CS 468 Lecture 16: Isometry Invariance and Spectral Techniques

Transcription

1 CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that s nvarant to sometres deformatons of the object that nvolve bendng wthout stretchng, thereby leavng ntrnsc dstances undsturbed. Examples where such characterzatons are useful nclude segmentaton, symmetry detecton, recognton, retreval, feature extracton, and algnment. Ths lecture ntroduces the mathematcal defnton of an sometry and descrbes several shape descrptors that are used to characterze geometres n an sometry-nvarant manner. Isometry. Let (X, d 1 ) and (Y, d 2 ) be metrc spaces. A map f : X Y s a global sometry f d 1 (x, y) = d 2 (f(x), f(y)) for every x X and y Y. A related concept apples to the case n whch X and Y are Remannan manfolds wth metrcs g 1 and g 2, and f s a dffeomorphsm. The map f s sad to be a local sometry f g 1 (v, w) = g 2 (f v, f w) for every par of vector felds v and w on X. Here, f : T X T Y denotes the push-forward. Shape Descrptors. A shape descrptor s an assgnment of a real number of tuple of real numbers h(x) R n to each pont x on a surface S R 3, 1

2 desgned n such a way that the tuple stored at each locaton characterzes the local geometry of the surface and descrbes the pont s role on the surface. We have already seen examples of shape descrptors: the Gaussan curvature K(x) = κ 1 (x)κ 2 (x) and the mean curvature H(x) = κ 1 (x) + κ 2 (x) are two such examples. Several ams should be kept n mnd when desgnng a good shape descrptor. Clearly, h(x) should provde useful nformaton about the pont x. It should be robust aganst nose n the trangulaton and aganst small deformatons, and t should be ntrnsc that s, ndependent of the manner n whch S s embedded n R 3. Fnally, t should be nvarant under rgd motons and other sometres. The Hodge Laplacan = δd + dδ, beng an ntrnsc operator, provdes a useful startng pont for the desgn of many shape descrptors. To ntut ts ntrnsc nature, note, for example, that solutons to the heat equaton u t = u on a surface S are unaltered by sometres. Global Pont Sgnature. An example of a shape descrptor that reles on the Hodge Laplacan s the Global Pont Sgnature (GPS). Ths shape descrptor assgns to each pont x S the sequence of real numbers ( ) GP S(x) = λ 1/2 1 ϕ 1 (x), λ 1/2 2 ϕ 2 (x), λ 1/2 3 ϕ 3 (x),..., where λ are the egenvalues of and ϕ are the correspondng egenfunctons. Beng derved solely from the Laplacan, the GPS s nvarant under sometres of S. Let us also note that the GPS, vewed as a map from S to the space of sequences of real numbers, s njectve, provded the surface S does not selfntersect. Ths follows from the fact the the egenfunctons ϕ 1, ϕ 2,... form a bass for the space of smooth functons on S. Abstractly, one can thnk of the mage of ths map as a surface n nfnte-dmensonal Eucldean space; njectvty mples that ths surface does not self-ntersect whenever S does not self-ntersect. 2

3 The GPS suffers from a few drawbacks. It assumes that the egenvalues of are unque, and can gve rse to abrupt changes n GPS values when a small deformaton of the surface leads to a reorderng of egenvalues. Fnally, t s a nonlocal feature snce the egenfunctons of the Laplacan generally have global support. Heat Kernel Sgnature and Wave Kernel Sgnature. Two other popular shape descrptors that derve from the Laplacan are the Heat Kernel Sgnature (HKS) and Wave Kernel Sgnature (WKS). To defne the HKS, let k t (x, y) denote the fundamental soluton to the heat equaton u t = u on S. That s, k t (x, y) s the value of the soluton to the heat equaton at tme t and poston x S, assumng the ntal condton s gven by a delta functon centered at y S. In terms of the egenfunctons ϕ and egenvalues λ of, k t (x, y) = e λt ϕ (x)ϕ (y). Fxng a tme t, the HKS s then defned as HKS(x) = k t (x, x) = e λ t ϕ (x) 2 In words, the Heat Kernel Sgnature measures the amount of heat left at x after t unts of tme have transpred, assumng the ntal heat dstrbuton was concentrated at x. An example of the HKS at four ponts on a trangulated surface s shown n Fg. 1. For short tmes, the four ponts have nearly dentcal heat kernel sgnatures k t (x, x) snce the local geometry (the tps of the dragon s feet) s roughly the same. At a later tme, the heat kernel sgnatures k t (x, x) capture more global nformaton about the surface s shape and dverge. In ths sense, the HKS s a multscale shape descrptor. The Wave Kernel Sgnature (WKS) s a shape descrptor of a smlar nature, except that t s based upon solutons to the Schrodnger wave equaton u tt = u 3

4 Fgure 1: Heat kernel sgnature. Upon selectng a famly of ntal energy dstrbutons f E (λ), E = 1, 2,..., the WKS s defned as ( W KS(x) = ϕ (x) 2 f 1 (λ ) 2, ϕ (x) 2 f 2 (λ ) 2, ) ϕ (x) 2 f 3 (λ ) 2,... The entres of ths vector correspond to the long-tme averages of the squared soluton to the Schrodnger wave equaton at poston x, gven the ntal energy dstrbutons f E. The HKS and WKS have smlar advantages and dsadvantages. Both are sometry-nvarant, easy to compute, and do not suffer from the danger of egenvalue swtchng under small deformatons that we observed for the GPS. Repeated egenvalues are stll an ssue, however, and the WKS can sometmes flter out large-scale features that mght be worth retanng. Shape descrptors lke those dscussed above have applcatons n a varety of contexts, ncludng feature extracton, correspondence between surfaces, matchng surfaces, and detectng dscrete symmetres. Contnuous symmetres Much of the machnery developed above s useful for detectng dscrete symmetres, such as symmetres under reflecton about an axs. A related noton s that of a contnuous symmetry e.g., rotatons and translatons that leave the geometry nvarant. A Kllng vector feld s a vector feld V along whch the metrc s nvarant. Informally, dstances between nearby ponts do not change when transported along the flow of the vector feld V. 4

5 0 (s) t (s) +s +t Fgure 2: Famly of curves γ t (s). To determne the condtons under whch a vector feld V qualfes as a Kllng vector feld, let γ 0 be curve on S parametrzed by arclength s, and let γ t (s) denote the locaton of γ 0 (s) after beng transported along the flow of V by t unts of tme, as n Fg. 2. If dstances are preserved, then the parameter s represents arclength along the deformed curve γ t ; hence γ 0(s) = γ t(s) = 1 for every t. Dfferentatng wth respect to tme and usng the symmetry of mxed partals gves 0 = t γ t(s) 1 = γ γ t(s), t(s) t γ t(s) = T (s), t γ t(s) = T (s), s t γ t(s) Now snce V (s) = γ t t(s), we obtan 0 = T (s), s V (s) 5

6 Fgure 3: Approxmate Kllng vector felds. Equvalently, 0 = T, DT V. Fnally, snce DT V has the same component n the T drecton as T V (the covarant dervatve of V n the drecton T ) we conclude that 0 = T, T V. (1) For V to be a Kllng vector feld, ths relaton must hold for every vector feld T on S. Except on surfaces wth hgh degrees of symmetry, t s often not possble to fnd a vector feld V for whch (1) holds exactly for every T. Instead, one can fnd approxmate Kllng vector felds va a least squares approach. Denotng P V = T, T V, we seek a V whch mnmzes the Kllng energy P V 2. S Some examples of approxmate Kllng vector felds are shown n Fg. 3. 6