the bezout number for linear piecewise algebraic curves a piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function this paper discusses the bezout number the maximum number of intersections between two linear piecewise algebraic curves whose intersections are finite on regular triangulations we give an upper bound of the bezout number for linear piecewise algebraic curves bn 1 0 1 0 d on the triangulation with an odd interior vertex for the triangulations which satisfy a vertex coloring condition we compute the exact value of the bezout number bn 1 0 1 0 d

discrete signal processing on graphs frequency analysis signals and datasets that arise in physical and engineering applications as well as social genetics biomolecular and many other domains are becoming increasingly larger and more complex in contrast to traditional time and image signals data in these domains are supported by arbitrary graphs signal processing on graphs extends concepts and techniques from traditional signal processing to data indexed by generic graphs this paper studies the concepts of low and high frequencies on graphs and low high and band pass graph signals and graph filters in traditional signal processing these concepts are easily defined because of a natural frequency ordering that has a physical interpretation for signals residing on graphs in general there is no obvious frequency ordering we propose a definition of total variation for graph signals that naturally leads to a frequency ordering on graphs and defines low high and band pass graph signals and filters we study the design of graph filters with specified frequency response and illustrate our approach with applications to sensor malfunction detection and data classification

progressive image denoising through hybrid graph laplacian regularization a unified framework recovering images from corrupted observations is necessary for many real world applications in this paper we propose a unified framework to perform progressive image recovery based on hybrid graph laplacian regularized regression we first construct a multiscale representation of the target image by laplacian pyramid then progressively recover the degraded image in the scale space from coarse to fine so that the sharp edges and texture can be eventually recovered on one hand within each scale a graph laplacian regularization model represented by implicit kernel is learned which simultaneously minimizes the least square error on the measured samples and preserves the geometrical structure of the image data space in this procedure the intrinsic manifold structure is explicitly considered using both measured and unmeasured samples and the nonlocal self similarity property is utilized as a fruitful resource for abstracting a priori knowledge of the images on the other hand between two successive scales the proposed model is extended to a projected high dimensional feature space through explicit kernel mapping to describe the interscale correlation in which the local structure regularity is learned and propagated from coarser to finer scales in this way the proposed algorithm gradually recovers more and more image details and edges which could not been recovered in previous scale we test our algorithm on one typical image recovery task impulse noise removal experimental results on benchmark test images demonstrate that the proposed method achieves better performance than state of the art algorithms

adaptive thinning algorithm of nonsingular triangulation this paper studies adaptive thinning strategies for the non singular triangulation of scattered data by c 1 rational spline function given a set of points in r 2 luo liu and chen have presented a triangulation algorithm which ensures the non singularity of s 2 1 and s 3 1 spaces in this paper we improve the algorithm to reduce the number of knots of the triangulation within a given tolerance while the non singularity of s 2 1 and s 3 1 spaces is ensured our strategies presented here depend on both the locations of the data points in the plane and the data values at these points we give the definition of discrete norm for c 1 rational spline function by using its coefficients then a weight is assigned to each knot which is a measure of the importance of knot in the representation of spline when the weight of the knot is less than the given tolerance its influence is regarded negligible then it can be removed it s a discrete method in the end of this paper several numerical examples are presented to show the feasibility and validity of our algorithm

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

a reliable estimation method of a dipole for three dimensional poisson equation we discuss an inverse source problem for three dimensional poisson equation the source term is expressed as a dipole that consists of two contiguous point sources our problem is estimating the dipole moment and its location under the condition that the potential and flux can be observed on the boundary we propose a numerical method based on a weighted residual approach which need not use direct analysis for the governing equation the method should be reliable since the error caused by the numerical integration and observation noise can be evaluated theoretically the effectiveness of the method is shown by numerical examples

