Moments and Moment Invariants – Theory and Applications

Abstract

This chapter aims to analyze the research field of moments and moment invariants in a holistic way. Initially, a literature analysis of the last 50 years is presented and discussed in order to highlight the potential of this topic and the increasing interest in many disciplines. A more in depth study of the issues addressed through the years by the researchers is next presented both in theory and applications of moments. The most representative works in each research direction are discussed in a chronological order to point out the progress in each specific field of action. This analysis concludes with the challenges and perspectives that should motivate researchers towards the promotion of the moments and their invariants to new scientific “horizons". For the first time, this chapter gives a global overview of what happened in the last 50 years in moments and moment invariants research field, but most of all it brings to light the open issues that should be addressed and highlights the rising topics that will occupy the scientists in the coming years. This chapter serves as a guide to those who find the field of moments and moment invariants a “brilliant field of action" since it encloses all the milestones of this field.

Abstract

Moment methods have been the subject of intensive research since the concept of image moments was introduced by Hu in 1962 [6]. Different types of conventional continuous orthogonal moments, defined in the rectangular region and circular domain, have been investigated as the unique image features for applications in fields of pattern recognition and image analysis. For a general study of continuous orthogonal moments, please refer to [11, 12, 4].
In this chapter, we will conduct the accuracy analysis of continuous moment functions defined in both the rectangular region and circular domain, analyze the computational errors of those moment functions, and propose solutions to improve the computing accuracy of moments, especially for the higher order moment functions.

Abstract

In most computer vision applications, the extraction of key image features, whatever the transformations applied and the image degradations observed, is of major importance. A large diversity of approaches has been reported in the literature. This chapter concentrates on one particular processing frame: the moment-based methods.
It provides a survey of methods proposed so far for the derivation of moment invariants to geometric transforms and blurring effects. Theoretical formulations and some selected examples dealing with very different problems are given.

Abstract

We give a general framework of statistical aspects of the problem of understanding and a description of image symmetries, by utilizing the theory of moment invariants.
In particular, we examine the issues of joint symmetry estimation and detection. These questions are formulated as the statistical decision and estimation problems since we cope with images observed in the presence of noise. The estimation/detection procedures are based on the minimum L2-distance between the reconstructed image function and the reconstruction of its hypothesized symmetrical version. Our reconstruction algorithms are relying on a class of radial orthogonal moments. The proposed symmetry estimation and detection techniques reveal some statistical optimality properties. Our technical developments are based on the statistical theory of nonparametric testing and semi-parametric inference.

Abstract

We propose a novel algorithm for image deconvolution from the geometric moments (GMs) of a degraded image by a circular or elliptical Gaussian point-spread function (PSF). In the proposed scheme, to show the invertibility of the moment equation in a closed form, we establish a relationship between the moments of the degraded image and the moments of the original image and the Gaussian PSF. The proposed inverted formula paves the way to reconstruct the original image using the Stirling numbers of the first kind. We validate the theoretical analysis of the proposed scheme and confirm its feasibility through the comparative studies.

Abstract

Orthogonal moment functions based on Tchebichef polynomials have found several applications in the field of image analysis because of their superior feature representation capabilities. Local features represented by such moments could also be used in the design of efficient texture descriptors. This chapter introduces a novel method of constructing feature vectors from orthonormal Tchebichef moments evaluated on 5 X 5 neighborhoods of pixels, and encoding the texture information as a Lehmer code that represents the relative strengths of the evaluated moments. The features will be referred to as Local Tchebichef Moments (LTMs). The encoding scheme provides a byte value for each pixel, and generates a gray-level “LTM-image” of the input image. The histogram of the LTM-image is then used as the texture descriptor for classification.
The theoretical framework as well as the implementation aspects of the descriptor are discussed in detail.

Abstract

This chapter introduces 2D and 3D Gaussian – Hermite moments and rotation invariants constructed from them. Thanks to their numerical stability, Gaussian – Hermite moments provide better reconstruction and recognition power than the geometric and most of other orthogonal moments while keeping the simplicity of design of the invariants. This is illustrated by experiments on real 2D and 3D data.

