计算机视觉课件(2)

*,单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,Course 7 Motion,Course 7 Motion,3D motion expression,:,Motion of a rigid object in 3D is usually expressed,as a rotation around system origin followed by a,translation.,Let be a 3D point of an object at time,Let be the point after motion at,be rotation,a 3x3 orthonormal matrix.,be translation vector.,Then,from time,t,1,to,t,2,:,r,11,2,+,r,12,2,+,r,13,2,=1,r,21,2,+,r,22,2,+,r,23,2,=1,r,31,2,+,r,32,2,+,r,33,2,=1,r,11,r,21,+r,12,r,22,+r,13,r,23,=,0,r,21,r,31,+r,22,r,32,+r,21,r,33,=,0,r,31,r,11,+r,32,r,12,+r,33,r,13,=,0,Note:,There are only 3 independent parameters in,R,.,Properties of Rotation matrix:,(1),(2),(3),(4),2),R,is expressed by rotation axis and,rotation angle.,Let rotation axis be,where:,n,1,2,+,n,2,2,+,n,3,2,=1.,Rotation is made by a rotating with an angle,around a rotation axis.,Then the elements of R are:,3)Express,R,by 3 rotation angles.,4)Quaternion form of rotation:,Quaternion is a four-element vector,which can be used to express a rotation:,Let a rotation around axis,by angle,Then:,Where,-scalar part,-vector part,Quaternion conjugation,Quaternion product:,A 3D vector can be expressed as a quaternion with scalar part being zero:,Pure rotation in 3D:,Express the rotation by quaternion:,5)Homogenous coordinate systems:,Then,in 3D coordinate system is written as,in homogenous coordinate system.,2.Motion from 3D PCs:,(1)method 1:,3 or more point on a rigid object(at least 3 point not collinear)can be seen over two-time instances t,1,and t,2,:,(1),(2),(3),Subtract(1)from(2),(3)respectively:,(4),(5),(5):,(6),From(4),(5)and(6),R,can be solved(never forget rotation matrix constraints).,After finding,R,(2)method 2:,From,eq,(4),vector to,subjects to a pure rotation,The vector,is perpendicular to rotation axis,The same is true for,Thus,And rotation angle can be determined by the angle of two planes,At least 3 not collinear 3D points on rigid object is needed to determine 3D motion!,3.Motion from 2D PC:,-From 2D images to determine 3D motion.,-At least 5-point correspondences over two-image view are required.,-3D translation can only be determined over a scale factor.,-Degeneration case:3D points are on a quadratic surface.,Assume:,-Single stationary camera.,-Central projection model.,-Rigid moving object.,-Focus length,f,=1,thus,3D point:,2D image:,Let 3D motion from of time to of,(1),Where,From equ(1),(2),Apply to both sides of equ(2),(3),Apply to both sides of equ(3),(4),Let we define,(5),The eq(4)can be rewritten as,(6),Note:eq.(6)is a homogeneous scalar equation.,is a matrix containing only motion parameters,8 or more PCs,can uniquely determine E,subject to:,After matrix E is found,translation can be solved:,Or,(7),can be determined from eq(7)subject to,Once is obtained,rotation,R,can be obtained by least-square method:,(8),Or let,Note,180,o,reflection of motion is still a solution of equ(7)(homogeneous equation).In this case,object is moving behind the camera.To check for a real solution,we apply to both sides of equ(2).,Therefore if,z 0,it must hold that,Thus if,let,Remark:,at least 8 non-degenerated PCs over two image frames are needed to solve for a 3D motion using a linear method!,4.Motion from LCs:,-from two image frames of a single camera,3D motion can never be solved.,-over 3 frames,at least 6 LCs are required.,-motion models,Model A,Model B,Relation between model A and B:,Now,let we consider model the,case B,:,At time t,1,:(10),At time t,2,:(11),At time t,3,:,(12),From equ.(11),(13),Applying to both sides of equ.(13)and notice that,We get,(14),In the same way,(15),Eliminate from eqs(14)and(15),we obtain:,(16),If we define (17),F,G,H,are,3x3,matrices.,Then equ.(16)can be written in compact form:,(18),Where,Note:,equ.(18)is a vector equation containing 3 linear homogeneous equations.And only two of them are linear independent(prove it).,Therefore,13 LCs over 3 frames are needed to linearly solve for,F,G,and,H,.,Let we define:,We have,After,E,is found,translation can be solved by:,Subject to ,let be,Similarly,we define,Then,And,Subject to ,let the solution be,In solving for,R,12,and,R,13,we rather reconstruct,E,and,E,for consistence(remember,E,and,E,were column by column).,are chosen such that,Rotations can then be solved by:,Remark:check revered rotations:,if,if,Next,we determine relative amplitudes of translations,Let,substitute them into eqs(17):,when,m,n,are solved,translations are:,Build structures of 3D lines:,Direction of 3D line :,Position of :,Choose sign to make,3D Line,Check translation reflection.Evidently,So for and,For and,If,Else if,5.Motion from other image clues:,-Optical flow,-Texture,-Other clues,Will be discussed later.,