On the Hessian of Shape Matching Energy

Transcription

1 On the Hessan of Shape Matchng Energy Yun Fe 1 Introducton In ths techncal report we derve the analytc form of the Hessan matrx for shape matchng energy. Shape matchng (Fg. 1) s a useful technque for meshless deformaton, whch can be easly combned wth multple technques n real-tme dynamcs (refer to [MHTG05, BMM15] for more detals). Nevertheless, t has been rarely appled n scenaros where mplct (such as backward dfferentaton formulas) ntegrators are requred, and hence strong vscous dampng effect, though popular n smulaton systems nowadays, s forbdden for shape matchng. The reason les n the dffculty to derve the Hessan matrx of the shape matchng energy. Computng the Hessan matrx correctly, and stably, s the key to more broadly applcaton of shape matchng n mplctly-ntegrated systems. 2 Shape Matchng Energy for Pont Cloud Gven the world-space postons of the r-th vertex on the obect as q r, and ts local coordnate of the rest pose as q 0 r, we defne the shape matchng potental as V = 1 2 r k r d T r d r (1) and where d r = q r B(q r )q 0 r t (2) t = 1 M r m r q r (3) s the center of mass and B(q r ) = γa a (q r )A 1 s + (1 γ)[r, 0](q r ) (4) s the blendng between the covarance matrx A and the best-ft rotaton matrx R. A a s the asymmetrc part where A a = 1 M r m r (q r t)(q 0 r ) T (5) 1

2 Fgure 1: Deformaton done wth mplct shape matchng. and A s = 1 M r m r q 0 r (q 0 r ) T (6) Usual methods extractng the rotaton nclude usng the sngular value decomposton (SVD), polar decomposton or QR decomposton to factorze the covarance matrx. The choce between these methods have been extensvely dscussed n prevous lterature, where the polar decomposton has been proven to be numercally stable aganst small perturbaton [SD92], whch fts our need to compute the Jacoban and Hessan of the rotaton matrx. Gven the asymmetrc part of a covarance matrx, A a, ts polar decomposton can be wrtten as A a (q) = R(q)S(q) (7) where R s the rotatonal part. Takng the dervatve of the potental energy over q, we get the potental gradent to be appled on vertex, whch has the followng form: V = k r ( d r ) T d r (8) r where and Bq 0 r d r = I Bq0 r Bq0 r m M I = r m M I = r (9) = γ A a A 1 s q 0 r + (1 γ)[ R, 0]q 0 r (10) 2

3 whle A a = m M (q0 m r r M q0 r ) T I (11) where s the tensor product. One navely usng symbolc dfferentaton to solve for Jacoban of the rotaton RT q 0 r as well as the Hessan wll sooner or later meet lots of numercal sngulartes, even for some very smple cases. In lteratures such as [Mat97, PL00, TKA10] the Jacoban of SVD s derved but not for ts Hessan; besdes several degenerated cases need to be specfcally handled. In [BZ11] the authors derved the Hessan of polar decomposton for a sngle temporal dervatve. Here we generalze ther results to partal dervatves, over postons, whch are also specfcally smplfed to avod tensor algebra. Multplyng both sdes of Equ. 7 wth R T (p) fxed at some pont p, we have R T (p)a a(q) = (R T (p)r(q))s(q) (12) where A a(q) s the frst two columns (n 2D, or three columns n 3D) of A a (the zero-order term). Snce R T (p)r(q) s dentty for p = q, ts dervatve at p = q must be some skew-symmetrc matrx ω for some vector ω. We have R(q) = R(q) ω (13) where we defne ωx = ω x and skew(a) as the unque vector ω such that ω = (A A T )/2. Then by dfferentatng both sde of Equ. 12, we have R T (p) A a(q) = ωs(p) + S(q) (14) Now replacng p wth q, and applyng the skew operator to both sdes, we drop the symmetrc term S(q), and we have where we defne and equaton 1 2 (tr(s)i S)RT ω = skew(r T A a ) (15) G = (tr(s)i S)R T, (16) Gω = 2skew(R T A a ) (17) can be solved for ω. We also defne e as the vector where the -th element s 1 and all zeros for other elements, we have A a = m M e (q 0 )T (18) 3

