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There is an unblurred image $g$ and a blurred image $x$.

Their relationship is expressed by the following formula using $psf$(point spread fucntion, size is $5×5$ kernel).

$g = x \otimes psf\tag 1$

where $\otimes$ is 2D convoluition. If I write the above formula by matrix operation, also, I am thinking about blind deconvolution

$\nabla^2\mathbf{g} = \nabla^2\mathbf{x\;PSF^{(k)}}\tag 2$

where, $\mathbf{g}$ and $\mathbf{x}$ is $N×N$ matrices, and $\mathbf{PSF}$ is $N^2×N^2$ Toeplitz matrices,$\nabla^2$ is the second order differential operator.

I want to solve an optimization problem that incorporates Equation $(2)$.

In the process of optimizing $\mathbf{x}$, I also want to optimize $\mathbf{PSF}$ by the conjugate gradient method.

$$\mathbf{PSF}^{(k+1)} \leftarrow \text{Conjugate Gradient Method}(\mathbf{\nabla^2g}, \;\mathbf{\nabla^2x}^{(k+1)},\; \mathbf{PSF}^{(k)})$$

[What I want to do]

In the process of optimizing $\mathbf{x}$, I also want to optimize $\mathbf{PSF^{(k)}}$ by the conjugate gradient method.

[What I want you to tell me]

From the following questions, [How to estimate filters using conjugate gradient?]

should I think of deconvolution in frequency space?

Or should I update psf by matrix operation in a different way?

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  • $\begingroup$ Why would you want to solve this that way? $\endgroup$
    – Royi
    Jun 28 at 3:31
  • $\begingroup$ I want to solve $\min\frac{1}{2}\Vert{x^{(k)} \otimes psf^{(k)}-g}\Vert_2^2+\Vert w\circ (D x^{(k)})\Vert_1 \tag 1$. $\endgroup$ Jun 28 at 9:33
  • $\begingroup$ I thought so but you write it not well. Edit your question with what you wrote on the comment. $\endgroup$
    – Royi
    Jun 28 at 9:40
  • $\begingroup$ Sorry, Comment editing has expired. $\endgroup$ Jun 28 at 9:41
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    $\begingroup$ You may +1 the links. If you focus the question here I will try answer. $\endgroup$
    – Royi
    Jul 4 at 13:12

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