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In this study, we focus on the extension of the trigonometric F-transform for functions in one variable to: (i) a larger domain, (ii) a higher degree of the F m -transform, and (iii) many-variable functions to improve its approximation properties over the entire domain and especially at its boundaries. In addition, the properties of approximation and convergence of direct and inverse extended and multidimensional trigonometric F m -transforms are discussed. Then, direct formulas for partial derivatives of functions of several variables are obtained in terms of trigonometric F m -transforms, which are used to solve the Cauchy problem for the transport equation. A new image compression method is proposed and compared with well-established compression methods such as JPEG, JPEG 2000 and their less complex variations JPEG(APDCBT), JPEG(APUBT3), APUBT3-NUP, and JPEG-FT. We have shown that this t F ¯ 11 -transform image compression method has high accuracy and reasonably low (irreducible) complexity. [ABSTRACT FROM AUTHOR]

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]

In image processing, the rapid approximate solution of variational problems involving generic data-fitting terms is often of practical relevance, for example in real-time applications. Variational solvers based on diffusion schemes or the Euler-Lagrange equations are too slow and restricted in the types of data-fitting terms. Here, we present a filter-based approach to reduce variational energies that contain generic data-fitting terms, but are restricted to specific regularizations. Our approach is based on reducing the regularization part of the variational energy, while guaranteeing non-increasing total energy. This is applicable to regularization-dominated models, where the data-fitting energy initially increases, while the regularization energy initially decreases. We present fast discrete filters for regularizers based on Gaussian curvature, mean curvature, and total variation. These pixel-local filters can be used to rapidly reduce the energy of the full model. We prove the convergence of the resulting iterative scheme in a greedy sense, and we show several experiments to demonstrate applications in image-processing problems involving regularization-dominated variational models. [ABSTRACT FROM AUTHOR]

In this work, we revisit fast dimension reduction approaches, as with random projections and random sampling. Our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis. Such dimension reduction can be very efficient when the signals of interest have a strong structure, such as with images. We focus on this setting and investigate feature clustering schemes for data reductions that capture this structure. An impediment to fast dimension reduction is then that good clustering comes with large algorithmic costs. We address it by contributing a linear-time agglomerative clustering scheme, Recursive Nearest Agglomeration (ReNA). Unlike existing fast agglomerative schemes, it avoids the creation of giant clusters. We empirically validate that it approximates the data as well as traditional variance-minimizing clustering schemes that have a quadratic complexity. In addition, we analyze signal approximation with feature clustering and show that it can remove noise, improving subsequent analysis steps. As a consequence, data reduction by clustering features with ReNA yields very fast and accurate models, enabling to process large datasets on budget. Our theoretical analysis is backed by extensive experiments on publicly-available data that illustrate the computation efficiency and the denoising properties of the resulting dimension reduction scheme. [ABSTRACT FROM AUTHOR]

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]

The bilateral filter is a non-linear filter that uses a range filter along with a spatial filter to perform edge-preserving smoothing of images. A direct computation of the bilateral filter requires $O(S)$ operations per pixel, where $S$ is the size of the support of the spatial filter. In this paper, we present a fast and provably accurate algorithm for approximating the bilateral filter when the range kernel is Gaussian. In particular, for box and Gaussian spatial filters, the proposed algorithm can cut down the complexity to $O(1)$ per pixel for any arbitrary $S$ . The algorithm has a simple implementation involving $N+1$ spatial filterings, where $N$ is the approximation order. We give a detailed analysis of the filtering accuracy that can be achieved by the proposed approximation in relation to the target bilateral filter. This allows us to estimate the order $N$ required to obtain a given accuracy. We also present comprehensive numerical results to demonstrate that the proposed algorithm is competitive with the state-of-the-art methods in terms of speed and accuracy. [ABSTRACT FROM AUTHOR]

