Differential Proximity Condition for Manifold-Valued Data Subdivision

Transcription

1 Differential Proximity Condition for Manifold-Valued Data Subdivision Tom Duchamp 1 Gang Xie 2 Thomas Yu 3 1 University of Washington, Seattle 2 East China University of Science and Technology, Shanghai 3 Drexel University, Philadelphia Papers available at math.drexel.edu/ tyu

2 amples of multiscale refinement of non re 1 and also [7]), (ii) aircraft (pitch,roll,yaw) (see Figure 2), (iii) diffusion tensor imaging (see Figure 3), design on surfaces (e.g. [83]), (v) array signal processing, (vi) numerical solution of y Manifold-Valued Data linear range data. (t) =F (y(t)), t 0, 0) M and F is a vector field on M; M is a Lie group in many applications, see [66, 58, 57] and the there-in. Figure 3: Diffusion tensor imaging (DTI): instead of traditional pixel values, a diffusion tensor image has a diffusion tensor (a 3 3 symmetric positive definite matrix) associated with each pixel. DTI is a breakthrough in magnetic resonance imaging (MRI) which also gives rise to nonlinear range data. Figure 1: Motion capture (or MoCap) gives rise to time series taking values on a product Lie group, this is an example of nonlinear range data coming from an important real-life application. Mo-Cap: [0, T ] SO(3) SO(3) SO(3) SO(2) SO(2) these applications, we are faced with the problem of finding efficient way to approximate functions with a dean domain (that usually We need models technology, time }{{ and/or } technology space) } but needs a{{ co-domain us? As } being technology } a nonlinear advances, {{ manifold. such} manifold-valued As data w motion capture gives acquired rise to time at higher series and neck ofhigher the form resolutions, shoulders and the multiscale methodselbows we explore in this proposal will become and more practically relevant. This is saying that the mathematics developed here can only be useful if techno :[0,T] SO(3) SO(3) SO(3) SO(3) SO(3) SO(2) SO(2) SO(3) SO(3) SO(2) SO(3) SO(2) }{{}}{{ such as {{ motion }} capturing }{{ and {{ MRI } diffusion }}{{ tensor } imaging }{{ really do} advance. However, things may even g opposite way: the mathematics developed in this proposal may one day lead to faster and finer acquisiti neck wrists shoulders elbows wrists manifold-valued signals. hips knees (1) SO(3) SO(2) SO(2) SO(3) SO(3) SO(3). SO(3) {{ SO(3) } We } now elaborate {{ SO(3) on } this } convoluted SO(3) {{ claim. } SO(3) } The multiscale SO(3) {{ methods developed } in this proposal will le } hips {{ novel methods } knees for} efficiently representing ankles complex {{ geometric data. spine For traditional } signals, such kind of parsimo ankles representation is the basic building blockspine of a compressed sensing [31, 13] acquisition system. In a nutshel pplication areas are growing revelationby of leaps compressed and bounds, sensing iswe that believe once we that know many thatmore the signals such applications of interest canare be efficiently going represented in the future. For instance, certain mathematical we are even basis seeing (wavelets, orbifold-valued curvelets, data for example) arising then fromathe subtle musical but efficient field [89, procedure 12]. of random DTI: [0, L] 3 SPD(3) ion also the recent series projections, of papers combined [3, 2, with 4] afrom nonlinear Farhart s reconstruction aeronautics algorithm, grouphas at the Stanford, magical which power of show directly sensing no but the sparse and useful components of the signal. It means that it may no longer be necessary to use lots an iques for Grassmannian-valued data interpolation are used in building reduced-order models for real-time of costly sensors to acquire a signal at a high resolution, and subsequently realize in a transform coder that d aeroelastic Tom Duchamp, computations. Gang Xie, Thomas Yu Differential Proximity Conditions August 29, 2013, Calgary 2 / 1

3 Subdivision of SO(3) Data M = SO(3) Tom Duchamp, Gang Xie, Thomas Yu Differential Proximity Conditions August 29, 2013, Calgary 3 / 1

4 Linear Subdivision of Real Data (S lin x) 2h+σ = l a 2l+σ x h l, σ = 0, 1, h Z e.g. = { 1/8xh 1 + 3/4x h + 1/8x h+1, σ = 0 1/2x h + 1/2x h+1, σ = 1

5 Subdivision of SO(3) Data Tom Duchamp, Gang Xie, Thomas Yu Differential Proximity Conditions August 29, 2013, Calgary 5 / 1

