Simple and fast computation of moments

Discrete Chebyshev moments (due to discrete polynomial basis) do not have the discretization errors that continuous-domain Legendre and Zernike moments contain. Calculation of polynomial basis coefficients of discrete moments is generally performed using recurrence relationships. Such recurrence equations cause numerical error accumulation especially for calculation of higher-order moments and for larger image sizes, causing significant degradation of image reconstruction from these moments. A method for better image reconstruction from high orders of discrete Chebyshev moments is demonstrated. This is accomplished by calculating Chebyshev polynomial coefficients directly from their definition formulas using arbitrary precision arithmetic and by forming lookup tables from these coefficients.

One of the main contributions of this paper is to show that contrary to the common belief Tchebichef moments are more sensitive to noise added to the image than Zernike moments. We show this by imposing different types and levels of noise on various images and measuring the error due to the added noise alone after image reconstruction. To calculate the error due to added noise alone, the image reconstruction error is subtracted from the total error, which initially includes both reconstruction and added noise errors. Another contribution of this paper is to present some analysis and compare the reconstruction capability of Tchebichef moments to that of Zernike and Legendre moments in noise free environments independent of the image used. We show this by spectrally analyzing the performance of Tchebichef, Legendre, and Zernike moments using single frequency images. Here the total reconstruction error of an arbitrary image is equal to the integration of the spectral errors, whose transfer functions are calculated using single frequency images, for all frequencies.

The fast computation of Zernike moments from normalized geometric moments has been developed in this paper. The computation is multiplication free and only additions are needed to generate Zernike moments. Geometric moments are generated using Hatamian's filter up to high orders by a very simple and straightforward computation scheme. Other kinds of moments (e.g., Legendre, pseudo Zernike) can be computed using the same algorithm after giving the proper transformations that state their relations to geometric moments. Proper normalizations of geometric moments are necessary so that the method can be used in the efficient computation of Zernike moments. To ensure fair comparisons, recursive algorithms are used to generate Zernike polynomials and other coefficients. The computational complexity model and test programs show that the speed-up factor of the proposed algorithm is superior with respect to other fast and/or direct computations. It perhaps is the first time that Zernike moments can be computed in real time rates, which encourages the use of Zernike moment features in different image retrieval systems that support huge databases such as the XM experimental model stated for the MPEG-7 experimental core. It is concluded that choosing direct computation would be impractical.

Agekyan's statistical criterion (Agekyan, 1982) was used for the calculation of higher order moments and finding corresponding confidence intervals.

Feature descriptors for 3-D images have recently gained considerable attention in application for games, virtual reality environment and solid modeling. Numerous research had been introduced for 3-D invariants of geometric moments, complex moments and Zernike moments. In this paper, we present a theoretical framework to derive translation and scale invariants for 3-D Legendre moments, by using indirect and direct methods. Indirect method generates 3-D Legendre invariants from the existing 3-D geometric moment invariants. Direct method, on the other hand, eliminates the displacement and scale factors from Legendre polynomials to generate translation and scale invariants. Experiment using 3-D binary images are carried out to verify the proposed feature descriptors.

A comprehensive treaty on the theory and applications of moment functions in image analysis. Moment functions are widely-used in various realms of computer vision and image processing. Numerous algorithms and techniques have been developed using image moments, in the areas of pattern recognition, object identification, three-dimensional object pose estimation, robot sensing, image coding and reconstruction. This is a compilation of the theoretical aspects related to different types of moment functions, and their applications in the above areas. The book is organized in two parts. The first part discusses the fundamental concepts behind important moments such as geometric moments, complex moments, Legendre moments, Zernike moments, and moment tensors. Most of the commonly-used properties of moment functions and the mathematical framework for the derivation of basic theorems and results are discussed. This includes the derivation of moment invariants, implementation aspects of moments, transport properties, and fast methods for computing the moment functions for both binary and gray-level images. The second part presents the key application areas of moments, such as pattern recognition, object identification, image-based pose estimation, edge detection, clustering, segmentation, coding and reconstruction. Important algorithms in each of these areas are discussed. A list of bibliographic references on image moments is also included.

ABSTRACTFor the exponential families normal, gamma, beta, Poisson, and negative binomial, there exists an expectation identity for each of the family. For the binomial family, we discover an expectation identity, which is useful in analytical calculations of its high-order moments.