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

learning spectral descriptors for deformable shape correspondence informative and discriminative feature descriptors play a fundamental role in deformable shape analysis for example they have been successfully employed in correspondence registration and retrieval tasks in recent years significant attention has been devoted to descriptors obtained from the spectral decomposition of the laplace beltrami operator associated with the shape notable examples in this family are the heat kernel signature hks and the recently introduced wave kernel signature wks the laplacian based descriptors achieve state of the art performance in numerous shape analysis tasks they are computationally efficient isometry invariant by construction and can gracefully cope with a variety of transformations in this paper we formulate a generic family of parametric spectral descriptors we argue that to be optimized for a specific task the descriptor should take into account the statistics of the corpus of shapes to which it is applied the signal and those of the class of transformations to which it is made insensitive the noise while such statistics are hard to model axiomatically they can be learned from examples following the spirit of the wiener filter in signal processing we show a learning scheme for the construction of optimized spectral descriptors and relate it to mahalanobis metric learning the superiority of the proposed approach in generating correspondences is demonstrated on synthetic and scanned human figures we also show that the learned descriptors are robust enough to be learned on synthetic data and transferred successfully to scanned shapes

compression artifact reduction by overlapped block transform coefficient estimation with block similarity block transform coded images usually suffer from annoying artifacts at low bit rates caused by the coarse quantization of transform coefficients in this paper we propose a new method to reduce compression artifacts by the overlapped block transform coefficient estimation from non local blocks in the proposed method the discrete cosine transform coefficients of each block are estimated by adaptively fusing two prediction values based on their reliabilities one prediction is the quantized values of coefficients decoded from the compressed bitstream whose reliability is determined by quantization steps the other prediction is the weighted average of the coefficients in nonlocal blocks whose reliability depends on the variance of the coefficients in these blocks the weights are used to distinguish the effectiveness of the coefficients in nonlocal blocks to predict original coefficients and are determined by block similarity in transform domain to solve the optimization problem the overlapped blocks are divided into several subsets each subset contains nonoverlapped blocks covering the whole image and is optimized independently therefore the overall optimization is reduced to a set of sub optimization problems which can be easily solved finally we provide a strategy for parameter selection based on the compression levels experimental results show that the proposed method can remarkably reduce compression artifacts and significantly improve both the subjective and objective qualities of block transform coded images

graph based compression of dynamic 3d point cloud sequences this paper addresses the problem of compression of 3d point cloud sequences that are characterized by moving 3d positions and color attributes as temporally successive point cloud frames share some similarities motion estimation is key to effective compression of these sequences it however remains a challenging problem as the point cloud frames have varying numbers of points without explicit correspondence information we represent the time varying geometry of these sequences with a set of graphs and consider 3d positions and color attributes of the point clouds as signals on the vertices of the graphs we then cast motion estimation as a feature matching problem between successive graphs the motion is estimated on a sparse set of representative vertices using new spectral graph wavelet descriptors a dense motion field is eventually interpolated by solving a graph based regularization problem the estimated motion is finally used for removing the temporal redundancy in the predictive coding of the 3d positions and the color characteristics of the point cloud sequences experimental results demonstrate that our method is able to accurately estimate the motion between consecutive frames moreover motion estimation is shown to bring a significant improvement in terms of the overall compression performance of the sequence to the best of our knowledge this is the first paper that exploits both the spatial correlation inside each frame through the graph and the temporal correlation between the frames through the motion estimation to compress the color and the geometry of 3d point cloud sequences in an efficient way

nonlocally centralized sparse representation for image restoration sparse representation models code an image patch as a linear combination of a few atoms chosen out from an over complete dictionary and they have shown promising results in various image restoration applications however due to the degradation of the observed image e g noisy blurred and or down sampled the sparse representations by conventional models may not be accurate enough for a faithful reconstruction of the original image to improve the performance of sparse representation based image restoration in this paper the concept of sparse coding noise is introduced and the goal of image restoration turns to how to suppress the sparse coding noise to this end we exploit the image nonlocal self similarity to obtain good estimates of the sparse coding coefficients of the original image and then centralize the sparse coding coefficients of the observed image to those estimates the so called nonlocally centralized sparse representation ncsr model is as simple as the standard sparse representation model while our extensive experiments on various types of image restoration problems including denoising deblurring and super resolution validate the generality and state of the art performance of the proposed ncsr algorithm