6 Subdivision Schemes for Manifold-Valued Data based on a linear scheme S lin with mask (a 2l+σ ) Donoho s single base-point ( scheme ) (Sx) 2h+σ = Exp xh l a 2l+σ Log xh (x h l ), σ = 0, 1, h Z Related to a non-redundant wavelet-like transform for manifold-valued data due to Donoho et al Two base-point scheme (with b h±1/2 judiciously chosen from data (x h )) ( ) (Sx) 2h = Exp bh 1/2 a 2l+σ Log bh 1/2 (x h l ) Many others... (Sx) 2h+1 = Exp bh+1/2 ( l l ) a 2l+σ Log bh+1/2 (x h l )

7 Log and Exp maps T p (M) 0 0 v 0 θ(t 1 ) θ(t) Exp p (θ(t 1 )) 0 0 Log p (p(t 1 )) p 0 =p(0) p(t 1 ) N p 0 M γ(t)=exp p (θ(t)) 0 Figure 3.5: A manifold, its tangent plane, and the correspondence between a line in the tangent plane and a geodesic in the manifold. Tom Duchamp, Gang Xie, Thomas Yu Differential Proximity Conditions August 29, 2013, Calgary 7 / 1

8 Subdivision Curve on Poincaré Disk 6 Escher Tom Duchamp, Gang Xie, Thomas Yu J. Wallner and H. Pottmann Fig. 5. Hyperbolic swordfish (Geodesic analogue of B-Spline subdivision B3 with C 2 limit curves). One hyperbolic control polygon is also shown. J. Wallner The Poincare disk model of hyperbolic geometry (here thought see Fig. 5) is a fascinating geometry which appears in Computer visualization of surface-like data which will not fit well into the Differential Proximity Conditions[Alekseevskij et al. August 2013, Calgary /1 computable 1993]29, they appear as circles8which

9 Smooth Compatibility Condition Nonlinear subdivision scheme: (Sx) 2i+σ = q σ (x i m+1,, x i m+l ), σ = 0, 1, i Z. q 0, q 1 : M M M nonlinear averaging rules q σ (x,..., x) = x Def.: S is smoothly compatible with S lin if dq σ (x,...,x) (X 1,..., X L ) = q lin,σ (X 1,..., X L ), σ = 0, 1. Smooth compatibility condition = Sx = S lin x + O( x 2 ) Wallner-Dyn 2005 C 1 -equivalence condition

10 Proximity Condition Smoothness Equivalence Definition S and S lin satisfy the (strong) order k proximity condition if j 1 (Sx S lin x) Ω j (x), j = 1,..., k, where Ω j (x) := j (γ 1,,γ j ) i=1 i x γ i, j i=1 i γ i = j + 1. Order k Condition: Order k 1 proximity conditions + 1 Sx S lin x x Sx 1 S lin x x 3 + x 2 x 3 2 Sx 2 S lin x x 4 + x 3 x + 2 x 2 Theorem (Xie and Y.) Strong Order k proximity condition? = = C k equivalence

11 Confusing state of affairs of the necessity question Years of usage of the (strong) proximity condition never failed to predict the exact order of smoothness equivalence. This strongly suggests necessity. Embarrassment: The Xie-Y. theorem only needs the following weak proximity condition: j 1 /// (Sx S lin x) Ω j (x), j = 1,..., k, And we didn t notice that for years! = =!! Aside: Is there a easy way to check either proximity condition? What does either proximity condition really mean afterall?

12 Overhaul: Differential proximity condition Restrict S to a minimal invariant neighborhood, call its size K + 1, and introduce coordinates, Q : R n R n dq (x,...,x) = Q lin, x Define, Σ = 1 : R n R n by (x 0, x 1,..., x K ) Σ (δ 0 = x 0, δ 1,..., δ K ), δ k := k-th order difference of x 0, x 1,..., x k Ψ := Q Σ : R } n {{ R n } (K + 1 copies) Ψ = [Ψ 0, Ψ 1,..., Ψ k,..., Ψ K ]

13 Q := S restricted to a minimal invariant neighborhood Assume support of the scheme same as degree k + 1 B-Spline (C k ) k = 1 = K k = 2 = K [ ] q q1 (x Q(x 0, x 1 ) = 0, x 1 ) 1 (x 0, x 1 ) Q(x q 0 (x 0, x 1 ) 0, x 1, x 2 ) = q 0 (x 0, x 1, x 2 ) q 1 (x 1, x 2 ) k = 3 = K q 0 (x 0, x 1, x 2 ) Q(x 0, x 1, x 2, x 3 ) = q 1 (x 0, x 1, x 2 ) q 0 (x 1, x 2, x 3 ) q 1 (x 1, x 2, x 3 )

