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We introduce a new type of subdivision-based multiresolution representation for triangle meshes, kite trees, which has the flexibility to represent arbitrary triangle meshes losslessly. We also develop an algorithm for extracting a balanced kite tree representation of an arbitrary input mesh that preserves mesh topology and regularities in the subdivision structure. Our scheme allows us to combine surface information automatically extracted from input data with algorithmically-generated information in a single multiresolution representation and to represent the results of adaptive refinement of regular subdivision surfaces.

In this paper, we are concerned with spectral-theoretic features of general iterated function systems (IFS). Such systems arise from the study of iteration limits of a finite family of maps τ_i, i=1,…,N, in some Hausdorff space Y. There is a standard construction which generally allows us to reduce to the case of a compact invariant subset X⊂Y. Typically, some kind of contractivity property for the maps τ_i is assumed, but our present considerations relax this restriction. This means that there is then not a natural equilibrium measure μ available which allows us to pass the point-maps τ_i to operators on the Hilbert space L²(μ). Instead, we show that it is possible to realize the maps τ_i quite generally in Hilbert spaces ℋ(X) of square-densities on X. The elements in ℋ(X) are equivalence classes of pairs (φ,μ), where φ is a Borel function on X, μ is a positive Borel measure on X, and ∫_X|φ|² dμ<∞. We say that (φ,μ)~(ψ,ν) if there is a positive Borel measure λ such that μ≪λ, ν≪λ, and

We further prove that this representation in the Hilbert space ℋ(X) has several universal properties.

A discrete wavelet multiresolution approach to the problem of lossy compression of random data is described. Also, numerical and analytical estimations of the compression rate and of the relative reconstruction error are given.

The community is the dominant structure that exhibits different features and multifold functions of complex networks from different levels; accordingly, multiresolution community detection is of critical importance in network science. In this paper, inspired by the ideas of the network flow, we propose an intensity-based community detection algorithm, i.e. ICDA, to detect multiresolution communities in weighted networks. First, the edge intensity is defined to quantify the relationship between each pair of connected nodes, and the vertices connected by the edges with higher intensities are denoted as core nodes, while the others are denoted as marginal nodes. Second, by applying the expansion strategy, the algorithm merges the closely connected core nodes as the initial communities and attaches marginal nodes to the nearest initial communities. To guarantee a higher internal density for the ultimate communities, the captured communities are further adjusted according to their densities. Experimental results of real and synthetic networks illustrate that our approach has higher performance and better accuracy. Meanwhile, a multiresolution investigation of some real networks shows that the algorithm can provide hierarchical details of complex networks with different thresholds.

We present a family of methods, analytical and numerical, which can describe behaviour in (non) equilibrium ensembles, both classical and quantum, especially in the complex systems, where the standard approaches cannot be applied. We demonstrate the creation of nontrivial (meta) stable states (patterns), localized, chaotic, entangled or decoherent, from basic localized modes in various collective models arising from the quantum hierarchy of Wigner-von Neumann-Moyal-Lindblad equations, which are the result of "wignerization" procedure of classical BBGKY hierarchy. We present the explicit description of internal quantum dynamics by means of exact analytical/numerical computations.

Edge detection is an active and critical topic in the field of image processing, and plays a vital role for some important applications such as image segmentation, pattern classification, object tracking, etc. In this paper, an edge detection approach is proposed using local edge pattern descriptor which possesses multiscale and multiresolution property, and is named varied local edge pattern (VLEP) descriptor. This method contains the following steps: firstly, Gaussian filter is used to smooth the original image. Secondly, the edge strength values, which are used to calculate the edge gradient values and can be obtained by one or more groups of VLEPs. Then, weighted fusion idea is considered when multiple groups of VLEP descriptors are used. Finally, the appropriate threshold is set to perform binarization processing on the gradient version of the image. Experimental results show that the proposed edge detection method achieved better performance than other state-of-the-art edge detection methods.

In the syntactic approach to pattern recognition, patterns are represented as strings, where each pattern is expressed as a composition of its component patterns, called subpatterns and pattern primitives. This approach draws an analogy between the structure of patterns and the syntax of a language. The patterns are considered at a single resolution level, and the recognition of each pattern is usually made by parsing the pattern structure according to a given set of syntax rules, obtained in the first stage of the analysis. This paper describes a pyramidal approach to 2-D object representation which relies on a linguistic description of the object contour at different resolution levels. Our approach, which is parallel and context-sensitive, implies the use of higher dimensional grammars for defining production rules which describe the evolution of the contour between levels. This method is computationally efficient (due to the fast coarse-fine search), rotationally invariant, as well as rather insensitive to noise.