a vector quantizer for the laplace source the low complexity nearly optimal vector quantizer vq is a generalization of t r fischer s 1986 pyramid vq and is similar in structure to the unrestricted polar quantizers previously presented for the independent gaussian source an analysis of performance is presented with results for both the product code pyramid vq and the unrestricted version this analysis although asymptotic in nature helps to demonstrate the performance advantages of the vq implementation issues of the vq are discussed nonasymptotic results are considered in particular the author presents an approximate design algorithm for finite bit rate and demonstrates the usefulness of this vq through several example designs with monte carlo simulations of performance for the restricted form the pyramid vq the author provides further implementational information and low dimension analytical results

cayley bacharach theorem of piecewise algebraic curves the piecewise algebraic curve determined by a bivariate spline function is a generalization of the classical algebraic curve in this paper by using bezout s theorem and nother type theorem of piecewise algebraic curves the cayley bacharach theorem and hilbert function of c0 piecewise algebraic curves are presented

random walk graph laplacian based smoothness prior for soft decoding of jpeg images given the prevalence of joint photographic experts group jpeg compressed images optimizing image reconstruction from the compressed format remains an important problem instead of simply reconstructing a pixel block from the centers of indexed discrete cosine transform dct coefficient quantization bins hard decoding soft decoding reconstructs a block by selecting appropriate coefficient values within the indexed bins with the help of signal priors the challenge thus lies in how to define suitable priors and apply them effectively in this paper we combine three image priors laplacian prior for dct coefficients sparsity prior and graph signal smoothness prior for image patches to construct an efficient jpeg soft decoding algorithm specifically we first use the laplacian prior to compute a minimum mean square error initial solution for each code block next we show that while the sparsity prior can reduce block artifacts limiting the size of the overcomplete dictionary to lower computation would lead to poor recovery of high dct frequencies to alleviate this problem we design a new graph signal smoothness prior desired signal has mainly low graph frequencies based on the left eigenvectors of the random walk graph laplacian matrix lerag compared with the previous graph signal smoothness priors lerag has desirable image filtering properties with low computation overhead we demonstrate how lerag can facilitate recovery of high dct frequencies of a piecewise smooth signal via an interpretation of low graph frequency components as relaxed solutions to normalized cut in spectral clustering finally we construct a soft decoding algorithm using the three signal priors with appropriate prior weights experimental results show that our proposal outperforms the state of the art soft decoding algorithms in both objective and subjective evaluations noticeably

sparsity based image error concealment via adaptive dual dictionary learning and regularization in this paper we propose a novel sparsity based image error concealment ec algorithm through adaptive dual dictionary learning and regularization we define two feature spaces the observed space and the latent space corresponding to the available regions and the missing regions of image under test respectively we learn adaptive and complete dictionaries individually for each space where the training data are collected via an adaptive template matching mechanism based on the piecewise stationarity of natural images a local correlation model is learned to bridge the sparse representations of the aforementioned dual spaces allowing us to transfer the knowledge of the available regions to the missing regions for ec purpose eventually the ec task is formulated as a unified optimization problem where the sparsity of both spaces and the learned correlation model are incorporated experimental results show that the proposed method outperforms the state of the art techniques in terms of both objective and perceptual metrics

lagrange interpolation by bivariate splines on cross cut partitions the piecewise algebraic curve is a generalization of the classical algebraic curve in this paper we give the nother type theorem of piecewise algebraic curves on cross cut partitions by using interpolation along a piecewise algebraic curve we also present a method to construct lagrange interpolation sets for bivariate spline spaces on cross cut partitions

graph based representation for multiview image geometry in this paper we propose a new geometry representation method for multiview image sets our approach relies on graphs to describe the multiview geometry information in a compact and controllable way the links of the graph connect pixels in different images and describe the proximity between pixels in 3d space these connections are dependent on the geometry of the scene and provide the right amount of information that is necessary for coding and reconstructing multiple views our multiview image representation is very compact and adapts the transmitted geometry information as a function of the complexity of the prediction performed at the decoder side to achieve this our graph based representation gbr carefully selects the amount of geometry information needed before coding this is in contrast with depth coding which directly compresses with losses the original geometry signal thus making it difficult to quantify the impact of coding errors on geometry based interpolation we present the principles of this gbr and we build an efficient coding algorithm to represent it we compare our gbr approach to classical depth compression methods and compare their respective view synthesis qualities as a function of the compactness of the geometry description we show that gbr can achieve significant gains in geometry coding rate over depth based schemes operating at similar quality experimental results demonstrate the potential of this new representation

