Laplace Operator and Heat Kernel for Shape Analysis

Laplace Operator and Heat Kernel for Shape Analysis Jian Sun Mathematical Sciences Center, Tsinghua University

R kf := 2 f x 2 1 Laplace Operator on R k, the standard Laplace operator: R kf := div f + + 2 f x 2 k

R kf := 2 f x 2 1 Laplace Operator on R k, the standard Laplace operator: R kf := div f + + 2 f x 2 k on a Riemannian manifold (M, g), Laplace-Beltrami operator: M f := div f M f := 1 det g i,j x j (g ij det g f x i )

Laplace Operator on a Riemannian manifold, Laplace-Beltrami operator: Exponential map: exp : V R k U by exp(x) = γ(p, x, 1) f(x) = f(exp(x)) M f := R k f = 2 f x 2 1 + + 2 f x 2 k Laplace-Beltrami operator is invariant under the map preserving geodesics V x x U R k exp

M φ = λφ Eigenvalues and eigenfunctions For compact manifold, M is compact 0 λ 0 λ 1 λ 2, is the only accumulating point

M φ = λφ Eigenvalues and eigenfunctions For compact manifold, M is compact 0 λ 0 λ 1 λ 2, is the only accumulating point Spectrum: eigenvalues λ n 4π 2 ( heat trace: i eλ it = n w d V ol(m) )2/d as n 1 (4πt) d/2 i c it. - c 0 = vol(m), c 1 = 1 3 s.

isospectrality Spectrum Can you hear the shape of a drum [Kac 1966] Does the spectrum determines the shape upto isometry

isospectrality Spectrum Can you hear the shape of a drum [Kac 1966] Does the spectrum determines the shape upto isometry negative [Gordon et al. 1992, Buser et al. 1992]

Spectrum Spectrum: shape DNA [Reuter et al. 2006]

Eigenfunctions M φ = λφ, (M, g) is C -manifold nodal set: φ 1 (0), nodal domain: the connected component of M \ φ 1 (0) Nodal domain theorem [Courant and Hilbert 1953, Cheng 1976]: # of nodal domains of the i-th eigenfunction i + 1 Properties of Nodal Set [Cheng 1976]: Except on a closed set of lower dimension(i.e., dim < d 1) the nodal set off forms an (d - 1)-dim C -manifold. + - - +

sphere (x 2 + y 2 + z 2 = 1) Examples λ 1 = 2(1.91) λ 2 = 2(1.92) λ 1 = 2(1.93)

star Examples

human Examples

Intrinsic Symmetry Detection

Intrinsic Symmetry intrinsic symmetry: a self map preserving geodesic distances

Intrinsic Symmetry intrinsic symmetry: a self map preserving geodesic distances invariant under non-rigid transformations

Intrinsic Symmetry intrinsic symmetry: a self map preserving geodesic distances invariant under non-rigid transformations extrinsic symmetry: rotation and reflection preserve Euclidean distances invariant only under rigid transformations

extrinsic symmetry Related Work [Mitra et al. 06] [Podolak et al. 06] [Shimari et al. 06] [Martinet et al. 07] intrinsic symmetry difficulty: no simple characterization

Global Point Signature our strategy: reduce intrinsic to extrinsic our tool: eigenfunctions φ i and eigenvalues λ i of M

Global Point Signature our strategy: reduce intrinsic to extrinsic our tool: eigenfunctions φ i and eigenvalues λ i of M for each point p on M, its GPS [Rustamov 07] s(p) = φ 1 (p) λ1 φ 2 (p) λ2. φ i (p) λi.

Global Point Signature our strategy: reduce intrinsic to extrinsic our tool: eigenfunctions φ i and eigenvalues λ i of M for each point p on M, its GPS [Rustamov 07] T s(p) = φ 1 (p) λ1 φ 2 (p) λ2. φ i (p) λi. relation between s(p) and s(t (p))? s(t (p)) = φ 1 (T (p)) λ1 φ 2 (T (p)) λ2. φ i (T (p)) λi.

Eigenfunctions and Intrinsic Symmetry 1. φ = φ T : positive eigenfunction 2. φ = φ T : negative eigenfunction 3. λ is a repeated eigenvalue φ(t (p)) = φ(p) q T (q) o φ p T (p)

Eigenfunctions and Intrinsic Symmetry 1. φ = φ T : positive eigenfunction 2. φ = φ T : negative eigenfunction 3. λ is a repeated eigenvalue φ(p) φ(t (p)) q T (q) o φ reflection p T (p)

Eigenfunctions and Intrinsic Symmetry T (p) 1. φ = φ T : positive eigenfunction 2. φ = φ T : negative eigenfunction 3. λ is a repeated eigenvalue φ 2 (φ 1 (p), φ 2 (p)) p T (p) φ(p) o φ 1 φ 1 p (φ 1 (T (p)), φ 2 (T (p))) rotation φ 2

