Recognition of 3-D Objects Using the Extended Gaussian Image

We propose to use an extended Gaussian imap,e (EGI) fo r i n t e r p r e t i n g 2-1/2-D representat ions for recogni t ion of 3-D ob jec ts . The EGI is constructed by mapping each surface normals of an object to the Gaussian sphere. The freedom in viewer d i rec t i ons caused by incomplete observat ion Is great ly reduced by applying cons t ra in ts derived from a global d i s t r i b u t i o n of surface normals on the EGI. One const ra in t on the viewer d i r e c t i o n is derived from the r a t i o of the projected area to the o r i g i n a l surface area. The other const ra in t comes from the d i r e c t i o n of the p r i nc i pa l a x i s . A f te r reducing the possible viewing d i rec t i ons w i th these cons t ra in t s , we w i l l apply a matching funct ion to ESls of a candidate set fo r a f i n a l dec is ion . We also propose an a lgor i thm for reconst ruct ion of the o r i g i n a l shape of a convex polyhedron from i t s EGI. This a lgor i thm is based on the analys is-by-synthes is method. 1 WHAT IS THE EXTENDED GAUSSIAN IMAGE A c o l l e c t i o n of l oca l surface normals [ 1 , 2 , 3 , 4 , 5 ] , sometimes re fe r red to as a 2-1/2-D representat ion of an object [ 6 ] , is o f ten provided by machine v i s i on at the low l e v e l . For example, an a lgor i thm based on the propagat ion-o f -const ra in ts technique [2] provides l oca l surface o r i en ta t i on from shading and occluding in fo rmat ion . The same a lgor i thm can also produce surface o r i en ta t i on from apparent d i s t o r t i o n of known patterns based on a regu la r -pa t te rn gradient map [A ] , The d i s t o r t i o n of these small c i r c l e s on the go l f b a l l in F i g . 1 can be used to recover loca l surface o r i e n t a t i o n . In the above cases, the next problem encountered is how to in terpre t these loca l representat ions for a global recogn i t ion of an ob jec t . One of the most important issues in t h i s process is how to convert a l oca l representat ion based on the viewer-centered coordinate system in to a g lobal descr ip t ion based on the object-centered coordinate system such as the generalized cy l inder [7 ,8 ] or the spike model [1] . Each l o c a l representat ion depicted as a needle in the above example is obtained at a pa r t i cu la r point expressed in viewer-centered coordinate system on the image plane. On the other hand, an object model is expressed in a p a r t i c u l a r coordinate system usual ly based on an object center and natura l axis of the ob jec t . These two coordinate systems are independent each other . I t is qu i te d i f f i c u l t to recover the o r i g i n a l object-centered coordinate system from the 2-1/2-D representat ion observed in the viewer-centered coordinate system. We propose to use the extended Gaussian image (EGI) as a t oo l fo r t h i s conversion process. An EGI of an object may be derived from a spike model of the object [1] . A spike model is a c o l l e c t i o n of surface normals on each surface patch in the 3-D wor ld . Let us assume that there is a f i xed number of surface patches per un i t surface area and that a un i t normal is erected on each patch. The c o l l e c t i o n of these normals is ca l led a spike model of the ob jec t . These normals are l i k e porcupines's q u i l l s [ 1 ] . These normals on surface patches in a spike model are moved to a common point of a p p l i c a t i o n . The locus of points cons is t ing of t he i r end points l i e s on the surface of a uni t sphere. This mapping is ca l led the Gauss map; the un i t sphere is ca l led the Gaussian sphere. If we at tach a un i t mass to each end point of the normal vec tor , we w i l l observe a d i s t r i b u t i o n of mass on the Gaussian sphere. This d i s t r i b u t i o n of mass w i l l be normalized. The resu l t i ng d i s t r i b u t i o n of mass on the Gaussian sphere is ca l led the extended Gaussian image (EGI) of the object [1 ,9 ] . An EGI is independent on both the pos i t ion of the o r i g i n and the scale of axes of the coordinate system. A coordinate system may be characterized using three components; the pos i t ion of o r i g i n , the d i r e c t i o n of coordinate axes, and a scale factor of a coordinate ax i s . Among these three components, an extended Gaussian image is independent on both the pos i t i on of the o r i g i n and the scale fac tor of coordinate ax i s . An EGI is independent on the pos i t i on of the o r i g i n , because Surface normals on patches w i l l be projected on the Gaussian sphere in p a r a l l e l t ransformation without regarding the pos i t i on of the o r i g i n . Also the representat ion is independent on the scale of the coordinate. For example, a 2x2x2-inches cube has the same representat ion as a l x l x l inches cube, provided that t o t a l mass of d i s t r i b u t i o n on the sphere is