Elsevier

Pattern Recognition

Volume 34, Issue 5, May 2001, Pages 1119-1126
Pattern Recognition

Fast computation of Legendre moments of polyhedra

https://doi.org/10.1016/S0031-3203(00)00049-2Get rights and content

Abstract

The three-dimensional (3D) orthogonal moments are an efficient tool for object reconstruction and 3D image analysis. However, until now, 3D orthogonal moments have not been analysed in detail from the point of view of reducing the computational complexity. In this paper, we present a recursive algorithm for fast computation of Legendre moments of polyhedra. First, a Gaussian theorem is employed to transform the volume integral into a surface one. The double integral can then be deduced from the simple integral by a Green's theorem. Finally, the recursive relationship is investigated. As one can see, the proposed method decreases the computational complexity tremendously.

Introduction

Since Hu introduced moment invariants in 1962 [1], moment functions such as geometric moments, orthogonal moments, rotational moments and complex moments, have played an important role in pattern recognition, image analysis and object representation [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. A survey of these applications can be found in a paper by Prokop and Reeves [12]. Among all the moments mentioned above, orthogonal moments can be used to represent an image with a minimum amount of information redundancy. Furthermore, because orthogonal moments have the simply inverse transform, they may be employed to determine the minimum number of moments necessary to appropriately reconstruct a given image.

Since a direct computation of moments requires a large number of operations, a lot of research works have been realised to decrease the moment computational complexity, and most of them focus on geometric moments [13], [14], [15], [16], [17]. Orthogonal moments, however, have not been analysed in detail in point of reducing the computational complexity. Two recent publications concentrate on the fast computation of 2D Legendre moments [18], [19]. Because the world around us is generally three-dimensional, therefore, the representation and description of 3D features and the reconstruction of 3D objects are very important. Like 2D moments, the 3D moments were also successfully used in 3D image analysis. Some fast algorithms have been proposed to speed up the 3D geometric moment calculation. Li and Shen applied a Pascal triangle transform to compute the 3D monomials so that no multiplication is required in the calculation of moments [15]. Li used a Gaussian theorem, and then proposed a recursive method to calculate the geometric moments of polyhedra [16]. Yang et al. proposed to use a discrete divergence theorem for fast computation of moments of an arbitrary 3D object [17]. However, until now, no effort has been made to improve the calculation of 3D orthogonal moments.

In this paper, by extending an algorithm recently developed by Shu et al. [19], we present a fast method for computing Legendre moments of polyhedra.

Section snippets

3D Legendre moments and Gaussian theorem

The 2D Legendre moment of an image intensity function f(x,y) is defined asLpq=(2p+1)(2q+1)4∫−11∫−11Pp(x)Pq(y)f(x,y)dxdy,where Pp(x) denotes the pth order Legendre polynomial, and p+q is the order of the moment.

Similarly, the 3D Legendre moment of order p+q+r of an image intensity function f(x,y,z) is defined asLpqr=(2p+1)(2q+1)(2r+1)8∫−11∫−11∫−11Pp(x)Pq(y)Pr(z)f(x,y,z)dxdydz.

Since the Legendre polynomials Pp(x), p=0,1,2,…, are a complete orthogonal basis set on the interval [−1, 1], a cubic

Algorithm

Based on the method for computing 3D Legendre moments Lpqr described in Section 2.3, the corresponding algorithms can be given as follows.

(1) Calculation of the Legendre polynomial values

As mentioned earlier, we need to calculate the Legendre polynomial values of order up to M+3 at each vertices of polyhedra. For this purpose, an iterative method is used in Fig. 3.

  • Denote J the total number of the edges of polyhedra, then we have 2J=∑Ii=1Ji. Therefore, algorithm 1 needs

  • Na1=3(M+2)∑i=1IJi=6(M+2)J

Discussions and conclusions

The result obtained in the previous section shows that the computational complexity of the method presented in this paper depends no longer on N3, but only on I and J where I and J denote, respectively, the boundary surface number and edge number of the polyhedra under consideration. These two numbers are substantially smaller than N3, so our method reduces significantly the computational complexity in comparison with the direct computation method.

As 2D Legendre moments, by expressing the

About the Author—HUAZHONG SHU received the B.S. degree in Applied Mathematics from Wuhan University, China, in 1987, and a Ph.D. degree in Numerical Analysis from the University of Rennes (France) in 1992. He was a postdoctoral fellow with the Department of Biology and Medical Engineering, Southeast University, from 1995 to 1997. His recent work concentrates on the treatment planning optimization, medical imaging, and pattern recognition.

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About the Author—HUAZHONG SHU received the B.S. degree in Applied Mathematics from Wuhan University, China, in 1987, and a Ph.D. degree in Numerical Analysis from the University of Rennes (France) in 1992. He was a postdoctoral fellow with the Department of Biology and Medical Engineering, Southeast University, from 1995 to 1997. His recent work concentrates on the treatment planning optimization, medical imaging, and pattern recognition.

About the Author—LIMIN LUO obtained his Ph.D. degree in 1986 from the University of Rennes (France). Now he is a professor and the chairman of the Department of Biology and Medical Engineering, Southeast University, Nanjing, China. He is the author and co-author of over 60 papers. His current research interests include medical imaging, image analysis, computer-assisted systems for diagnosis and therapy in medicine, and computer vision. Dr Luo is a senior member of the IEEE. He is an associate editor of IEEE Eng. Med. Biol. Magazine and Innovation et Technologie en Biologie et Medecine (ITBM).

About the Author—WENXUE YU received his B.S. degree in Machine Engineering from Shandong Engineering College in 1992, the M.E. degree in Mechanics from the Southeast University in 1997. He is now a Ph.D. student at the Department of Biology and Medical Engineering of Southeast University. His current research interests include image processing and analysis, radiosurgery treatment planning and pattern recognition.

About the Author—JINDAN ZHOU received the B.S. degree in BioMedical Engineering in 1999 from Southeast University, Nanjing, China. She is now a graduate student of the Department of Biology and Medical Engineering of Southeast University. Her current research is mainly focused on pattern recognition and image processing.

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