Pseudo-Zernike moments have better feature representation capability, and are more robust to image noise than those of the conventional Zernike moments. However, due to the computation complexity of pseudo-Zernike polynomials, pseudo-Zernike moments are yet to be extensively used as feature descriptors as compared to Zernike moments. In this paper, we propose two new algorithms, namely coefficient method and p-recursive method, to accelerate the computation of pseudo-Zernike moments. Coefficient method calculates polynomial coefficients recursively. It eliminates the need of using factorial functions. Individual order or index of pseudo-Zernike moments can be derived independently, which is useful if selected orders or indices of moments are needed as pattern features. p-recursive method uses a combination of lower order polynomials to derive higher order polynomials with the same index q. Fast computation is achieved because it eliminates the requirements of calculating polynomial coefficients, Bpqk, and power of radius, rk, in each polynomial. The performance of the proposed algorithms on moment computation and image reconstruction, as compared to those of the present methods, are experimentally verified using a set of binary and grayscale images.

Many financial crises have occurred in recent decades, such as the International Debt Crisis of 1982, the East Asian Economic Crisis of 1997–2001, the Russian economic crisis of 1992–1997, the Latin American debt Crisis of 1994–2002, the Global Economic Recession of 2007–2009, which had a strong impact on international relations. The aim of this article is to create an econometric model of the indicator for identifying crisis situations arising in stock markets. The approach under consideration includes data for preprocessing and assessing the stability of the trend of time series using higher-order moments. The results obtained are compared with specific practical situations. To test the proposed indicator, real data of the stock indices of the USA, Germany and Hong Kong in the period World Financial Crisis are used. The scientific novelty of the results of the article consists in the analysis of the initial and given initial moments of high order, as well as the central and reduced central moments of high order. The econometric model of the indicator for identifying crisis situations arising considered in the work, based on high-order moments plays a pivotal role in crisis detection in stock markets, influencing financial innovations in managing the national economy. The findings contribute to the resilience and adaptability of the financial system, ultimately shaping the trajectory of the national economy. By facilitating timely crisis detection, the model supports efforts to maintain economic stability, thereby fostering sustainable growth and resilience in the face of financial disruptions. The model’s insights can shape the national innovation ecosystem by guiding the development and adoption of monetary and financial innovations that are aligned with the economy’s specific needs and challenges.

The nodal integral method (NIM) has been widely used to solve multidimensional steady-state convection-diffusion problems. However, unphysical oscillating behavior arises when NIM is applied to steep-gradient problems and discontinuous problems. In this paper, a new nodal expansion method (NEM) with high-order moments (NEM_HM) is developed to reduce the numerical oscillation drawback of NIM. High-order moments of transverse-integrated variables are introduced. Based on the definition of Legendre moments, all the expansion coefficients of NEM_HM can be defined as shared moments and unshared moments. Then, the calculation framework of the traditional NEM is extended to include the high-order moments. Additional nodal balance equations are introduced to ensure the uniqueness of all the shared variables such as node-average variables. Finally, coupled discrete equations are obtained in terms of various order moments on the surfaces of the nodes. The classical Smith-Hutton problem and a cross-flow problem are chosen to test the effectiveness of NEM_HM. Numerical results show that the accuracy of NEM_HM outperforms NIM for steep-gradient problems and discontinuous cases.

An Efficient Method for Computation of Legendre Moments

Efficient Legendre moment computation for grey level images

This paper proposes an approach combining the set of Krawtchouk moments and HMMs (Hidden Markov Models) for unconstrained offline handwritten Chinese character recognition. The Krawtchouk moments can be effectively used as pattern features in analysis of two-dimensional images. Since the basis set is orthogonal in the discrete domain of the image coordinate space, the implementation of Krawtchouk moments doesn't involve any numerical approximation or coordinate transformation. This property makes Krawtchouk moments superior to conventional, more continuous moments such as Legendre moments and Zernike moments. In the field of character recognition, HMMs architecture has been used widely because of its strong adaptability to written distortion and variability. In this paper, Krawtchouk moments and HMMs are combined to recognize unconstrained offline handwritten Chinese amount in words. Experiments show that the average recognition of Krawtchouk is higher than Zernike and Legendre moments within the HMM framework.

Computation of higher-order cross moments based on matrix multiplication

Higher-order Statistics (HOS), also known as cumulants, cross moments and their frequency domain counterparts have emerged as a powerful signal processing tool for the synthesis and analysis of signals and systems. The computation of higher-order cross moments is a necessary step towards the estimation of other statistics such as the higher-order cumulants and Wigner Spectra. In this paper, we present FPGA based design for the computation of third-order cross moments. The proposed design is coded in VHSIC Hardware Description Language (VHDL) and functionally verified by synthesizing it on Xilinx Spartan-3 XC3S2000FG900-4 FPGA chip. FPGA implementation results are presented. The proposed design operates at a maximum frequency of 86.618 MHz.