:N-1)∑(n2=0:N-1)[X(n1,n2)*(WN^(n1k1+n2k2))] (1)
0≤k1,k2≤N-1
Where WN^k=e^((-j2πk)/N)
Since the twiddle factor WN is periodic, (1) can be expressed as
Y(k1,k2)=∑(n1=0:N-1)∑(n1=0:N-1)[X(n1,n2)*(WN^([(n1k1+n2k2)mod N) ] (2)
The exponent ((n1k1 +n2k2)) N = p is satisfied by a set of (n1,n2) for a given (k1,k2). Hence, by grouping such data and applying the property that WN^(p+N /2) = -(WN^P),
(2) can be expressed as
Y(k1,k2)= ∑(p=0:M-1)[Y(k1,k2,p)*(WN^p)] (3)
Where
Y(k1,k2,p)= ∑(∀(n1,n2)|z=p)X(n1,n2) - ∑(∀(n1,n2)|z=p+M)X(n1,n2) (4)
z=[(n1k1+n2k2)mod N] (5)
I am coding a program to find Y(k1,k2,p).ie I need to create slices of 2d matrices(ie a 3D matrix in which each slice is a 2D matrix )from a given 2D square matrix (which is the matrix X)..Dimensions of X can be upto 512.
Based on the above equations,I have written a code as follows.I need to vectorise it.
N=size(X,1);
M=N/2;
Y(1:N,1:N,1:M)=0;
for k1 = 1:N
for k2 = 1:N
for p= 1:M
for n1=1:N
for n2=1:N
N1=n1-1; N2=n2-1; P=p-1; K1=k1-1; K2=k2-1;
z=mod((N1*K1+N2*K2),N);
if (z==P)
Y(k1,k2,p)= Y(k1,k2,p)+ X(n1,n2);
elsif (z==(P+M))
Y(k1,k2,p)= Y(k1,k2,p)- X(n1,n2);
end
end
end
end
end
As there is 5 FOR loops, the execution time is very large for large dimensions of N. Hence please provide me a solution for eliminating the FOR loops and vectorising the code..I need to make the code execute in maximum speed...Thanks Again..

Answer 1

Here is a first hint to vectorize the most inner loop.
From your code, we can notice that n1, N1, P, K1 and K2 are constant in this loop.
So we can rewrite z as a mask vector as follows:
z = mod(N1*K1+K2*(0:N-1));
Then your if-statement is equivalent to adding the sum of all elements in X so that z==P minus the sum of all elements in X so that z==P+M. Rewriting this is straightforward:
Y(k1,k2,p)= Y(k1,k2,p)+sum(X(n1,z==P))-sum(X(n1,z==P+M));
So your program can be first written as follows:
N=size(X,1);
M=N/2;
Y(1:N,1:N,1:M)=0;
for k1 = 1:N
for k2 = 1:N
for p= 1:M
for n1=1:N
N1=n1-1; P=p-1; K1=k1-1; K2=k2-1;
z=mod(N1*K1+K2*(0:N-1),N);
Y(k1,k2,p) = sum(X(n1,z==P))-sum(X(n1,z==P+M));
end
end
end
end
Then you can do the same thing with n1; for that, you need to construct a 2D array for z, such as:
z = mod(K1*repmat(0:N-1,N,1)+K2*repmat((0:N-1).',1,N));
Notice that size(z)==size(X).Then the 2D sum for Y becomes:
Y(k1,k2,p) = Y(k1,k2,p)+sum(X(z==P))-sum(X(z==P+M));
The += operation is here no longer needed, since you access only once to each element of Y:
Y(k1,k2,p)= sum(X(n1,z==P))-sum(X(n1,z==P+M));
And so we discard one more loop:
N=size(X,1);
M=N/2;
Y(1:N,1:N,1:M)=0;
for k1 = 1:N
for k2 = 1:N
for p= 1:M
P=p-1; K1=k1-1; K2=k2-1;
z = mod(K1*repmat(0:N-1,N,1)+K2*repmat((0:N-1).',1,N));
Y(k1,k2,p) = sum(X(z==P))-sum(X(z==P+M));
end
end
end
Concerning the other loops, I don't think it worths it to vectorize them, as you have to build a 5D array, which could be very huge in memory. My advise is to keep z as a 2D array, as it is of the size of X. If it does not fit well in memory, just vectorize the most inner loop.