Fast Exact Area Image Upsampling with Natural Biquadratic Histosplines

Interpreting pixel values as averages over abutting squares mimics the image capture process. Average Matching (AM) exact area resampling involves the construction of a surface with averages given by the pixel values; the surface is then averaged over new pixel areas. AM resampling approximately preserves local averages (error bounds are given). Also, original images are recovered by box filtering when themagnification factor is an integer in both directions. Natural biquadratic histosplines, which satisfy aminimal norm property like bicubic splines, are used to construct the AM surface. Recurrence relations associated with tridiagonal systems allow the computation of tensor B-Spline coefficients at modest cost and their storage in reduced precision with little accuracy loss. Pixel values are then obtained by multiplication by narrow band matrices computed from B-Spline antiderivatives. Tests involving the re-enlargement of images downsampled with box filtering suggest that natural biquadratic histopolation is the best linear upsampling reconstructor. 1 From Point Values to Pixel Averages Image upsampling is most commonly implemented as a two step process [1]. First, interpolation is used to construct a continuous version of the image: a surface f (x y) such that f (x j yi) pi j (reconstruction). (1) Here, pi j is the pixel value with index (i j), and (x j yi) is the position of the corresponding pixel. The reconstructed surface is then resampled at the desired rate, that is, the pixel values PI J of the upsampled image are given by PI J f (XJ YI) (sampling), (2) where (XJ YI) is the position of the corresponding pixel in the enlarged image. 1.1 Average Matching (AM) Image Resampling Making the reconstructed light intensity surface have point values matching the pixel values as in Eq. (1) ignores the fact that image sensors count incoming A. Campilho and M. Kamel (Eds.): ICIAR 2008, LNCS 5112, pp. 85–96, 2008. c © Springer-Verlag Berlin Heidelberg 2008 86 N. Robidoux et al. photons over small non-overlapping areas, so that pixel values are better interpreted as averages than point values [2]. This is a gross simplification of the image capture process [3]. In addition, raw digital images are usually further processed prior to magnification. It is nonetheless reasonable to expect the average value interpretation to yield better resampling schemes than the point value interpretation [4]. We define average matching (AM) resampling to be exact area resampling in which the pixels of the input image are assumed to be abutting squares, and those of the output image, abutting rectangles [1]. In an AM method, the reconstructed intensity surface is defined by