Deblurring and Denoising of Maps between Shapes

Abstract

Shape correspondence is an important and challenging problem in geometry processing. Generalized map representations, such as functional maps, have been recently suggested as an approach for handling difficult mapping problems, such as partial matching and matching shapes with high genus, within a generic framework. While this idea was shown to be useful in various scenarios, such maps only provide low frequency information on the correspondence. In many applications, such as texture transfer and shape interpolation, a high quality pointwise map that can transport high frequency data between the shapes is required. We name this problem map deblurring and propose a robust method, based on a smoothness assumption, for its solution. Our approach is suitable for non‐isometric shapes, is robust to mesh tessellation and accurately recovers vertex‐to‐point, or precise, maps. Using the same framework we can also handle map denoising, namely improvement of given pointwise maps from various sources. We demonstrate that our approach outperforms the state‐of‐the‐art for both deblurring and denoising of maps on benchmarks of non‐isometric shapes, and show an application to high quality intrinsic symmetry computation.

Elastic Correspondence between Triangle Meshes

“…We compare our results with the initial map computed using HOT and with the shape correspondence method that was recently suggested by Manded et al [MCSK∗17] (VMTP). Additionally, we compare with two other methods that improve input correspondences, where we use the same initialization as our method (HOT): one method is based on functional maps [EBC17] (DDM) and another optimizes the reversible harmonic energy [ESBC19] (RHM). The quantitative comparison is shown in Fig.…”

Unsupervised cycle‐consistent deformation for shape matching

“…Shape matching is a long‐standing problem in shape analysis [vKZHCO11]. It is often done explicitly, by deforming a source shape to a target [RL01, BR07, LSP08, HAWG08, ZSCO*08], or implicitly, by mapping points [KLF11, CK15, OMMG10, BBK06] or functions [OBCS*12, RPWO18, EBC17] on one shape to another. The deformation‐based methods typically aim to minimize the amount of distortion introduced by the deformation, and the mapping‐based approaches often assume that shapes to be near‐isometric.…”

Structured Regularization of Functional Map Computations

“…Functional Maps. Our approach fits within the functional map framework, which was originally introduced in [OBCS*12] for solving non‐rigid shape matching problems, and extended significantly in follow‐up works, including [KBB*13, ADK16, KBBV15, RCB*17, EBC17, BDK17] among many others (see also [OCB*17] for an overview). The key observation in these techniques is that it is often easier to estimate correspondences between real‐valued functions, rather than points on the shapes.…”

“…As mentioned above, the most common approach consists of penalizing the failure of the unknown functional map to commute with the Laplace‐Beltrami operators, which can be written as: where Δ 1 and Δ 2 are the Laplace‐Beltrami operators of the two shapes expressed in the respective bases. Here, and throughout the rest of our paper, unless stated otherwise || · || denotes the matrix Frobenius norm. Convert the functional map C to a point‐to‐point map, for example using an iterative approach, such as the Iterative Closest Point (ICP) in the spectral embedding, or using other more advanced techniques [RMC15, EBC17]. …”