Page - 56 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

This joint energy functional is then minimised by an alternating minimisation. In the PSF estimation step,Rh(h) is representedasaquadraticform,Rh(h)= ∑ i,j,i′,j′Hi,j;i′,j′hi,jhi′,j′,with thecoefficient matrix (Hessian)H=(Hi,j;i′,j′)i,j,i′,j′ givenby H= sxsy∑ k=1 A vk(L(f)) mx,my TA vk(L(f)) mx,my σk(L(f))2 , (5) and this iscombinedwith thedata termfrom(3) toestablishaquadraticminimisationproblemforh. In our re-implementation of the PSF estimation from [5], this quadratic minimisation problem is solvedvia thecorresponding linearequationsystemandanLUdecomposition[9,pp.52p.], followed by a projection step that eliminates negative entries inh and normalisesh to unit total weight. As a refinement of the projection step, it turned out useful to cut off even small positive entries inhby a threshold, thus additionally enforces sparsity of the PSF. Experiments indicate that the threshold is best adaptedasamultipleof somequantile, e.g.0.1 times the95%-quantileof theentriesofh. The image estimation step that alternates with PSF estimation comes down to a TV-regularised non- blinddeconvolutionproblemforwhichseveral approachesexist. In [5] themethod from[4] isused. 3. Robust ImageandPSFEstimation As demonstrated in e.g. [1, 8, 14, 13], robust data terms allow to achieve favourable deconvolution results even with imprecise estimates of the PSF or slight deviations from the spatial invariant blur model. While the latter is generally relevant in deconvolution of real-world images, robustness to imprecise PSF estimates is particularly useful in blind deconvolution. This makes it attractive to incorporate robustdata terms into the frameworkof [5],which isour goal in this section. Dueto thealternatingminimisationstructureof themethod,weconsider the twostepsseparately. We start with the image estimation, which is tantamount to non-blind deconvolution. Thus, we simply have to replace theTVdeconvolutionmodelwithasuitable robustapproach. In thiswork,wechoose RRRL[13] for thispurpose,which is afixed point iteration associated to theenergy function E(u)= ∑ i,j Φ ( [u∗h]i,j−fi,j−fi,j ln [u∗h]i,j fi,j ) dx+αRu(u) . (6) We prefer this method for efficiency reasons; note that the non-blind deconvolution step is needed in each iteration of the alternating minimisation. RRRL is known to evolve fast toward a good solution during the first few iterations, see also [14], whereas methods based on approaches as in [1] tend to require more iterations. Following [13], the data term penaliser in the RRRL method is chosen as Φ(z)=2 √ z,whereas the image regulariserRu is chosen as totalvariationas inSection2. For the PSF estimation, we insert a penaliser functionΦas mentioned above into the discretised data term from (3), which is then combined with the unaltered regulariserRh from (4) to yield a (partial) discreteenergy function for theestimationofh: E(h)= ∑ i,j Φ ( (fi,j− [u∗h]i,j)2 ) dx+αRh(h) . (7) Unlike itscounterpart inSection2., thisenergyfunction isno longerquadratic. Equating thegradient (i.e. the derivatives w.r.t.hi,j) to zero now yields a system of nonlinear equations for the PSF entries. 56