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ADAPTIVE MULTISCALE DETECTION OF FILAMENTARY STRUCTURES IN A BACKGROUND OF UNIFORM RANDOM POINTS 1
, 2003
"... We are given a set of n points that might be uniformly distributed in the unit square [0,1] 2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C αnorm bounded by β. An ..."
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Cited by 14 (2 self)
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We are given a set of n points that might be uniformly distributed in the unit square [0,1] 2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C αnorm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α,β)> 0, if the number of points sampled from the curve is smaller than T−(α,β)n 1/(1+α) , reliable detection is not possible for large n. We describe a multiscale significantruns algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α,β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will
Multiscale Detection of Filamentary Features
 in Image Data. SPIE WaveletX
, 2003
"... Taking advantage of the new developments in mathematical statistics, a multiscale approach is designed to detect filament or filamentlike features in noisy images. The major contribution is to introduce a general framework in cases when the data is digital. Our detection method can detect the prese ..."
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Cited by 6 (4 self)
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Taking advantage of the new developments in mathematical statistics, a multiscale approach is designed to detect filament or filamentlike features in noisy images. The major contribution is to introduce a general framework in cases when the data is digital. Our detection method can detect the presence of an underlying curvilinear feature with the lowest possible strength that are still detectible in theory. Simulation results on synthetic data will be reported to illustrate its effectiveness in finite digital situations.
Interpolation of Random Hyperplanes
, 2006
"... Let {(Zi,Wi) : i = 1,...,n} be uniformly distributed in [0,1] d × G(k,d), where G(k,d) denotes the space of kdimensional linear subspaces of R d. For a differentiable function f: [0,1] k → [0,1] d, we say that f interpolates (z,w) ∈ [0,1] d × G(k,d) if there exists x ∈ [0,1] k such that f(x) = z ..."
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Cited by 2 (0 self)
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Let {(Zi,Wi) : i = 1,...,n} be uniformly distributed in [0,1] d × G(k,d), where G(k,d) denotes the space of kdimensional linear subspaces of R d. For a differentiable function f: [0,1] k → [0,1] d, we say that f interpolates (z,w) ∈ [0,1] d × G(k,d) if there exists x ∈ [0,1] k such that f(x) = z and ⃗ f(x) = w, where ⃗ f(x) denotes the tangent space at x defined by f. For a smoothness class F of Hölder type, we obtain probability bounds on the maximum number of points a function f ∈ F interpolates. 1
Adaptive Multiscale Detection of Smooth Curves Embedded in a Background of Uniform Random Points
, 2003
"... We are given a set of n points that appears uniformly distributed in the unit square [0, 1] 2. We wish to test whether the set actually is generated from a nonuniform distribution having a small fraction of points concentrated on some (a priori unknown) curve with C αnorm bounded by β. An asymptot ..."
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We are given a set of n points that appears uniformly distributed in the unit square [0, 1] 2. We wish to test whether the set actually is generated from a nonuniform distribution having a small fraction of points concentrated on some (a priori unknown) curve with C αnorm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α, β)> 0, if the number of points on the curve is smaller than T−(α, β)n 1/(1+α) , reliable detection is not possible for large n. We describe a Multiscale SignificantRuns Algorithm; it can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α, β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips have an area of C/n and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices are strips all of the same anisotropy and whose edges link such strips to their ‘good continuations’. The point cloud data are reduced to counts measuring membership in strips. Each anisotropy graph is reduced to a subgraph consisting of strips with ‘significant ’ counts. The algorithm rejects H0 whenever some such subgraph contains a path connecting many consecutive ‘significant ’ counts.
Adaptive Multiscale Detection of Filamentary Structures Embedded in a Background of Uniform Random Points
, 2003
"... We are given a set of n points that appears uniformly distributed in the unit square [0, 1] 2. We wish to test whether the set actually is generated from a nonuniform distribution having a small fraction of points concentrated on some (a priori unknown) curve with C αnorm bounded by β. An asymptot ..."
