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This paper is aimed at 3D object understanding from 2D images, including articulated objects in active vision environment, using interactive, and internet virtual reality techniques. Generally speaking, an articulated object can be divided into two portions: main rigid portion and articulated portion. It is more complicated that "rigid" object in that the relative positions, shapes or angles between the main portion and the articulated portion have essentially infinite variations, in addition to the infinite variations of each individual rigid portions due to orientations, rotations and topological transformations. A new method generalized from linear combination is employed to investigate such problems. It uses very few learning samples, and can describe, understand, and recognize 3D articulated objects while the objects status is being changed in an active vision environment.

In dealing with large volume image data, sequential methods usually are too slow and unsatisfactory. This paper introduces a new system employing parallel matching in high-level recognition of 3D articulated objects. A new structural strategy using linear combination and parallel graphic matching techniques is presented for 3D polyhedral objects representable by 2D line-drawings. It solves one of the basic concerns in diffusion tomography complexities, i.e. patterns can be reconstructed through fewer projections, and 3D objects can be recognized by a few learning sample views. It also improves some of the current methods while overcoming their drawbacks. Furthermore, it can distinguish very similar objects and is more accurate than other methods in the literature. An online webpage system for understanding and recognizing 3D objects is also illustrated.

On the basis of previous studies, we explore the approximation of continuous functions with fractal structure. We first give the calculation of fractal dimension of the linear combination of continuous functions with different Hausdorff dimension. Fractal dimension estimation of the linear combination of continuous functions with the same Hausdorff dimension has also been discussed elementary. Then, based on Weierstrass Theorem and the related results of Weierstrass function, we give the conclusion that the linear combination of polynomials with the same Hausdorff dimension approximates the objective function. The corresponding results with noninteger and integer Hausdorff dimensions have been investigated. We also give the preliminary applications of the theory in the last section.

Missing data is a usual drawback in many real-world applications of pattern classification. Methods of pattern classification with missing data are grouped into four types: (a) deletion of incomplete samples and classifier design using only the complete data portion, (b) imputation of missing data and learning of the classifier using the edited set, (c) use of model-based procedures and (d) use of machine learning procedures. These methods can be useful in case of small amount of missing values, but they may be unsuitable in case of relatively large amount of missing values. We proposed a method to design pattern classification model with block missing training data. First, we separated submatrices from the block missing training data. Second, we designed classification submodels using each submatrix. Third, we designed final classification model using a linear combination of these submodels. We tested the classifying accuracy rate and data usage rate of the classification model designed by means of the proposed method by simulation experiments on some datasets, and verified that the proposed method was effective from the viewpoint of classifying accuracy rate and data usage rate.

This paper investigates the regularity of a class of differential operators generated by Szász–Mirakjan operators. With the aid of the regularity, the relation between the uniform saturated approximation order of linear combinations of Szász–Mirakjan operators and the smoothness of the approximated functions is studied. The saturation class for linear combinations of Szász–Mirakjan operators is characterized.

In this paper, we investigate the $L_p$-saturation of linear combinations of Szász–Mirakjan–Kantorovich operators. The main difficulty is how to tackle the differential operators deduced by Szász–Mirakjan–Kantorovich operators. We firstly investigate the regularity of these differential operators. By virtue of the regularity, the saturation theorem for linear combinations of Szász–Mirakjan–Kantorovich operators is ultimately obtained. Namely, the $L_p$-saturation class of these combinations is characterized.

The paper presents an elementary and unified approach to vector spaces over fields of order greater than or equal to one (the latter reducing to sets), based on three key principles. Firstly, use of quasigroups enables the field concept to be redefined in a way that admits a field of order one. Secondly, use of hyperquasigroups provides a recursive definition of linear combination that applies equally well to vector spaces over fields of order greater than or equal to one. Thirdly, it is recognized that relations rather than functions provide the correct morphisms for a category of sets to behave like categories of vector spaces over fields of order greater than one.

Let be a frame for Hilbert space H. The purpose of this paper is to present an approximation formula of any f ∊ H by a linear combination of finitely many frame elements in the frame and show that the obtained approximation error depends on the bounds of frame and the convergence rate of frame coefficients of f as well as the relation among frame elements.

This paper is aimed at 3D object understanding from 2D images, including articulated objects in active vision environment, using interactive, and internet virtual reality techniques. Generally speaking, an articulated object can be divided into two portions: main rigid portion and articulated portion. It is more complicated that “rigid” object in that the relative positions, shapes or angles between the main portion and the articulated portion have essentially infinite variations, in addition to the infinite variations of each individual rigid portions due to orientations, rotations and topological transformations. A new method generalized from linear combination is employed to investigate such problems. It uses very few learning samples, and can describe, understand, and recognize 3D articulated objects while the objects status is being changed in an active vision environment.

The improved resampling algorithms commonly in particle filter (PF), increase particles’ diversity by making new particles with various methods, and thus improve PF’s accuracy. However, they also increase the distance of particle probability distribution before resampling and reduce theactual estimation accuracy. To solve this problem, thispaper proposes an improved Gaussian resampling(IGR) algorithm, based on Gaussian Resampling (GR) Algorithm. Under the premise of maintaining the diversity of particles, we enable new particles to contain part of the low-weighted particles’ information by conducting proper linear combination with low-weighted particles. Simulation experiments conducted on single variable non-growth model suggest that the improved algorithm reduces the particles’ Kullback-Leibler(K-L) distance and improves the final tracking accuracy of PF.