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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 →

For the purposes of ‘sharpness’ spatial resolution measurement in photography cameras can be considered shift-invariant, linear systems when capturing scene detail of random size and direction such as one often finds in landscapes.

Shift invariant means that the imaging system should respond exactly the same way no matter where light from the scene falls on the sensing medium . We know that in a strict sense this is not true because for instance pixels tend to have squarish active areas so their response cannot be isotropic by definition. However when using the slanted edge method of linear spatial resolution measurement we can effectively make it shift invariant by careful preparation of the testing setup. For example the edges should be slanted no more than this and no less than that. Continue reading Linearity in the Frequency Domain →

Musings about Photography