Abstract: Bilateral filtering is a method often used in image processing applications. It is specifically useful for HDR algorithms. A novel approach to a fast and close approximation of bilateral filtering is presented. The method is designed especially with a focus on HDR image conversion into a normal color space processing. This paper presents the methods itself, describes the sources of acceleration and discusses the results of the method. [Copyright &y& Elsevier]

Clustering is central to many image processing and remote sensing applications. ISODATA is one of the most popular and widely used clustering methods in geoscience applications, but it can run slowly, particularly with large data sets. We present a more efficient approach to ISODATA clustering, which achieves better running times by storing the points in a kd-tree and through a modification of the way in which the algorithm estimates the dispersion of each cluster. We also present an approximate version of the algorithm which allows the user to further improve the running time, at the expense of lower fidelity in computing the nearest cluster center to each point. We provide both theoretical and empirical justification that our modified approach produces clusterings that are very similar to those produced by the standard ISODATA approach. We also provide empirical studies on both synthetic data and remotely sensed Landsat and MODIS images that show that our approach has significantly lower running times. [ABSTRACT FROM AUTHOR]

Abstract: Crystal unbending, the process that aims to recover a perfect crystal from experimental data, is one of the more important steps in electron crystallography image processing. The unbending process involves three steps: estimation of the unit cell displacements from their ideal positions, extension of the deformation field to the whole image and transformation of the image in order to recover an ideal crystal. In this work, we present a systematic analysis of the second step oriented to address two issues. First, whether the unit cells remain undistorted and only the distance between them should be changed (rigid case) or should be modified with the same deformation suffered by the whole crystal (elastic case). Second, the performance of different extension algorithms (interpolation versus approximation) is explored. Our experiments show that there is no difference between elastic and rigid cases or among the extension algorithms. This implies that the deformation fields are constant over large areas. Furthermore, our results indicate that the main source of error is the transformation of the crystal image. [Copyright &y& Elsevier]

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]

Abstract: We study input sensitive algorithms for point pattern matching under various transformations and the Hausdorff metric as a distance function. Given point sets P and Q in the plane, the problem of point pattern matching is to determine whether P is similar to some portion of Q, where P may undergo transformations from a group G of allowed transformations. All algorithms are based on methods for extracting small subsets from Q that can be matched to a small subset of P. The runtime is proportional to the number k of these subsets. Let d be the number of points in P that are needed to define a transformation in G. The key observation is that for some set of cardinality larger than d, the number of subsets of Q of this cardinality that match B, is practically small, as the problem becomes more constrained. We present methods to extract efficiently all these subsets in Q. We provide algorithms for homothetic, rigid and similarity transformations in the plane and give a general method that works for any dimension and for any group of transformations. The runtime of our algorithms depends roughly linearly on the number of subsets k, in addition to an factor. Thus our approximate matching algorithms run roughly in time , where m and n are the number of points in P and Q, respectively. The constants hidden in the big O vary depending on the group of transformations G. [Copyright &y& Elsevier]

Effectiveness of one-dimensional Bezier–Bernstein polynomial has been studied for compression of graylevel images. Both raster and Hilbert scan have been used for this purpose. Two different algorithms for one-dimensional approximation and encoding of Hilbert images provide evidence of better compression ratio compared to that for raster scanned images. Comparison with the result of an existing algorithm has also been performed to examine the effectiveness of our methods. [Copyright &y& Elsevier]

Multiple-view triangulation by $\ell _\infty$ minimisation has become established in computer vision. State-of-the-art $\ell _\infty$ triangulation algorithms exploit the quasiconvexity of the cost function to derive iterative update rules that deliver the global minimum. Such algorithms, however, can be computationally costly for large problem instances that contain many image measurements, e.g., from web-based photo sharing sites or long-term video recordings. In this paper, we prove that $\ell _\infty$ triangulation admits a coreset approximation scheme, which seeks small representative subsets of the input data called coresets. A coreset possesses the special property that the error of the $\ell _\infty$ solution on the coreset is within known bounds from the global minimum. We establish the necessary mathematical underpinnings of the coreset algorithm, specifically, by enacting the stopping criterion of the algorithm and proving that the resulting coreset gives the desired approximation accuracy. On large-scale triangulation problems, our method provides theoretically sound approximate solutions. Iterated until convergence, our coreset algorithm is also guaranteed to reach the true optimum. On practical datasets, we show that our technique can in fact attain the global minimiser much faster than current methods. [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: Neural networks are widely used in many applications including astronomical physics, image processing, recognition, robotics and automated target tracking, etc. Their ability to approximate arbitrary functions is the main reason for this popularity. The main result of this paper is a constructive proof of a formula for the upper bound of the approximation error by feedforward neural networks with one hidden layer of sigmoidal units and a linear output. The result can also be used to estimate complexity of the maximum error network. An example to demonstrate the theoretical result is given. [Copyright &y& Elsevier]

