Elsevier

Pattern Recognition

Volume 36, Issue 7, July 2003, Pages 1519-1528
Pattern Recognition

A fast algorithm for the computation of axial moments and its application to the orthogonal fitting of curves

https://doi.org/10.1016/S0031-3203(03)00003-7Get rights and content

Abstract

This paper describes a fast algorithm to compute local axial moments used in the detection of objects of interest in images. The basic idea is the elimination of redundant operations while computing axial moments for two neighboring angles of orientation. The main result is that the complexity of the recursive computation of axial moments becomes independent of the total number of computed moments at a given point, i.e., it is of the order O(N) where N is the size of the data set. This result is of great importance in computer vision since many feature extraction methods rely on the computation of axial moments. The use of this algorithm for fast object skeletonization in images by orthogonal regression fitting is described in detail, with the experimental results confirming the theoretical computational complexity.

Introduction

Image analysis algorithms are mainly based on the detection and processing of structural features that are capable of providing a concise description of objects in a scene. For example, boundaries of objects, snakes and skeletons are often used as reliable shape descriptors. Moments of an image object are applied due to their invariance to geometrical transformations [1], [2], [3], [4], [5]. Axial moments, in particular, can be used for the determination of object orientation or piecewise-linear representation of object boundaries and skeletons.

A relatively new approach to image analysis based on moment calculation is the detection of local symmetries. Symmetry is useful to develop intermediate and high-level vision algorithms. For example, Kelly and Levine [4] used local central moments in a symmetry operator for object location and shape description. This approach was extended to other symmetries by Reisfeld et al. [5] in their generalized symmetry transform. Di Gesù and Valenti [6] proposed a new measure of image symmetry, called discrete symmetry transform, which involves the computation of local axial moments. Usually, the time complexity of algorithms for symmetry analysis in images is of O(I2×L2), where I×I is the size of input image, and L×L is the window size in the definition of local moments.

Computational complexity is an important consideration in real-time applications of image analysis algorithms based on axial moments. In order to reduce the computational complexity, recursive implementations of the algorithms, which provide significant computational speedup, are often used in practice. For example, Zakaria et al. [7] proposed a fast algorithm that decreases the cost of moment calculations to O(I2) multiplication and O(I4) addition operations. A linear digital filter based on moment computations has been proposed by Hatamian [8] for fast image filtering. Li and Shen [9] developed another algorithm for moment calculations that involves a constant number of multiplications, and O(I2) additions. However, the algorithm works only on binary images and the application of this method to multi-level images requires a preliminary segmentation procedure. Fast computation of local moments for the case of gray-scale images and its application to fitting of intensity surfaces was presented in Ref. [10]. For example, Yang and Albregsten [11] developed a fast scheme for exact computation of Cartesian geometric moments using a discrete Green's theorem.

Recently, a fast algorithm was developed by Di Gesù and Palenichka [12] to compute local axial moments in a running manner, i.e., in each image point (x,y) with respect to a window W(x,y) centered at that point. Its computational complexity does not depend on the window size, and is of the order O(I2). However, it computes the axial moments only with respect to four main directions and requires image rotation for moment calculation if an arbitrary orientation is requested. The image rotation using interpolation schemes may result in a loss of accuracy and implies additional computation costs.

In this paper, a new fast algorithm to compute axial moments for an arbitrary angle of orientation is presented, which is based on recursive image processing. It computes all axial moments, for arbitrary angles, at a given image point, with O(L2) operations per point in a window of size L×L. So, the computational complexity does not depend on the total number K of computed moments. In contrast, the direct computation of K axial moments of the first order requires O(K×L2) operations per pixel for gray-level images. In order to achieve a sufficient accuracy, the value of K would have to be of the order O(L2). The existing fast algorithms can compute only axial moments of one orientation (or a few basic orientations) in each image point to achieve the comparable performance.

Another advantage of the proposed algorithm is its versatility due to its capability of fast moment computation with respect to all given axes at a given pixel. For example, such a recursive approach can be incorporated successfully for fast regression fitting of curves or piecewise-linear skeletonization of objects in binary images. Algorithms for fast polygonal approximation of digitized planar curves are presented in [13], [14]. However, all these algorithms assume that the curves are one pixel wide, with a given connectivity pattern (four-point connectivity or eight-point connectivity). The case addressed in this paper concerns thick curves. On the other hand, the thinning algorithms, which are working on binary images, require some pre-processing steps such as median filtering or morphological filtering that in turn introduce certain distortions and hence reduce the precision of the skeletonization [15]. Another new approach to the curve skeletonization is a direct skeletonization of gray-scale images omitting the binarization step (see, e.g., [16], [17], [18]). The method of self-organizing maps was successfully applied to piecewise-linear skeletonization of objects with sparse shapes in binary images [19]. After the proper self-organization process, the neural units corresponding to the skeleton vertices are located on or near the medial axes of the objects.

The sparseness of shape usually occurs as a result of binarization by comparing the gray-scale intensity with a threshold. In some cases, such as the multi-spectral imagery in remote sensing, only binary images are available when object points are segmented from the background by a clustering procedure, not by a thresholding operation. The main goal of the presented application was to speed up the skeletonization of objects in binary images obtained by segmentation of multi-spectral images. In combination with the calculation of other local image properties, the proposed algorithm can be used in image analysis where local image properties are estimated based on more complex criteria.

