Vectorizing sums of different diagonals in a matrix

Question

I want to vectorize the following MATLAB code. I think it must be simple but I\'m finding it confusing nevertheless.

r = some constant less than m or n
[m,n         


        

Answer 1

If I understand correctly, you're trying to calculate the diagonal sum of every subarray of C, where you have removed the last row and column of C (if you should not remove the row/col, you need to loop to m-r+1, and you need to pass the entire array C to the function in my solution below).

You can do this operation via a convolution, like so:

S = conv2(C(1:end-1,1:end-1),eye(r),'valid');

If C and r are large, you may want to have a look at CONVNFFT from the Matlab File Exchange to speed up calculations.

Answer 2

Based on the idea of JS, and as Jonas pointed out in the comments, this can be done in two lines using IM2COL with some array manipulation:

B = im2col(C, [r r], 'sliding');
S = reshape( sum(B(1:r+1:end,:)), size(C)-r+1 );

Basically B contains the elements of all sliding blocks of size r-by-r over the matrix C. Then we take the elements on the diagonal of each of these blocks B(1:r+1:end,:), compute their sum, and reshape the result to the expected size.

---

Comparing this to the convolution-based solution by Jonas, this does not perform any matrix multiplication, only indexing...

Answer 3

Is this what you're looking for? This function adds the diagonals and puts them into a vector similar to how the function 'sum' adds up all of the columns in a matrix and puts them into a vector.

function [diagSum] = diagSumCalc(squareMatrix, LLUR0_ULLR1)
% 
% Input: squareMatrix: A square matrix.
%        LLUR0_ULLR1:  LowerLeft to UpperRight addition = 0     
%                      UpperLeft to LowerRight addition = 1
% 
% Output: diagSum: A vector of the sum of the diagnols of the matrix.
% 
% Example: 
% 
% >> squareMatrix = [1 2 3; 
%                    4 5 6;
%                    7 8 9];
% 
% >> diagSum = diagSumCalc(squareMatrix, 0);
% 
% diagSum = 
% 
%       1 6 15 14 9
% 
% >> diagSum = diagSumCalc(squareMatrix, 1);
% 
% diagSum = 
% 
%       7 12 15 8 3
% 
% Written by M. Phillips
% Oct. 16th, 2013
% MIT Open Source Copywrite
% Contact mphillips@hmc.edu fmi.
% 

if (nargin < 2)
    disp('Error on input. Needs two inputs.');
    return;
end

if (LLUR0_ULLR1 ~= 0 && LLUR0_ULLR1~= 1)
    disp('Error on input. Only accepts 0 or 1 as input for second condition.');
    return;
end

[M, N] = size(squareMatrix);

if (M ~= N)
    disp('Error on input. Only accepts a square matrix as input.');
    return;
end

diagSum = zeros(1, M+N-1);

if LLUR0_ULLR1 == 1
    squareMatrix = rot90(squareMatrix, -1);
end

for i = 1:length(diagSum)
    if i <= M
        countUp = 1;
        countDown = i;
        while countDown ~= 0
            diagSum(i) = squareMatrix(countUp, countDown) + diagSum(i);
            countUp = countUp+1;
            countDown = countDown-1;
        end
    end
    if i > M
        countUp = i-M+1;
        countDown = M;
        while countUp ~= M+1
            diagSum(i) = squareMatrix(countUp, countDown) + diagSum(i);
            countUp = countUp+1;
            countDown = countDown-1;
        end
    end
end

Cheers

Answer 4

I would think you might need to rearrange C into a 3D matrix before summing it along one of the dimensions. I'll post with an answer shortly.

EDIT

I didn't manage to find a way to vectorise it cleanly, but I did find the function accumarray, which might be of some help. I'll look at it in more detail when I am home.

EDIT#2

Found a simpler solution by using linear indexing, but this could be memory-intensive.

At C(1,1), the indexes we want to sum are 1+[0, m+1, 2*m+2, 3*m+3, 4*m+4, ... ], or (0:r-1)+(0:m:(r-1)*m)

sum_ind = (0:r-1)+(0:m:(r-1)*m);

create S_offset, an (m-r) by (n-r) by r matrix, such that S_offset(:,:,1) = 0, S_offset(:,:,2) = m+1, S_offset(:,:,3) = 2*m+2, and so on.

S_offset = permute(repmat( sum_ind, [m-r, 1, n-r] ), [1, 3, 2]);

create S_base, a matrix of base array addresses from which the offset will be calculated.

S_base = reshape(1:m*n,[m n]);
S_base = repmat(S_base(1:m-r,1:n-r), [1, 1, r]);

Finally, use S_base+S_offset to address the values of C.

S = sum(C(S_base+S_offset), 3);

You can, of course, use bsxfun and other methods to make it more efficient; here I chose to lay it out for clarity. I have yet to benchmark this to see how it compares with the double-loop method though; I need to head home for dinner first!