14 Differential proximity condition cont d Ψ := Q Σ : R } n {{ R n } (K + 1 copies) Ψ l (x 0, δ 1,..., δ K ) = Ψ lin,l (x 0, δ 1,..., δ K ) + 1 ν! Dν Ψ l (x0,0,...,0)δ ν 1 1 δν K K ν 2 Definition Differential order k proximity condition if x 0, D ν Ψ l (x0,0,...,0) = 0, when weight(ν) := jν j l, 1 l k. j 1

15 Dynamical System Interpretation j 1 δ /2 δ /4 δ /8 δ 3 = 2 j 4 j 8 j /16 δ 4 16 j }{{} Ψ lin,from S lin Same asymptotic behavior when iterating: δ 0 δ 0 δ 1 Ψ δ 2 δ 3 = Ψ δ 1 δ1 2 + δ 1δ 2 + lin δ 2 δ 3 + δ δ 1δ 2 + δ 1 δ 3 + δ δ 1 δ 2 + δ 1 δ 3 + δ1 3 + δ 4 δ 4 δ δ1 3 + δ 1δ 2 + δ δ 3 + δ2 2 + δ 1δ 4 + differential proximity condition resonant and stronger-than-resonant terms disappear

16 Main Result Theorem (Duchamp, Xie, Y.) Assume smooth compatibility condition between S and S lin, Order k differential proximity condition C k equivalence Three technical difficulties/innovations: Dynamical system interpretation suggests a slowdown of asymptotic decay in the presence of resonance term, proving such a lower bound is difficult Lack of stability condition in nonlinear subdivision, need to venture into super-convergence properties. Use of minimal invariant neighborhood does not sound sufficient

17 Caution Necessity part of our main result: intuitively obvious, but seems technically difficult, for two reasons Sufficiency part of our main result: seems too good to be true! Tom Duchamp, Gang Xie, Thomas Yu Differential Proximity Conditions August 29, 2013, Calgary 17 / 1

18 Difficulty I: lower bound E.g. With non-vanishing curvature, resonance term shows up when k = 4 or 5 for Donoho s scheme Appearance of resonant term δ ν δν k 1 k 1, ν 1 + 2ν 2 + = k, in Ψ k = k S j x j2 kj k 1 S j x (2 j ) k 1 j2 j ( expect (k 1)-th derivative not even Lipschitz) Difficulty of proving this lower bound: hard to guarantee that resonance would not dissipate Requires subtle choice of initial data

19 Difficulty II: non-linear stability? With the slowed decay rate proved, we expect S is at best C k 1,1 ɛ smooth, hence not C k. Can we really conclude this? Answer: Not yet Recall from linear theory: k S j lin c 2 j(k+α) = S lin is C k 1,α k S j lin c 2 j(k+α) = S lin is C k 1,α + S lin is stable Underlying reason: S j x (S x) 2 j Z Replacement for the linear stability condition? Obvious: rate of convergence is relevant Not so obvious: need super-convergence property to break a logical deadlock

20 Difficulty III: apparent insufficiency of min. invar. nhbd. E.g. Decay of 2 Q j (x 0, x 1, x 2 ) should not capture 2 S j x l! In fact, need decay of 3 S j x, not 2, to prove C 2! The order k differential proximity condition appears to say nothing about decay of (k + 1)-th order differences. Remedy: use a bigger invariant neighborhood, impose a similar differential proximity condition on Q big (x 0, x 1, x 2, x 3 ) = q 1 (x 0, x 1 ) q 0 (x 0, x 1, x 2 ) q 0 (x 0, x 1, x 2 ) q 1 (x 1, x 2 ), or q 1 (x 1, x 2 ) q 0 (x 1, x 2, x 3 ), Ψ big = Q big Σ. q 0 (x 1, x 2, x 3 ) q 1 (x 1, x 2 ) Recall: Joint Spectral Radius of a matrix pair from linear theory

21 Difficulty III: apparent insufficiency of minimal invariant neighborhood Surprise: diff. prox. cond. on Ψ Big diff. prox. cond. on Ψ A conspiracy between subdivision structure and compatibility condition This unexpected algebraic structure also explains the strong-weak proximity condition embarrassment.

22 Conclusion New differential proximity condition: easier to interpret easier to check necessary and sufficient for C k -equivalence must also be coordinate independent sufficiency seems too good to be true, but is true dissolves a confusing + embarrassing state of affairs Icing on the cake: An intrinsic, coordinate free definition? Applications to many schemes in the literature