Separating points by parallel hyperplanes--characterization problem.

This paper deals with partitions of a discrete set S of points in a d-dimensional space, by h parallel hyperplanes. Such partitions are in a direct correspondence with multilinear threshold functions which appear in the theory of neural networks and multivalued logic. The characterization (encoding) problem is studied. We show that a unique characterization (encoding) of such multilinear partitions of S = {0, 1,..., m-1}d is possible within theta(h x d2 x log m) bit rate per encoded partition. The proposed characterization (code) consists of (d + 1) x (h + 1) discrete moments having the order no bigger than 1. The obtained bit rate is evaluated depending on the mutual relations between h, d, and m. The optimality is reached in some cases.     

Separating Points by Parallel Hyperplanes— Characterization Problem

This paper deals with partitions of a discrete set S of points in a d-dimensional space, by h parallel hyperplanes. Such partitions are in a direct correspondence with multilinear threshold functions which appear in the theory of neural networks and multivalued logic. The characterization (encoding) problem is studied. We show that a unique characterization (encoding) of such multilinear partitions of S = {0,1,..., m - 1}^d is possible within O(h ldr d² ldr log to) bit rate per encoded partition. The proposed characterization (code) consists of(d+l)ldr(h+l) discrete moments having the order no bigger than 1. The obtained bit rate is evaluated depending on the mutual relations between h, d, and m. The optimality is reached in some cases.     

基于两变量Krawtchouk矩的图像重建

传统的离散正交Krawtchouk矩的基函数由两个单变量的Krawtchouk多项式乘积构成,它割裂平面两个方向之间的联系。提出了一种新的、以两变量Krawtchouk正交多项式为基函数的图像矩,并推导了正则化后两变量多项式的简单的计算方法。重建实验结果表明,相对于同系数的单变量的离散正交矩,两变量离散正交矩的重建误差更小。     

基于两变量Hahn矩的图像分析

提出了一种新的、以两变量离散正交Hahn多项式为核函数的图像矩,推导了正则化后,两变量离散正交Hahn多项式的简单的计算方法。对二值图像、灰度图像以及噪声图像的重建实验表明：相对于同系数的单变量的Hahn矩,两变量Hahn矩的重建误差更小。因此,它们能够更好地提取图像的特征。     

Analysis of large deviations behavior of multi-GPU memory access in deep learning

The unpredictable nature of irregular memory accesses in a mixed memory applications such as deep learning application poses many challenges due to the communication issues. Typically, a multi-GPU node that has a large number of simultaneous memory requests consumes almost 80% of the processing time for memory mapping. This calls for characterization of mixed regular and irregular memory accesses so that memory divergence can be simplified to improve performance. In this paper, using large deviations principle, it is shown that the mixed regular and irregular memory accesses can be viewed as a combination of continuous and discrete functions. This view point is proved to give better performance through characterization of memory divergence in multi-GPU node using the sub-additivity property. Further, a detection test procedure based on quenched large deviations model is proposed which generates threshold values for optimizing the memory mapping in data intensive applications and hence it will improve the performance.     

基于Radon变换不变矩和小波提升的水印算法

提出了一种基于Radon变换不变矩和提升小波的抗几何攻击水印算法。该方法首先对图像进行一次提升小波分解,然后计算其低频成分的Radon变换不变矩来构建水印系统。水印提取过程简单,只需计算所得图像的几个Radon变换不变矩不变量。文章给出了实验结果,并与基于几何矩不变量的算法进行了比较。经过仿真实验证明,该方法对于旋转,缩放,平移等攻击具有很好鲁棒性的同时,对于普通的加噪,滤波,JPEG压缩攻击也具有很好的鲁棒性,且具有极低误检率。     