image interpolation via graph based bayesian label propagation in this paper we propose a novel image interpolation algorithm via graph based bayesian label propagation the basic idea is to first create a graph with known and unknown pixels as vertices and with edge weights encoding the similarity between vertices then the problem of interpolation converts to how to effectively propagate the label information from known points to unknown ones this process can be posed as a bayesian inference in which we try to combine the principles of local adaptation and global consistency to obtain accurate and robust estimation specially our algorithm first constructs a set of local interpolation models which predict the intensity labels of all image samples and a loss term will be minimized to keep the predicted labels of the available low resolution lr samples sufficiently close to the original ones then all of the losses evaluated in local neighborhoods are accumulated together to measure the global consistency on all samples moreover a graph laplacian based manifold regularization term is incorporated to penalize the global smoothness of intensity labels such smoothing can alleviate the insufficient training of the local models and make them more robust finally we construct a unified objective function to combine together the global loss of the locally linear regression square error of prediction bias on the available lr samples and the manifold regularization term it can be solved with a closed form solution as a convex optimization problem experimental results demonstrate that the proposed method achieves competitive performance with the state of the art image interpolation algorithms

counting positive solutions for polynomial systems with real coefficients based on the subresultant polynomial chain and the method presented by hongguang fu and lu yang et al we propose a practical algorithm for computing the number of sign changes of the leading coefficients and the constant terms of a generalized sturm sequence gss with the principal minors sequence of the discrimination matrix of the polynomial and the number of sign changes of the gss a formula determining the number of positive solutions of a given bivariate polynomial systems with real coefficients with a finite number of complex solutions and its algorithm are presented using the techniques of the b net form of bivariate splines function this algorithm can be used for computing the number of real intersection points of two piecewise algebraic curves and its distributions in all cells when common points are finite a piecewise algebraic curve is defined by a bivariate spline function

homology of smooth splines generic triangulations and a conjecture of strang for a a triangulated d dimensional region in rd let sr a denote the vector space of all cr functions f on a that restricted to any simplex in a are given by polynomials of degree at most m we consider the problem of computing the dimension of such spaces we develop a homological approach to this problem and apply it specifically to the case of triangulated manifolds a in the plane getting lower bounds on the dimension of sr a for all r for r 1 we prove a conjecture of strang concerning the generic dimension of the space of cl splines over a triangulated manifold in r2 finally we consider the space of continuous piecewise linear functions over nonsimplicial decompositions of a plane region

the maximum number and its distribution of singular points for parametric piecewise algebraic curves abstract the piecewise algebraic curve as the zero set of a bivariate spline function is a generalization of the classical algebraic curve based on the previous method presented by lai et al 2009 we show that computing singular points of parametric piecewise algebraic curves amounts to solving parametric piecewise polynomial systems in this article we give a method to compute the maximum number and its distribution of singular points for a given parametric piecewise algebraic curve this method also produces necessary and sufficient conditions of its parameters must be satisfied an illustrated example shows that our method is flexible

optimized update prediction assignment for lifting transforms on graphs transformations on graphs can provide compact representations of signals with many applications in denoising feature extraction or compression in particular lifting transforms have the advantage of being critically sampled and invertible by construction but the efficiency of the transform depends on the choice of a good bipartition of the graph into update mathcal u and prediction mathcal p nodes this is the update prediction mathcal u mathcal p assignment problem which is the focus of this paper we analyze this problem theoretically and derive an optimal mathcal u mathcal p assignment under assumptions about signal model and filters furthermore we prove that the best mathcal u mathcal p partition is related to the correlation between nodes on the graph and is not the one that minimizes the number of conflicts connections between nodes of same label or maximizes the weight of the cut we also provide experimental results in randomly generated graph signals and real data from image and video signals that validate our theoretical conclusions demonstrating improved performance over state of the art solutions for this problem

