# Poisson Noise Removal from an Image

What type of filters can be used to remove Poisson noise in an image? I tried applying Gaussian filter to few points in the spectrum.

In image denoising far more important then the noise distribution is the noise spatial correlation properties and the prior about the image.

Let's try building some cases and dealing with them.
The model is:

$$y = x + n$$

Where $$x$$ is the clan image, $$n$$ is the Poisson Noise (With mean $$\lambda$$) and $$y$$ is the noisy image.

## Noise Is Poisson Distributed and Spatially White

In this case each pixel $${y}_{i}$$ is distributed with Poisson with mean $${x}_{i} + \lambda$$.

Deriving the the estimator for $${x}_{i}$$ will yield it is $${y}_{i}$$. So no gain here what so ever.

## Noise Is Poisson Distributed and Spatially White with Image Prior

If we assume some kind of a prior on the image (Piece Wise Smooth) we can do something.
The simplest idea is use the mean of a neighborhood we believe should have the same value (Think of capturing a wall with the same color all over it).

A very popular and effective way to enforce Piece Wise model is using the Total Variation prior on the Gradients of the image.
Then the MAP becomes something to work with (See Noise, Image Reconstruction with Noise (EE367/CS448I: Computational Imaging and Display, Class 10, Gordon Wetzstein, Stanford University)).

## Noise Is Poisson Distributed and Spatially Correlated

The usual technique to deal with spatially correlated noise is working in a Multi Scale manner.

## Noise Is Poisson Distributed and Spatially Correlated with Image Prior

Now combine all methods and voila, you have a real world denoiser.
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