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We are given a set of n points that appears uniformly distributed in the unit square [0, 1] 2. We wish to test whether the set actually is generated from a nonuniform distribution having a small fraction of points concentrated on some (a priori unknown) curve with C αnorm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α, β)> 0, if the number of points on the curve is smaller than T−(α, β)n 1/(1+α) , reliable detection is not possible for large n. We describe a Multiscale SignificantRuns Algorithm; it can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α, β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips have an area of C/n and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices are strips all of the same anisotropy and whose edges link such strips to their ‘good continuations’. The point cloud data are reduced to counts measuring membership in strips. Each anisotropy graph is reduced to a subgraph consisting of strips with ‘significant ’ counts. The algorithm rejects H0 whenever some such subgraph contains a path connecting many consecutive ‘significant ’ counts.
Dynamic Programming Methods for“Connect the Dots” in Scattered Point Clouds
"... ConnectTheDots (CTD) is our name for a class of geometric optimization problems in which, given a point cloud, one finds the curve f from a given class F which passes through the maximum possible number of points in the cloud. Examples in 2D include monotone increasing graphs, curves of bounded cu ..."
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ConnectTheDots (CTD) is our name for a class of geometric optimization problems in which, given a point cloud, one finds the curve f from a given class F which passes through the maximum possible number of points in the cloud. Examples in 2D include monotone increasing graphs, curves of bounded curvature or length, and so on. In higher dimensions we may generalize graphs/curves to surfaces of fixed codimension, or bounded surface area. Other generalizations include replacing the “passing through points ” condition by “passing through points in specified directions. ” Potential applications arise in, for example, image processing (tracking amid clutter) and understanding human vision (perceptual psychophysics), and many more. This report describes a family of dynamic programming algorithms for various CTD problems. Some are their first appearance: e.g., the algorithms for the Hölder2 functions and the nondecreasing function in high dimension. A software library—CTDLab—that implements many of these algorithms is available on the Internet. Simulations that are based on these tools give insights and new conjectures regarding the asymptotic behavior of solutions when the point (or vector) clouds are i.i.d. uniform.
Detection
, 2004
"... I would like to express my sincere gratitude to my advisor, Dr. Xiaoming Huo for his great help and support during my graduate studies. His patience and insight make it a pleasure to work with him. Thanks are also due to other members of my committee, Professor JyeChyi Lu, Xiaoping Hu, KwokLeung T ..."
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I would like to express my sincere gratitude to my advisor, Dr. Xiaoming Huo for his great help and support during my graduate studies. His patience and insight make it a pleasure to work with him. Thanks are also due to other members of my committee, Professor JyeChyi Lu, Xiaoping Hu, KwokLeung Tsui, Brani Vidakovic and Guotong Zhou. On a more personal level, I would like to thank my close friends, Qian Li, Xiantian Dai, Ping Jing and Yun Xing. I am also grateful to my family for their love and support throughout. Finally, my husband, Wuqin Lin, has been a tremendous source of encouragement in the past three years. His genius and creativity has given me great help in my research.
Average
"... case analysis of disk scheduling, increasing subsequences and spacetime geometry ..."
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case analysis of disk scheduling, increasing subsequences and spacetime geometry
Contemporary Mathematics Discrete spacetime and its applications
, 2007
"... A discrete spacetime is a random partially ordered set. It is obtained by sampling n points from a compact domain in a spacetime manifold w.r.t. the volume form. The partial order is given by restricting the causal structure on the domain to the sampled points. These objects were introduced by physi ..."
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A discrete spacetime is a random partially ordered set. It is obtained by sampling n points from a compact domain in a spacetime manifold w.r.t. the volume form. The partial order is given by restricting the causal structure on the domain to the sampled points. These objects were introduced by physicists in an attempt to reconcile gravity with quantum mechanics, [40, 45, 19]. They received very little direct attention in the mathematics literature. Recently, discrete spacetimes have appeared in very diverse applications which are very far removed from their original motivation. In this selfcontained survey we will present some results and numerical calculations regarding the asymptotic behavior of these objects. Some of the more basic results are stated as theorems since they are particularly useful in the applications. Numerical calculations are presented with the hope that they will incite some theoretical work, aimed at proving some related assertions. We then present a few of the applications by shamelessly following the pattern of the recent survey paper [21], which describes some related material. We will present a short list of seemingly unrelated problems and show