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Are there any differences if you use Maximum Likelihood or Maximum a Posteriori to estimate the Point Spread Function for image deconvolution?

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Maximum a Posteriori (MAP) is the same as Maximum Likelihood Estimation (MLE) except with a Bayesian prior distribution on whatever it is that you're trying to estimate. So if you have prior information on the distribution of point spread functions then MAP will work better.

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  • $\begingroup$ Could you give a concrete example in this case? And how would you estimate the prior if you have little information and a very complicated function? What kind of prior information do you suppose to get in a case of a deconvolution? And could you illustrate it with an image? $\endgroup$ – Marka Dec 13 '13 at 17:28
  • $\begingroup$ I don't know much about your specific application. If you had some information about what the PSF typically looks like then that distribution would be your prior. If you have no prior information about the average PSF then MAP is equivalent to MLE. $\endgroup$ – Aaron Dec 13 '13 at 20:45
  • $\begingroup$ " some information about what the PSF typically looks like" <-- Do you have an example of this? $\endgroup$ – Marka Dec 13 '13 at 20:58
  • $\begingroup$ I assume that if you are working with them that you might have an idea of what they look like. Like if they tend to be vertical lines or circles or whatever. $\endgroup$ – Aaron Dec 13 '13 at 23:22
  • $\begingroup$ To be clear: You don't have any images to illustrate this? $\endgroup$ – Marka Dec 14 '13 at 12:10
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You can think of MAP as a regularization of the ML.
Just like you have regularization for Least Squares Problem (They can be built, mostly, as MAP problem).

The nice thing is that, as always, the best regularization is more data, namely, in most case when there is a lot of data they collide (Namely, low sensitivity fir the Posterior PDF).

So they differ mainly when there is (Relatively) low amount of data.
Now, when you have low amount of data and you know nothing about the parameters you're trying to estimate, ML is the way to go.
If you have low amount of data yet some prior knowledge about it, or reasonable assumption, make those assumption as regularization.
It even good to make the effort and describe this knowledge in Posterior PDF form.

Update

Found really nice tutorial about the topic - The Truth About Priors and Overfitting.

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