Intro. I Motivation (Pose est )Newton's method I Iterative method SFM1: find pose(R 3x313X1t Pose estimation One image (taken at time t) is enough for finding the pose at time t, if the model is known Pose estimation vo.a 11
Intro. | Motivation | Pose est.| Newton’s method | Iterative method SFM1 : find pose (R3x3,T3x1) t Pose estimation One image (taken at time t) is enough for finding the pose at time t, if the model is known. Pose estimation V0.a 11
Intro. I Motivation (Pose est )Newton's method I Iterative method Problem setting Model m at t-I Mi=/Xi Yi, zil V-axis Image YY-axis aXI u-aXIs focal leng th X-axis Fig. 1. Perspective projection of an object onto an image Pose estimation vo.a
Intro. | Motivation | Pose est.| Newton’s method | Iterative method Problem setting Pose estimation V0.a 12 •
Intro. I Motivation I Pose est. I Newton's method I Iterative method matlab: object rotation cam is stationary at world center; object and cam are all in world coord object rotation center is the center of the object %seehttp://inside.minesedu/fshome/gmurray/arbitraryaxisRotation %cos x=cos(theta x); cos_y=cos(theta y); cos_z=cos(theta z); theta x=object rotate around x-axis etc. %sin x=sin(theta x); sin _y=cos(theta y); sin_z=cos(theta z); Object rotation object syms Rx Ry Rz cos x cos_y cos_ z sin x sin y sin_z rotates around x first, then y, X 0 en z .sin x SIn x Ry=[ cos_ y sin y sin y Cos y sin Z sIn z 0 Bx R object(rotates around x first, then y then z =Rz *Ry*RX= Cam is at world center >>Answer in matlab: r object=Rz Ry *RXE cos_y*cos_Z, CoS_z*sin x *sin_y-COS_X*sin z, sin x*sin_Z+COS_x*cos Z*sin y] World coord. sys cos y*sin Z, cos x*coS z+sin x*sin_y*sin Z, cos x*sin y*sin z-COS z*sin x -sin y, cos y sIn X, cos x*cos y Pose estimation vo.a
Intro. | Motivation | Pose est.| Newton’s method | Iterative method • % matlab: object rotation: cam is stationary at world center; object and cam are all in world coord; • % object rotation center is the center of the object. • %see http://inside.mines.edu/fs_home/gmurray/ArbitraryAxisRotation/ • %cos_x=cos(theta_x);cos_y=cos(theta_y);cos_z=cos(theta_z); theta_x = object rotate around x-axis etc. • %sin_x=sin(theta_x);sin_y=cos(theta_y);sin_z=cos(theta_z); • syms Rx Ry Rz cos_x cos_y cos_z sin_x sin_y sin_z • Rx=[ 1 0 0 • 0 cos_x -sin_x • 0 sin_x cos_x]; • Ry=[ cos_y 0 sin_y • 0 1 0 • -sin_y 0 cos_y]; • Rz=[ cos_z -sin_z 0 • sin_z cos_z 0 • 0 0 1]; • ‘R_object(rotates around x first, then y then z) =Rz*Ry*Rx=' • Rz*Ry*Rx • >>Answer in matlab: R_object=Rz*Ry*Rx= • [ cos_y*cos_z, cos_z*sin_x*sin_y - cos_x*sin_z, sin_x*sin_z + cos_x*cos_z*sin_y] • [ cos_y*sin_z, cos_x*cos_z + sin_x*sin_y*sin_z, cos_x*sin_y*sin_z - cos_z*sin_x] • [ -sin_y, cos_y*sin_x, cos_x*cos_y] Pose estimation V0.a 13 Xw Zw Yw y Object rotation Robject: rotates around x first, then y, then z Cam is at world center z x World coord. sys
Intro. I Motivation (Pose est )Newton's method I Iterative method Pose estimation problem definition (given model points and one image at t, find pose) The definition of rt There are n3-D feature can be found in the points on the model. the relative positions of the -(21) appendix of chapter2 (camera models and 3D features are known i=1..N through measurements There are N feature points in the model/parameters) At time t(t=1,…,) there X are N image features F|+|T i=2,1i=2,bi=2 Assume you know the correspondences for all i=2,1i=2,bj=2 ⅰ=1,…,Ny, That means (X, Y,Zi=N F>qi=lNI Total R(rotation), Modek The target is to find, T N-features T(translation from l=1.N/t Only one image at t is Ⅸl=1,Y1=1,z1=7 needed i=1,j=1,2j= World cos(吗)cos()cos(嗔)sn()sm呜)-COS()sn吗)sm(吗)sm吗)+cos()os(吗sn( coordinates R=co吗in()cos()os()+sm(n(吗n()cos()n鸟)n(鸟)-cos(s(啊) sn(吗) cos(, )sin( cos( )cos(o,) Bi, 8. B are angles rotated aginst X, Y, Z axes resp Pose estimation vo.a
Intro. | Motivation | Pose est.| Newton’s method | Iterative method Pose estimation problem definition (given model points and one image at t, find pose ) • There are N 3-D feature points on the model. The relative positions of the 3D features are known through measurements. • At time t (t=1,..,)there are N image features {qi=1..,N }t • Assume you know the correspondences for all i=1,…,N, That means: (X,Y,Z)i=1,..N → {qi=1..,N }t • The target is to find R,T from {qi=1..,N }t • Only one image at t is needed. + = = = = 3 2 1 ' ' ' There are N feature points in the model 1,..., ' ' , ' ' T T T Z Y X R Z Y X i N Z Y f Z X f v u q T i i i i i i i Pose estimation V0.a 14 Model R (rotation), T (translation) [Xi=1,Yi=1,Zi=1] T [Xi=2,Yi=2,Zi=2] T World coordinates [X’i=1,Y’i=1,Z’i=1] T i=1,…,N Total N-features [X’ i=2,Y’ i=2,Z’ i=2] T , , are angles rotated aginst axes resp. -sin( ) cos( )sin( ) cos( )cos( ) cos( )sin( ) cos( )cos( ) sin( )sin( )sin( ) cos( )sin( )sin( )- cos( )sin( ) cos( )cos( ) cos( )sin( )sin( ) - cos( )sin( ) sin( )sin( ) + cos( )cos( )sin( ) 1 2 3 2 2 1 1 2 2 3 1 3 1 2 3 1 2 3 3 1 2 3 3 1 2 1 3 1 3 1 3 2 X,Y,Z R = + The definition of R,T can be found in the appendix of chapter2 (camera models and parameters)
Intro. I Motivation (Pose est )Newton's method I Iterative method Example of pose estimation We know the 3-D positions of the features on this box. ( e.g. 4 points as shown corners of a 10cm 3 cube) In the image at time t,we know the correspondences of which corners appear in the image and their image q=1:=0 positions.image correspondences ° We can find r, T from thi qi =2,t=0 尺,7 g=2.t1 image at time t1 Time t=0 Time t=t1 15 Pose estimation vo.a
Intro. | Motivation | Pose est.| Newton’s method | Iterative method Example of pose estimation • We know the 3-D positions of the features on this box. (e.g. 4 points as shown, corners of a 10cm^3 cube) • In the image at time t, we know the correspondences of which corners appear in the image and their image positions. (image correspondences) • We can find R,T from this image at time t1. Pose estimation V0.a Time t=0 Time t=t1 15 R,T qi=1,t=0 qi=2,t=0 qi=2,t=t1 qi=1,t=t1