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On a variational theory of image amodal completion. (English) Zbl 1150.49023
Digital images can be represented as gray level functions \(u (x, y)\) defined on a simple open subset of \(\mathbb R^ 2\). Digital images are given as a discrete set of samples, but there are standard interpolation methods to get back to a smooth image, e.g. a trigonometric polynomial by Shannon interpolation. There is no substantial difference between digital images and what we know of retinal images as rough data: in both cases, images are band-limited by an optical device and then sampled on a grid. So most questions in visual perception theory are easily translated into computer vision problems. This opens the way to a mathematical formalization and numerical experiments. In this paper, the authors study a variational model for image amodal completion, i.e., the recovery of missing or damaged portions of a digital image by technics inspired by the well–known amodal completion process in human vision. Representing the image by a real-valued function, the authors find a set of interpolating level lines, the approach that is optimal with respect to an appropriate criterion. It is proven that this method is theoretically well–founded and equivalent to a more classical method based on a direct interpolation of the function.

MSC:
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
49J45 Methods involving semicontinuity and convergence; relaxation
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
92C99 Physiological, cellular and medical topics
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