nonlinear wavelet transforms for image coding via lifting we investigate central issues such as invertibility stability synchronization and frequency characteristics for nonlinear wavelet transforms built using the lifting framework the nonlinearity comes from adaptively choosing between a class of linear predictors within the lifting framework we also describe how earlier families of nonlinear filter banks can be extended through the use of prediction functions operating on a causal neighborhood of pixels preliminary compression results for model and real world images demonstrate the promise of our techniques

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

efficient quadtree coding of images and video the quadtree data structure is commonly used in image coding to decompose an image into separate spatial regions to adaptively identify the type of quantizer used in various regions of an image the authors describe the theory needed to construct quadtree data structures that optimally allocate rate given a set of quantizers a lagrange multiplier method finds these optimal rate allocations with no monotonicity restrictions they use the theory to derive a new quadtree construction method that uses a stepwise search to find the overall optimal quadtree structure the search can be driven with either actual measured quantizer performance or ensemble average predicted performance they apply this theory to the design of a motion compensated interframe video coding system using a quadtree with vector quantization

sparse geometric image representations with bandelets this paper introduces a new class of bases called bandelet bases which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow this geometric flow indicates directions in which the image gray levels have regular variations the image decomposition in a bandelet basis is implemented with a fast subband filtering algorithm bandelet bases lead to optimal approximation rates for geometrically regular images for image compression and noise removal applications the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion comparisons are made with wavelet image compression and noise removal algorithms

new tight frames of curvelets and optimal representations of objects with piecewise c2 singularities this paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise c 2 edges conceptually the curvelet transform is a multiscale pyramid with many directions and positions at each length scale and needle shaped elements at fine scales these elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets for instance curvelets obey a parabolic scaling relation which says that at scale 2 j each element has an envelope that is aligned along a ridge of length 2 j 2 and width 2 j we prove that curvelets provide an essentially optimal representation of typical objects f that are c 2 except for discontinuities along piecewise c 2 curves such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object for instance the n term partial reconstruction f c n obtained by selecting the n largest terms in the curvelet series obeys f f c n 2 l2 c n 2 log n 3 n this rate of convergence holds uniformly over a class of functions that are c 2 except for discontinuities along piecewise c 2 curves and is essentially optimal in comparison the squared error of n term wavelet approximations only converges as n 1 as n which is considerably worse than the optimal behavior

the contourlet transform an efficient directional multiresolution image representation the limitations of commonly used separable extensions of one dimensional transforms such as the fourier and wavelet transforms in capturing the geometry of image edges are well known in this paper we pursue a true two dimensional transform that can capture the intrinsic geometrical structure that is key in visual information the main challenge in exploring geometry in images comes from the discrete nature of the data thus unlike other approaches such as curvelets that first develop a transform in the continuous domain and then discretize for sampled data our approach starts with a discrete domain construction and then studies its convergence to an expansion in the continuous domain specifically we construct a discrete domain multiresolution and multidirection expansion using nonseparable filter banks in much the same way that wavelets were derived from filter banks this construction results in a flexible multiresolution local and directional image expansion using contour segments and thus it is named the contourlet transform the discrete contourlet transform has a fast iterated filter bank algorithm that requires an order n operations for n pixel images furthermore we establish a precise link between the developed filter bank and the associated continuous domain contourlet expansion via a directional multiresolution analysis framework we show that with parabolic scaling and sufficient directional vanishing moments contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves finally we show some numerical experiments demonstrating the potential of contourlets in several image processing applications

image coding by adaptive tree structured segmentation a new algorithmic approach to segmentation based image coding is proposed a good compromise is achieved between segmentation by quadtree based decomposition and by free region growing in terms of time complexity and scene adaptability encoding is to recursively partition an image into convex n gons 3

frequency domain system identification using arbitrary signals it is the common conviction that frequency domain system identification suffers from the drawback that it cannot handle arbitrary signals without introducing systematic errors this paper shows that it is possible to deal with nonperiodic signals without any approximation and under the same assumptions as in the time domain by estimating simultaneously some initial conditions and the system model parameters

algorithms for nonlinear piecewise polynomial approximation theoretical aspects in this article algorithms are developed for nonlinear n term courant element approximation of functions in lp 0 p oo on bounded polygonal domains in r 2 redundant collections of courant elements which are generated by multilevel nested triangulations allowing arbitrarily sharp angles are investigated scalable algorithms are derived for nonlinear approximation which both capture the rate of the best approximation and provide the basis for numerical implementation simple thresholding criteria enable approximation of a target function f to optimally high asymptotic rates which are determined and automatically achieved by the inherent smoothness of f the algorithms provide direct approximation estimates and permit utilization of the general jackson bernstein machinery to characterize n term courant element approximation in terms of a scale of smoothness spaces b spaces which govern the approximation rates

