Tag Archives: Metric Geometry

On the best lattice quantizers [CL]

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http://arxiv.org/abs/2202.09605


A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization noise: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any locally optimal lattice quantizer and (ii) for an optimal product lattice, if the component lattices are themselves locally optimal. We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the Zador upper bound.

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E. Agrell and B. Allen
Thu, 24 Feb 22
47/52

Comments: N/A

Super-resolving star clusters with sheaves [IMA]

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http://arxiv.org/abs/2106.08123


This article explains an optimization-based approach for counting and localizing stars within a small cluster, based on photon counts in a focal plane array. The array need not be arranged in any particular way, and relatively small numbers of photons are required in order to ensure convergence. The stars can be located close to one another, as the location and brightness errors were found to be low when the separation was larger than $0.2$ Rayleigh radii. To ensure generality of our approach, it was constructed as a special case of a general theory built upon topological signal processing using the mathematics of sheaves.

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M. Robinson and C. Capraro
Wed, 16 Jun 21
33/57

Comments: arXiv admin note: text overlap with arXiv:2106.04445

papaya2: 2D Irreducible Minkowski Tensor computation [CL]

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http://arxiv.org/abs/2010.15138


A common challenge in scientific and technical domains is the quantitative description of geometries and shapes, e.g. in the analysis of microscope imagery or astronomical observation data. Frequently, it is desirable to go beyond scalar shape metrics such as porosity and surface to volume ratios because the samples are anisotropic or because direction-dependent quantities such as conductances or elasticity are of interest. Minkowski Tensors are a systematic family of versatile and robust higher-order shape descriptors that allow for shape characterization of arbitrary order and promise a path to systematic structure-function relationships for direction-dependent properties. Papaya2 is a software to calculate 2D higher-order shape metrics with a library interface, support for Irreducible Minkowski Tensors and interpolated marching squares. Extensions to Matlab, JavaScript and Python are provided as well. While the tensor of inertia is computed by many tools, we are not aware of other open-source software which provides higher-rank shape characterization in 2D.

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F. Schaller, J. Wagner and S. Kapfer
Fri, 30 Oct 20
69/74

Comments: 5 pages, 3 figures, published in the Journal of Open Source Software, code available at this https URL