Page - 54 - in Proceedings - OAGM & ARW Joint Workshop 2016 on "Computer Vision and Robotics“

and second, performing non-blind deconvolution with that PSF, see e.g. [3, 16]. Merging both steps, uandhcanbe estimatedsimultaneouslybyminimising a joint energy functional suchas E[u,h] := ∫ Ω (f−u∗h)2dx+αRu[u]+βRh[h] , (2) see e.g. [2, 11, 18], which combines the data term that integrates the squared model error (f−u∗ h)2 over the image domainΩwith regularisation functionalsRu andRh for the image and PSF, respectively, using regularisation weightsα,β. Note that there is a formal symmetry betweenu and h in the data term, coming from the blur model (1); however, this symmetry usually does not extend toRu andRh – regularisers that work well for images do generally not perform favourably in PSF estimation, and vice versa. This is because the regularisers express model requirements for sharp images and for PSFs, respectively, and these model requirements differ substantially. For example, sharp edges are important foruwhich makes total variation regularisers a good candidate, whereas forh rather locality andsparse support maybemeaningful requirements. Motivated by the separation of regularisers in (2), one often separatesu andh again in the minimi- sation, by using iterative methods that alternatingly updateu andh. Each cycle then comprises an image estimation step, which is a non-blind deconvolution, and a PSF estimation step. Whereas the latter can formally be considered as non-blind deconvolution of the blurred image with respect to the sharpened image as convolution kernel, the dissimilarity of regularisers in fact often implies that substantiallydifferent algorithmshave tobeused for imageand PSFestimation. A refinement of (2) results from applying to the squared model error (f−u∗h)2 a functionΦwith less-than-linear growth, yielding a sub-quadratic data term ∫ Ω Φ((f−u∗h)2)dx. Data terms of this kind have been proven useful in various image processing tasks in order to reduce sensitivity to (particularly,heavy-tailed)noiseandmeasurementerrorsaswellas tominordeviations fromthedata model, and are therefore known as robust data terms, see e.g. [1, 19] in the deconvolution context. A similar modification of the objective function underlying the Richardson-Lucy deconvolution (the information divergence) has been introduced in [13], leading to a non-blind deconvolution method called robust and regularised Richardson-Lucydeconvolution (RRRL). Ourcontribution. In this paper, important parts of which are based on the thesis [7], we review a recent blind deconvolution approach from [5] that is based on alternating minimisation of an energy in the sense of (2) with a PSF regulariser constructed from so-called convolution eigenvalues and eigenvectors. We then modify both the PSF and image estimation components of this approach by using robust data terms. To this end, we adopt in the image estimationcomponent the RRRL method from [13]; regarding the PSF estimation component, we introduce a subquadratic data term. The so modified PSF estimation component has to the best of our knowledge not been studied before. We presentexperimentsonaproof-of-concept level that support theconclusion thatourmodifiedmethod achieveshigher reconstruction quality than itspredecessor. 2. PSFEstimationUsingSpectraofConvolutionOperators In this section, we review the approach from [5] which forms the basis for our further work in this paper. Astheconstructionof theregulariserRh from[5]reliesonspectraldecompositionsformulated inmatrix language,weswitchour notations tousediscrete images from hereon. 54