f 1 2

face。 Let us also assume a weak perspective camera the corresponding centroids of & : pc and pc , from

approximation [16], and fix an object-centered frame Eq。 (2) we obtain p2 − pc2 D A12。p1 − pc1/, with

hoi D fio; j o; kog whose origin is the point c Pc 。 The A12 D T2。T1/− 1 : (4)

tangent  plane  equation  is  expressed  in  the  object

frame simply as oz D 0。 The following equation re- It follows that changes in shape are modeled as 2D

lates camera coordinates and object frame coordinates

for a given projected point p D [px ; py ] T : affine transformations, which form a sub-group of 2D

projective transformations (isomorphic to SL(3))。

T T o o T The dynamic interaction between camera and ob-

[px ; py ] D   [xc ; yc ] C T [ x;  y] ; (2)

ject can be expressed, at a generic image point p, in

where   D f =zc and T  is the weak perspective pro-

terms of the 2D motion field  v。px py / D p arising

jection matrix。 It holds that P

in the image plane due to both surface shape and 3D

c c  c' C s s' c c  s' − s c' (3) relative velocity twist。 We show below that with our

T ; assumptions the motion field has a first-order spatial

D s c  c' − c s' s c  s' C c c' structure around the centroid pc  of the image patch

where the notation c#  D cos # and s#  D sin # is used & 。 At time t, for a generic image point results p。t / D

and the angles   2 [0;  =2] (slant),   2 [− ;  ] (tilt) pc 。t / C A。t /。p。0/ − pc 。0// which, differentiated with

and ' 2 [− ;  ] (orientation) define a minimal repre- respect to time, yields:

246 F。 Conticelli et al。 / Robotics and Autonomous Systems 29 (1999) 243–256

p。t / p 。t / A。t /A。t /−1 。p。t /