MAP_K approach to kernel estimation under Gaussian prior

The following note excerpts from “Understanding and Evaluating Blind Deconvolution Algorithms” by Anat Levin.

Let $X_w$ , $Y_w$ and $K_w$ be the Fourier transform of original image $x$ , blurry image $x$ and blurring kernel $k$ .
Assuming Gaussian prior on $X_w$ , i.e.
$X_w \sim N(0, \frac{1}{\sigma_w^2})$ ,
where $\sigma_w^2 = \beta (|G_{xw}|^2 + |G_{yw}|^2)^{-1} = \beta |G_w|^{-2}$ .

Here $|G_{xw}|$ and $|G_{yw}|$ are Fourier transform of image derivative operators $g_x$ and $g_y$ .
Then $Y_w$ also follows a Gaussian distribution:
$Y_w \sim N(0, \frac{1}{\beta} |K_w|^2|G_{w}|^2 + \eta^2)$

The log likihood of $Y_w$ is therefore
$C-\frac{1}{2}\left(\frac{|Y_w|^2}{|K_wG_w|^2/\beta + \eta^2} + log({|K_wG_w|^2/\beta + \eta^2})\right)$
where $C$ is a constant.
The second term is the likelihood when $X_w$ takes its MAP estimate, and the log term is proportional to the volume of X.

If no assumption is made about the kernel, this constrains the squared MTF of the kernel since the log likelihood is maximized when
$|K_w|^2 = \beta \frac{max(|Y_w|^2 - \eta^2, 0)}{|G_w|^2}.$ (1)

Note that squared MTF of a signal is its spatial auto-correlation. ThereforeEq.(1) can be interpreted as “The auto-correlation in the (denoised) blurry image is the convolution of kernel auto-correlation and sharp image’s auto-correlation.”

However, it also shows that at least under Gaussian prior:

Levin’s approach does not seem to provide enough constraint on free form kernels. Additional prior on kernel (e.g. non-negative, continuous) might be necessary. However, if the kernel is limited to certain classes, e.g. disk, box filter, the approach seems to be sufficient.
Although it might not be important to use L-2 norm rather than L-1 norm, the choice of prior $G_w$ is essential.

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MAP_K approach to kernel estimation under Gaussian prior

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