1 Derivation of Point-to-Plane Minimization

Transcription

1 1 Dervaton of Pont-to-Plane Mnmzaton Consder the Chen-Medon (pont-to-plane) framework for ICP. Assume we have a collecton of ponts (p, q ) wth normals n. We want to determne the optmal rotaton and translaton to be appled to the frst collecton of ponts (.e., the p ) to brng them nto algnment wth the second (.e., the q ). Thus, we want to mnmze the algnment error [ ] E (Rp + t q ) n 2 (1) wth respect to the rotaton R and translaton t. The rotaton s a nonlnear functon, ncorporatng snes and cosnes of the rotaton angles. If, however, we assume that ncremental rotatons wll be small, t s possble to lnearze the rotatons, approxmatng cos θ by 1 and sn θ by θ. For example, n the case of rotaton n x, R x,α Thus, the full rotaton may be approxmated as R cos α sn α 0 sn α cos α α 0 α 1. 1 γ β γ 1 α β α 1 for rotatons α, β, and γ around the x, y, and z axes, respectvely. Substtutng Equaton (2) nto (1) we obtan E [ (p,x γp,y + βp,z + t x q,x )n,x + (γp,x + p,y αp,z + t y q,y )n,y + ] ( βp,x + αp,y + p,z + t z q,z )n 2,z,, (2) whch may be rewrtten as Defnng E [ (p q ) n + t n + α(p,y n,z p,z n,y ) + β(p,z n,x p,x n,z ) + γ(p,x n,y p,y n,x ) ]2. c p n 1

2 and r α β γ, the algnment error may be wrtten as E (p q ) n + t n + r c 2. We now mnmze E wth respect to α, β, γ, t x, t y, and t z by settng the partal dervatves to zero: α [ ] 2 c,x (p q ) n + t n + r c β [ ] 2 c,y (p q ) n + t n + r c γ t x t y t z 2 c,z (p q ) n + t n + r c 2 n,x (p q ) n + t n + r c 2 n,y (p q ) n + t n + r c 2 n,z (p q ) n + t n + r c These equatons may be collected and wrtten n matrx form: c,x c,x c,x c,y c,x c,z c,x n,x c,x n,y c,x n,z c,y c,x c,y c,y c,y c,z c,y n,x c,y n,y c,y n,z c,z c,x c,z c,y c,z c,z c,z n,x c,z n,y c,z n,z n,x c,x n,x c,y n,x c,z n,x n,x n,x n,y n,x n,z n,y c,x n,y c,y n,y c,z n,y n,x n,y n,y n,y n,z n,z c,x n,z c,y n,z c,z n,z n,x n,z n,y n,z n,z α β γ t x t y t z c,x (p q ) n c,y (p q ) n c,z (p q ) n n,x (p q ) n n,y (p q ) n n,z (p q ) n Ths s a lnear matrx equaton of the form Cx b, where C s the 6 6 covarance matrx accumulated from the c and n, x s a 6 1 vector of unknowns, and b s a 6 1 vector that also depends on the data ponts. The equaton may be solved usng standard methods (A s symmetrc, so Cholesky decomposton s the preferred algorthm), yeldng the optmal ncremental rotaton and translaton.. 2

3 2 Analyss of Stablty The above 6 6 covarance matrx also encodes the ncrease n the algnment error when the transformaton s moved away from ts optmum: de 1 2 ( dα dβ dγ dtx dt y dt z ) The larger ths ncrease the greater the stablty of ICP, snce the error landscape wll have a deep, well-defned mnmum. On the other hand, f there are ncremental transformatons that cause only a small ncrease n algnment error, ICP wll be relatvely unstable wth respect to these degrees of freedom. By expandng C n terms of ts egenvectors we may see drectly the effect of varous ncremental transformatons. If all egenvalues of C are large, any transformaton away from the mnmum wll result n a large ncrease n algnment error. If, on the other hand, one or more egenvalues are small, the correspondng egenvectors are transformatons that do not ncrease error much, and therefore represent drectons n transformaton space along whch the error landscape s shallow. 3 Applcatons of Egenvalue Analyss The most obvous applcaton of the above analyss s to evaluate the stablty of algnng two meshes together. Ths nvolves computng the matrx C, summed over the entre regon of overlap, and fndng ts egenvalues. Any small egenvalues ndcate a low-confdence algnment. Ths has mplcatons on whch parngs to use for global regstraton. A second potental applcaton nvolves lookng at small patches on a sngle mesh, computng the egenvalues of C over each patch, and thus determnng the potental stablty of ICP on each regon. Applcatons of ths mght be: Usng the local stablty to assgn weghts durng ICP. Ths would help to prevent nose n mostly-flat regons from swampng the sgnal avalable near good features. Usng local stablty to determne the best places to compute and store surface sgnatures for structural ndexng. Buldng a dctonary of local surface shapes and the number of small egenvalues each produces. For example: C dα dβ dγ dt x dt y dt z. 3

4 Planar patch 3 small egenvalues 1 rotaton and 2 translatons unstable Sphercal patch 3 small egenvalues 3 rotatons unstable Cylndrcal patch 2 small egenvalues 1 translaton and 1 rotaton unstable Patch w. sphercal bump 1 small egenvalue 1 rotaton unstable Patch w. groove 1 small egenvalue 1 translaton unstable Corner of a cube 0 small egenvalues No unstable components of algnment Ths table s probably not complete t would be nce to come up wth a systematc way of classfyng the possbltes. 4

5 Neghborhood sze: 1 Neghborhood sze: 3 Neghborhood sze: 6 Neghborhood sze: 10 Fgure 1: Bunny model color coded accordng to the magntude of the three smallest egenvalues on local neghborhoods. The four rows of the table correspond to evaluatng C over neghborhoods of radus 1, 3, 6, and 10 edges. The color codng s such that black regons correspond to 3 small egenvalues, blue to 2 small egenvalues, green to 1 small egenvalue, and red to no small egenvalues. 5