iterative | Strolls with my Dog

Deconvolution by the Richardson-Lucy algorithm is achieved by minimizing the convex loss function derived in the last article

(1) $\begin{equation*} J(O) = \sum \bigg (O**PSF - I\cdot ln(O**PSF) \bigg) \end{equation*}$

with

$J$ , the scalar quantity to minimize, function of ideal image $O(x,y)$
$I(x,y)$ , linear captured image intensity laid out in $M$ rows and $N$ columns, corrupted by Poisson noise and blurred by the $PSF$
$PSF(x,y)$ , the known two-dimensional Point Spread Function that should be deconvolved out of $I$
$O(x,y)$ , the output image resulting from deconvolution, ideally without shot noise and blurring introduced by the $PSF$
** two-dimensional convolution
$\cdot$ element-wise product
$ln$ , element-wise natural logarithm

In what follows indices $x$ and $y$ , from zero to $M$ -1 and $N$ -1 respectively, are dropped for readability. Articles about algorithms are by definition dry so continue at your own peril.

So, given captured raw image $I$ blurred by known function $PSF$ , how do we find the minimum value of $J$ yielding the deconvolved image $O$ that we are after?

Continue reading The Richardson-Lucy Algorithm →

Musings about Photography