Creating a differential matrix in Matlab - matlab

Question

I need to create matrices Dx and Dy that give the partial diferentials in the x and y directions respectivly when multipled by column stack representation of 2D area (x) so that Dx*x will give me the partial derivetives in the x direction in vector form as well (the same dimentions as x).
The derivitives are aproximated by:
f'(x,y) = 0.5*(f(x+1,y)-f(x-1,y)) for all x > 1 and x < n (the number of rows)
f'(1,y) = f(2,y)-f(1,y)
f'(n,y) = f(n,y)-f(n-1,y)
for Dx and similarly for Dy.
I tried creating the matrices using the fallowing code:
[X,Y] = meshgrid(1:n,1:m); % m,n - the dimentions of x
X = X(:)';
Y = Y(:)';
Dx = sparse(m*n,m*n);
Dy = sparse(m*n,m*n);
for x = 1:n
for y = 1:m
idx = y + (x-1)*m;
if (x == 1)
Dx(idx,:) = ((X == x+1) .* (Y == y)) - ((X == x) .* (Y == y));
elseif (x == n)
Dx(idx,:) = ((X == x) .* (Y == y)) - ((X == x-1) .* (Y == y));
else
Dx(idx,:) = 0.5*(((X == x+1) .* (Y == y)) - ((X == x-1) .* (Y == y)));
end
if (y == 1)
Dy(idx,:) = ((X == x) .* (Y == y+1)) - ((X == x) .* (Y == y));
elseif (y == m)
Dy(idx,:) = ((X == x) .* (Y == y)) - ((X == x) .* (Y == y-1));
else
Dy(idx,:) = 0.5*(((X == x) .* (Y == y+1)) - ((X == x) .* (Y == y-1)));
end
end
end
But this takes too much time, I feel that there should be a faster to create Dy and Dx?
About x: for a 2D function represented by X so that size(X) is [n m], then x=X(:) and size(x) is [n*m 1]

Answer 1

Follow those steps:
Represent the operation using a Filter. Derivative Filters are easy the way you defined the above.
Create the Convolution Matrix using MATLAB's convmtx2() Function.
This will get you what you wanted.
Derivative Filters in Image Processing
Horizontal Derivative:
* Forward Finite Difference - $ [-1, 1] $.
* Backward Finite Difference - $ [1, -1] $.
Vertical Derivative:
* Forward Finite Difference - $ [-1; 1] $.
* Backward Finite Difference - $ [1; -1] $.