How to count the elements of a sparse matrix in a certain region? - matlab

Question

I have a sparse matrix and want to divide the region into 4 parts, dividing x and y in 2 equidistant pieces and want to calculate the sum of the corresponding values.
For the example below, the coordinates x-y each corresponds to [0,16] so the region is a square. There is a sparse matrix in this square, which is symmetrical. I would like to divide the region into smaller squares and sum up the sparse values. Region 0:8,0:8 has 2 elements and their values are both (2,3)=(3,2)=8 so the sum is 16.
Summation of the 1st region should give 16, 2nd and 3rd are 36 and the 4th one is 26.
x = sparse(16,16);
x (3,2) = 8;
x (10,2) = 8;
x (13,2) = 8;
x (14,2) = 4;
x (15,2) = 4;
x (2,3) = 8;
x (10,3) = 4;
x (13,3) = 4;
x (14,3) = 2;
x (15,3) = 2;
x (2,10) = 8;
x (3,10) = 4;
x (13,10) = 4;
x (14,10) = 2;
x (15,10) = 2;
x (2,13) = 8;
x (3,13) = 4;
x (10,13) = 4;
x (14,13) = 2;
x (15,13) = 2;
x (2,14) = 4;
x (3,14) = 2;
x (10,14) = 2;
x (13,14) = 2;
x (15,14) = 1;
x (2,15) = 4;
x (3,15) = 2;
x (10,15) = 2;
x (13,15) = 2;
x (14,15) = 1;
i would rather appriciate a shorter way, rather than writing a line for each sub-square. lets say for 6000 sub-squares one should write 6000 lines?

Answer 1

Let's define the input in a more convenient way:
X = sparse([...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 8, 0, 0, 0, 0, 0, 0, 8, 0, 0, 8, 4, 4
0, 8, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 2, 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 2, 2
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
0, 8, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 2, 2
0, 4, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 1
0, 4, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 1, 0]);
For convenience, we first make the array dimensions even. We don't use padarray() for this because this makes the sparse matrix full!
sz = size(X);
newX = sparse(sz(1)+1,sz(2)+1);
padTopLeft = true; % < chosen arbitrarily
if padTopLeft
newX(2:end,2:end) = X;
else % bottom right
newX(1:sz(1),1:sz(2)) = X;
end
%% Preallocate results:
sums = zeros(2,2,2);
Method #1: accumarray
We create a mask of the form:
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4
and then use it to sum the appropriate elements of newX:
sums(:,:,1) = reshape(...
accumarray(reshape(repelem([1,2;3,4], ceil(sz(1)/2), ceil(sz(2)/2)),[],1),...
reshape(newX, [],1),...
[],#sum) ,2,2);
Method #2: blockproc (requires the Image Processing Toolbox)
sums(:,:,2) = blockproc(full(newX), ceil(sz/2), #(x)sum(x.data(:)));
Several notes:
I also tried histcounts2, which is very short, but it only tells you the amount of values in each quadrant, not their sum:
[r,c] = find(newX);
histcounts2(r,c,[2,2])
I might've overcomplicated the accumarray solution.

Answer 2

Although your question is not very precise and you don't made any efford to find a solution, here is what you are asking..
clear;clc;close;
Matrix=rand(20,20);
Acc=zeros(1,4);
Acc(1)=sum(sum( Matrix(1:size(Matrix,1)/2,1:size(Matrix,2)/2) ));
Acc(2)=sum(sum( Matrix((size(Matrix,1)/2)+1:end,1:size(Matrix,2)/2)));
Acc(3)=sum(sum( Matrix(1:size(Matrix,1)/2,((size(Matrix,2)/2)+1):end )));
Acc(4)=sum(sum( Matrix((size(Matrix,1)/2)+1:end,((size(Matrix,2)/2)+1):end)));
% Verification
sum(sum(Matrix)) % <- is the same with
sum(Acc) % <- this

Answer 3

You can define any rectangle within the matrix by defining the 4 corners of it. Then use a for loop to process all rectangles.
regions = [
1 8 1 8
9 16 1 8
1 8 9 16
9 16 9 16
];
regionsum = zeros(size(regions,1),1);
for rr = 1:size(regions,1)
submat = x(regions(rr,1):regions(rr,2),regions(rr,3):regions(rr,4));
regionsum(rr) = sum(submat(:));
end
>> regionsum
regionsum =
16
36
36
26
If you mean you want to divide the square matrix into 2^N (N>2) squares of the same size then you can write regions with a for loop.
N = 1; % 2^N-by-2^N sub-squares
L = size(x,1);
dL = L/(2^N);
assert(dL==int32(dL),'Too many divisions')
segments = zeros(2^N,2);
for nn = 1:2^N
segments(nn,:) = [1,dL]+dL*(nn-1);
end
regions = zeros(2^(2*N),4);
for ss = 1:2^N
for tt = 1:2^N
regions((2^N)*(ss-1) + tt,:) = [segments(ss,:),segments(tt,:)];
end
end
example output with dividing into 16 (N=2) square submatrices:
>> regions
regions =
1 4 1 4
1 4 5 8
1 4 9 12
1 4 13 16
5 8 1 4
5 8 5 8
5 8 9 12
5 8 13 16
9 12 1 4
9 12 5 8
9 12 9 12
9 12 13 16
13 16 1 4
13 16 5 8
13 16 9 12
13 16 13 16
>> regionsum
regionsum =
16
0
12
24
0
0
0
0
12
0
0
8
24
0
8
10
>>