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Abstract: We study a spline-based approximation of vector fields in the conservative case (the gradient vector field derives from a potential function). We introduce a minimization problem on a Hilbert space for which the existence and uniqueness of the solution is given. We apply this approach to a registration process in image processing. [Copyright &y& Elsevier]

Fuzzy transforms (F-transforms) have shown to be a powerful technique to deal with several tasks of image processing and time series. However, F-transforms have an important handicap in order to be applied to data in a broader sense: They are by definition applicable only to functions. This paper presents an extension of the definition of F-transforms to a more general structure of data than functions. Moreover, we provide a theoretical study by showing how the fundamental properties of F-transforms are extended to this new approach. Finally, we include also a section to explore potential applications of this novel extension. [ABSTRACT FROM AUTHOR]

In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and second-order wedgelets. Unlike all known geometrical methods used in adaptive image approximation, smoothlets are continuous functions. They can adapt to location, size, rotation, curvature, and smoothness of edges. The M-term approximation of smoothlets is O(M^-3). In this paper, an image compression scheme based on the smoothlet transform is also presented. From the theoretical considerations and experiments, both described in the paper, it follows that smoothlets can assure better image compression than the other known adaptive geometrical methods, namely, wedgelets and second-order wedgelets. [ABSTRACT FROM AUTHOR]

Recently, weighted local mean operators are widely used in image processing, compressive sensing and other areas. A weighted local mean operator changes its characteristics depending on a function content within a local area in order to preserve the function features. The directional diffusion filter and Yaroslavsky neighbourhood filter (also called the sigma filter) are discrete versions of such operators. Although these operators are not convolution ones, due to their sparsity, the corresponding numerical algorithms have simple structure and fast performance. In this paper, we study the approximate properties of the weighted local mean operators, particularly focus on their asymptotic expansions, which are related to non-linear diffusion equations. [ABSTRACT FROM AUTHOR]

Abstract: Our aim is to discuss the problem of approximating the boundary of an object in a binary image. In contrast to earlier papers on similar topics, we avoid finding contours and then their approximations. Instead, we sample an image in a relatively small number of points (1%–3% of pixels) at random. The collected data is used for imposing constraints on parameters of a radial basis functions (RBF) neural net. It is proved that if the RBF net structure is sufficiently rich to approximate an object boundary, then estimates of RBF net parameters tend to their true values, as the number of samples approaches to infinity. Then, a seemingly linear algorithm of estimating linear weights is proposed and its consistency is also proved. Its implementation, which is based on an iterative use of the linear programming, is briefly discussed and the results of its testing are shortly reported. [Copyright &y& Elsevier]

Analysis of functions on combinatorial graphs is an emerging field attracting more and more attention. In this paper, we study the approximation of functions defined on combinatorial graphs by functions in Paley–Wiener spaces. First, we use a family of graph translation operators to define the modulus of smoothness, which has several properties similar to their counterparts in the classical approximation theory. Next, we establish Jackson's and Bernstein's inequalities for functions defined on graphs. Finally, we provide an estimation on the decay of graph Fourier coefficients in terms of the modulus of smoothness. These results lead to a theory of approximation of functions on combinatorial graphs and have potential applications to filtering, denoising, data dimension reduction, image processing and learning theory. [ABSTRACT FROM AUTHOR]

Abstract: A nondeterministic weighted finite automaton (WFA) maps an input word to a numerical value. Applications of weighted automata include formal verification of quantitative properties, as well as text, speech, and image processing. Many of these applications require the WFAs to be deterministic, or work substantially better when the WFAs are deterministic. Unlike NFAs, which can always be determinized, not all WFAs have an equivalent deterministic weighted automaton (DWFA). In Mohri (1997) [22], Mohri describes a determinization construction for a subclass of WFA. He also describes a property of WFAs (the twins property), such that all WFAs that satisfy the twins property are determinizable and the algorithm terminates on them. Unfortunately, many natural WFAs cannot be determinized. In this paper we study approximated determinization of WFAs. We describe an algorithm that, given a WFA and an approximation factor , constructs a DWFA that -determinizes . Formally, for all words , the value of in is at least its value in and at most times its value in . Our construction involves two new ideas: attributing states in the subset construction by both upper and lower residues, and collapsing attributed subsets whose residues can be tightened. The larger the approximation factor is, the more attributed subsets we can collapse. Thus, -determinization is helpful not only for WFAs that cannot be determinized, but also in cases determinization is possible but results in automata that are too big to handle. We also describe a property (the -twins property) and use it in order to characterize -determinizable WFAs. Finally, we describe a polynomial algorithm for deciding whether a given WFA has the -twins property. [Copyright &y& Elsevier]

The spatial-based method has become the most widely used method in improving the visibility of images. The visibility improving is mainly to remove the noise in the image, in order to trade off denoising and detail maintaining. A novel adaptive non-local means-based nonlinear fitting method is proposed in this paper. Firstly, according to the smoothness of the intensity around the central pixel, eight kinds of templates with different precision are exploited to approximate the central pixel through a novel adaptive non-local means filter design; the approximate weight coefficients of templates are derived from the approximation credibility. Subsequently, the fractal correction is used to smooth the denoising results. Eventually, the Rockafellar multiplier method is employed to generalize the smooth plane fitting to any geometric surface, thus yielding the optimal fitting of the center pixel approximation. Through a large number of experiments, it is clearly elucidated that compared with the classical spatial iteration-based methods and the recent denoising algorithms, the proposed algorithm is more robust and has better effect on denoising, while keeping more original details during denoising. [ABSTRACT FROM AUTHOR]

Abstract: Metric space is a universal and versatile model of similarity that can be applied in various areas of information retrieval. However, a general, efficient, and scalable solution for metric data management is still a resisting research challenge. We introduce a novel indexing and searching mechanism called Metric Index (M-Index) that employs practically all known principles of metric space partitioning, pruning, and filtering, thus reaching high search performance while having constant building costs per object. The heart of the M-Index is a general mapping mechanism that enables to actually store the data in established structures such as the B+-tree or even in a distributed storage. We implemented the M-Index with the B+-tree and performed experiments on two datasets—the first is an artificial set of vectors and the other is a real-life dataset composed of a combination of five MPEG-7 visual descriptors extracted from a database of up to several million digital images. The experiments put several M-Index variants under test and compare them with established techniques for both precise and approximate similarity search. The trials show that the M-Index outperforms the others in terms of efficiency of search-space pruning, I/O costs, and response times for precise similarity queries. Further, the M-Index demonstrates excellent ability to keep similar data close in the index which makes its approximation algorithm very efficient—maintaining practically constant response times while preserving a very high recall as the dataset grows and even beating approaches designed purely for approximate search. [Copyright &y& Elsevier]

Some topics related to scattered data approximation and function approximation by linear superposition of basis functions are outlined. A new method to speed up the evaluation of the approximation is presented which is particularly useful when very large sets of scattered data are involved as in hypersurface reconstruction, image recognition, speech, and processing. This method is based on the technique of principal components analysis and allows us to select and use only the salient features needed to correctly classify patterns. The error that this technique introduces is analyzed in the context of the application to sigmoidal and radial neural networks and overestimations for it are given. The new method is also compared with some other feature selection techniques to illustrate how to apply it and to show its effectiveness. [Copyright &y& Elsevier]