Page - 57 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

a b c d Figure 1. Blinddeconvolutionof a synthetically blurred image. (a) Input image,289×289pixels, blurredwith thePSF,13×13pixels, shownas insert. From[5], adapted. – (b)Reconstructed imageandPSF(inserted)by the method from [5],mx=my=13,β=5050,K=200. – (c) Sameas (b) butwithRRRLused in the image estimationstep,α=0.0018,Ku=30. – (d)Sameas(b)butwithRRRLfor imageestimation,andthenonlinear PSF estimationmethod fromSection 3.,α=0.0018,β=5050,Ku=30,Kh=20. – For τ, the quantile criterion(seeSection2.)wasused in (c,d)andyieldedvalues in therange0.11. ..0.12. (b)–(d) from[7],adapted. By a standard procedure of lagged weights (analogous to the lagged diffusivity method or Kacˇanov method) we transform the nonlinear equation system into a sequence of linear ones. Note that the nonlinearities result from the terms ∂hp,qΦ ( (fi,j− [u∗h]i,j)2 ) =−2Φ′((fi,j− [u∗h]i,j)2)(fi,j− [u ∗h]i,j)ui−p,j−q. Starting with some initial approximationh0 forh, we proceed therefore for l= 0,1,2, . . . as follows: Compute the weightsϕli,j := Φ ′((fi,j− [u∗hl]i,j)2) and replaceΦ′(·) in the equation system with the fixedϕi,j. This gives a linear equation system forh. Applying LU decomposition as in Section 2. one computes the solutionhl+1 of this system, which is the starting point for the next iteration. A more spelled-out derivation of the sequence of linear equation systems is found in [7]. Experimental evidence in [7], see also Section 4., confirms the quick convergence of the sequence (hl); inpractical cases, often10 . . .20 iterationsare sufficient. Toend thedescriptionofour robustblinddeconvolutionmethod,wesummarise itsparameterswhich will also be referred to in Section 4. The original method from [5] and the version with RRRL and linear PSF estimation use obvious subsets of these parameters. We start by the model parameters. First, thereare thePSFsizesmx,my thatneed tobechosensomewhat larger than theactualPSF.For the sample sizessx, sywe adopt the heuristic choicesx,y≈ 1.5mx,y from [5]. Regarding the image regularisation weightα in (6), a continuation strategy that starts with a largerα in the first iterations of the alternating minimisation and reducesα during the alternating minimisation process helps to speed up convergence; the final values ofα lie in the rangeα≈ 0.001 . . .0.002 proposed in [13]. The PSF regularisation weightβ is set manually; if it is too small, the blur will be underestimated (with a point kernel as extreme); too largeβ leads to oversharpening, compare [5]. Finally, there is the threshold τ for the PSF entries. The essential numerical parameters are three iteration counts:K for the alternating minimisation,Ku for RRRL, andKh for the iterated linearisation of the nonlinear equationsystemin thePSF estimation. 4. Experiments Asaproofofconcept,wepresent twoexperimentshere; furtherexperimentscanbefoundin[7]. Our first experiment is based on a synthetically blurred image, Fig. 1(a), that was already used in [5] to exemplify themethodreviewedinSection2.Theresultof thismethodisshowninFigure1(b). Frame (c) has been obtained by replacing the image estimation component with RRRL, whereas in frame (d) also the robust PSF estimation from Section 3. has been employed. Comparing (b) and (c), it is 57