Image quality assessment by discrete orthogonal moments

This paper proposes a novel full-reference quality assessment (QA) metric that automatically assesses the quality of an image in the discrete orthogonal moments domain. This metric is constructed by representing the spatial information of an image using low order moments. The computation, up to fourth order moments, is performed on each individual (8×8) non-overlapping block for both the test and reference images. Then, the computed moments of both the test and reference images are combined in order to determine the moment correlation index of each block in each order. The number of moment correlation indices used in this study is nine. Next, the mean of each moment correlation index is computed and thereafter the single quality interpretation of the test image with respect to its reference is determined by taking the mean value of the computed means of all the moment correlation indices. The proposed objective metrics based on two discrete orthogonal moments, Tchebichef and Krawtchouk moments, are developed and their performances are evaluated by comparing them with subjective ratings on several publicly available databases. The proposed discrete orthogonal moments based metric performs competitively well with the state-of-the-art models in terms of quality prediction while outperforms them in terms of computational speed.     

Asymptotics of the Moments of Extreme-Value Related Distribution Functions

The asymptotic cost of many algorithms and combinatorial structures is related to the extreme-value Gumbel distribution exp(-exp(-x)). The following list is not exhaustive: Trie, Digital Search Tree, Leader Election, Adaptive Sampling, Counting Algorithms, trees related to the Register Function, Composition of Integers, some structures represented by Markov chains (Column-Convex Polyominoes, Carlitz Compositions), Runs and number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Sometimes we can start from an exact (discrete) probability distribution, sometimes from an asymptotic analysis of the discrete objects (e.g., urn models) before establishing the relationship with the Gumbel distribution function. Also some Markov chains are either exactly and directly given by the structure itself, or as a limiting Markov process. The main motivation of the paper is to compute the asymptotic distribution and the moments of the random variables in question. The moments are usually given by a dominant part and a small fluctuating part. We use Laplace and Mellin transforms and singularity analysis and aim for a unified treatment in all cases. Furthermore, our goal is a purely mechanical computation of dominant and fluctuating components, with the help of a computer algebra system. We provide each time the first three moments, but the treatment is (almost) completely automatic. We need some real analysis for the approximations and apart from that only easy complex analysis; simple poles and a few special functions.     

Autonomous Path Estimation for a Descent Vehicle Using Recursive Gaussian Filters

This paper considers a refinement problem for the flight altitude, speed, and flight-path angle of an aircraft during aerodynamic deceleration in the atmosphere of Mars using inaccurate discrete g-load measurements. For solving this problem, several suboptimal discrete nonlinear filters of higher accuracy than the extended Kalman filter based on the Taylor linearization procedure with static (Gaussian) linearization of the nonlinearities are applied and compared with each other. The equations of these filters are derived, the Gaussian moments of their structural functions are calculated, and the resulting accuracy of these filters is analyzed in statistical terms.     

Image reconstruction from continuous Gaussian–Hermite moments implemented by discrete algorithm

The problem of image reconstruction from its statistical moments is particularly interesting to researchers in the domain of image processing and pattern recognition. Compared to geometric moments, the orthogonal moments offer the ability to recover much more easily the image due to their orthogonality, which allows reducing greatly the complexity of computation in the phase of reconstruction. Since the 1980s, various orthogonal moments, such as Legendre moments, Zernike moments and discrete Tchebichef moments have been introduced early or late to image reconstruction. In this paper, another set of orthonormal moments, the Gaussian–Hermite moments, based on Hermite polynomials modulated by a Gaussian envelope, is proposed to be used for image reconstruction. Especially, the paper's focus is on the determination of the optimal scale parameter and the improvement of the reconstruction result by a post-processing which make Gaussian–Hermite moments be useful and comparable with other moments for image reconstruction. The algorithms for computing the values of the basis functions, moment computation and image reconstruction are also given in the paper, as well as a brief discussion on the computational complexity. The experimental results and error analysis by comparison with other moments show a good performance of this new approach.     