Medical imaging is a powerful mean to access dynamic function of 3D deformable organs of the body. Due to the flow of incoming data, multiresolution methods on parallel computers is the only way to achieve complex processings in reasonable time. We present an active pyramid to model dynamic volumes. This pyramid is built on the first volume of a sequence and contains a binary model of the objects of interest. Previous knowledge is introduced within this binary model. The structure of the pyramid is rigid, but its main interest is that its components are deformed to fit the data using energetical constraints. A multiresolution algorithm based on self-organizing maps is then applied to deform the model through time. This algorithm matches the different levels of the pyramid in a coarse to fine approach. The output of the matching process is the field of deformation, modeling the transformations. This pyramid is applied to real data in the result section. The rigid structure of the pyramid is suitable for massively parallel architectures.

This paper presents a system based on new operators for handling sets of propositional clauses compactly represented by means of ZBDDs. The high compression power of such data structures allows efficient encodings of structured instances. A specialized operator for the distribution of sets of clauses is introduced and used for performing multiresolution on clause sets. Cut eliminations between sets of clauses of exponential size may then be performed using polynomial size data structures. The ZRES system, a new implementation of the Davis-Putnam procedure of 1960, solves two hard problems for resolution, that are currently out of the scope of the best SAT provers.

This paper presents two self-similar models that allow the control of curves and surfaces. The first model is based on IFS (Iterated Function Systems) theory and the second on subdivision curve and surface theory. Both of these methods employ the detail concept as in the wavelet transform, and allow the multiresolution control of objects with control points at any resolution level.

In the first model, the detail is inserted independently of control points, requiring it to be rotated when applying deformations. In contrast, the second method describes details relative to control points, allowing free control point deformations.

Modeling examples of curves and surfaces are presented, showing manipulation facilities of the models.

This paper describes a new and fast method for reconstructing a 3D computerized model from a cloud of points sampled from the object's surface. The proposed method aggregates very large scale 3D scanning data into a Hierarchical Space Decomposition Model (HSDM), realized by the Octree data structure. This model can represent both the boundary surface and the interior volume of an object. Based on the proposed volumetric model, the boundary reconstruction process becomes more robust and stable with respect to sampling noise. The hierarchical structure of the proposed volumetric model enables data reduction, while preserving critical geometrical features and object topology. As a result of data reduction, the execution time of the reconstruction process is significantly reduced. Moreover, the proposed model naturally allows multiresolution boundary extraction, represented by a mesh with regular properties.

The proposed surface reconstruction approach is based on Connectivity Graph extraction from HSDM, and facet reconstruction. This method's feasibility will be presented on a number of complex objects.

Geometry editing operations commonly use mesh encodings which capture the shape properties of the models. Given modified positions for a set of anchor vertices, the encoding is used to compute the positions for the rest of the mesh vertices, preserving the model shape as much as possible. In this paper, we introduce a new shape preserving and rotation invariant mesh encoding. We use this encoding for a variety of mesh editing applications: deformation, morphing, blending and motion reconstruction from Mocap data. The editing algorithms based on our encoding and decoding mechanism generate natural looking models that preserve the shape properties of the input.

Elsewhere we have introduced a construction to produce biorthogonal multiresolutions from given subdivisions. This construction was formulated in matrix terms, which is appropriate for curves and tensor-product surfaces. For mesh surfaces of non-tensor connectivity, however, matrix notation is inconvenient. This work presents the construction for regular meshes using diagrams (stencils, masks) and interactions between diagrams to replace matrices and matrix multiplication. Regular triangular meshes with butterfly subdivision and a variant of Loop subdivision due to Litke, et al. are used as examples.

Shape design is often performed by starting from a basic surface and by refining it afterward by adding details. In order to construct this first approximation surface, we present in this article a method to generate a basic polyhedron from a volumic voxel-based skeleton. This approach preserves the topology described by the discrete skeleton in a 3D grid considering the 26-adjacency: if a cycle is sketched, then there is a hole in the resulting surface, and if a closed hull is designed, then the output has a cavity. We verify the same properties for connected components. This surrounding basic polyhedron is computed with simple geometrical rules, and it is a good starting point for 3D shape design from a discrete voxel skeleton. In order to add multiresolution features to our approach, we use this rough mesh as the control polyhedron of a subdivision surface, according to the Loop scheme dedicated to triangulated surfaces. We show that the resulting set of smooth refined meshes is well suited for further modifications in the frame of a 3D modeling software.