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: Neural networks are widely used in many applications including astronomical physics,image processing, recognition, robotics, and automated target tracking, etc. Their ability to approximate arbitrary functions is the main reason for this popularity. In this paper, we discuss the constructive approximation on the whole real line by a neural networks with a sigmoidal activation function and a fixed weight. Using the convolution method, we show neural network approximation with a fixed weight to a continuous function on a compact interval. Also, we demonstrate a computational work that shows good agreement with theory. [Copyright &y& Elsevier]

Development of energy-efficient and high-performance bioreactors requires progress in methods for assessing the key parameters of the biosynthesis process. With a wide variety of approaches and methods for determining the phase contact area in gas–liquid flows, the question of obtaining its accurate quantitative estimation remains open. Particularly challenging are the issues of getting information about the mass transfer coefficients instantly, as well as the development of predictive capabilities for the implementation of effective flow control in continuous fermentation both on the laboratory and industrial scales. Motivated by the opportunity to explore the possibility of applying classical and non-classical computer vision methods to the results of high-precision video records of bubble flows obtained during the experiment in the bioreactor vessel, we obtained a number of results presented in the paper. Characteristics of the bioreactor's bubble flow were estimated first by classical computer vision (CCV) methods including an elliptic regression approach for single bubble boundaries selection and clustering, image transformation through a set of filters and developing an algorithm for separation of the overlapping bubbles. The application of the developed method for the entire video filming makes it possible to obtain parameter distributions and set dropout thresholds in order to obtain better estimates due to averaging. The developed CCV methodology was also tested and verified on a collected and labeled manual dataset. An onwards deep neural network (NN) approach was also applied, for instance the segmentation task, and has demonstrated certain advantages in terms of high segmentation resolution, while the classical one tends to be more speedy. Thus, in the current manuscript both advantages and disadvantages of the classical computer vision method (CCV) and neural network approach (NN) are discussed based on evaluation of bubbles' number and their area defined. An approach to mass transfer coefficient estimation methodology in virtue of obtained results is also represented. [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]

As a family of multiresolution adaptive transforms and a generalization of wedgelets, smoothlets are more efficient in representing images with various sharp edges than other “X-lets”. Smoothlets use a horizon function to model an edge and define transition in a fixed axis direction. Furthermore, the conventional elliptical smoothlets (ES) use half of an ellipse to model edges. Various sharp edges of images may not be expressed well by half of an ellipse with transition only in a fixed axis direction. In this paper, we propose extended smoothlets (ExSmoothlets) framework by using more general characteristic functions with adaptive transition directions. Specifically, two methods of ExSmoothlets, the elliptical ExSmoothlets (EES) and homocentric elliptical ExSmoothlets (HEES) are developed and evaluated in the experiments against the ES. The results show that the proposed EES and HEES can effectively improve the image quality in image approximation and denoising in terms of PSNR. [ABSTRACT FROM AUTHOR]