adaptive multiresolution analysis based on anisotropic triangulations a simple greedy refinement procedure for the generation of data adapted triangulations is proposed and studied given a function of two variables the algorithm produces a hierarchy of triangulations dj j 0 and piecewise polynomial approximations of on these triangulations the refinement procedure consists in bisecting a triangle in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between and its piecewise polynomial approximation after is bisected the hierarchical structure allows us to derive various approximation tools such as multiresolution analysis wavelet bases adaptive triangulations based either on greedy or optimal cart trees as well as a simple encoding of the corresponding triangulations we give a general proof of convergence in the lp norm of all these approximations numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio which is dictated by the local hessian of in case of c2 functions

model selection through a statistical analysis of the global minimum of a weighted nonlinear least squares cost function this paper presents a model selection algorithm for the identification of parametric models that are linear in the measurements it is based on the mean and variance expressions of the global minimum of a weighted nonlinear least squares cost function the method requires the knowledge of the noise covariance matrix but does not assume that the true model belongs to the model set unlike the traditional order estimation methods available in literature the presented technique allows to detect undermodeling the theory is illustrated by simulations on signal modeling and system identification problems and by one real measurement example

wedgelets nearly minimax estimation of edges we study a simple horizon model for the problem of recovering an image from noisy data in this model the image has an edge with fi holder regularity adopting the viewpoint of computational harmonic analysis we develop an overcomplete collection of atoms called wedgelets dyadically organized indicator functions with a variety of locations scales and orientations the wedgelet representation provides nearly optimal representations of objects in the horizon model as measured by minimax description length we show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet decorated recursive partition which minimizes a complexity penalized sum of squares this estimate using su cient sub pixel resolution achieves nearly the minimax mean squared error in the horizon model in fact the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon 1 fi 2 wedgelet analysis and de noising may be used successfully outside the horizon model we study images modelled as indicators of star shaped sets with smooth bound aries and show that complexity penalized wedgelet partitioning achieves nearly the minimax risk in that setting also

general framework for asymptotic properties of generalized weighted nonlinear least squares estimators with deterministic and stochastic weighting this paper studies the asymptotic properties strong consistency convergence rate asymptotic normality of a generalized weighted nonlinear least squares estimator under weak noise assumptions both deterministic and stochastic weighting are handled and the presence of model errors is considered for particular models estimators and noise assumptions the general framework boils down to known time and frequency domain estimators

preserving step edges in low bit rate progressive image compression with the growing importance of low bandwidth applications such as wireless access to the internet images are often sent or received at low bit rates at these bit rates they suffer from significant distortion and artifacts making it difficult for those viewing the images to understand them we present two progressive compression algorithms that focus on preserving the clarity of important image features such as edges at compression ratios of 80 1 and more both algorithms capture and encode the locations of important edges in the images the first algorithm then transmits a standard spiht set partitioning in hierarchical trees bit stream and at the decoder applies a nonlinear edge enhancement procedure to improve the clarity of the encoded edges the second approach uses a modified wavelet transform to remove the edges and encodes the remaining texture information using spiht with both approaches features in the images that may be important for recognition are well preserved even at low bit rates

generalized g h convexity and minimax theorems in g h spaces a new class of minimax theorems with their applications to saddle points based on an r g h kkm selection theorem is presented in a generalized h space setting

on rate distortion models for natural images and wavelet coding performance operational rate distortion rd functions of most natural images when compressed with state of the art wavelet coders exhibit a power law behavior dpropr gamma at moderately high rates with gamma being a constant depending on the input image deviating from the well known exponential form of the rd function dprop2 xir for bandlimited stationary processes this paper explains this intriguing observation by investigating theoretical and operational rd behavior of natural images we take as our source model the fractional brownian motion fbm which is often used to model nonstationary behaviors in natural images we first establish that the theoretical rd function of the fbm process both in 1 d and 2 d indeed follows a power law then we derive operational rd function of the fbm process when wavelet encoded based on water filling principle interestingly both the operational and theoretical rd functions behave as dpropr gamma for natural images the values of gamma are found to be distributed around 1 these results lend an information theoretical support to the merit of multiresolution wavelet compression of self similar processes and in particular natural images that can be modelled by such processes they may also prove useful in predicting performance of rd optimized image coders

rate distortion optimized tree structured compression algorithms for piecewise polynomial images this paper presents novel coding algorithms based on tree structured segmentation which achieve the correct asymptotic rate distortion r d behavior for a simple class of signals known as piecewise polynomials by using an r d based prune and join scheme for the one dimensional case our scheme is based on binary tree segmentation of the signal this scheme approximates the signal segments using polynomial models and utilizes an r d optimal bit allocation strategy among the different signal segments the scheme further encodes similar neighbors jointly to achieve the correct exponentially decaying r d behavior d r spl sim c sub 0 2 sup c1r thus improving over classic wavelet schemes we also prove that the computational complexity of the scheme is of o nlogn we then show the extension of this scheme to the two dimensional case using a quadtree this quadtree coding scheme also achieves an exponentially decaying r d behavior for the polygonal image model composed of a white polygon shaped object against a uniform black background with low computational cost of o nlogn again the key is an r d optimized prune and join strategy finally we conclude with numerical results which show that the proposed quadtree coding scheme outperforms jpeg2000 by about 1 db for real images like cameraman at low rates of around 0 15 bpp

image compression using binary space partitioning trees for low bit rate compression applications segmentation based coding methods provide in general high compression ratios when compared with traditional e g transform and subband coding approaches in this paper we present a new segmentation based image coding method that divides the desired image using binary space partitioning bsp the bsp approach partitions the desired image recursively by arbitrarily oriented lines in a hierarchical manner this recursive partitioning generates a binary tree which is referred to as the bsp tree representation of the desired image the most critical aspect of the bsp tree method is the criterion used to select the partitioning lines of the bsp tree representation in previous works we developed novel methods for selecting the bsp tree lines and showed that the bsp approach provides efficient segmentation of images in this paper we describe a hierarchical approach for coding the partitioning lines of the bsp tree representation we also show that the image signal within the different regions resulting from the recursive partitioning can be represented using low order polynomials furthermore we employ an optimum pruning algorithm to minimize the bit rate of the bsp tree representation for a given budget constraint while minimizing distortion simulation results and comparisons with other compression methods are also presented

the curvelet transform for image denoising we describe approximate digital implementations of two new mathematical transforms namely the ridgelet transform and the curvelet transform our implementations offer exact reconstruction stability against perturbations ease of implementation and low computational complexity a central tool is fourier domain computation of an approximate digital radon transform we introduce a very simple interpolation in the fourier space which takes cartesian samples and yields samples on a rectopolar grid which is a pseudo polar sampling set based on a concentric squares geometry despite the crudeness of our interpolation the visual performance is surprisingly good our ridgelet transform applies to the radon transform a special overcomplete wavelet pyramid whose wavelets have compact support in the frequency domain our curvelet transform uses our ridgelet transform as a component step and implements curvelet subbands using a filter bank of a spl grave trous wavelet filters our philosophy throughout is that transforms should be overcomplete rather than critically sampled we apply these digital transforms to the denoising of some standard images embedded in white noise in the tests reported here simple thresholding of the curvelet coefficients is very competitive with state of the art techniques based on wavelets including thresholding of decimated or undecimated wavelet transforms and also including tree based bayesian posterior mean methods moreover the curvelet reconstructions exhibit higher perceptual quality than wavelet based reconstructions offering visually sharper images and in particular higher quality recovery of edges and of faint linear and curvilinear features existing theory for curvelet and ridgelet transforms suggests that these new approaches can outperform wavelet methods in certain image reconstruction problems the empirical results reported here are in encouraging agreement

on the relation between weighted frequency domain maximum likelihood power spectral estimation and the prefiltered covariance extension approach the aim of this correspondence is to study the connection between weighted frequency domain maximum likelihood power spectral estimation and the time domain prefiltered covariance extension approach weighting and prefiltering are introduced to emphasize the model fit in a certain frequency range the main result is that these two methods are very closely related for the case of autoregressive ar model estimation which implies that both can be formulated as convex optimization problems examples illustrating the methods and the effect of prefiltering weighting are provided

wavelet decompositions of random forests smoothness analysis sparse approximation and applications in this paper we introduce in the setting of machine learning a generalization of wavelet analysis which is a popular approach to low dimensional structured signal analysis the wavelet decomposition of a random forest provides a sparse approximation of any regression or classification high dimensional function at various levels of detail with a concrete ordering of the random forest nodes from significant elements to nodes capturing only insignificant noise motivated by function space theory we use the wavelet decomposition to compute numerically a weak type smoothness index that captures the complexity of the underlying function as we show through extensive experimentation this sparse representation facilitates a variety of applications such as improved regression for difficult datasets a novel approach to feature importance resilience to noisy or irrelevant features compression of ensembles etc

hierarchical segmentation based image coding using hybrid quad binary trees a novel segmentation based image approximation and coding technique is proposed a hybrid quad binary qb tree structure is utilized to efficiently model and code geometrical information within images compared to other tree based representation such as wedgelets the proposed qb tree based method is more efficient for a wide range of contour features such as junctions corners and ridges especially at low bit rates

optimal pruning with applications to tree structured source coding and modeling an algorithm introduced by l breiman et al 1984 in the context of classification and regression trees is reinterpreted and extended to cover a variety of applications in source coding and modeling in which trees are involved these include variable rate and minimum entropy tree structured vector quantization minimum expected cost decision trees variable order markov modeling optimum bit allocation and computer graphics and image processing using quadtrees a concentration on the first of these and a detailed analysis of variable rate tree structured vector quantization are provided it is found that variable rate tree structured vector quantization outperforms not only the fixed rate variety but also full search vector quantization the successive approximation character of variable rate tree structured vector quantization permits it to degrade gracefully if the rate is reduced at the encoder this has applications to the problem of buffer overflow

on the importance of combining wavelet based nonlinear approximation with coding strategies this paper provides a mathematical analysis of transform compression in its relationship to linear and nonlinear approximation theory contrasting linear and nonlinear approximation spaces we show that there are interesting classes of functions random processes which are much more compactly represented by wavelet based nonlinear approximation these classes include locally smooth signals that have singularities and provide a model for many signals encountered in practice in particular for images however we also show that nonlinear approximation results do not always translate to efficient compress on strategies in a rate distortion sense based on this observation we construct compression techniques and formulate the family of functions stochastic processes for which they provide efficient descriptions in a rate distortion sense we show that this family invariably leads to besov spaces yielding a natural relationship among besov smoothness linear nonlinear approximation order and compression performance in a rate distortion sense the designed compression techniques show similarities to modern high performance transform codecs allowing us to establish relevant rate distortion estimates and identify performance limits

an optimal instrumental variable method for arma spectral estimation a multistep iterative and fast recursive algorithm for autoregressive moving average arma spectral estimation is presented the ar parameters of an arma process are estimated using the extended instrumental variable eiv method the optimal choice of instruments prefilter and weighting matrix is investigated a bootstrapping procedure that has computational convenience is proposed for the algorithm the statistical analysis and experiments show that the optimal iv estimate is unbiased consistent efficient asymptotically normal and equivalent to the maximum likelihood ml estimate and the prediction error pe estimate and the proposed algorithm has the advantages of sharper resolution less frequency bias and better efficiency of convergence

image compression with variable block size segmentation high quality variable rate image compression is achieved by segmenting an image into regions of different sizes classifying each region into one of several perceptually distinct categories and using a distinct coding procedure for each category segmentation is performed with a quadtree data structure by isolating the perceptually more important areas of the image into small regions and separately identifying larger random texture blocks since the important regions have been isolated the remaining parts of the image can be coded at a lower rate than would be otherwise possible high quality coding results are achieved at rates between 0 35 and 0 7 b p depending on the nature of the original image and satisfactory results have been obtained at 0 25 b p

multiresolution graph fourier transform for compression of piecewise smooth images piecewise smooth pws images e g depth maps or animation images contain unique signal characteristics such as sharp object boundaries and slowly varying interior surfaces leveraging on recent advances in graph signal processing in this paper we propose to compress the pws images using suitable graph fourier transforms gfts to minimize the total signal representation cost of each pixel block considering both the sparsity of the signal s transform coefficients and the compactness of transform description unlike fixed transforms such as the discrete cosine transform we can adapt gft to a particular class of pixel blocks in particular we select one among a defined search space of gfts to minimize total representation cost via our proposed algorithms leveraging on graph optimization techniques such as spectral clustering and minimum graph cuts furthermore for practical implementation of gft we introduce two techniques to reduce computation complexity first at the encoder we low pass filter and downsample a high resolution hr pixel block to obtain a low resolution lr one so that a lr gft can be employed at the decoder upsampling and interpolation are performed adaptively along hr boundaries coded using arithmetic edge coding so that sharp object boundaries can be well preserved second instead of computing gft from a graph in real time via eigen decomposition the most popular lr gfts are pre computed and stored in a table for lookup during encoding and decoding using depth maps and computer graphics images as examples of the pws images experimental results show that our proposed multiresolution gft scheme outperforms h 264 intra by 6 8 db on average in peak signal to noise ratio at the same bit rate

