MADMM: A Generic Algorithm for Non-smooth Optimization on Manifolds

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.

Section: Related Workmentioning

confidence: 99%

Deblurring and Denoising of Maps between Shapes

Ezuz

Ben‐Chen

2017

“…In particular, it has a clear Fourier‐like meaning that makes its use well interpretable. Differently from [NVT*14, KGB16, BCKS16, Rus11] which impose locality through an L 1 constraint, we allow an explicit indication of the local support of each function. This improves the versatility in controlling the local analysis, especially for semantically guided interventions.…”

Section: Related Workmentioning

confidence: 99%

“…In many applications, one wishes to have a local basis that allows to limit the analysis to specific parts of the shape. The recently proposed compressed manifold harmonics [OLCO13, NVT*14, KGB16, BCKS16] attempt to construct local orthogonal bases that approximately diagonalize the Laplace–Beltrami operator. The main disadvantage of this framework is the inability to explicitly control the localization of the basis functions.…”

Section: Introductionmentioning

confidence: 99%

Localized Manifold Harmonics for Spectral Shape Analysis

Melzi

Rodolà

Castellani

et al. 2017

The use of Laplacian eigenfunctions is ubiquitous in a wide range of computer graphics and geometry processing applications. In particular, Laplacian eigenbases allow generalizing the classical Fourier analysis to manifolds. A key drawback of such bases is their inherently global nature, as the Laplacian eigenfunctions carry geometric and topological structure of the entire manifold. In this paper, we introduce a new framework for local spectral shape analysis. We show how to efficiently construct localized orthogonal bases by solving an optimization problem that in turn can be posed as the eigendecomposition of a new operator obtained by a modification of the standard Laplacian. We study the theoretical and computational aspects of the proposed framework and showcase our new construction on the classical problems of shape approximation and correspondence. We obtain significant improvement compared to classical Laplacian eigenbases as well as other alternatives for constructing localized bases.

“…The matrices P and Q are found to make sure that the Fourier coefficients of corresponding functions F and G in the respective bases

\overset{Φ}{true}

and

\overset{Ψ}{true}

are approximately equal, making sure at the same time that

\overset{Φ}{true}

and

\overset{Ψ}{true}

approximately diagonalize the respective Laplacians, by solving the optimization problem of the form

\begin{matrix} true \\ right min_{P, Q} & left ∥ G^{⊤} Ψ Q - F^{⊤} {Φ P ∥}_{F}^{2} + μ_{1} off (P^{⊤} Λ_{X} P) + μ_{2} off (Q^{⊤} Λ_{Y} Q) \\ left normals . normalt . P^{⊤} P = Q^{⊤} Q = I, \end{matrix}

where

off (A) = \sum_{i \neq j} a_{i j}^{2}

is a penalty on a non‐diagonal structure, and

Λ_{X}

Λ_{Y}

are diagonal matrices containing the first Laplacian eigenvalues. This optimization is carried out efficiently using manifold optimization techniques [KGB15]. If X and Y are nearly isometric, the correspondence matrix C represented in the joint basis is approximately diagonal, allowing to reduce the system of

q K

equations in K 2 variables to K variables considering only the diagonal elements of C .…”

Section: Evaluating Similarity Between Shapesmentioning

confidence: 99%

Recent Trends, Applications, and Perspectives in 3D Shape Similarity Assessment

Biasotti

Cerri

Bronstein

et al. 2015

123

The recent introduction of 3D shape analysis frameworks able to quantify the deformation of a shape into another in terms of the variation of real functions yields a new interpretation of the 3D shape similarity assessment and opens new perspectives. Indeed, while the classical approaches to similarity mainly quantify it as a numerical score, map‐based methods also define (dense) shape correspondences. After presenting in detail the theoretical foundations underlying these approaches, we classify them by looking at their most salient features, including the kind of structure and invariance properties they capture, as well as the distances and the output modalities according to which the similarity between shapes is assessed and returned. We also review the usage of these methods in a number of 3D shape application domains, ranging from matching and retrieval to annotation and segmentation. Finally, the most promising directions for future research developments are discussed.