4 n 2D) of therefore Gω = 2m M skew(rt e (q 0 )T ) (19) Solvng for ω, then we can compute and correspondngly for 3D case R = ω R (20) Rq 0 r = [ ω 0 Rq 0 r, ω 1 Rq 0 r, ω 2 Rq 0 r ] (21) Ths process s smple. Note for 2D case, ths s even smpler snce there s no need for the equaton solve n Equ. 19 snce G 1 = tr(s) 1 I. To compute the Hessan of the shape matchng potental energy, we take dervatves over Equ. 8, where 2 l V = k r ( (1 γ)( 2 Rq 0 r ) T d r + ( d r ) T d r ) (22) r l l To compute the frst term of l V, we calculate the dervatve of S as S ls = R T ( m l M e s(q 0 l )T R ls S) (23) We take dervatves on both sdes of Equ. 19 and after rearrangng (also usng the property of antsymmetrc matrx where ω T = ω), we have Gω ls, = 2m M skew(rt ω ls e (q 0 )T ) (tr( S )I S )R T ω ls + (tr(s)i S)R T ω ls ω ls (24) For 2D case the last term of the rght hand sde can be dropped snce we have ω ls ω = 0 for any l, s,,, where we have Gω ls, = 2m M skew(rt ω ls e (q 0 )T ) (tr( S ls )I S ls )R T ω (25) After solvng for ω ls,, we compute 2 Rq 0 r ls as and for 3D, 2 Rq 0 r ls = ( ω ls, + ω ω ls )Rq 0 r (26) ( 2 Rq 0 r ) T d r = l q 0 T r R T ( ω l0 ω 0 ω l0,0 )d r q 0 T r R T ( ω l1 ω 0 ω l1,0 )d r q 0 T r R T ( ω l2 ω 0 ω l2,0 )d r q 0 T r R T ( ω l0 ω 1 ω l0,1 )d r q 0 T r R T ( ω l1 ω 1 ω l1,1 )d r q 0 T r R T ( ω l2 ω 1 ω l2,1 )d r q 0 T r R T ( ω l0 ω 2 ω l0,2 )d r q 0 T r R T ( ω l1 ω 2 ω l1,2 )d r q 0 T r R T ( ω l2 ω 2 ω l2,2 )d r (27) 4

5 3 Shape Matchng wth Vscous Dampng Now we have the equatons for undamped moton. Next we derve the force and Hessan of the vscous dampng occurred n shape matchng. We defne = α 2 r k r ( V da = α 2 r k r ḋ T r ḋr d r q ) T d r ( q ) as the stffness dampng energy, where q s the velocty of the -th partcle, and (28) V db = β 2 r m r q T r q r (29) as the mass dampng energy. Followng the Raylegh dampng model we have the total dampng energy as Hence the potental gradent of the -th partcle s Correspondngly, the postonal Hessan s V d = V da + V db. (30) V d = βm q q + α k r ( d r ) T ḋ r (31) r 2 V d = α(1 γ) l q k r (( 2 Rq 0 r ) T ḋ r + ( d r ) T r l ( 2 Rq 0 r ) T q l ) (32) whle the velocty Hessan s 4 Total Energy 2 V d = βm q l q I + α k r ( d r ) T d r (33) r l By combnng Equ. 8 and Equ. 31 we have the Jacoban of shape matchng energy as V total = βm q q + k r ( d r ) T (d r + αḋ r ) (34) r Smlarly, by combnng Equ. 22 and Equ. 32 we have the postonal Hessan of shape matchng energy as 2 V total l q = k r ( (1 γ)( 2 Rq 0 r ) T (d r + αḋ r ) + ( d r ) T ( d r (1 γ)α r l l ( 2 Rq 0 r ) T q l )) (35) 5

6 References [BMM15] [BZ11] [Mat97] Jan Bender, Matthas Müller, and Mles Mackln. Poston-based smulaton methods n computer graphcs. EUROGRAPHICS 2015 Tutoral Notes, Jerne Barbč and Yl Zhao. Real-tme large-deformaton substructurng. In ACM transactons on graphcs (TOG), volume 30, page 91. ACM, Arakaparampl M Matha. Jacobans of matrx transformatons and functons of matrx argument. World Scentfc, [MHTG05] Matthas Müller, Bruno Hedelberger, Matthas Teschner, and Markus Gross. Meshless deformatons based on shape matchng. ACM Trans. Graph., 24(3): , July [PL00] [SD92] [TKA10] Théodore Papadopoulo and Manols IA Louraks. Estmatng the acoban of the sngular value decomposton: Theory and applcatons. In Computer Vson-ECCV 2000, pages Sprnger, Ken Shoemake and Tom Duff. Matrx anmaton and polar decomposton. In Proceedngs of the Conference on Graphcs Interface 92, pages , San Francsco, CA, USA, Morgan Kaufmann Publshers Inc. Chrstopher D Twgg and Zoran Kačć-Alesć. Pont cloud glue: constranng smulatons usng the procrustes transform. In Proceedngs of the 2010 ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton, pages Eurographcs Assocaton,