Image representation using separable two-dimensional continuous and discrete orthogonal moments

This paper addresses bivariate orthogonal polynomials, which are a tensor product of two different orthogonal polynomials in one variable. These bivariate orthogonal polynomials are used to define several new types of continuous and discrete orthogonal moments. Some elementary properties of the proposed continuous Chebyshev–Gegenbauer moments (CGM), Gegenbauer–Legendre moments (GLM), and Chebyshev–Legendre moments (CLM), as well as the discrete Tchebichef–Krawtchouk moments (TKM), Tchebichef–Hahn moments (THM), Krawtchouk–Hahn moments (KHM) are presented. We also detail the application of the corresponding moments describing the noise-free and noisy images. Specifically, the local information of an image can be flexibly emphasized by adjusting parameters in bivariate orthogonal polynomials. The global extraction capability is also demonstrated by reconstructing an image using these bivariate polynomials as the kernels for a reversible image transform. Comparisons with the known moments are performed, and the results show that the proposed moments are useful in the field of image analysis. Furthermore, the study investigates invariant pattern recognition using the proposed three moment invariants that are independent of rotation, scale and translation, and an example is given of using the proposed moment invariants as pattern features for a texture classification application.     

A novel approach to the fast computation of Zernike moments

This paper presents a novel approach to the fast computation of Zernike moments from a digital image. Most existing fast methods for computing Zernike moments have focused on the reduction of the computational complexity of the Zernike 1-D radial polynomials by introducing their recurrence relations. Instead, in our proposed method, we focus on the reduction of the complexity of the computation of the 2-D Zernike basis functions. As Zernike basis functions have specific symmetry or anti-symmetry about the x-axis, the y-axis, the origin, and the straight line y=x, we can generate the Zernike basis functions by only computing one of their octants. As a result, the proposed method makes the computation time eight times faster than existing methods. The proposed method is applicable to the computation of an individual Zernike moment as well as a set of Zernike moments. In addition, when computing a series of Zernike moments, the proposed method can be used with one of the existing fast methods for computing Zernike radial polynomials. This paper also presents an accurate form of Zernike moments for a discrete image function. In the experiments, results show the accuracy of the form for computing discrete Zernike moments and confirm that the proposed method for the fast computation of Zernike moments is much more efficient than existing fast methods in most cases.     

Perfect discrete Morse functions on 2-complexes

This paper is focused on the study of perfect discrete Morse functions on a 2-simplicial complex. These are those discrete Morse functions such that the number of critical i-simplices coincides with the ith Betti number of the complex. In particular, we establish conditions under which a 2-complex admits a perfect discrete Morse function and conversely, we get topological properties necessary for a 2-complex admitting such kind of functions. This approach is more general than the known results in the literature (Lewiner et al., 2003), since our study is not restricted to surfaces. These results can be considered as a first step in the study of perfect discrete Morse functions on 3-manifolds.     

Moment-preserving piecewise linear approximations of signals andimages

Approximation techniques are an important aspect of digital signal and image processing. Many lossy signal compression procedures such as the Fourier transform and discrete cosine transform are based on the idea that a signal can be represented by a small number of transformed coefficients which are an approximation of the original. Existing approximation techniques approach this problem in either a time/spatial domain or transform domain, but not both. This paper briefly reviews various existing approximation techniques. Subsequently, we present a new strategy to obtain an approximation fˆ(x) of f(x) in such a way that it is reasonably close to the original function in the domain of the variable x, and exactly preserves some properties of the transformed domain. In this particular case, the properties of the transformed values that are preserved are geometric moments of the original function. The proposed technique has been applied to single-variable functions, two-dimensional planar curves, and two-dimensional images, and the results obtained are demonstrative     

Robust H∞ control for discrete-time polytopic uncertain systems with linear fractional vertices

The robust H∞ control problem for discrete-time uncertain systems is investigated in this paper. The uncertain systems are modelled as a polytopic type with linear fractional uncertainty in the vertices. A new linear matrix inequality (LMI) characterization of the H∞ performance for discrete systems is given by introducing a matrix slack variable which decouples the matrix of a Lyapunov function candidate and the parametric matrices of the system. This feature enables one to derive sufficient conditions for discrete uncertain systems by using parameter-dependent Lyapunov functions with less conservativeness. Based on the result, H∞ performance analysis and controller design are carried out. A numerical example is included to demonstrate the effectiveness of the proposed results.     

