Computing Minimum Complexity 1D Curve Simplifications under the Fréchet Distance

We consider the problem of simplifying curves under the Fréchet distance. Let P be a curve and ε ≥ 0 be a distance threshold. An ε-simplification is a curve within Fréchet distance ε of P . We consider ε-simplifications of minimum complexity (i.e. minimum number of vertices). Parameterized by ε, we define a continuous family of minimum complexity ε-simplifications P ε of a curve P inone dimension. We present a data structure that after linear preprocessing time can report the ε-simplification in linear output-sensitive time. Moreover, for k ≥ 1, we show how this data structure can be used to report a simplification P ε with at most k vertices that is closest to P in O(k) time

A quantitative version of Krein's theorems for Fréchet spaces

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being &#963;(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) &#8722; 2)k(H) + 1/2^n for all n &#8712; N. Consequently, this yields also the following formula k(coH) &#8804; (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions.The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.Angosto Hernández, C.; Kakol, J.; Kubzdela, A.; López Pellicer, M. (2013). A quantitative version of Krein's theorems for Fréchet spaces. Archiv der Mathematik. 101(1):65-77. https://doi.org/10.1007/s00013-013-0513-4S65771011Angosto C., Cascales B.: Measures of weak noncompactness in Banach spaces. Topology Appl. 156, 1412–1421 (2009)C. Angosto, Distance to spaces of functions, PhD thesis, Universidad de Murcia (2007).C. Angosto and B. Cascales, A new look at compactness via distances to functions spaces, World Sc. Pub. Co. (2008).Angosto C., Cascales B.: The quantitative difference between countable compactness and compactness. J. Math. Anal. Appl. 343, 479–491 (2008)Angosto C., Cascales B., Namioka I.: Distances to spaces of Baire one functions. Math. Z. 263, 103–124 (2009)C. Angosto, J. Ka̧kol, and M. López-Pellicer, A quantitative approach to weak compactness in Fréchet spaces and spaces C(X), J. Math. Anal. Appl. 403 (2013), 13–22.Cascales B., Marciszesky W., Raja M.: Distance to spaces of continuous functions. Topology Appl. 153, 2303–2319 (2006)M. Fabian et al. Functional Analysis and Infinite-dimensional geometry, CMS Books in Mathematics, Canadian Math. Soc., Springer (2001).M. Fabian et al. A quantitative version of Krein’s theorem, Rev. Mat. Iberoam. 21 (2005), 237–248Granero A. S.: An extension of the Krein-Smulian Theorem. Rev. Mat. Iberoam. 22, 93–110 (2006)Granero A. S., Hájek P., Montesinos V.: Santalucía, Convexity and ω*-compactness in Banach spaces. Math. Ann. 328, 625–631 (2004)Grothendieck A.: Criteres de compacité dans les spaces fonctionnelles généraux. Amer. J. Math. 74, 168–186 (1952)Khurana S. S.: Weakly compactly generated Fréchet spaces. Int. J. Math. Math. Sci. 2, 721–724 (1979

Efficient Fréchet distance queries for segments

We study the problem of constructing a data structure that can store a two-dimensional polygonal curve P, such that for any query segment ab one can efficiently compute the Fréchet distance between P and ab. First we present a data structure of size O(n log n) that can compute the Fréchet distance between P and a horizontal query segment ab in O(log n) time, where n is the number of vertices of P. In comparison to prior work, this significantly reduces the required space. We extend the type of queries allowed, as we allow a query to be a horizontal segment ab together with two points s, t ∈ P (not necessarily vertices), and ask for the Fréchet distance between ab and the curve of P in between s and t. Using O(n log2 n) storage, such queries take O(log3 n) time, simplifying and significantly improving previous results. We then generalize our results to query segments of arbitrary orientation. We present an O(nk3+ϵ + n2) size data structure, where k ∈ [1, n] is a parameter the user can choose, and ϵ > 0 is an arbitrarily small constant, such that given any segment ab and two points s, t ∈ P we can compute the Fréchet distance between ab and the curve of P in between s and t in O((n/k) log2 n + log4 n) time. This is the first result that allows efficient exact Fréchet distance queries for arbitrarily oriented segments. We also present two applications of our data structure. First, we show that our data structure allows us to compute a local δ-simplification (with respect to the Fréchet distance) of a polygonal curve in O(n5/2+ϵ) time, improving a previous O(n3) time algorithm. Second, we show that we can efficiently find a translation of an arbitrary query segment ab that minimizes the Fréchet distance with respect to a subcurve of P

Systematic approach to nonlinear filtering associated with aggregation operators. Part 2. Frechet MIMO-filters

Median filtering has been widely used in scalar-valued image processing as an edge preserving operation. The basic idea is that the pixel value is replaced by the median of the pixels contained in a window around it. In this work, this idea is extended onto vector-valued images. It is based on the fact that the median is also the value that minimizes the sum of distances between all grey-level pixels in the window. The Frechet median of a discrete set of vector-valued pixels in a metric space with a metric is the point minimizing the sum of metric distances to the all sample pixels. In this paper, we extend the notion of the Frechet median to the general Frechet median, which minimizes the Frechet cost function (FCF) in the form of aggregation function of metric distances, instead of the ordinary sum. Moreover, we propose use an aggregation distance instead of classical metric distance. We use generalized Frechet median for constructing new nonlinear Frechet MIMO-filters for multispectral image processing. (C) 2017 The Authors. Published by Elsevier Ltd.This work was supported by grants the RFBR No 17-07-00886, No 17-29-03369 and by Ural State Forest University Engineering's Center of Excellence in "Quantum and Classical Information Technologies for Remote Sensing Systems"

Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fr\'echet distance between surfaces. While the Fr\'echet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality