Fast Computation of Three-Dimensional Geometric Moments Using a Discrete Divergence Theorem and a Generalization to Higher Dimensions
✍ Scribed by Luren Yang; Fritz Albregtsen; Torfinn Taxt
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 370 KB
- Volume
- 59
- Category
- Article
- ISSN
- 1077-3169
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✦ Synopsis
three-dimensional Cartesian geometric moment (for short 3-D moment) of order p ϩ q ϩ r of a 3-D object is defined
The three-dimensional Cartesian geometric moments (for short 3-D moments) are important features for 3-D object recogas [2] nition and shape description. To calculate the moments of objects in a 3-D image by a straightforward method requires m pqr ϭ ͵ ͵ ͵ R x p y q z r g(x, y, z) dz dy dx,
(2)
large number of computing operations. Some authors have proposed fast algorithms to compute the 3-D moments. However, the problem of computation has not been well solved since where R is a 3-D region. In a binary 3-D image, the moment all known methods require computations of order N 3 , assuming that the object is represented by an N ؋ N ؋ N voxel image. of a 3-D homogeneous object represented by voxels (vol-In this paper, we present a discrete divergence theorem which ume elements) is often evaluated as [3] can be used to compute the sum of a function over an ndimensional discrete region by a summation over the discrete m pqr ϭ R x p y q z r (3) surface enclosing the region. As its corollaries, we give a discrete Gauss's theorem for 3-D discrete objects and a discrete Green's theorem for 2-D discrete objects. Using a fast surface tracking algorithm and the discrete Gauss's theorem, we design a new which is a sum of monomials over a discrete 3-D region R. method to compute the geometric moments of 3-D binary ob-2-D moments are important shape features of a 2-D jects as observed in 3-D discrete images. This method reduces object, and have been widely used in image processing and the computational complexity significantly, requiring computaanalysis. Applications of 2-D moments can be found in tion of O(N 2 ). This reduction is demonstrated experimentally object recognition [4, 5], shape description [6], object repon two 3-D objects. We also generalize the method to deal with resentation [7], edge detection [8], image alignment [9], high-dimensional images. Some 3-D moment invariants and motion estimation [10], document image analysis [11], and shape features are also discussed. © 1997 Academic Press texture analysis [12, 13]. Many of these applications were surveyed by Prokop and Reeves [1]. Since the 2-D moments are very useful, many efforts [14-21] have been