Abstract

We present a detailed analysis of the Jacobi-Fourier moments and their applications in digital image processing. In order to reach numerical stability during the computation of the Jacobi radial polynomials a recursive approach is described. Also, some discussions are done about the best values of the parameters α and β in terms of its performance. Moreover, the digital image applications studied here are divided in low or high orders n of the polynomials. Typically, the pattern recognition applications are based in low order polynomials whilst image reconstruction can be achieved by using high order polynomials. On the other hand, the polar pixel approach is taken into account, in order to increase the numerical accuracy in the calculation of the moments, also some ad hoc cases using this polar geometry are studied. Experiments and results are presented.

Abstract

Methods based on moments and moment invariants have been extensively used in image analysis tasks but rarely in audio applications. However, while images are typically two-dimensional (2D) and audio signals are one-dimensional (1D), many studies have showed that image analysis techniques can be successfully applied on audio after 1D audio signal is converted into a 2D time-frequency auditory image. Motivated by these observations, in this chapter we propose using moments to solve an important problem of audio analysis, i.e., music identification. Especially, we focus on music identification in the compressed domain since nowadays compressed-format audio has grown into the dominant way of storing and transmitting music.
There have been different types of moments defined in the literature, among which we choose to use Zernike moments to derive audio feature for music identification. Zernike moments are stable under many image transformations, which endows our music identification system with strong robustness against various audio distortions. Experiments carried out on a database of 21,185 MP3 songs show that even when the music queries are seriously distorted, our system can still achieve an average top-5 hit rate of up to 90% or above.

Abstract

The rapid growth of the Internet and multimedia information has generated a need for technical indexing and searching of multimedia information, especially in image retrieval. Image searching systems have been developed to allow searching in image databases. However, these systems are still inefficient in terms of semantic image searching by textual query. To perform semantic searching, it is necessary to be able to transform the visual content of the images (colours, textures, shapes) into semantic information. This transformation, called image annotation, assigns a legend or keywords to a digital image. The traditional methods of image retrieval rely heavily on manual image annotation which is very subjective, very expensive and impossible given the size and the phenomenal growth of currently existing image databases. Therefore it is quite natural that the research has emerged in order to find a computing solution to the problem. It is thus that research work has quickly bloomed on the automatic image annotation, aimed at reducing both the cost of annotation and the semantic gap between semantic concepts and digital low-level features. One of the approaches to deal with image annotation is image classification. From the segmented image, the feature vector is calculated and fed to the classifier in order to choose the appropriate keyword for each region that represents the image content. In this chapter, the use of Hu, Zernike and Legendre moments as feature extraction will be presented.

Abstract

The term adaptivity is absent from the state-of-the-art moment-based image water-marking methods. A question to be answered is whether adaptive watermark insertion will guide to the enhancement of image’s security (concerning a number of requirements such as robustness, imperceptibility, complexity and capacity). Initially, the term adaptivity is being unfold from different perspectives; the selection of the most qualified coefficients (considering their order and magnitude) for carrying the watermark information; the selection of the most qualified image region for hosting the watermark information; and finally the optimum calibration of the quantizer parameters for embedding the watermark information. An experimental justification of the need for adaptivity is being presented, highlighting also the classic tradeoff between imperceptibility and robustness. Furthermore, a number of solutions for each adaptivity perspective are presented along with its corresponding analysis (limitations and future work). To the best of our knowledge, the current chapter constitutes the primary attempt for highlighting/justifying the significance of adaptivity during moment-based watermarking process providing also the readers with a number of tools (adaptivity solutions) that function in gray-scale and color space. Next generation moment-based image watermarking algorithms should consider and benefit from the current adaptivity solutions regarding a high quality security result.

Abstract

Image features play a vital role in image retrieval. This chapter presents the use of Zernike moment features for retrieving the binary and gray level images from established image databases. To retrieve a set of similar category of images from an image database, up to 25 Zernike moment features from order zero to order 8 were utilized and experimented in this chapter. A total of 1400 binary images from MPEG-7 dataset and 1440 images from a COIL-20 dataset were used to evaluate the capability of Zernike moments features for image retrieval. The experimental results show that Zernike moments implementation is suitable for image retrieval due to rotation invariance and fast computation.