# tian - X0 =(X0 Y0 Z0 3-D coordinates of a point at time t0...

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X 0 = (X 0 , Y 0 , Z 0 ) > : 3-D coordinates of a point at time t 0 T = (T 1 , T 2 , T 3 ) > : Translation vector spherical coordinates: slant, θ T , and tilt, φ T : T = (sin θ T cos φ T , sin θ T sin φ T , cos θ T ) > . (1) q = (q 0 , q 1 , q 2 , q 3 ) > : Quaternion q = (sin α 2 n , cos α 2 ) > Rotation Matrix: R = q 2 0 - q 2 1 - q 2 2 + q 2 3 2( q 0 q 1 + q 2 q 3 ) 2( q 0 q 2 - q 1 q 3 ) 2( q 0 q 1 - q 2 q 3 ) - q 2 0 + q 2 1 - q 2 2 + q 2 3 2( q 1 q 2 + q 0 q 3 ) 2( q 0 q 2 + q 1 q 3 ) 2( q 1 q 2 - q 0 q 3 ) - q 2 0 - q 2 1 + q 2 2 + q 2 3 , (2) q (t) = w x w sin wt 2 , w y w sin wt 2 , w z w sin wt 2 , cos wt 2 > , (3) w = q w 2 x + w 2 y + w 2 z , and w 6 = 0.

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X t = R ( q (t)) X 0 + t T . (4) Perspective projection: X = (X 1 , X 2 , X 3 ), x = f X 1 X 3 , y = f X 2 X 3 Unknowns: a = ( θ T , φ T , w x , w y , w z , Z 0 ) > . Problem: determine a , given { x s , x s+1 , ··· , x s+N } , where x t = (x t , y t ) > , t = s , ··· , s + N, s 0 Let h t ( a ) be the 3-D coordinates of a point at time t , given parameter a : h t ( a ) = R ( q (t)) X 0 + t T . (5) Given image coordinates x t , and depth Z t , its corresponding 3-D coordinates are: X t = (Z t x t / f , Z t y t / f , Z t ) > . Let p = (0, 0, 1), then Z t = p h t ( a ). Now, X t becomes X t = p h t ( a ) x t /f y t /f 1 . 2
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tian - X0 =(X0 Y0 Z0 3-D coordinates of a point at time t0...

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