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Multiscale Covariance Fields, Local Scales, and Shape Transforms

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Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Covariance Fields, Local Scales, and Shape Transforms Diego Hernan Diaz Martinez1 Facundo Memoli2 Washington Mio1 1 Department Of Mathematics - Florida State University, USA 2 Department Of Mathematics - Ohio State University, USA August 2013, Geometric Science of Information Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Outline Kernel Functions and Multiscale Covariance Tensor Fields. Data geometry and topology from covariance fields. Anisotropy Measures for Covariance Tensors. Local Scales: scales at which key features around a point are revealed. Shape Transforms: identification of geometrically salient points. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Outline Kernel Functions and Multiscale Covariance Tensor Fields. Data geometry and topology from covariance fields. Anisotropy Measures for Covariance Tensors. Local Scales: scales at which key features around a point are revealed. Shape Transforms: identification of geometrically salient points. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Outline Kernel Functions and Multiscale Covariance Tensor Fields. Data geometry and topology from covariance fields. Anisotropy Measures for Covariance Tensors. Local Scales: scales at which key features around a point are revealed. Shape Transforms: identification of geometrically salient points. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Outline Kernel Functions and Multiscale Covariance Tensor Fields. Data geometry and topology from covariance fields. Anisotropy Measures for Covariance Tensors. Local Scales: scales at which key features around a point are revealed. Shape Transforms: identification of geometrically salient points. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Outline Kernel Functions and Multiscale Covariance Tensor Fields. Data geometry and topology from covariance fields. Anisotropy Measures for Covariance Tensors. Local Scales: scales at which key features around a point are revealed. Shape Transforms: identification of geometrically salient points. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Analysis Shape data as a probability distribution. Kernel Function K(x, y, σ) ≥ 0 controls scale dependence through σ. Kernel delimits the horizon of an observer at x. Variation of data is measured relatively to every point, not just mean. Benefits: gain additional insight on local and regional data geometry. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Analysis Shape data as a probability distribution. Kernel Function K(x, y, σ) ≥ 0 controls scale dependence through σ. Kernel delimits the horizon of an observer at x. Variation of data is measured relatively to every point, not just mean. Benefits: gain additional insight on local and regional data geometry. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Analysis Shape data as a probability distribution. Kernel Function K(x, y, σ) ≥ 0 controls scale dependence through σ. Kernel delimits the horizon of an observer at x. Variation of data is measured relatively to every point, not just mean. Benefits: gain additional insight on local and regional data geometry. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Analysis Shape data as a probability distribution. Kernel Function K(x, y, σ) ≥ 0 controls scale dependence through σ. Kernel delimits the horizon of an observer at x. Variation of data is measured relatively to every point, not just mean. Benefits: gain additional insight on local and regional data geometry. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Kernel Function Two types of kernels will be discussed. Isotropic Gaussian kernel: Kgauss(x, y, σ) = 1 (2πσ2)d/2 exp − x − y 2 2σ2 (1) Truncation kernel Ktrunc(x, y, σ) = χσ(x − y) Vσ (2) Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Kernel Function Two types of kernels will be discussed. Isotropic Gaussian kernel: Kgauss(x, y, σ) = 1 (2πσ2)d/2 exp − x − y 2 2σ2 (1) Truncation kernel Ktrunc(x, y, σ) = χσ(x − y) Vσ (2) Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Multiscale Covariance Field Multiscale Covariance Σ(x, σ) = Rd (y − x) ⊗ (y − x)K(x, y, σ)dµ(y) (3) If y1, ..., yn ∈ Rd , the estimator of the multiscale covariance is Σn(x, σ) = 1 n n i=1 (yi − x) ⊗ (yi − x)K(x, yi , σ) (4) where x ⊗ y(v, w) = x, v · y, w is a bilinear form and ·, · is the Euclidean inner product. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Geometry Underlying Data Spectrum of covariance field gives information of the geometry at x with horizon defined by σ. Small scales give local information on the geometry near x. Analysis at different scales can help identify points of geometric interest. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Planar Curves in R2 Planar Curves in R2 Parametrize curve C using coordinate system defined by tangent and normal at for some x0 ∈ C using curvature κ. Covariance field for truncation kernel at x0 σ2 3 − 7κ2 σ4 180 + O(σ6 ) κsσ4 30 + O(σ6 ) κsσ4 30 + O(σ6 ) κ2 σ4 20 + O(σ6 ) Trace is σ2 3 + 2κ2 σ4 180 + O(σ6 ). We can recover the curvature at x0 for σ > 0 small. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Surfaces in R3 Surfaces in R3 Let x0 ∈ S a non-umbilic point. Coordinate system at x0 defined by 1st principal direction at x0 2nd principal direction at x0 Normal to S at x0 using principal curvatures κ1 and κ2. Covariance field for truncation kernel at x0    σ2 4 − (3κ1+κ2)2 384 σ4 + O(σ5 ) O(σ5 ) O(σ4 ) O(σ5 ) σ2 4 − (κ1+3κ2)2 384 σ4 + O(σ5 ) O(σ4 ) O(σ4 ) O(σ4 ) 3κ2 1+2κ1κ2+3κ2 2 96 σ4 + O(σ5 )    The diagonal is dominant for σ small. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Surfaces in R3 Invariant polynomials It follows that for σ > 0 small Trace σ2 2 − (κ1 − κ2)2 192 σ4 + O(σ5 ) Second Order σ4 16 + 14κ2 1 + 4κ1κ2 + 14κ2 2 1536 σ6 + O(σ7 ) Determinant 3κ2 1 + 2κ1κ2 + 3κ2 2 1536 σ8 + O(σ9 ) So, from the spectrum of ˆΣ(x0, σ) we are able to recover the principal curvatures of S at x0. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Anisotropy Measures Anisotropy Measures Find eigenvalues of covariance tensor. Order them λ1(x, σ) ≥ λ2(x, σ) ≥ ... ≥ λd (x, σ) ≥ 0. Define anisotropy measures h(x, σ) based on eigenvalues. Fix x, vary σ, define a function hx (σ), behavior at fixed point. Fix σ, vary x, define hσ(x), behavior at fixed scale. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Anisotropy Measures Anisotropy Measures Find eigenvalues of covariance tensor. Order them λ1(x, σ) ≥ λ2(x, σ) ≥ ... ≥ λd (x, σ) ≥ 0. Define anisotropy measures h(x, σ) based on eigenvalues. Fix x, vary σ, define a function hx (σ), behavior at fixed point. Fix σ, vary x, define hσ(x), behavior at fixed scale. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Anisotropy Measures Anisotropy Measures Find eigenvalues of covariance tensor. Order them λ1(x, σ) ≥ λ2(x, σ) ≥ ... ≥ λd (x, σ) ≥ 0. Define anisotropy measures h(x, σ) based on eigenvalues. Fix x, vary σ, define a function hx (σ), behavior at fixed point. Fix σ, vary x, define hσ(x), behavior at fixed scale. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Anisotropy Measures Anisotropy Measures Find eigenvalues of covariance tensor. Order them λ1(x, σ) ≥ λ2(x, σ) ≥ ... ≥ λd (x, σ) ≥ 0. Define anisotropy measures h(x, σ) based on eigenvalues. Fix x, vary σ, define a function hx (σ), behavior at fixed point. Fix σ, vary x, define hσ(x), behavior at fixed scale. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 2D Shapes 2D Shapes For 2D shapes, we define the anisotropy function h(x, σ) : R2 × (0, ∞) → R as h(x, σ) = 1 − λ2(x, σ) λ1(x, σ) Figure : h ≈ 1 and h ≈ 0 Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 2D Shapes 2D Shapes Data sampled uniformly from the circle Figure : Truncation kernel and Gaussian kernel Ellipses very close to data are highly anisotropic and its major axis is nearly tangent to the curve. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 2D Shapes 2D Shapes: Local Scales For fixed x, rapid growth or decay of hx reveals scales at which large changes to the field occur. Values of σ at which local maxima and minima of dhx dσ occur represent geometrically important scales at x. Figure : Local scales for a point on the Apple shape Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 2D Shapes 2D Shapes: Shape Transforms Figure : hσ for Truncation kernel and points for hx Figure : hx at different points Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 2D Shapes 2D Shapes: Shape Transform Define TVA : R2 → R as the Total Variation of hx over a fixed finite interval. TVA gives summary of complexity of geometry of data from the perspective of an observer positioned at x. Figure : TVA for Apple and Star shapes As expected, TVA is able to detect points with most salient local-global geometry. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 3D Shapes 3D Shapes For 3D shapes, we define the analogous anisotropy function h(x, σ) : R3 × (0, ∞) → R3 as h(x, σ) = λ3 − λ2 λ1 + λ2 + λ3 , 2(λ2 − λ1) λ1 + λ2 + λ3 , 3λ1 λ1 + λ2 + λ3 which can be mapped to points on an equilateral triangle, describing the shape of the ellipsoid. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 3D Shapes 3D Shapes: Examples (fixed scale, Gaussian kernel) Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 3D Shapes 3D Shapes: Local Scales In this case, hx can be thought of as a path on the equilateral triangle, as the scale varies. Local scales are found in an analogous way. Figure : Spheres represent 1, 1.5 and 2 times the local scale for Gaussian kernel Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary 3D Shapes 3D Shapes: Shape Transforms The analogous to the 2D TVA function is a function that calculates the length of the path on the triangle for a fixed finite interval. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary Summary Multiscale covariance fields modulated by a kernel function as a tool for analyzing data sets. Curvature can be recovered from small scale covariance fields. Anisotropy measures help to interpret covariance fields at different scales and points. Local scales represent important scales from the perspective of a fixed point. Shape Transforms help locate geometrically salient points by summarizing the behavior of points at different scales. Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms Introduction Multiscale Covariance Fields Geometry of Curves and Surfaces Local Scales and Shape Transforms Summary THANK YOU FOR YOUR ATTENTION! Diaz Martinez, Memoli, Mio FSU Math, OSU Math Multiscale Covariance Fields, Local Scales, and Shape Transforms