Page - 134 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

Superresolution Alignment with Innocence Assumption: Towards a Fair Quality Measurement for Blind Deconvolution Martin Welk1 Abstract—Quantitative measurements of restoration quality in blind deconvolution are complicated by the necessity to compensate for opposite shifts of reconstructed image and point-spread function. Alignment procedures mentioned for this purpose in the literature are sometimes not exactly enough specified; alignment-free approaches sometimes do not take into account the full variability of possible shifts. We investigate by experiments on a simple test case the errors induced by interpolation-based alignment procedures. We propose a new method for MSE/PSNR measurement of image pairs involving non-integer displacements that is based on a superresolution approach. We introduce an innocence assumption in order to keep deviations that can be explained by shifted sampling grids out of the error measurement. In our test case, the new measurement procedure reduces the variations in MSE/PSNR measurements substantially, creating the hope that it can be used for valid comparisons of blind deconvolution methods. I. INTRODUCTION The removal of blur in images by blind image deconvo- lution has been studied for many years [2], [3], [4], [5], [6], [10], [16], [21], and received increasing interest during the last years [1], [7], [8], [9], [11], [12], [14]. A frequently used simplifying assumption is that the blur is spatially invariant, i.e. the redistribution of intensity is described by the same point-spread function (PSF) h at each image location. Blur is then described by a convolution between the unobserved sharp image g and the PSF h; incorporating additive noise n, the observed image f is given by the blur model f =g∗h+n . (1) Whereas for non-blind deconvolution one assumes that f and h are known, and aims at an estimate u for the sharp image g, the knowledge of h is often not available in practice, thus necessitating blind deconvolution where the estimate u of the sharp image is to be obtained along with the PSF h, using only f as input image. A variety of approaches to solve this task have been de- veloped, creating the need for quality comparisons. Besides visual assessment, one is interested in quantitative measure- ments of reconstruction quality versus a known ground truth. Frequently used standard measures for image reconstruc- tion methods include the mean-square error (MSE) as well as the signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR) both of which are closely related to the MSE; furthermore, sometimes the average absolute error (AAE) is *This work was not supported by any organization 1Martin Welk is with Department of Biomedical Informatics and Mechatronics, Private University for Health Sciences, Medical Informatics and Technology (UMIT), 6060 Hall/Tyrol, Austria martin.welk@umit.at advocated. Another measure that puts some more emphasis on important structural details of images such as contrast edges is the structural similarity index (SSIM), see [17]. Let us shortly recall the first three measures. For a reference (ground-truth) image g and degraded (or reconstructed) image u, both of size n×m pixels, their MSE is given by MSE(u,g)= 1 nm n−1 ∑ i=0 m−1 ∑ j=0 (ui,j−gi,j)2 . (2) Provided that u and g have equal mean intensityµ (which we will assume in the following), this is the variance var(u−g) of u−g. Using the variance of g given by var(g)= 1 nm n−1 ∑ i=0 m−1 ∑ j=0 (gi,j−µ)2 , (3) and the range R(g) :=maxi,jgi,j−mini,jgi,j (255 for satu- rated 8-bit images), one can compute the SNR SNR(u,g)=10log var(g) var(u−g)dB (4) and PSNR PSNR(u,g)=10log R(g)2 var(u−g)dB . (5) For non-blind deconvolution, both MSE/(P)SNR and SSIM are frequently used to assess reconstruction quality. Although these quantitative measures are not always in good agreement with visual assessments by humans, they are generally accepted as simple and objective measures. For a recent study on measures that approximate better the human perception of restoration quality see [13]. In blind deconvolution, however, their application meets a difficulty: If the reconstructed image u is translated by an arbitrary, often non-integer, displacement d, and the point- spread function h is translated by −d, these translations cancel in the convolution u∗h. Blind deconvolution results that differ just by such opposite translations of u and h must therefore be considered equally valid reconstructions. An ex- ample of such shifts that indeed occur in blind deconvolution results is shown in Fig. 1. This precludes a straightforward (P)SNR or SSIM comparison of blind deconvolution results with ground truth. Obviously, some kind of alignment – rigid registration restricted to translations as transformations – has to be applied. Nevertheless, blind deconvolution results are compared by PSNR and other quantitative measures in a number of works, e.g. [6], [7], [8], [9], [10], [14]. In many of these, 134