24,556 research outputs found
Functional Maps Representation on Product Manifolds
We consider the tasks of representing, analyzing and manipulating maps
between shapes. We model maps as densities over the product manifold of the
input shapes; these densities can be treated as scalar functions and therefore
are manipulable using the language of signal processing on manifolds. Being a
manifold itself, the product space endows the set of maps with a geometry of
its own, which we exploit to define map operations in the spectral domain; we
also derive relationships with other existing representations (soft maps and
functional maps). To apply these ideas in practice, we discretize product
manifolds and their Laplace--Beltrami operators, and we introduce localized
spectral analysis of the product manifold as a novel tool for map processing.
Our framework applies to maps defined between and across 2D and 3D shapes
without requiring special adjustment, and it can be implemented efficiently
with simple operations on sparse matrices.Comment: Accepted to Computer Graphics Foru
Functional maps representation on product manifolds
We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices
Algebraic Rieffel Induction, Formal Morita Equivalence, and Applications to Deformation Quantization
In this paper we consider algebras with involution over a ring C which is
given by the quadratic extension by i of an ordered ring R. We discuss the
*-representation theory of such *-algebras on pre-Hilbert spaces over C and
develop the notions of Rieffel induction and formal Morita equivalence for this
category analogously to the situation for C^*-algebras. Throughout this paper
the notion of positive functionals and positive algebra elements will be
crucial for all constructions. As in the case of C^*-algebras, we show that the
GNS construction of *-representations can be understood as Rieffel induction
and, moreover, that formal Morita equivalence of two *-algebras, which is
defined by the existence of a bimodule with certain additional structures,
implies the equivalence of the categories of strongly non-degenerate
*-representations of the two *-algebras. We discuss various examples like
finite rank operators on pre-Hilbert spaces and matrix algebras over
*-algebras. Formal Morita equivalence is shown to imply Morita equivalence in
the ring-theoretic framework. Finally we apply our considerations to
deformation theory and in particular to deformation quantization and discuss
the classical limit and the deformation of equivalence bimodules.Comment: LaTeX2e, 51pages, minor typos corrected and Note/references adde
Deep Functional Maps: Structured Prediction for Dense Shape Correspondence
We introduce a new framework for learning dense correspondence between
deformable 3D shapes. Existing learning based approaches model shape
correspondence as a labelling problem, where each point of a query shape
receives a label identifying a point on some reference domain; the
correspondence is then constructed a posteriori by composing the label
predictions of two input shapes. We propose a paradigm shift and design a
structured prediction model in the space of functional maps, linear operators
that provide a compact representation of the correspondence. We model the
learning process via a deep residual network which takes dense descriptor
fields defined on two shapes as input, and outputs a soft map between the two
given objects. The resulting correspondence is shown to be accurate on several
challenging benchmarks comprising multiple categories, synthetic models, real
scans with acquisition artifacts, topological noise, and partiality.Comment: Accepted for publication at ICCV 201
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