相关论文: Can you hear your location on a manifold?

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…

Knowing the geometry of a space is desirable for many applications, e.g. sound source localization, sound field reproduction or auralization. In circumstances where only acoustic signals can be obtained, estimating the geometry of a room is…

Inspired by the role geometric structures play in our understanding of surfaces and three-manifolds, and Berger's observation that a surface of constant sectional curvature is determined up to local isometry by its Laplace spectrum, we…

To unify general relativity and quantum theory is hard in part because they are formulated in two very different mathematical languages, differential geometry and functional analysis. A natural candidate for bridging this language gap, at…

Perceptual manifolds arise when a neural population responds to an ensemble of sensory signals associated with different physical features (e.g., orientation, pose, scale, location, and intensity) of the same perceptual object. Object…

We have recently seen great progress in learning interpretable music representations, ranging from basic factors, such as pitch and timbre, to high-level concepts, such as chord and texture. However, most methods rely heavily on music…

The images and sounds that we perceive undergo subtle but geometrically consistent changes as we rotate our heads. In this paper, we use these cues to solve a problem we call Sound Localization from Motion (SLfM): jointly estimating camera…

In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace-Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such…

``Can one hear the shape of a drum?'' was a question posed (and made famous) by mathematician Mark Kac in the mid-1960s. It addresses whether a deeper connection exists between the resonance modes (eigenmodes) of a drum and its shape. Here…

We study the problem of localizing a configuration of points and planes from the collection of point-to-plane distances. This problem models simultaneous localization and mapping from acoustic echoes as well as the notable "structure from…

In this talk, I will discuss the use of harmonic functions to study the geometry and topology of complete manifolds. In my previous joint work with Luen-fai Tam, we discovered that the number of infinities of a complete manifold can be…

Which properties of an orbifold can we ``hear,'' i.e., which topological and geometric properties of an orbifold are determined by its Laplace spectrum? We consider this question for a class of four-dimensional K\"{a}hler orbifolds:…

Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image…

Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator.…

We show that on an a-priori unknown Riemannian manifold $(M,g)$, measuring the source-to-solution map for the semilinear wave equation at a single point determines the topological, differential, and geometric structure.

We reexamine the proofs of isospectrality of the counterexample domains to Kac' question `Can one hear the shape of a drum?' from an analytical viewpoint. We reformulate isospectrality in a more abstract setting as the existence of a…

We consider the sound ranging, or source localization, problem - find the source-point from the moments when the wave-sphere of linearly, with time, increasing radius reaches the sensor-points - in the proper metric spaces (any closed ball…

Assume that a ground-based vehicle moves in a room with walls or other planar surfaces. Can the vehicle reconstruct the positions of the walls from the echoes of a single sound event? We assume that the vehicle carries some microphones and…

In this thesis I demonstrate that isospectral domains, that is domains of differing geometric shapes that possess identical spectra, do not remain isospectral when subject to uniform rotation. One thus *can* hear the shape of a rotating…

The spectral action in noncommutative geometry naturally implements an ultraviolet cut-off, by counting the eigenvalues of a (generalized) Dirac operator lower than an energy of unification. Inverting the well known question "how to hear…