a direct identification method for current dipoles from electromagnetic fields we discuss the identification problem for current dipoles in a spherically symmetric conductor this mathematical model is used for a biomedical inverse problem such as the source current identification for the human brain activity we have already proposed a direct identification method for this inverse source problem using observations of the magnetic fields outside of the conductor one of the difficulties of current dipole identification using the magnetic fields is caused by the fact that magnetic field does not include any information about the radial component of dipole moments in this paper we consider an improvement of the direct method to identify both radial and tangential components of current dipole moments by combining electric and magnetic observation data furthermore our approach is effective in the case where the number of dipoles is unknown

the correspondence between multivariate spline ideals and piecewise algebraic varieties as a piecewise polynomial with a certain smoothness the spline plays an important role in computational geometry the algebraic variety is the most important subject in classical algebraic geometry as the zero set of multivariate splines the piecewise algebraic variety is a generalization of the algebraic variety in this paper the correspondence between piecewise algebraic varieties and spline ideals is discussed furthermore hilbert s nullstellensatz for the piecewise algebraic variety is also studied

a general framework for regularized similarity based image restoration any image can be represented as a function defined on a weighted graph in which the underlying structure of the image is encoded in kernel similarity and associated laplacian matrices in this paper we develop an iterative graph based framework for image restoration based on a new definition of the normalized graph laplacian we propose a cost function which consists of a new data fidelity term and regularization term derived from the specific definition of the normalized graph laplacian the normalizing coefficients used in the definition of the laplacian and associated regularization term are obtained using fast symmetry preserving matrix balancing this results in some desired spectral properties for the normalized laplacian such as being symmetric positive semidefinite and returning zero vector when applied to a constant image our algorithm comprises of outer and inner iterations where in each outer iteration the similarity weights are recomputed using the previous estimate and the updated objective function is minimized using inner conjugate gradient iterations this procedure improves the performance of the algorithm for image deblurring where we do not have access to a good initial estimate of the underlying image in addition the specific form of the cost function allows us to render the spectral analysis for the solutions of the corresponding linear equations in addition the proposed approach is general in the sense that we have shown its effectiveness for different restoration problems including deblurring denoising and sharpening experimental results verify the effectiveness of the proposed algorithm on both synthetic and real examples

graph laplacian regularization for image denoising analysis in the continuous domain inverse imaging problems are inherently underdetermined and hence it is important to employ appropriate image priors for regularization one recent popular prior the graph laplacian regularizer assumes that the target pixel patch is smooth with respect to an appropriately chosen graph however the mechanisms and implications of imposing the graph laplacian regularizer on the original inverse problem are not well understood to address this problem in this paper we interpret neighborhood graphs of pixel patches as discrete counterparts of riemannian manifolds and perform analysis in the continuous domain providing insights into several fundamental aspects of graph laplacian regularization for image denoising specifically we first show the convergence of the graph laplacian regularizer to a continuous domain functional integrating a norm measured in a locally adaptive metric space focusing on image denoising we derive an optimal metric space assuming non local self similarity of pixel patches leading to an optimal graph laplacian regularizer for denoising in the discrete domain we then interpret graph laplacian regularization as an anisotropic diffusion scheme to explain its behavior during iterations e g its tendency to promote piecewise smooth signals under certain settings to verify our analysis an iterative image denoising algorithm is developed experimental results show that our algorithm performs competitively with state of the art denoising methods such as bm3d for natural images and outperforms them significantly for piecewise smooth images

fast graph sampling set selection using gershgorin disc alignment graph sampling set selection where a subset of nodes are chosen to collect samples to reconstruct a smooth graph signal is a fundamental problem in graph signal processing gsp previous works employ an unbiased least squares ls signal reconstruction scheme and select samples via expensive extreme eigenvector computation instead we assume a biased graph laplacian regularization glr based scheme that solves a system of linear equations for reconstruction we then choose samples to minimize the condition number of the coefficient matrix specifically maximize the smallest eigenvalue lambda min circumventing explicit eigenvalue computation we maximize instead the lower bound of lambda min designated by the smallest left end of all gershgorin discs of the matrix to achieve this efficiently we first convert the optimization to a dual problem where we minimize the number of samples needed to align all gershgorin disc left ends at a chosen lower bound target t algebraically the dual problem amounts to optimizing two disc operations i shifting of disc centers due to sampling and ii scaling of disc radii due to a similarity transformation of the matrix we further reinterpret the dual as an intuitive disc coverage problem bearing strong resemblance to the famous np hard set cover sc problem the reinterpretation enables us to derive a fast approximation scheme from a known sc error bounded approximation algorithm we find an appropriate target t efficiently via binary search extensive simulation experiments show that our disc based sampling algorithm runs substantially faster than existing sampling schemes and outperforms other eigen decomposition free sampling schemes in reconstruction error

