But I want to get the matrix operator corresponding to the above such that D*u = u_x Is it possible?
To get a general linear operation in matrix form, you can use my func2mat utility ( Download ). This covers 1D, 2D, or whatever. However, some of the operations you describe are separable in the x,y components of the image and that can be exploited to make things more efficient.
Your example with imfilter along the x direction is a particular case of this. The x-direction operator Dx can be obtained as,
[m,n]=size(yourImage);
fun = @(u) imfilter(u, [1;-1], 'corr','symmetric','same');
Dx = func2mat(fun,yourImage(:,1));
and now the operator for the total 2D image is
D=kron(speye(n),Dx);
In other words
ux=imfilter(yourImage, [1 -1], 'corr','symmetric','same');
is now equivalent to
ux=D*YourImage(:)
ux=reshape(ux,m,n);
Of course, once you have the 1D operator Dx, you can also just as easily do,
ux=Dx*yourImage;
(Additional question) Actually, what I want to do is to get transpose of D, i.e., D'. Is there a way to directly apply D' through imfilter?
It can largely be obtained by switching 'corr' to 'conv' and off-centering the filter kernel,
imfilter(u, [0;1;-1],'conv','symmetric','same');
except there may be some discrepancy at the boundaries of the image due to nonlinear edge handling done by imfilter. If the edge voxel are zero, it won't matter.
