Seite - 136 - in Proceedings of the OAGM&ARW Joint Workshop - Vision, Automation and Robotics

a b c d Fig. 2. (a) Ground truth image, 128×128 pixels. (Clipped, downscaled and converted to greyscale from a photograph of the building of TU Vienna. Source of original image: https://upload.wikimedia. org/wikipedia/commons/e/e9/TU Bibl 01 DSC1099w.jpg, Author: Peter Haas. Available under licence CC BY-SA 3.0.) – (b) 16 PSFs, 10×10 pixels each, subsampled from the same high-resolution input. The shift from row to row and from column to column is 0.25 pixels. – (c) Image (a) blurred by convolution with PSF from (b), first row, second column. – Bottom right: Image (c) deblurred with PSF from fourth row, third column, resulting in a shift relative to ground truth of (0.25,0.75) pixels. reconstructed image to match the ground-truth image; (b) warping the ground-truth image to match the reconstructed image; (c) applying half the shift to each of the ground-truth and reconstructed image. Statistics of the resulting PSNR values are presented in Table I. To bring the previous procedure closer to a true blind deconvolution setting, we now switch to determining also the displacement from a minimisation of the MSE (or max- imisation of the PSNR). To avoid analysing possible multiple optima, we employ here a brute-force optimisation varying the displacements in x and y direction in 0.01 steps from−1 to 1; note that the exact displacements occur in the sequence TABLE I PSNR STATISTICS FOR 256 RECONSTRUCTED IMAGES WITH ALIGNMENT BY THE KNOWN (GROUND-TRUTH) SHIFT USING BILINEAR OR BICUBIC INTERPOLATION; (A) WARPING THE RECONSTRUCTED IMAGE, (B) WARPING THE GROUND TRUTH, (C) HALF-WAY WARPING GROUND TRUTH AND RECONSTRUCTED IMAGE. Interpolation bilinear bicubic Alignment (a) (b) (c) (a) (b) (c) min 27.47 29.74 29.74 28.35 29.74 29.39 max 30.41 33.54 33.55 30.41 31.84 31.55 (max−min) 2.94 3.80 3.81 2.06 2.10 2.16 mean 28.57 32.18 31.25 29.23 30.85 30.05 standard dev. 0.711 0.970 0.805 0.474 0.489 0.459 of displacements sampled thereby. Table II contains statistics of the misestimations δx, δy of the x and y displacements, and the resulting PSNR. The latter values are slightly higher than in Table I but not seriously so. As can be expected, warping the reconstructed image to match the ground truth (see columns marked (a) in Tables I and II) leads to lower PSNR values for image pairs with non- integer displacements. The variation is about 3dB with bi- linear interpolation; bicubic interpolation reduces it to about 2dB which is still likely to warp comparisons substantially. When aligning instead the ground truth to the reconstructed images (columns (b) in Tables I and II) PSNR values are sur- prisingly higher for non-integer displacements, which means by comparison to the no-shift cases a clear overestimation of reconstruction quality. Apparently the warping of the ground truth image introduces some blur which matches well the remaining blur in the deconvolution results. Inspection of the detail results corroborates that for the same image pair the choice which image is aligned to which one leads to discrepancies in PSNR of 4dB and more with bilinear, and still about 3dB with bicubic interpolation. Distributing the shift to both images (columns (c) in Table I) yields similar results as shifting the ground truth. As this proceeding does not offer an advantage, we do not pursue it further in the computationally more expensive scenario of Table II where also the displacements are optimised. III. ALIGNMENT BY SUPERRESOLUTION We turn now to designing a procedure for image re- construction error measurement with alignment. We give preference to the MSE as basis of our considerations because unlike the (P)SNRit treats the two imagesbeingcompared in a completely symmetric way. We want to keep this symmetry also in the alignment procedure, thereby removing one of the arbitrarities of interpolation-based alignment procedures. For easier comparison to usual PSNR figures we will neverthe- less report in the experiments later PSNR values computed from our MSE measurements. An obvious requirement is that for perfectly aligned images the standard MSE measure has to be reproduced. Whereas the procedure will be described for prescribed TABLE II STATISTICS OF DISPLACEMENT MISESTIMATIONSδx,δy AND PSNR FOR 256 RECONSTRUCTED IMAGES WITH ALIGNMENT ESTIMATED BY MSE MINIMISATION. Interpolation bilinear bicubic Alignment (a) (b) (a) (b) max|δx| 0.18 0.17 0.07 0.16 std.dev. δx 0.079 0.081 0.028 0.080 max|δy| 0.17 0.15 0.06 0.14 std.dev. δy 0.064 0.067 0.024 0.072 min PSNR 27.47 31.42 28.37 29.89 max PSNR 30.41 33.61 30.41 31.86 (max−min) PSNR 2.94 2.19 2.04 1.97 mean PSNR 28.67 32.67 29.25 31.10 std.dev. PSNR 0.697 0.527 0.471 0.465 136