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### Discrete Analytical Hyperplanes

Eric Andres, Raj Acharya, Claudio Sibata
1997 Graphical Models and Image Processing
We show that the discrete scanner for a 3D DR, a PET scanner for a 4D DR, etc. hyperplane is a generalization of the classical digital hyper-The second one is by digitizing the primitives forming planes  ...  There are two main ways of obtaining a DR of a real world object. One is by acquisition with a This paper presents the properties of the discrete analytical hyperplanes.  ...  . Ⅲ flatness is a digital segment of a digital hyperplane.  ...

### Hierarchical Classification using Binary Data [article]

Denali Molitor, Deanna Needell
2018 arXiv   pre-print
In classification problems, especially those that categorize data into a large number of classes, the classes often naturally follow a hierarchical structure.  ...  Here, we extend a recent simple classification approach on binary data in order to efficiently classify hierarchical data.  ...  of hyperplanes enables one to make more accurate predictions.  ...

### Page 2227 of Mathematical Reviews Vol. , Issue 92d [page]

1992 Mathematical Reviews
Using these schemes, algorithms are presented that determine whether or not a finite set S of n digital points in a k-dimensional space is a digital line segment, digital hyperplane, or a digital m-flat  ...  , hyperplanes, and flats in arbitrary dimensions.  ...

### Digital Geometry from a Geometric Algebra Perspective [chapter]

Lilian Aveneau, Laurent Fuchs, Eric Andres
2014 Lecture Notes in Computer Science
Focusing on the Conformal Geometric Algebra, the claim of the paper is that this framework is useful in digital geometry too.  ...  Moreover, the notion of duality is an inherent part of the Geometric Algebra. This is particularly useful since many algorithms are based on this notion in digital geometry.  ...  Thus, for a point x not on the hyperplane, we can determine the relative positions of x and π using the sign of x · π. Flats.  ...

### Page 1905 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews
In this paper we generalize hyperplane regression depth to k-flats for any k between 0 and d —1. The k = 0 case gives the classical notion of center points.  ...  Summary: “The regression depth of a hyperplane with respect to a set of » points in R? is the minimum number of points the hyperplane must pass through in a rotation to vertical.  ...

### Hierarchical Classification Using Binary Data

Denali Molitor, Deanna Needell
2019 The AI Magazine
Motivated by a recent simple classification approach on binary data, we propose a variant that is tailored to efficient classification of hierarchical data.  ...  In classification problems, especially those that categorize data into a large number of classes, the classes often naturally follow a hierarchical structure.  ...  Acknowledgments The authors were partially supported by NSF CAREER Grant 1348721 and NSF BIGDATA Grant 1740325. Note 1. yann.lecun.com/exdb/mnist  ...

### Output-sensitive algorithm for generating the flats of a matroid [article]

A. Montina
2011 arXiv   pre-print
We present an output-sensitive algorithm for generating the whole set of flats of a finite matroid.  ...  In the case of vectorial matroids, a specific algorithm is reported whose time complexity is equal to O(N^2 M d^2), d being the rank of the matroid.  ...  Research at Perimeter Institute for Theoretical Physics is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MRI.  ...

### Digital Analytical Geometry: How Do I Define a Digital Analytical Object? [chapter]

Eric Andres
2015 Lecture Notes in Computer Science
This paper is meant as a short survey on analytically dened digital geometric objects. We will start by giving some elements on digitizations and its relations to continuous geometry.  ...  We will end with open problems and challenges for the future.  ...  An interesting sequence of papers has focused on the connectivity of digital analytical hyperplanes [51, 45, 46] .  ...

### A Note on Stabbing Convex Bodies with Points, Lines, and Flats [article]

Sariel Har-Peled, Mitchell Jones
2022 arXiv   pre-print
Specifically, a (k,)-net is a set of k-flats, such that any convex body in [0,1]^d of volume larger than is stabbed by one of these k-flats.  ...  Consider the problem of constructing weak -nets where the stabbing elements are lines or k-flats instead of points.  ...  We also thank the anonymous reviewers for detailed comments that improved the paper.  ...

### Page 6215 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews
For n > 2, an n-dimensional linear space S is defined to be a simple matroid of rank n. The flats of rank 1, 2 and n in S are called points, lines and hyperplanes, respectively.  ...  The reference list in this paper contains about 50 articles on digital metrics.  ...

### The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations [article]

Zhengwei Liu, William Norledge, Adrian Ocneanu
2022 arXiv   pre-print
We study a certain discrete differentiation of piecewise-constant functions on the adjoint of the braid hyperplane arrangement, defined by taking finite-differences across hyperplanes.  ...  In terms of Aguiar-Mahajan's Lie theory of hyperplane arrangements, we show that this structure is equivalent to the action of Lie elements on faces.  ...  This made possible the visiting appointment of Adrian Ocneanu and the postdoctoral fellowship of William Norledge for the academic year 2017-2018.  ...

### Iteratively decodable codes on m flats for WDM high-speed long-haul transmission

S. Sankaranarayanan, I.B. Djordjevic, B. Vasic
2005 Journal of Lightwave Technology
The codes on m flats also provide a greater selection of structured LDPC codes of rate 0.8 or higher.  ...  As a natural extension of the prior work, in this paper, we consider LDPC codes on m flats derived from projective and affine geometries, which outperform codes from finite planes.  ...  Each one of the first v powers of α can be taken as the basis of one of the 0 flats in PG(n, q).  ...

### Hyperplane Arrangements and Locality-Sensitive Hashing with Lift [article]

Makiko Konoshima, Yui Noma
2012 arXiv   pre-print
This hashing can be seen as a discretization of the feature space by hyperplanes. If labels for data are given, one can determine the hyperplanes by using learning algorithms.  ...  There is a hashing scheme that maps feature vectors to bit arrays depending on the signs of the inner products between feature vectors and the normal vectors of hyperplanes placed in the feature space.  ...  This proposed method is a higher dimensional analog of the horizontal lift of curves on the base space with a flat connection on a trivial fiber bundle  .  ...

### A toolkit to describe and interactively display three-manifolds embedded in four-space [article]

Don V. Black
2012 arXiv   pre-print
By intersecting a 3-flat with this 3-manifold, the algorithm will extract the requested closed pure simplicial 2-complex surface enclosing the desired 3D slice.  ...  or asymmetries in the world of 3-manifolds in 4-space.  ...  Richard Palais for his three-torus definition; Dr James Arvo for his Toytracer Raytrace code; Digital ChoreoGraphics and the Edutech Foundation for their computer hardware and infrastructure.  ...

### Polynomial multiplicities over finite fields and intersection sets

A.A Bruen
1992 Journal of combinatorial theory. Series A
In the special case when the subspaces are in fact hyperplanes we offer two different proofs, one based on the method in Cl] (Sections 1,2) the other on the ideas in  (Sections 3,4].  ...  Our results in Section 1 connect the t-intersection question with a general lower bound, obtained by elementary methods, on the degree of a polynomial in several variables over GF(q) having certain multiplicity  ...  Some of the details in  were discussed in a series of lectures that I presented at the University of Bari in September 1984, following the "Riva de1 Sole" conference at Bari entitled "Combinatorics  ...
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