# How to update point spread function of blind deconbolution by conjugate gradient?

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?

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