point cloud denoising via feature graph laplacian regularization point cloud is a collection of 3d coordinates that are discrete geometric samples of an object s 2d surfaces imperfection in the acquisition process means that point clouds are often corrupted with noise building on recent advances in graph signal processing we design local algorithms for 3d point cloud denoising specifically we design a signal dependent feature graph laplacian regularizer sdfglr that assumes surface normals computed from point coordinates are piecewise smooth with respect to a signal dependent graph laplacian matrix using sdfglr as a signal prior we formulate an optimization problem with a general ell p norm fidelity term that can explicitly remove only two types of additive noise small but non sparse noise like gaussian using ell 2 fidelity term and large but sparser noise like laplacian using ell 1 fidelity term to establish a linear relationship between normals and 3d point coordinates we first perform bipartite graph approximation to divide the point cloud into two disjoint node sets red and blue we then optimize the red and blue nodes coordinates alternately for ell 2 norm fidelity term we iteratively solve an unconstrained quadratic programming qp problem efficiently computed using conjugate gradient with a bounded condition number to ensure numerical stability for ell 1 norm fidelity term we iteratively minimize an ell 1 ell 2 cost function using accelerated proximal gradient apg where a good step size is chosen via lipschitz continuity analysis finally we propose simple mean and median filters for flat patches of a given point cloud to estimate the noise variance given the noise type which in turn is used to compute a weight parameter trading off the fidelity term and signal prior in the problem formulation extensive experiments show state of the art denoising performance among local methods using our proposed algorithms

structure and dimension of multivariate spline space of lower degree on arbitrary triangulation in this paper we discuss the structure of multivariate spline spaces on arbitrary triangulation by using the methods and results of smoothing cofactor and generator basis of modules on the base of analyzing the algebraic and geometric results about singularity of s21 ms we build the structure of triangulation and give some useful geometric conditions such that s 1 space is singular and we obtain an algebraic condition which is necessary and sufficient for the singularity of s 1 spaces as well as their dimension formulae moreover the structure matrix of spline spaces over any given partition is defined which has been used to discuss the structure of s31 and s21 spaces over arbitrary triangulation and to prove the nonsingularity of s31 spaces this partially settles a conjecture on the singularity of spline spaces in wang et al multivariate spline and its applications kluwer press dordrecht 2002 academic press beijing 1994 in chinese meanwhile the dimension formulae of s31 s21 spaces and the dimension formulae of s 1 1 spaces are also given in this paper

recent researches on multivariate spline and piecewise algebraic variety the multivariate splines as piecewise polynomials have become useful tools for dealing with computational geometry computer graphics computer aided geometrical design and image processing it is well known that the classical algebraic variety in algebraic geometry is to study geometrical properties of the common intersection of surfaces represented by multivariate polynomials recently the surfaces are mainly represented by multivariate piecewise polynomials i e multivariate splines so the piecewise algebraic variety defined as the common intersection of surfaces represented by multivariate splines is a new topic in algebraic geometry moreover the piecewise algebraic variety will be also important in computational geometry computer graphics computer aided geometrical design and image processing the purpose of this paper is to introduce some recent researches on multivariate spline piecewise algebraic variety curve and their applications

image bit depth enhancement via maximum a posteriori estimation of ac signal when images at low bit depth are rendered at high bit depth displays missing least significant bits needs to be estimated we study the image bit depth enhancement problem estimating an original image from its quantized version from a minimum mean squared error mmse perspective we first argue that a graph signal smoothness prior one defined on a graph embedding the image structure is an appropriate prior for the bit depth enhancement problem we next show that directly solving for the mmse solution is in general too computationally expensive to be practical we then propose an efficient approximation strategy in particular we first estimate the ac component of the desired signal in a maximum a posteriori formulation efficiently computed via convex programming we then compute the dc component with an mmse criterion in a closed form given the computed ac component experiments show that our proposed two step approach has improved performance over the conventional bit depth enhancement schemes in both objective and subjective comparisons

