We first determine the asymptotes of the $\epsilon$-covering numbers of Hölder-Zygmund type spaces on data-defined manifolds. Secondly, a fully discrete and finite algorithmic scheme is developed providing explicit $\epsilon$-coverings whose cardinality is asymptotically near the $\epsilon$-covering number. Given an arbitrary Hölder-Zygmund type function, the nearby center of a ball in the $\epsilon$-covering can also be computed in a discrete finite fashion.

Matrix completion that estimates missing values in visual data is an important topic in computer vision. Most of the recent studies focused on the low rank matrix approximation via the nuclear norm. However, the visual data, such as images, is rich in texture which may not be well approximated by low rank constraint. In this paper, we propose a novel matrix completion method, which combines the nuclear norm with the local geometric regularizer to solve the problem of matrix completion for redundant texture images. And in this paper we mainly consider one of the most commonly graph regularized parameters: the total variation norm which is a widely used measure for enforcing intensity continuity and recovering a piecewise smooth image. The experimental results show that the encouraging results can be obtained by the proposed method on real texture images compared to the state-of-the-art methods.

Ant colony optimization (ACO) is often used to solve optimization problems, such as traveling salesman problem (TSP). When it is applied to TSP, its runtime is proportional to the squared size of problem $N$ so as to look less efficient. The following statistical feature is observed during the authors’ long-term gene data analysis using ACO: when the data size $N$ becomes big, local clustering appears frequently. That is, some data cluster tightly in a small area and form a class, and the correlation between different classes is weak. And this feature makes the idea of divide and rule feasible for the estimate of solution of TSP. In this paper an improved ACO algorithm is presented, which firstly divided all data into local clusters and calculated small TSP routes and then assembled a big TSP route with them. Simulation shows that the presented method improves the running speed of ACO by 200 factors under the condition that data set holds feature of local clustering.

This paper describes a method for nonrigid registration of monomodal MRI based on physical laws. The proposed method assumes that the properties of image deformations are like those of viscoelastic matter, which exhibits the properties of both an elastic solid and a viscous fluid. Therefore, the deformation fields of the deformed image are constrained by both sets of properties. After global registration, the local shape variations are assumed to have the properties of the Maxwell model of linear viscoelasticity, and the deformation fields are constrained by the corresponding partial differential equations. To speed up the registration, an adaptive force is introduced according to the maximum displacement of each iteration. Both synthetic datasets and real datasets are used to evaluate the proposed method. We compare the results of the linear viscoelastic model with those of the fluid model on the basis of both the standard and adaptive forces. The results demonstrate that the adaptive force increases in both models and that the linear viscoelastic model improves the registration accuracy.

We propose a new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with and median correction. Analyzing the difficulties to choose the fractional orders and unsatisfied processing results in signal processing using fractional-order partial differential equations and related methods; we think that the nonzero values of the truncated fractional order derivatives in the smooth regions are major causes for these situations. In order to resolve the problem, the absolute values of truncated parts of the G-L formula are estimated by the median of signal values of the remainder parts, and then the truncated G-L formula is modified by replacing each of the original signal value to the differences of the signal value and the median. Since the sum of the coefficients of the G-L formula is zero, the median correction can reduce the truncated errors greatly to proximate G-L formula better. We also present some simulation results and experiments to support our theory analysis.

We study fractional-order derivatives of left-handed Grünwald-Letnikov formula with to detect and locate singularities in theory. The widely used four types of ideal singularities are analyzed by deducing their fractional derivative formula. The local extrema of fractional derivatives are used to locate the singularities. Theory analysis indicates that fractional-order derivatives of left-handed Grünwald-Letnikov formula with can detect and locate four types of ideal singularities correctly, which shows better performance than classical 1-order derivatives in theory.

The forest new gap models via local fractional calculus are investigated. The JABOWA and FORSKA models are extended to deal with the growth of individual trees defined on Cantor sets. The local fractional growth equations with local fractional derivative and difference are discussed. Our results are first attempted to show the key roles for the nondifferentiable growth of individual trees.

We propose a numerical reconstruction method for solving a time-fractional backward heat conduction problem. Based on the idea of reproducing kernel approximation, we reconstruct the unknown initial heat distribution from a finite set of scattered measurements of transient temperature at a fixed final time. The standard Tikhonov regularization technique using the norm of reproducing the kernel Hilbert space as the penalty term is adopted to provide a stable solution when the measurement data contains noise. Numerical results indicate that the proposed method is efficient.

We first investigate the construction of a fusion frame system in a finite-dimensional Hilbert space ${𝔽}^n$ when its fusion frame operator matrix is given and provides a corresponding algorithm. The matrix representations of its local frame operators and inverse frame operators are naturally obtained. We then study the related properties of the constructed fusion frame systems. Finally, we implement the construction of fusion frame systems which behave optimally for erasures in some special sense in signal transmission.

Fractional diffusion equation in fractal media is an integropartial differential equation parametrized by fractal Hausdorff dimension and anomalous diffusion exponent. In this paper, the similarity solution of the fractional diffusion equation was considered. Through the invariants of the group of scaling transformations we derived the integro-ordinary differential equation for the similarity variable. Then by virtue of Mellin transform, the probability density function $p(r,t)$, which is just the fundamental solution of the fractional diffusion equation, was expressed in terms of Fox functions.

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integers p and q, we wish to introduce an alternative definition of relative $(p,q)$th order which improves the earlier definition of relative $(p,q)$th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative $(p,q)$th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.

The $Z$-transform has played an important role in signal processing. In this paper the $Z$-transform has been generalized by the coupling of both the $Z$-transform and the local fractional complex calculus. In the literature the local fractional $Z$-transform is applied to analyze signals, in the following it will be used to analyze signals on Cantor sets. Some examples are also given to show the efficiency and accuracy for handling the signals on Cantor sets.

The discrimination of seismic event and nuclear explosion is a complex and nonlinear system. The nonlinear methodologies including Random Forests (RF), Support Vector Machines (SVM), and Naïve Bayes Classifier (NBC) were applied to discriminant seismic events. Twenty earthquakes and twenty-seven explosions with nine ratios of the energies contained within predetermined “velocity windows” and calculated distance are used in discriminators. Based on the one out cross-validation, ROC curve, calculated accuracy of training and test samples, and discriminating performances of RF, SVM, and NBC were discussed and compared. The result of RF method clearly shows the best predictive power with a maximum area of 0.975 under the ROC among RF, SVM, and NBC. The discriminant accuracies of RF, SVM, and NBC for test samples are 92.86%, 85.71%, and 92.86%, respectively. It has been demonstrated that the presented RF model can not only identify seismic event automatically with high accuracy, but also can sort the discriminant indicators according to calculated values of weights.

We introduce a new combination of Bernstein polynomials (BPs) and Block-Pulse functions (BPFs) on the interval [0, 1]. These functions are suitable for finding an approximate solution of the second kind integral equation. We call this method Hybrid Bernstein Block-Pulse Functions Method (HBBPFM). This method is very simple such that an integral equation is reduced to a system of linear equations. On the other hand, convergence analysis for this method is discussed. The method is computationally very simple and attractive so that numerical examples illustrate the efficiency and accuracy of this method.

The Fourier law of one-dimensional heat conduction equation in fractal media is investigated in this paper. An approximate solution to one-dimensional local fractional Volterra integral equation of the second kind, which is derived from the transformation of Fourier flux equation in discontinuous media, is considered. The Picard successive approximation method is applied to solve the temperature field based on the given Mittag-Leffler-type Fourier flux distribution in fractal media. The nondifferential approximate solutions are given to show the efficiency of the present method.