Image Deconvolution with Sparse Priors

Optical systems used for image acquisition are usually not perfect and
leading to degraded images. A typical degradation is image blur. Building
perfect optics is not always possible due to physical limitations, cost,
size or weight. Therefore, there is interest in computational solutions to
remove these degradations. By knowing the sources of distortion it is
possible to remove them.
Image blur can be removed by deconvolution, however, the problem
which has to be solved is underdetermined. For solving these ill-posed
problems additional assumptions have to be considered. Recently, many
advances were made in the investigation of underdetermined systems of
equations [1] in cases where the solution can be sparsely encoded. The
sparseness constraint is used to select a plausible solution out of an innite
set of possible solutions. This method is applied to the deconvolution
problem.
Similar to other approaches to deconvolution based on sparse coding,
for speed and memory eciency we apply the fast Fourier transform and
the fast wavelet transform to model the convolution and provide a sparse
basis [2]. For the convolution, boundary areas are cut to avoid wrong
modelling due to the cyclic nature of the Fourier transform. By cutting
the boundary areas the system of equations becomes underdetermined.
We apply this approach to a pinhole camera setting. Using a simulated
pinhole camera, we look at the in
uence of sparseness and the robustness
to noise. First tests have also been made using a real pinhole camera.