3d Zernike Moments and Zernike Aane Invariants for 3d Image Analysis and Recognition

Guided by the results of much research work done in the past on the performance of 2D image moments and moment invariants in the presence of noise, suggesting that by using orthogonal 2D Zernike rather than regular geometrical moments one gets many advantages regarding noise eeects, information suppression at low radii and redundancy , we have worked out and introduce a complete set of 3D polynomials orthonormal within the unit sphere that exhibits a "form invariance" property under 3D rotation like the 2D Zernike polyno-mials do in the plane. For that reason we call this set 3D Zernike polynomials. The role of the angular exponential function in the 2D Zernike polyno-mials set is now played by the spherical harmonics on the surface of the unit sphere. Spherical harmonics and spherical moments are introduced in a very succinct, self-contained and compact way using algebraically powerful tools like 'power substi-tutions' and generating functions. Unambiguous aane normalization and unique aane pose determination using 3D image moments of degree not greater than three as well as derivation of complete , uncorrelated aane invariants are naturally accomplished using the concepts we introduce in the present paper.