a mathematical analysis of the dct coefficient distributions for images over the past two decades there have been various studies on the distributions of the dct coefficients for images however they have concentrated only on fitting the empirical data from some standard pictures with a variety of well known statistical distributions and then comparing their goodness of fit the laplacian distribution is the dominant choice balancing simplicity of the model and fidelity to the empirical data yet to the best of our knowledge there has been no mathematical justification as to what gives rise to this distribution we offer a rigorous mathematical analysis using a doubly stochastic model of the images which not only provides the theoretical explanations necessary but also leads to insights about various other observations from the literature this model also allows us to investigate how certain changes in the image statistics could affect the dct coefficient distributions

a reliable identification of electric current dipoles using harmonic functions we consider an identification problem for electric current dipoles in a spherically symmetric conductor from observation data of magnetic fields outside of the conductor this problem can be expressed as an inverse source problem for three dimensional poisson equation an application of this model is the analysis of magnetoencephalography we propose a reliable numerical method for the above inverse source problem without using a priori information of the locations moments and number of dipoles for two dimensional case an identification method of dipoles using harmonic functions has been proposed we extend this method to three dimensional problem and also give error estimates for identified locations and moments the effectiveness of our method is shown by numerical examples

learning parametric dictionaries for signals on graphs in sparse signal representation the choice of a dictionary often involves a tradeoff between two desirable properties the ability to adapt to specific signal data and a fast implementation of the dictionary to sparsely represent signals residing on weighted graphs an additional design challenge is to incorporate the intrinsic geometric structure of the irregular data domain into the atoms of the dictionary in this work we propose a parametric dictionary learning algorithm to design data adapted structured dictionaries that sparsely represent graph signals in particular we model graph signals as combinations of overlapping local patterns we impose the constraint that each dictionary is a concatenation of subdictionaries with each subdictionary being a polynomial of the graph laplacian matrix representing a single pattern translated to different areas of the graph the learning algorithm adapts the patterns to a training set of graph signals experimental results on both synthetic and real datasets demonstrate that the dictionaries learned by the proposed algorithm are competitive with and often better than unstructured dictionaries learned by state of the art numerical learning algorithms in terms of sparse approximation of graph signals in contrast to the unstructured dictionaries however the dictionaries learned by the proposed algorithm feature localized atoms and can be implemented in a computationally efficient manner in signal processing tasks such as compression denoising and classification

on structure of bivariate spline space of lower degree over arbitrary triangulation the structure of bivariate spline space over arbitrary triangulation is complicated because the dimension of a multivariate spline space depends not only on the topological property of the triangulation but also on its geometric property a new vertex coding method to a triangulation is introduced in this paper to further study structure of the spline spaces the upper bound of the dimension of spline spaces over triangulation given by l l schumaker is slightly improved via the new vertex coding method the structure of multivariate spline spaces s 2 1 and s 3 1 over arbitrary triangulation are studied via the method of smoothness cofactor and the structure matrix of multivariate spline ring by luo and wang a kind of sufficient conditions on judging non singularity of the s 2 1 and s 3 1 spaces over arbitrary triangulation is given which only depends on the topological property of the triangulation from the sufficient conditions a triangulation strategy is presented at the end of the paper the strategy ensures that the constructed triangulation is non singular or generic for s 2 1 and s 3 1

image denoising by sparse 3 d transform domain collaborative filtering we propose a novel image denoising strategy based on an enhanced sparse representation in transform domain the enhancement of the sparsity is achieved by grouping similar 2d image fragments e g blocks into 3d data arrays which we call groups collaborative altering is a special procedure developed to deal with these 3d groups we realize it using the three successive steps 3d transformation of a group shrinkage of the transform spectrum and inverse 3d transformation the result is a 3d estimate that consists of the jointly filtered grouped image blocks by attenuating the noise the collaborative filtering reveals even the finest details shared by grouped blocks and at the same time it preserves the essential unique features of each individual block the filtered blocks are then returned to their original positions because these blocks are overlapping for each pixel we obtain many different estimates which need to be combined aggregation is a particular averaging procedure which is exploited to take advantage of this redundancy a significant improvement is obtained by a specially developed collaborative wiener filtering an algorithm based on this novel denoising strategy and its efficient implementation are presented in full detail an extension to color image denoising is also developed the experimental results demonstrate that this computationally scalable algorithm achieves state of the art denoising performance in terms of both peak signal to noise ratio and subjective visual quality

