Moment functions have a broad spectrum of applications in image analysis, such as invariant pattern recognition, object classification, pose estimation, image coding and reconstruction. A set of moments computed from a digital image, generally represents global characteristics of the image shape, and provides a lot of information about different types of geometrical features of the image. The feature representation capability of image moments has been widely used in object identification techniques in several areas of computer vision and robotics. Geometric moments were the first ones to be applied to images, as they are computationally very simple. With the progress of research in image processing, many new types of moment functions have been introduced in the recent past, each having its own advantages in specific application areas. For example, moment invariants with respect to image rotation can be easily derived from complex moments. Orthogonal moments, on the other hand, characterize independent features of the image and thus have minimum information redundancy in a set.
The properties of image moments have the following analogies in statistics and mechanics. Moments of orders zero, one, and two of a probability density function represent the total probability, the expectation, and variance respectively. In mechanics, these moments of a spatial distribution of mass give the total mass, the centroid position, and the inertia values respectively. Considering an image as a two-dimensional intensity distribution, the geometric moment functions of the pixel values with respect to their spatial locations in the image, can similarly provide the shape information such as the total image area, the coordinates of the image centroid, and the orientation. These shape characteristics can be further used to construct feature vectors that are invariant with respect to image translation, rotation and scaling. While moments of orders zero up to three are used to represent gross level image features, higher order moments contain finer details about the image and are often more sensitive to image noise.
This book presents a survey of image moment functions, their properties, computational aspects and applications. The purpose of this work is (i) to provide a global view of the utility of moment functions in image analysis, (ii) to serve as a compilation of fundamental concepts and applications of moments reported in literature, and (iii) to aid researchers and academicians in the area of computer vision and image analysis with adequate reference material on moment functions. The book is organized into two parts. The mathematical framework underlying basic theoretical concepts on image moments is presented in Part 1. This includes moment definitions, derivations of important formulae and the description of significant results and algorithms. Several applications of image moments in the field of image analysis, and the description of related algorithms are given in Part 2.
The first part of this monograph introduces different types of moment functions that are commonly used in image analysis, beginning with a detailed description of geometric moments and their invariant functions (Chapter 2). This chapter also gives numerical algorithms for fast computation of geometric moments of binary images. The concept of complex moments and their properties are presented in Chapter 3. These are moments with complex kernels such as radial, Fourier-Mellin and complex-domain functions. Two important orthogonal moments which have found several applications in image representation are Legendre moments and Zernike moments. The properties of Legendre polynomials, and the algorithms related to the computation of Legendre moments are given in Chapter 4. An introduction of the radial polynomials of Zernike, and the characteristics of the associated moments (Zernike moments and Pseudo-Zernike moments), and Zernike moment invariants are included in Chapter 5. Fast methods to compute Zernike moments of binary and gray-level images are also discussed. The properties of geometric moments when viewed as contravariant symmetric tensors are given in Chapter 6.
The second part of this book describes the main application areas of moment functions in image analysis. The capability of moments to provide shape characteristics of an image has been widely put to use in many pattern recognition applications. Moment based algorithms in pattern recognition are briefly discussed in Chapter 7. Moment invariants have also been used as feature descriptors to identify objects, independent of the translation, rotation and scale factors of the image introduced by the camera view geometry. Object identification and classification methods are also included in Chapter 7. Object pose recovery using image moments has found several applications in computer vision. The problem of estimating the object orientation and position parameters with the help of moments computed from images is detailed in Chapter 8, together with the mathematical derivations leading to the solutions of object pose parameters for various object-camera configurations. Miscellaneous applications of moment functions in image analysis are described in Chapter 9. These include moment based algorithms for edge detection, texture segmentation, image reconstruction, clustering and thresholding.
An exhaustive list of invariant functions of geometric moments is given in Appendix 1. Analytical expressions of geometric and Zernike moments of a general elliptical shape are provided in Appendix 2. The bibliographical list given in the end of this book, contains references primarily on the theory and applications of moment functions, reported in the open literature, and covers most of the important technical journals and conference proceedings in the areas of computer vision, image processing, pattern recognition, optical engineering, and robotics.
The motivation for this work came from the need for a survey and compilation of various aspects of moment functions, considering their potential uses and applications in various realms of computer vision and image analysis. An attempt is made to present all important theoretical and application oriented details on most of the commonly used types of image moments. Due to obvious reasons, every method and theory on moments reported in literature could not be included. Only representative algorithms in primary application areas are discussed in detail. Among the many schemes and diverse moment based techniques developed over the past as well as in recent years, a majority of the work has been referred in bibliography, and fundamental concepts and methods have been outlined in the text.