In this paper, we present a new approach to geometrical modeling which allows the user to easily characterize and control the shape defined to a closed surface. We will focus on dealing with the shape's topological, morphological and geometrical properties separately. To do this, we have based our work on the following observations concerning surfaces defined by control-points, and implicit surfaces with skeleton. They both provide complementary approaches to the surface's deformation, and both have specific advantages and limits. We thus attempted to conceive a model which integrates the local and geometrical characterization induced by the control points, as well as the representation of the morphology given by the skeleton. Knowing that the lattice of control points is close to the surface and that the skeleton is centered in the related shape, we thought of a 3-layer model. The transition layer separates the local geometrical considerations from those linked to the global morphology. We apply our model to shape design in order to modify an object in an interactive and ergonomic way, as well as to reconstruction which allows better shape understanding. To do so, we present the algorithms related to these processes.

A novel three-dimensional gray-level interpolation method called the Directional Coherence Interpolation (DCI) is presented in the paper. The principal advantage of the proposed approach is that it leads to significantly higher visual quality in 3D rendering when compared with traditional image interpolation methods. The basis of DCI is a form of directional image-space coherence. DCI interpolates the missing image data along the maximum coherence directions (MCD), which are estimated from the local image intensity yet constrained by a generic smoothness term. In order to further improve both the algorithm efficiency and robustness, we also propose to apply a pyramidal search strategy for MCD estimation. This coarse-to-fine scheme requires less computation time by starting with the reduced amount of data and propagating searching results to finer resolutions. DCI can incorporate image shape and structure information without the prior requirement of explicit representation of object boundary/surface. Extensive experiments were performed on both synthetic and real medical images to evaluate the proposed approaches. The experimental results show that the proposed methods are able to handle general object interpolation, while achieving both accuracy and efficiency in interpolation compared with the existing techniques.

The RR and RT time intervals extracted from the electrocardiogram measure respectively the duration of cardiac cycle and repolarization. The series of these intervals recorded during the exercise test are characterized by two trends: A decreasing one during the stress phase and an increasing one during the recovery, separated by a global minimum. We model these series as a sum of a deterministic trend and random fluctuations, and estimate the trend using methods of curve extraction: Running mean, polynomial fit, multi scale wavelet decomposition. We estimate the minimum location from the trend. Data analysis performed on a group of 20 healthy subjects provides evidence that the minimum of the RR series precedes the minimum of the RT series, with a time delay of about 19 seconds.

Recently we developed a subdivision scheme for Powell–Sabin splines. It is a triadic scheme and it is general in the sense that it is not restricted to uniform triangles, the vertices must not have valence six and there are no restrictions on the initial triangulation. A sequence of nested spaces or multiresolution analysis can be associated with the base triangulation. In this paper we use the lifting scheme to construct basis functions for the complement space that captures the details that are lost when going to a coarser resolution. The subdivision scheme appears as the first lifting step or prediction step. A second lifting step, the update, is used to achieve certain properties for the complement spaces and the wavelet functions such as orthogonality and vanishing moments. The design of the update step is based on stability considerations. We prove stability for both the scaling functions and the wavelet functions.

We study the statistical performance of multiresolution-based estimation procedures for the scaling exponents of multifractal processes. These estimators rely on the computation of multiresolution quantities such as wavelet, increment or aggregation coefficients. Estimates are obtained by linear fits performed in log of structure functions of order q versus log of scale plots. Using various and recent types of multiplicative cascades and a large variety of multifractal processes, we study and benchmark, by means of numerical simulations, the statistical performance of these estimation procedures. We show that they all undergo a systematic linearisation effect: for a range of orders q, the estimates account correctly for the scaling exponents; outside that range, the estimates significantly depart from the correct values and systematically behave as linear functions of q. The definition and characterisation of this effect are thoroughly studied. In contradiction with interpretations proposed in the literature, we provide numerical evidence leading to the conclusion that this linearisation effect is neither a finite size effect nor an infiniteness of moments effect, but that its origin should be related to the deep nature of the process itself. We comment on its importance and consequences for the practical analysis of the multifractal properties of empirical data.

A discrete wavelet transform is one of the effective methodologies for compressing the image data and extracting the major characteristics from various data, but it always requires a number of target data composed of a power of 2. To overcome this difficulty without losing any original data information, we propose here a novel approach based on the Fourier transform. The key idea is simple but effective because it keeps all of the frequency components comprising the target data exactly. The raw data is firstly transformed to the Fourier coefficients by Fourier transform. Then, the inverse Fourier transform makes it possible to the number of data comprising a power of 2. We have applied this interpolation for the wind vector image data, and we have tried to compress the data by the multiresolution analysis by using the three-dimensional discrete wavelet transform. Several examples demonstrate the usefulness of our new method to work out the graphical communication tools.