Transforming Theorem Theorem: For a compact manifold M, T is an intrinsic symmetry if and only if there is a transformation R such that R(s(p)) = s(t (p)) for each point p M and R restricting to any eigenspace is a rigid transformation. one to one correspondence between T and R - T is an identity R is an identity detection of intrinsic symmetry reduced to that of extrinsic rigid transformation - detection of extrinsic symmetry [Rus07, PSG06, MGP06, MSHS06]

Results scans of a real person (SCAPE dataset)

Limitation of Global Point Signature For any x, its GPS [Rus07]: GPS x = ( φ 1 (x)/ λ 1, φ 2 (x)/ λ 2,, φ i (x)/ λ i, ) global not unique - orthonormal transformation within eigenspace - eigenfunction switching [GVL96] courtesy of Jain and Zhang

Heat Kernel Signature

Heat Kernel Signature define HKS for any point x as a function on R + : HKS x : R + R +

Heat Kernel Signature define HKS for any point x as a function on R + : HKS x : R + R + isometric invariant

HKS x : R + R + isometric invariant multi-scale Heat Kernel Signature define HKS for any point x as a function on R + :

HKS x : R + R + isometric invariant multi-scale informative Heat Kernel Signature define HKS for any point x as a function on R + : {HKS x } x M characterizes almost all shapes up to isometry.

Heat Diffusion Process how heat diffuses over time f H t (f) =? o t

Heat Diffusion Process how heat diffuses over time f H t (f) =? o x y t heat kernel k t (x, y) : R + M M R + heat transferred from y to x in time t H t f(x) = M k t(x, y)f(y)dy

Heat Diffusion Process how heat diffuses over time f H t (f) =? o x y t heat kernel k t (x, y) : R + M M R + heat transferred from y to x in time t H t f(x) = M k t(x, y)f(y)dy k t (x, y) = i e λ it φ i (x)φ i (y)

Heat Kernel characterize shape up to isometry T : M N is isometric iff k t (x, y) = k t (T (x), T (y)). heat kernel recovers geodesic distances. - d 2 M (x, y) = 4 lim t 0 t log k t (x, y) heat diffusion process governed by heat equation. - M u(t, x) = u(t,x) t generate a Brownian motion on a manifold.

multi-scale Heat Kernel for any x, a family of functions {k t (x, )} t x x x x x t = 0: δ funct t : const funct t local features in small t s; global summaries in big t s

multi-scale Heat Kernel for any x, a family of functions {k t (x, )} t x x x x x t = 0: δ funct t : const funct t local features in small t s; global summaries in big t s however, {k t (x, )} t s complexity is extremely high difficult to compare {k t (x, )} t with {k t (x, )} t

Heat Kernel Signature HKS is the restriction of {k t (x, )} t to the temporal domain HKS x : R + R + by HKS x (t) = k t (x, x)

Heat Kernel Signature HKS is the restriction of {k t (x, )} t to the temporal domain HKS x : R + R + by HKS x (t) = k t (x, x) concise and commensurable multi-scale isometric invariant

Heat Kernel Signature HKS is the restriction of {k t (x, )} t to the temporal domain HKS x : R + R + by HKS x (t) = k t (x, x) concise and commensurable multi-scale isometric invariant informative?

{HKS x } x M is informative Heat Kernel Signature Informative Theorem. If the eigenvalues of M and N are not repeated, a homeomorphism T : M N is isometric iff kt M (x, x) = kt N (T (x), T (x)) for any x M and any t > 0. almost all shapes have no repeated eigenvalues [BU82]

Relation to Curvature the polynomial expansion of HKS at small t: HKS x (t) = k t (x, x) = (4πt) d/2 (1 + 1 6 s(x)t + O(t2 )) plot of k t (x, x) for a fixed t

Relation to Diffusion Distance diffusion distance [Laf04] d 2 t (x, y) = k t (x, x) + k t (y, y) 2k t (x, y) eccentricity in terms of diffusion distance 1 ecc t (x) = d 2 t (x, y)dy A M M = k t (x, x) + H M (t) 2 A M, - ecc t (x) and k t (x, x) have the same level sets, in particular, extrema points - shape segmentation [dggv08]

Multi-Scale Matching scaled HKS: k t (x,x) M k t(x,x)dx

Multi-Scale Matching

Multi-Scale Matching maxima of k t (x, x) for a fixed t.

Thank you for your attention Questions?

Results

Laplace-Beltrami Operator: Computation based on its eigenfunctions and eigenvalues k t (x, y) = i e λ it φ i (x)φ i (y) HKS x (t) = k t (x, x) = i e λ it φ 2 i (x) discrete case: build the discrete Laplace operator L = A 1 W [BSW08] solve W φ = λaφ compute HKS x (t) = i e λ it φ 2 i (x)

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