How do I construct those gradient and divergent matrices in MATLAB? - matlab

Question

How can I construct the matrices G and D if I have h and w?
Please pay attention that in G there are some 0 rows and in D there are some 0 columns...

Answer 1

Even though i am very late with my solution for this, here it is for anybody wanting to implement it:
function g = gradientMat(w,h)
one = [-ones(w*h,1) ones(w*h,1)];
d = [0 1];
dx = spdiags(one,d,w*h,w*h);
dx(1:h:end,:) = zeros(w,w*h);
d = [0 h];
dy = spdiags(one,d,w*h,w*h);
dy(h*(w-1)+1:end,:) = zeros(h,w*h);
g = [dx;dy];
end
This solution has the advantage to use matlabs sparse diagonal matrices, which allow you to create the gradient matrix for large images and make calculation with these faster.
To obtain the divergence, simpley use:
g = gradientMat(size(image)(1),size(image)(2));
div = (-g')*g;

Answer 2

this is how I did it now, but there must be a more elegant way, right?!
%% construct G
% upper half
firstDiagG = ones(h, 1);
firstDiagG(h) = 0;
firstDiagG = -firstDiagG;
firstDiagG = repmat(firstDiagG,w,1);
% first = diag(firstDiagG);
firstG = spdiags(firstDiagG,0,n,n);
secondDiagG = ones(h, 1);
secondDiagG(h) = 0;
secondDiagG = repmat(secondDiagG,w,1);
% second = diag(secondDiagG);
secondG = spdiags(secondDiagG,0,n,n);
secondG = [zeros(size(secondG,1),1) secondG];
secondG = secondG(:,1:end-1);
upperHalf = firstG + secondG;
% lower half
thirdDiagG = ones(n-h, 1);
thirdDiagG = [thirdDiagG; zeros(h, 1)];
thirdDiagG = -thirdDiagG;
% thirdG = diag(thirdDiagG);
thirdG = spdiags(thirdDiagG,0,n,n);
fourthDiagG = [ones(n-2*h, 1); zeros(n-2*h, 1)];
% fourthG = diag(fourthDiagG);
fourthG = spdiags(fourthDiagG,0,n,n);
fourthG = [zeros(size(fourthG,1), h) fourthG];
fourthG = fourthG(:,1:end-h);
% fourthG = [fourthG zeros(size(fourthG,1),h)];
% fourthG = [fourthG; zeros(size(fourthG,1), size(fourthG,2))];
lowerHalf = thirdG + fourthG;
G = [upperHalf; lowerHalf];
%% construct D
% left half
firstD = firstG; % is the same as in G
secondDiagD = secondDiagG; % is the same as in G
% secondD = diag(secondDiagD);
secondD = spdiags(secondDiagD,0,n,n);
secondD = [zeros(1, size(secondD,1)); secondD];
secondD = secondD(1:end-1,:);
leftHalf = firstD + secondD;
% right half
thrirdDiagD = flip(thirdDiagG);
% thirdD = diag(thrirdDiagD);
thirdD = spdiags(thrirdDiagD,0,n,n);
fourthDiagD = ones(n, 1);
% fourthD = diag(fourthDiagD);
fourthD = spdiags(fourthDiagD,0,n,n);
fourthD = [zeros(h, size(fourthD,1)); fourthD];
fourthD = fourthD(1:end-h,:);
rightHalf = thirdD + fourthD;
D = [leftHalf rightHalf];