The conversion of traditional film into stereo 3D has become an important problem in the past decade. One of the main bottlenecks is a disocclusion step, which in commercial 3D conversion is usually done by teams of artists armed with a toolbox of inpainting algorithms. A current difficulty in this is that most available algorithms are either too slow for interactive use, or provide no intuitive means for users to tweak the output.In this paper we present a new fast inpainting algorithm based on transporting along automatically detected splines, which the user may edit. Our algorithm is implemented on the GPU and fills the inpainting domain in successive shells that adapt their shape on the fly. In order to allocate GPU resources as efficiently as possible, we propose a parallel algorithm to track the inpainting interface as it evolves, ensuring that no resources are wasted on pixels that are not currently being worked on. Theoretical analysis of the time and processor complexity of our algorithm without and with tracking (as well as numerous numerical experiments) demonstrate the merits of the latter.Our transport mechanism is similar to the one used in coherence transport [7,27], but improves upon it by correcting a "kinking" phenomenon whereby extrapolated isophotes may bend at the boundary of the inpainting domain. Theoretical results explaining this phenomena and its resolution are presented.Although our method ignores texture, in many cases this is not a problem due to the thin inpainting domains in 3D conversion. Experimental results show that our method can achieve a visual quality that is competitive with the state-of-the-art while maintaining interactive speeds and providing the user with an intuitive interface to tweak the results.

This paper aims to present a unified framework for deriving analytical formulas for smoothing factors in arbitrary dimensions, under certain simplifying assumptions. To derive these expressions we rely on complex analysis and geometric considerations, using the maximum modulus principle and Möbius transformations. We restrict our attention to pointwise and block lexicographic Gauss-Seidel smoothers on a d-dimensional uniform mesh, where the computational molecule of the associated discrete operator forms a (2d + 1)-point star. In the pointwise case, the effect of a relaxation parameter is analysed. Our results apply to any number of spatial dimensions and are applicable to high-dimensional versions of a few common model problems with constant coefficients, including the Poisson and anisotropic diffusion equations, as well as a special case of the convection-diffusion equation. We show that in most cases our formulas, exact under the simplifying assumptions of local Fourier analysis, form tight upper bounds for the asymptotic convergence of geometric multigrid in practice. We also show that there are asymmetric cases where lexicographic Gauss-Seidel smoothing outperforms red-black Gauss-Seidel smoothing; this occurs for certain model convection-diffusion equations with high mesh Reynolds numbers.

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In this paper we study a class of fast geometric image inpainting methods based on the idea of filling the inpainting domain in successive shells from its boundary inwards. Image pixels are filled by assigning them a color equal to a weighted average of their already filled neighbors. However, there is flexibility in terms of the order in which pixels are filled, the weights used for averaging, and the neighborhood that is averaged over. Varying these degrees of freedom leads to different algorithms, and indeed the literature contains several methods falling into this general class. All of them are very fast, but at the same time all of them leave undesirable artifacts such as "kinking" (bending) or blurring of extrapolated isophotes. Our objective in this paper is to build a theoretical model in order to understand why these artifacts occur and what, if anything, can be done about them. Our model is based on two distinct limits: a continuum limit in which the pixel width h → 0 and an asymptotic limit in which h > 0 but h 1. The former will allow us to explain "kinking" artifacts (and what to do about them) while the latter will allow us to understand blur. Both limits are derived based on a connection between the class of algorithms under consideration and stopped random walks. At the same time, we consider a semi-implicit extension in which pixels in a given shell are solved for simultaneously by solving a linear system. We prove (within the continuum limit) that this extension is able to completely eliminate kinking artifacts, which we also prove must always be present in the direct method. Finally, we show that although our results are derived in the context of inpainting, they are Communicated by Hans Munthe-kaas. CBS acknowledges support from the Leverhulme Trust project 'Breaking the non-in fact abstract results that apply more generally. As an example, we show how our theory can also be applied to a problem in numerical linear algebra. Mathematics Subject Classification

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