Page - 55 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

Givenaunsharpdiscretegrey-value imagef=(fi,j)i,j, the sharp imageuand thePSFhare sought asminimisersof the function (adiscreteversionof the functional (2)) E(u,h) := ∑ i,j (fi,j− [u∗h]i,j)2+αRu(u)+βRh(h) (3) where in the discretised data term [u∗h]i,j denotes the sampling value of the discrete convolution u∗hat location (i,j), and the regularisersRu,Rh are still to bespecified. For the image, [5] use a total variation regulariser, which is common in literature, and known to produce favourable results in non-blind deconvolution. In discretised form it reads asRu(u) =∑ i,j‖[∇u]i,j‖where [∇u]i,j denotesadiscretisationof∇uat location (i,j). Thecentral innovation of [5] lies in thePSFregulariserRhwhichisbuilt fromconvolutioneigenvaluesandeigenvectors, i.e. singularvalues andsingular vectors of a linearoperator associatedwith the imagef. Notefirst that anydiscrete imagew, actingbyconvolutionw∗hon thePSF,yieldsa linearoperator onh. In the discrete setting, we assume that the support ofh=(hi,j)i,j has sizemx×my, which is embedded in a larger areasx×sy ([5] suggestssx,y≈ 1.5mx,y), and the discrete imagew is of size nx×ny. By discrete convolution with zero-padding, one has a linear operatorAwsx,sy :h 7→w∗h mappingRsx×sy toR(sx+nx−1)×(sy+ny−1) which has sxsy (right) singular valuesσk(w)with singular vectorsvk(w)∈Rsx×sy, which arecalled theconvolutioneigenvaluesandeigenvectorsofw. Theoreticalanalysis in[5]hasbroughtout that, formeaningfulconvolutionkernelsh, theconvolution eigenvalues ofw ∗h are significantly smaller than those ofw; moreover, it is shown in [5] that particularly the convolution eigenvectors with smallest convolution eigenvalues ofw∗h are almost convolution-orthogonal toh, i.e.‖vk(w∗h)∗h‖ isalmostzero for thosek forwhichσk(w) is small enough. This motivates that for a given blurred imagef the underlying PSFh can be sought as a minimiserof ∑mxmy k=1 ‖vk(f)∗h‖2/σk(f)2whereplacing thesquaredconvolutioneigenvalues in the denominator ensures the higher influence of the convolution eigenvectors with smallest eigenvalues, andavoids introducinga threshold parameter to singleout the“small”convolutioneigenvalues. An additional degree of freedom in the procedure is that the imagef can be preprocessed by some linear filterL. Since convolution itself is a linear filter, and therefore commutes with any linear filter L, the above reasoning about singular values remains valid in this case; at the same time, a suitable choice ofL allows to reweight the influence of different parts off on the PSF estimation. Based on the well-known fact from literature, see e.g. [16], that edge regions are particularly well-suited to estimate blur, [5] suggest the use of a Laplacean-of-Gaussian (LoG) filter, thus leading to the final formulationof theobjective function Rh(h) := sxsy∑ k=1 ‖vk(L(f))∗h‖2 σk(L(f))2 (4) whereL is a LoG operator. Whereas the extended support size sx× sy forh is used inRh, its minimisation is constrained toPSFhof theactual support sizemx×my. UsingRh alone as objective function would already allow to estimate the PSF fairly accurate. How- ever, as discussed in [5] such a proceeding tends toward some over-sharpening of the image with visible artifacts. In order to achieve a good joint reconstruction of the sharp image and PSF that also takes into account regularity constraints on the image expressed byRu, and improves the treatment of imageswithmoderatenoise, [5] insertRh insteadasPSFregulariser into (3). 55