《图像处理与计算机视觉 Image Processing and Computer Vision》课程教学资源（PPT课件讲稿）Chapter 11 Bundle adjustment Structure reconstruction SFM from N-frames

Exercise11. 2: proves these partial derivatives: (Note: f= focal length) R 偿 R R R V Pose estimation vo.a

Exercise11.2: proves these partial derivatives: (Note: f= focal length) Pose estimation V0.a • 11 ( ) ( ) ( ) ( ) ( ) ( )       = −         = −         = −         = −         = −         = −   2 23 33 2 22 32 2 21 31 2 13 33 2 12 32 2 11 31 R i R i R i i i R i R i R i i i R i R i R i i i R i R i R i i i R i R i R i i i R i R i R i i i Z r X Z r f Z v Z r X Z r f Y v Z r Y Z r f X v Z r X Z r f Z u Z r X Z r f Y u Z r X Z r f X u

Reca Scalar functions Let f: R"-I be a function of n variables denoted by 1, 2,..., n Let a =(a1, 2, ., Cn). We present first order and second order Taylor expansions of the function f(r), without providing convergence results Let a=(an, 02, .,an)ER. The linear Taylor expansion of the function )around the point a f(a)≈fa)+∑( 05(o) ai)aci (5.25) Pose estimation vo.a

Recall • Pose estimation V0.a 12

Assume f=l, r are known, for the i model point: Scalar Functions Let f: R-R be n function of n variables denoted by r, T? Let r=(rh,. In). We present first order and second order Taylor M=(,Y, Zi )is a guessed 3-D model point expansions of the function f(r), without providing convergence results Let a=(a1, ag.,,,)ER". The linear Taylor expansion of the funetion f(r) around the point a It f(x)≈f(a)+∑(r-a)(a (5.25) gn(1,M1) aX (x1-X) ag,(e, M) ag, (0, M (Z1-Z)--4a) Y Z v1-g,(61,M,)≈ g40,M(x-x)+31,) o(-分)+g(,M (21-21)-(4b) OX aZ combine(4a)and(4b), put them in a matrix form X.-Ⅹ l1 2 jM Y-Y 21 ag,(0,M)agn (0, M) ag, (0, M) X-X g(C.Dg(,M)图,M aX ar aZ aX ar aZ 3x1 Pose estimation vO.a

Continue • Pose estimation V0.a 13 (5) ~ ~ ~ ) ~ ) ( , ~ ) ( , ~ ( , ) ~ ) ( , ~ ) ( , ~ ( , ~ ~ ~ ~ ~ combine (4a) and (4b), put them in a matrix form )--(4b) ~ ( ) ~ ( , ) ~ ( ) ~ ( , ) ~ ( ) ~ ( , ) ~ ( , )--(4a) ~ ( ) ~ ( , ) ~ ( ) ~ ( , ) ~ ( ) ~ ( , ) ~ ( , ~ is a guessed 3- D model point ~ ~ ~ ~ Assume are known, for the model point : 3 1 2 3 , , , , , 1,.., − − −       − − −                 =       − − − =      − − = −   − +   − +   −  −   − +   − +   −  − = =   =  i i i i i i v t v t v t u t u t u t i i i i i i M i i i i i i v t i i v t i i v t i t v i t t i i u t i i u t i i u t i t u t i i t i t i i i i th t Z Z Y Y X X Z g M Y g M X g M Z g M Y g M X g M Z Z Y Y X X j v v u u e Z Z Z g M Y Y Y g M X X X g M v g M Z Z Z g M Y Y Y g M X X X g M u g M u u M (X ,Y ,Z ) i               

tinue x, og, (0, M) dg, (0, M) g, (0, M) [ x-X jim2xl Y aX (1)2×1 () 21-21 ag,, m og (0, m. 0,, m) Z-2 (t)3 OX aZ If we have the i"model point viewed t=1, 2,. I times, stack the matrix relations u-u ji=lx -x (8, M ag, e, M)ag, (0, M) aX Y ag. 0, M)ag 0, M)ag(, M) aX aZ X-X ag, e,,m ag(e,M(,, M) 21-21 aX Y aZ ag e,, m),, M 8g(0,, M) Z X-X Setup the Jacobain for the model j(m)= △M= △M ---(7) Pose estimation vo

continue • Pose estimation V0.a 14 * (7) ~ ~ ~ Setup the Jacobain for the model : , (6) ~ ~ ~ ) ~ ) ( , ~ ) ( , ~ ( , ) ~ ) ( , ~ ) ( , ~ ( , : ) ~ ) ( , ~ ) ( , ~ ( , ) ~ ) ( , ~ ) ( , ~ ( , ~ ~ ~ : ~ ~ :~ ~ : If we have the model point viewed 1 2 times,stack the matrix relations (5) ~ ~ ~ ) ~ ) ( , ~ ) ( , ~ ( , ) ~ ) ( , ~ ) ( , ~ ( , ~ ~ ~ ~ ~ 3 1 2 3 ( ) 2 1 2 3 3 1 ( ) 1 ( ) ( ) 3 1 2 3 1 2 3 3 1 ( )1 1 1 2 1 3 1 ( )2 3 ( ) 2 3 ( ) ( ) ( )2 1 3 1 2 1 =  − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − −       − − −  =       = − − − − − − − − − − −       − − −                                     =       − − −       =           − −       − − =       = = − − −       − − −                 =       − − − =       − − =      = =   = =   = = = = = =        E J M Z Z Y Y X X M j j J Z Z Y Y X X Z g M Y g M X g M Z g M Y g M X g M Z g M Y g M X g M Z g M Y g M X g M Z Z Y Y X X j j v v u u v v u u e e E i t , ,,.Γ Z Z Y Y X X Z g M Y g M X g M Z g M Y g M X g M Z Z Y Y X X j v v u u e m i i i i i i t m t m m i i i i i i t v t v t v t u t u t u t t v t v t v t u t u t u t i i i i i i m tm t t i i i i t i i i i t t th i i i i i i t v t v t v t u t u t u t t i i i i i i m t t i i i i t                  

SFM2: Iteration for finding the model point i: In this algorithm each point i(i=1, 2,. N)is found independently, so the following algorithm will be run n times E=Jm)*△M,so((m)*E=M SFM2: This △M=△X△Y△Z algorithm is to find the Base on the first guess M-o, find model m EL and (J(mlM k=0 a because they both depend on Mk=o Iterate(k=0, k=k max, k=k+1) △M, k Break if AM is small enough Next guess is MK+I=MK+AMK After the end of the above loop Mk is the result

SFM2: Iteration for finding the model point i: In this algorithm each point i (i=1,2,..N) is found independently , so the following algorithm will be run N times. • Pose estimation V0.a 15 ( )   ( ( )) ( ( )) ( ( )) is the result ~ After the end of the above loop } ~ ~ Next guessis Break if issmall enough ~ // if is not a square matrx, use pseudo inverse ~ find and { Iterate 0 _ max 1 ~ ,because they both depend on ~ and , find ~ Base on the first guess * ,so * 1 1 ( ) 1 ( ) 0 1 0 ( ) 0 0 1 ( ) ( ) k k k k k k k m k k m k k k m k k T m m M M M M M M J M E E J M J (k ,k K ,k k ) E J M M M M X Y Z E J M J E ΔM = +    = = = = +  =    =  = + − − = − = = = − SFM2: This algorithm is to find the model M