The effects of limited-precision weights on the threshold Adaline

The effect of limited-precision weights on the functional capability of a threshold Adaline is examined. The number of logic functions which can be implemented by the threshold Adaline serves as the primary measure of functional capability. Closed-form expressions are provided for the number of logic functions which can be implemented by a threshold Adaline with four different levels of weight precision. In addition, realization tests and procedures (based on the characterizing numbers) are provided for each level of weight precision. The number of realizable logic functions is compared to the capacity of a threshold Adaline with limited precision weights and a relationship between the two measures is proposed.     

A unified methodology for the efficient computation of discrete orthogonal image moments

A novel methodology is proposed in this paper to accelerate the computation of discrete orthogonal image moments. The computation scheme is mainly based on a new image representation method, the image slice representation (ISR) method, according to which an image can be expressed as the outcome of an appropriate combination of several non-overlapped intensity slices. This image representation decomposes an image into a number of binary slices of the same size whose pixels come in two intensities, black or any other gray-level value. Therefore the image block representation can be effectively applied to describe the image in a more compact way. Once the image is partitioned into intensity blocks, the computation of the image moments can be accelerated, as the moments can be computed by using decoupled computation forms. The proposed algorithm constitutes a unified methodology that can be applied to any discrete moment family in the same way and produces similar promising results, as has been concluded through a detailed experimental investigation.     

A numerical recipe for accurate image reconstruction from discrete orthogonal moments

Recursive procedures used for sequential calculations of polynomial basis coefficients in discrete orthogonal moments produce unreliable results for high moment orders as a result of error accumulation. This paper demonstrates accurate reconstruction of arbitrary-size images using full-order (orders as large as the image size) Tchebichef and Krawtchouk moments by calculating polynomial coefficients directly from their definition formulas in hypergeometric functions and by creating lookup tables of these coefficients off-line. An arbitrary precision calculator is used to achieve greater numerical range and precision than is possible with software using standard 64-bit IEEE floating-point arithmetic. This reconstruction scheme is content and noise independent.     

Improved quality of reconstructed images using floating point arithmetic for moment calculation

Zernike moments which are superior to geometric moments because of their special properties of image reconstruction and immunity to noise, suffer from several discretization errors. These errors lead to poor quality of reconstructed image and wide variations in the numerical values of the moments. The predominant factor, as observed in this paper, is due to the discrete integer implementation of the steps involved in moment calculation. It is shown in this paper that by modifying the algorithms to include discrete float implementation, the quality of the reconstructed image improves significantly and the first-order moment becomes zero. Low-order Zernike moments have been found to be stable under linear transformations while the high-order moments have large variations. The large variations in high-order moments, however, do not greatly affect the quality of the reconstructed image, implying that they should be ignored when numerical values of moments are used as features. The 11 functions based on geometric moments have also been found to be stable under linear transformations and thus these can be used as features. Pixel level analysis of the images has been carried out to strengthen the results.     

基于Tchebichef矩的区域复制窜改检测算法*

提出了一种新的数字图像被动认证算法,用于检测同幅图像的区域复制窜改问题。算法首先利用离散小波变换提取图像的低频分量,再对低频分量进行分块并提取每一块的离散正交Tchebichef矩特征;然后将特征矢量进行字典排序,比较相邻两组特征矢量的相似性;最后利用阈值判别实现窜改伪造区域的检测和定位。实验结果证明,算法能较好地检测及定位出图像中复制与窜改区域,且具有运算量小、检测效率高、鲁棒性好等特点。