To the Graduate Council: I am submitting herewith a dissertation ...

Chapter 3: A Feature Saliency Descriptor for Registration and Pose Recovery 36D⎛⎜2⎞⎛⎟⎜i j2i j2⎜exp( − )( x − ) exp()( )2⎜−−= ixj ⎟y2iyjσσ⎟ ⎜⎜ ∑−22 ⎟⎜∑22 ⎟ ⎜ ∑iX ∈P( xi−xj)+ ( yi−yj)X ∈P( xi−xj)+ ( yi−yj)jjX j ∈P⎜⎜⎝X − X⎟⎜⎟⎜⎠⎝X − X2⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝2X⎞i− Xj⎟exp( − )( x )( )2i−xjyi−yjσ⎟22 ⎟( xi−xj)+ ( yi−yj)⎟⎟⎠2(3.11)For the 3D case, we derive the expression of the descriptor using a similar reasoning.Voting will be done using plate tensors encoding the uncertainty of normals (Fig. 3.8).Site XiTT= ( x , y , z ) ∈ P will vote at neighboring site X = ( x , y , z ) ∈ P in theiiijjjjdirection of the unit vectorX − Xt )X − Xij i j ij ij ij T= = ( tx, ty, tzwith the plate tensorijexpressed as:ij 2 ij ijij ij⎡1− ( t⎤x) − txty− txtz⎢ ij ijij 2 ij ij ⎥Tij= ⎢ − txty1−( ty) − tytz ⎥(3.12)⎢ij ijij ijij 2⎥⎣ − txtz− tytz1−( tz) ⎦=Xi1− Xj2⎡(⎢⎢⎢⎣y − y )i− ( xi− ( xij2− x− x+ ( zj− z ))( y − y )ji)( z − z )iijjj2− ( x− ( y− x( x − x )ii j2ji− y)( y − y )+ ( zjii− z ))( z − z )ijjj2− ( x⎤i− xj)( zi− zj)⎥− ( yi− yj)( zi− zj) ⎥22( x − ) + ( − ) ⎥ixjyiyj ⎦P 1P 2Fig. 3.8. Voting for feature saliency in 3D using plate tensors, the accumulated votes aredescribed by the ellipsoid bounding the two plates.