On High Precision Algorithm Of The Orthogonal Moments

Since the 80s of 20th century, orthogonal moments have been concerned and studied by many scholars immediately, which are widely utilized in various fields of image processing. In comparison to other moments, orthogonal moments set have several unique advantages as follows:1 They include inverse transform, which can be used to reconstruct the primary image completely in theory; 2 They include minimum information redundancy, which largely decreases the workload of image feature extraction. However, computational accuracy and algorithm efficiency of moments still restrict their development and application in the area of pattern recognition. Discrete errors and propagation errors have irreconcilable conflicts with calculation. Therefore, it becomes a blind spot on high-precision algorithm of orthogonal moments that how to properly resolve the contradiction between precision and efficiency of the algorithm. They should be divided into two categories:one is continuous orthogonal moments by representative as Zernike moments and the other is discrete orthogonal moments by Krawtchouk moments. In this paper, according to the development of the the orthogonal moments, theories of high-precision arithmetic of the two kinds moments have been studied respectively, and their fast algorithms under the model of high-precision orthogonal moments have been derived.On the basis of previous work, several key algorithms are classified, analyzed and evaluated and some common valuation systems of algorithm errors are introduced. Algorithms of continuous moments mainly include boundary method, transform method and iterative method, while those of discrete moments include classic recursive method and symmetric method.For continuous moments, a novel algorithm so called as triangle-integration algorithm is proposed to accurately calculate Zernike moments in Cartesian Coordinates. Firstly, Zernike moments with any order and repetition are expressed as a linear combination of the Fourier-Mellin moments. And the moment integration in a pixel domain is re-arranged as a summation of four integrations in four triangle domains respectively. Secondly, we convert the classical un-precise integration formula to a set of high-accurate analytic equations in Cartesian Coordinates based on the trigonometric functions. Finally, a set of high efficient computational recursive relations are proposed. Several experiments are designed to verify the performance of the proposed algorithm.For discrete moments, the accuracy of the Krawtchouk moments for the common case of p≠0 has been discussed and a novel symmetry and bi-recursive algorithm is proposed to accurately calculate the Krawtchouk moments for the case of p∈(0,1). Firstly, the x-n plane is divided into four parts by x=n and x+n=N-1. Then We use the n-ascending recurrence formula to calculate the polynomials in the domain of N-1-n≥x≤n and apply the n-descending recurrence relations in the domain of n≥x≥N-1-n. Finally, with the help of the diagonal symmetry property on x=n, the Krawtchouk polynomial values of high precision in the whole x-n coordinates are obtained. By limiting the maximum recursion times to N/2, the algorithm ensures that the maximum recursive numerical errors are within an acceptable range. An experiment on a large image of 400×400 pixels is designed to demonstrate the performance of the proposed algorithm against the classical method.