Lehrstuhl F Ur Mustererkennung Und Bildverarbeitung Fast 3d Zernike Moments and -invariants

The aim of this report is threefold: First we generalize to 3D a long ago known fast algorithm for the computation of ordinary geometrical moments of 2D elds starting from what could be named cumulative moments. This is done by rst reformulating the 2D algorithm in terms of matrix operations and subsequently extending the result straightforwardly to 3D. Second, guided by the results of much research work done in the past on the performance of 2D 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 e ects, 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 rotation like the 2D Zernike polynomials do. For that reason we call this set 3D Zernike polynomials. The role of the angular exponential function in the 2D Zernike polynomials set is now played by the spherical harmonics on the surface of the unit sphere. We show how to directly compute the associated 3D Zernike moments from the cumulative moments without the need to go through the geometrical moments, thus avoiding the enormous dynamic range required by the latter. In the third part it is shown that a set of complete moment invariants for orthogonal transformations of 3D objects that we had previously derived may be rebuilt to yield the 3D Zernike moments of the object in its standard position.