spectral partitioning works planar graphs and finite element meshes abstract spectral partitioning methods use the fiedler vector the eigenvector of the second smallest eigenvalue of the laplacian matrix to find a small separator of a graph these methods are important components of many scientific numerical algorithms and have been demonstrated by experiment to work extremely well in this paper we show that spectral partitioning methods work well on bounded degree planar graphs and finite element meshes the classes of graphs to which they are usually applied while naive spectral bisection does not necessarily work we prove that spectral partitioning techniques can be used to produce separators whose ratio of vertices removed to edges cut is o n for bounded degree planar graphs and two dimensional meshes and o n1 d for well shaped d dimensional meshes the heart of our analysis is an upper bound on the second smallest eigenvalues of the laplacian matrices of these graphs we prove a bound of o 1 n for bounded degree planar graphs and o 1 n2 d for well shaped d dimensional meshes

motion compensated compression of dynamic voxelized point clouds dynamic point clouds are a potential new frontier in visual communication systems a few articles have addressed the compression of point clouds but very few references exist on exploring temporal redundancies this paper presents a novel motion compensated approach to encoding dynamic voxelized point clouds at low bit rates a simple coder breaks the voxelized point cloud at each frame into blocks of voxels each block is either encoded in intra frame mode or is replaced by a motion compensated version of a block in the previous frame the decision is optimized in a rate distortion sense in this way both the geometry and the color are encoded with distortion allowing for reduced bit rates in loop filtering is employed to minimize compression artifacts caused by distortion in the geometry information simulations reveal that this simple motion compensated coder can efficiently extend the compression range of dynamic voxelized point clouds to rates below what intra frame coding alone can accommodate trading rate for geometry accuracy

an identification method of electric current dipoles in spherically symmetric conductor a numerical method is proposed for the identification of electric current dipoles in a spherically symmetric conductor we use observations of the magnetic induction outside of the conductor our idea is to probe electric current dipoles using an indicator function that satisfies the laplace equation in our method any a priori information for the number of dipoles and initial estimates are not required the effectiveness of the method is shown by numerical examples

multivariate spline and algebraic geometry abstract the purpose of this survey is to emphasize the special relationship between multivariate spline and algebraic geometry we will not only point out the algebraic geometric method of multivariate spline but also the algebraic geometric background and essence of multivariate spline especially we have made an introduction to the so called piecewise algebraic curve piecewise algebraic variety and some of their properties

low rank decomposition based restoration of compressed images via adaptive noise estimation images coded at low bit rates in real world applications usually suffer from significant compression noise which significantly degrades the visual quality traditional denoising methods are not suitable for the content dependent compression noise which usually assume that noise is independent and with identical distribution in this paper we propose a unified framework of content adaptive estimation and reduction for compression noise via low rank decomposition of similar image patches we first formulate the framework of compression noise reduction based upon low rank decomposition compression noises are removed by soft thresholding the singular values in singular value decomposition of every group of similar image patches for each group of similar patches the thresholds are adaptively determined according to compression noise levels and singular values we analyze the relationship of image statistical characteristics in spatial and transform domains and estimate compression noise level for every group of similar patches from statistics in both domains jointly with quantization steps finally quantization constraint is applied to estimated images to avoid over smoothing extensive experimental results show that the proposed method not only improves the quality of compressed images obviously for post processing but are also helpful for computer vision tasks as a pre processing method

the dimension and basis of spaces of multivariate splines abstract the purpose of this paper is to study the recent development of certain aspects of multivariate spline spaces more specifically we will survey the existence of bivariate spline the dimension and basis of the bivariate spline spaces s k with various partitions furthermore we will also introduce some results on the higher dimensional splines