Abstract: To automatically determine semantics of a shape or to generate a set of keywords that describe the content of a given image is a difficult problem due to: (a) the high-dimensional problem, (b) the unsolved automatic object segmentation (mis-segmentation), and (c) the lack of well-labeled large image database (mis-labeling). In order to tackle (a), despite (b), (c) and the expensive handy image segmentation and labeling, visual features should be automatically selected to convey the most robust and discriminant information without requiring too computational cost. Therefore, we propose a novel method: ‘Approximation of Linear Discriminant Analysis’ (ALDA), which is more generic than LDA: ALDA does not require explicit class labeling of each training samples. We theoretically show that under weak assumption, ALDA allows efficient ranking estimation of the discriminant powers of the visual features. We apply ALDA on COREL database (10K images, 267 words) with Normalized Cuts segmentation algorithm. First, we demonstrate an image classification gain of 43%, while reducing features set by a factor 10. Secondly, we demonstrate that for some words (like ‘Door’, ‘Flag’), even low-level shape features (convex hull, or moment of inertia) are more discriminant than any color or texture features. [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: We introduce concepts of minimal immersions and bandlimited (Paley–Wiener) immersions of combinatorial weighted graphs (finite or infinite) into Euclidean spaces. The notion of bandlimited immersions generalizes the known concept of eigenmaps of graphs. It is shown that our minimal immersions can be used to perform interpolation, smoothing and approximation of immersions of graphs into Euclidean spaces. It is proved that under certain conditions minimal immersions converge to bandlimited immersions. Explicit expressions of minimal immersions in terms of eigenmaps are given. The results can find applications for data dimension reduction, image processing, computer graphics, visualization and learning theory. [Copyright &y& Elsevier]

Abstract: The question of adaptive mesh generation for approximation by splines has been studied for a number of years by various authors. The results have numerous applications in computational and discrete geometry, computer aided geometric design, finite element methods for numerical solutions of partial differential equations, image processing, and mesh generation for computer graphics, among others. In this paper we will investigate the questions regarding adaptive approximation of functions with arbitrary but fixed throughout the domain signature by multilinear splines. In particular, we will study the asymptotic behavior of the optimal error of the weighted uniform approximation by interpolating and quasi-interpolating multilinear splines. [Copyright &y& Elsevier]

Previous research into the reliability of forensic craniofacial reconstruction (CFR) has focused primarily on the accuracy of reconstructed faces from European or African ancestry skulls. Moreover, the recognition of CFR in relation to the experience and ancestry of the practitioners and the assessors has not been previously considered. The cross-race effect is a recognised phenomenon in psychology studies, where familiar ancestry faces are recognised more readily than unfamiliar ancestry faces, but there is a paucity of research addressing the relationship between the accuracy of reconstructed faces and the familiarity with this ancestry by the practitioners/assessors. The aims of this research were to investigate whether 'unfamiliar-race effect' has any influence on the accuracy of CFR and to evaluate how much the correct recognition rate of CFR is affected by the cross-race effect. Eight CFRs from three ancestry groups were produced by experienced practitioners in order to explore the aims. The results demonstrated that practitioners produced more recognisable CFRs using skulls from a familiar ancestry than skulls from unfamiliar ancestries. [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]

Abstract: Given a triangular Bézier surface of degree , the problem of multi-degree reduction by a triangular Bézier surface of degree with boundary constraints is investigated. This paper considers the continuity of triangular Bézier surfaces at the three corners, so that the boundary curves preserve endpoints continuity of any order . The - and -norm combined with the constrained least-squares method are used to get the matrix representations for the control points of the degree reduced surfaces. Both methods can be applied to piecewise continuous triangular patches or to only a triangular patch with the combination of surface subdivision. And the resulting piecewise approximating patches are globally continuous. Finally, error estimation is given and numerical examples demonstrate the effectiveness of our methods. [Copyright &y& Elsevier]

Abstract: Recently, how to express human perception has become one of the most active research topics in image retrieval. In this paper, we propose a relevance feedback mechanism that can express objectively human perception by using rough set theory in retrieval system. The mechanism makes full use of the inherent advantages of rough set to solve the difficulty that the retrieval system cannot express human perception. In the end, an image retrieval system – “Basestar”, which is based on the technology, is introduced to illustrate the retrieval performance. [Copyright &y& Elsevier]