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MAX_LIKELIHOOD

MAX_LIKELIHOOD

## Purpose

Maximum likelihood deconvolution of an image or a spectrum.

## Explanation

Deconvolution of an observed image (or spectrum) given the
instrument point spread response function (spatially invariant psf).
Performs iteration based on the Maximum Likelihood solution for
the restoration of a blurred image (or spectrum) with additive noise.
Maximum Likelihood formulation can assume Poisson noise statistics
or Gaussian additive noise, yielding two types of iteration.

## Calling Sequence

for i=1,Niter do Max_Likelihood, data, psf, deconv, FT_PSF=psf_ft

## Inputs Parameters

data = observed image or spectrum, should be mostly positive,
with mean sky (background) near zero.
psf = Point Spread Function of the observing instrument,
(response to a point source, must sum to unity).

## Input/output Parameters

deconv = as input: the result of previous call to Max_Likelihood,
(initial guess on first call, default = average of data),
as output: result of one more iteration by Max_Likelihood.
Re_conv = (optional) the current deconv image reconvolved with PSF
for use in next iteration and to check convergence.

## Optional Input Keywords

/GAUSSIAN causes max-likelihood iteration for Gaussian additive noise
to be used, otherwise the default is Poisson statistics.
FT_PSF = passes (out/in) the Fourier transform of the PSF,
so that it can be reused for the next time procedure is called,
/NO_FT overrides the use of FFT, using the IDL function convol() instead.
POSITIVITY_EPS = value of epsilon passed to function positivity,
default = -1 which means no action (identity).
UNDERFLOW_ZERO = cutoff to consider as zero, if numbers less than this.

## External Calls

function convolve( image, psf ) for convolutions using FFT or otherwise.
function positivity( image, EPS= ) to make image positive.

## Method

Maximum Likelihood solution is a fixed point of an iterative eq.
(derived by setting partial derivatives of Log(Likelihood) to zero).
Poisson noise case was derived by Richardson(1972) & Lucy(1974).
Gaussian noise case is similar with subtraction instead of division.

## Notes

WARNING: The Poisson case may not conserve flux for an odd image size.
This behavior is being investigated.

## History

written: Frank Varosi at NASA/GSFC, 1992.
F.V. 1993, added optional arg. Re_conv (to avoid doing it twice).
Converted to IDL V5.0 W. Landsman September 1997
Use COMPLEMENT keyword to WHERE() W. Landsman Jan 2008

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