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ABSTACT There have been several investigations to perform image recognition tasks in a convenient algebraic environmen. In this paper we extend the polynomial approach which has been done so far for representing 2-D images only [2], to 3-D vision. Here the object is approximated by a volume occupancy array, each cell of the array is usually referrred to as a vozel. Our polynomial representation of a 3-D object A is then given as follows: A _= ~id, k Pi9~ X i Y j Z k where Pijk = 1 if the voxel (i, j , k) of the array lies completely inside A and otherwise we set pijk = 0. We describe some operations we have developed in 3-D vision. Rotation and coarse magnification. We transform a 3-D object represented by P(X, Y, Z) into p( xi* yi~ Zis, X JayJ2 ZJ,, X k' yk2 Z ks ). Then, by giving the suitable values to it,jt,kt,1 < t < 3,we can get the magnification and the rotation of original object. Morphological Operations. We show that the primitive morphological operations dilation and erosion can be carried out conveniently using the polynomial representation. Let X -Y'~(ij, k)ex X i y j z k , B -Y]~(i,j,k)eB XiYJZk, be the polynomials representing a 3-D object and a structuring element respectively. Let P = X • B = Y]~i,j,k Pifl* X i Y j Z k denote the usual (polynomial) product of X and B. Then, we have for example, the dilation $ is given by: X $ B ~-~(ijj,) cifl, X i y j z k, where cijk : 1 i f p i j k ~ 1 and 0 otherwise, and d/jh = 1 ifpifl, = n and 0 otherwise where n is the number of terms in the polynomial B. Shape Decomposition. We do successive erosions on the original solid object X by the structure dement B until the result of the erosion becomes an empty set at the (Nx + 1)th step of the execution of the algorithm where N1 is the size of the maximal object inscribable in X. We also have developed a method to determine the surface of a 3-D object.

3-D vision algorithms based on polynomial representation(abstract and references only)