How to produce self-adjoint filters using Gaussian and motion blur in Matlab?

Question

I am trying to write a Matlab code to produce motion and Gaussian blur. Here is my code:

f0=imread('cameraman.png');
[Nx,Ny,Nz]=size(f0);
 if Nz>1;f0=double(rgb2gray(f0));
 else
     f0=double(f0);
 end

blurfilter1 = fspecial('gaussian', [7 7], 1);
blurfilter2 = fspecial('motion',5,30);

Au = @(u) imfilter(u,blurfilter1,'symmetric');

Bu = @(u) imfilter(u,blurfilter2,'symmetric');

In above code, operators Au and Bu will produce Gaussian blur and motion blur respectively. However, I need these two operators to be self-adjoint. I have found the following code for producing self-adjoint opearots but I am not sure whether it works for both Gaussian and blur operators. Here is the code:

Au = @(u) imfilter(imfilter(u,blurfilter1,'symmetric'),blurfilter1','symmetric');
Bu = @(u) imfilter(imfilter(u,blurfilter2,'symmetric'),blurfilter2','symmetric');

I really appriciate it if anyone could please help me to build self-adjoint Au and Bu operators for Gaussian and motion blur.

Answer 1

Gaussian

To start, fspecial('gaussian' is deprecated; the fspecial documentation explains how to do the same thing, but better.

Why do you use an n-dimensional convolution (imfilter) if you're actually after 2 sequences of 1-dimensional convolutions?

Something like (excuse me, this is untested, I uninstalled my Matlab recently)

x = [-3:3]
sigma = 1
gaussian = exp(- x.^ 2 / (2 * sigma ^ 2));
self_adjoint_filter = @(img, impresp1d) conv2(impresp1d, impresp1d, img)
Au = @(img) self_adjoint_filter(img, gaussian)

should do the trick.

Motion blur

That's not a self-adjunct operation. It has a definite direction, so eigenvalues can't be the same magnitude.

Answer 2

First there are functions convmtx and convmtx2 which returns the matrix operator form of any convolution kernel.

Second, if you consider the convolution operator, the complex sinusoidal are their eigenfunctions and the amplitude of transfer function is their eigenvalues. So, if the operator is selfadjoint, it must have real eigenvalues and in order to have a selfadjoint convolution operator the Fourier transform of kernel must have real values. If a function is real valued in Fourier space then it must have even symmetry in spatial domain.