This paper is organized as follows. Upon a brief discussion of existing approaches to fast computation of moments in computer vision, our fast algorithm to compute local axial moments with respect to all given axes is presented in Section 2. Application of this algorithm to orthogonal regression fitting of curves in images is discussed in Section 3. Experimental results are given in Section 4, and concluding remarks in Section 5.

Section snippets

Definition of axial moments

Fast computation of axial moments in computer vision algorithms is necessary mainly with respect to two major problems: orthogonal projection regression and detection of objects and regions of interest. In turn, the method of orthogonal projection regression can be successfully used for the skeletonization of objects in images [20]. Skeletonization methods such as thinning and distance transform require a pre-processing procedure in order to cope with sparse shapes or occlusions present in

Polygonal line algorithm for piecewise-linear skeletonization of curves

There are several applications in image processing and computer vision for the proposed fast algorithm for axial moments. One application area will be discussed here in detail, namely, image skeletonization by the orthogonal regression fitting of curves. One example of the application is the tracing of river systems from remote-sensing imagery. Another application example is object skeletonization in character recognition, including automated recognition of handwriting. Image data are given in

Experimental results

The main goal of the experiments was to evaluate the execution time of the proposed fast recursive algorithm. Since, to our knowledge, there are no other fast algorithms for the case of an arbitrary angle of rotation, the proposed method was compared with the direct implementation. The applicability of this fast algorithm in the case of real-time computations was tested in an application of the skeletonization of sparse objects obtained from remote-sensing imagery.

An important issue in

Conclusions

A fast algorithm for the computation of axial moments was proposed and tested on synthetic and real images. The computational complexity of this algorithm does not depend on the total number K of computed moments, and it is of the order O(N), where N=L×L is the window size. In practice, the recursive algorithm has a certain implementation overhead, which results in a slowdown for lower values of K, but provides a significant speedup of the order N for higher values of K and for large window

Summary

Moments of an image object or functions of image moments are often involved in the description of object shape due to the invariance to geometrical transformations. Axial moments, in particular, can be used for the determination of object orientation or piecewise-linear representation of object boundaries and skeletons. However, the computational complexity is a serious concern in real-time applications of image analysis algorithms based on axial moments. In order to reduce the computational

About the Author—ROMAN M. PALENICHKA graduated in applied mathematics from Lviv Polytechnic Institute (Ukraine) with honor diploma in 1979. He obtained his Ph.D. degree in computer science in 1986 from the Glushkov Institute of Cybernetics in Kyiv, Ukraine. In 1990 he was granted the degree of senior research associate form the Ukrainian Academy of Sciences. Currently, he is employed as a researcher at the University of Quebec in Hull (Canada).

His major fields of expertise are image

References (24)

  • C. Teh et al.

    On image analysis by the method of moments

    IEEE Trans. Pattern Anal. Mach. Intell.

    (1988)
  • F. Kelly, M.D. Levine, Annular symmetry operators: a method for locating and describing objects, Proceedings of...
  • Cited by (0)

    About the Author—ROMAN M. PALENICHKA graduated in applied mathematics from Lviv Polytechnic Institute (Ukraine) with honor diploma in 1979. He obtained his Ph.D. degree in computer science in 1986 from the Glushkov Institute of Cybernetics in Kyiv, Ukraine. In 1990 he was granted the degree of senior research associate form the Ukrainian Academy of Sciences. Currently, he is employed as a researcher at the University of Quebec in Hull (Canada).

    His major fields of expertise are image segmentation and computer vision, especially detection of objects of interest in application to industrial and medical diagnostics imaging and remote sensing. Dr. Palenichka has published over 52 scientific papers in international journals and conference proceedings and is a co-author of three books including the Dictionary of Computer Graphics and Image Analysis. Dr. Palenichka holds about 48 filed certificates of inventions (patents) of the former USSR. He participated as the principal investigator or the project director in many national/international research projects or project with industry related to medical and industrial diagnostic imaging. Currently he is a member of the editorial board of International Journal “Machine Graphics & Vision”.

    About the Author—MAREK B. ZAREMBA is a Professor of Computer Engineering at Université du Québec in Hull, Quebec, Canada since 1985. He received M.Sc. and Ph.D. degrees in automatic control systems from Warsaw University of Technology. His primary area of expertise is the design of adaptive information and control systems, in particular the design of hybrid systems integrating computational intelligence paradigms, real-time intelligent systems, high-performance distributed computing, and learning control. His industrial experience includes the design, for a General Motors application, of a vision-based robot control system that employed the NRC photogrammetric technology used in the Canadarm project, the development of an adaptive technology for detection of leakages in subterranean energy transmission networks, and the development of a dynamic task distribution technology for rapid prototyping of distributed intelligent systems. Dr. Zaremba has authored or co-authored over 150 books and technical papers in scientific journals and conferences. He is currently Associate Editor of Control Engineering Practice and member of the Advisory Board of Concurrent Engineering: Research and Applications. He has been involved in the activities of IFAC (International Federation of Automatic Control) as Chairman of the Technical Committee on Advanced Manufacturing Technology (1993–1999), as a member of the IFAC Technical Board (since 1999), and on the Board of Directors of IFAC-Canada. Dr. Zaremba is a Senior Member of IEEE, Fellow of ISPE, and a Registered Professional Engineer in the Province of Ontario, Canada.

    View full text