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How to deal with this transposition in inverse problem ie. image

aheartthatlovesisalwaysyoung@gmail.com — Fri, 03 Apr 2009 06:04:12 -0400

How to deal with this transposition in inverse problem ie. image
restoration

For example, when minimizing || g - Hf ||^2 + =EB||Qf||^2 , where image
matrix f & g are the true and observed value in column-lexically
heaped vectors, H is block-toeplitz matrix of the convolution kernel.

Then the solution is f =3D H^Tg/(H^TH+ =EBQ^TQ) , here H^T means the
conjugate transpose matrix of H.

In programming, we usually compute it in the Fourier domain.

My question is how to deal with H^T in convolution ie H & x. conv2
(H,x) (here H, x is a image matrix)?

And what is the relation between (H^T x) and (H x) in frequency
domain?

ifftn( conj(fftn(H)) .* fftn(x) )?
or conv2(rot90(H,2), x)

this problem confused me very long long time!

Can you help me ?

Sincerely AHeartThatLovesIsAlwaysYoung
20090403

Re: How to deal with this transposition in inverse problem ie. image

Bruno Luong — Fri, 03 Apr 2009 07:02:03 -0400

aheartthatlovesisalwaysyoung@gmail.com wrote in message <471e1973-38cd-4ba9-b081-d9a74c7b8d9c@w35g2000prg.googlegroups.com>...

>
> My question is how to deal with H^T in convolution ie H & x. conv2
> (H,x) (here H, x is a image matrix)?

I believe H^T is convolution with the flipped kernel. Write down what is transpose operator. This relation might be useful to come back to the basic
dot(H^T y , x) = (y , conv2(H,x)) for all images x

>
> And what is the relation between (H^T x) and (H x) in frequency
> domain?
>

Multiplication of the flipped Fourier transform of the kernel, I believe. You have now all the idea to check by yourself.

Bruno

Re: How to deal with this transposition in inverse problem ie. image

timedcy Asayaku — Fri, 03 Apr 2009 09:15:04 -0400

"Bruno Luong" <b.luong@fogale.findmycountry> wrote in message <gr4c9a$c3c$1@fred.mathworks.com>...

Thank you, Bruno
you are right
but conv2(rot90( H,2) ,x ) equals fftshift ( ifftn(fftn(conj(H).* fftn(x) ) )

Thank you agian~~

> aheartthatlovesisalwaysyoung@gmail.com wrote in message <471e1973-38cd-4ba9-b081-d9a74c7b8d9c@w35g2000prg.googlegroups.com>...
>
> >
> > My question is how to deal with H^T in convolution ie H & x. conv2
> > (H,x) (here H, x is a image matrix)?
>
> I believe H^T is convolution with the flipped kernel. Write down what is transpose operator. This relation might be useful to come back to the basic
> dot(H^T y , x) = (y , conv2(H,x)) for all images x
>
> >
> > And what is the relation between (H^T x) and (H x) in frequency
> > domain?
> >
>
> Multiplication of the flipped Fourier transform of the kernel, I believe. You have now all the idea to check by yourself.
>
> Bruno