g h kkm type theorems and their applications to a new class of minimax inequalities abstract first some nonempty intersection theorems involving g h kkm type mappings are presented and then a related new class of minimax inequalities is given

a new recursive pseudo least squares algorithm for arma filtering and modeling ii this study is based on the observation that if the bootstrapping is combined with different parameterizations of the arma autoregressive moving average process then different linearized problems are obtained for the underlying nonlinear arma modeling problem in this part a specific parameterization termed the predictor space representation for an arma process which decouples the estimation for the ar and the ma parameters is used a vector space formalism for the given data case is then defined and the least squares arma filtering problem is interpreted in terms of projection operations on some linear spaces a new projection operator update formula which is particularly suited for the underlying problem is then used in conjunction with the vector space formalism to develop a computationally efficient pseudo least squares algorithm for arma filtering it is noted that these recursions can be put in the form of a filter structure

interpolation of besov spaces we investigate besov spaces and their connection with dyadic spline approximation in lp q 0 p oo our main results are the determination of the interpolation spaces between a pair of besov spaces an atomic decomposition for functions in a besov space the characterization of the class of functions which have certain prescribed degree of approximation by dyadic splines

on a generalized class of minimax inequalities abstract some results on a generalized class of minimax inequalities based on the r i g h kkm mapping theorems in a g h space setting are presented the r i g h kkm mappings represent a new class of kkm mappings in g h spaces as well as in the interval spaces

binary partition tree as an efficient representation for image processing segmentation and information retrieval this paper discusses the interest of binary partition trees as a region oriented image representation binary partition trees concentrate in a compact and structured representation a set of meaningful regions that can be extracted from an image they offer a multiscale representation of the image and define a translation invariant 2 connectivity rule among regions as shown in this paper this representation can be used for a large number of processing goals such as filtering segmentation information retrieval and visual browsing furthermore the processing of the tree representation leads to very efficient algorithms finally for some applications it may be interesting to compute the binary partition tree once and to store it for subsequent use for various applications in this context the paper shows that the amount of bits necessary to encode a binary partition tree remains moderate

an improved image compression algorithm using binary space partition scheme and geometric wavelets geometric wavelet is a recent development in the field of multivariate nonlinear piecewise polynomials approximation the present study improves the geometric wavelet gw image coding method by using the slope intercept representation of the straight line in the binary space partition scheme the performance of the proposed algorithm is compared with the wavelet transform based compression methods such as the embedded zerotree wavelet ezw the set partitioning in hierarchical trees spiht and the embedded block coding with optimized truncation ebcot and other recently developed sparse geometric representation based compression algorithms the proposed image compression algorithm outperforms the ezw the bandelets and the gw algorithm the presented algorithm reports a gain of 0 22 db over the gw method at the compression ratio of 64 for the cameraman test image

time series analysis in the frequency domain this correspondence presents a parametric frequency domain identification algorithm for autoregressive moving average arma processes that does not suffer from spectral leakage errors it is based on an extended transfer function model that takes into account the begin and end effect of the finite data record the relationship with the one step ahead prediction error method is established the advantages of the proposed method are easy prefiltering and leakage free spectral representation of the raw data

optimal approximations by piecewise smooth functions and associated variational problems abstract this reprint will introduce and study the most basic properties of three new variational problems which are suggested by applications to computer vision in computer vision a fundamental problem is to appropriately decompose the domain r of a function g x y of two variables this problem starts by describing the physical situation which produces images assume that a three dimensional world is observed by an eye or camera from some point p and that g1 rho represents the intensity of the light in this world approaching the point sub 1 from a direction rho if one has a lens at p focusing this light on a retina or a film in both cases a plane domain r in which we may introduce coordinates x y then let g x y be the strength of the light signal striking r at a point with coordinates x y g x y is essentially the same as sub 1 rho possibly after a simple transformation given by the geometry of the imaging syste the function g x y defined on the plane domain r will be called an image what sort of function is g the light reflected off the surfaces si of various solid objects o sub i visible from p will strike the domain r in various open subsets r sub i when one object o1 is partially in front of another object o2 as seen from p but some of object o2 appears as the background to the sides of o1 then the open sets r1 and r2 will have a common boundary the edge of object o1 in the image defined on r and one usually expects the image g x y to be discontinuous along this boundary jhd

best wavelet packet bases in a rate distortion sense a fast rate distortion r d optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented the scheme guarantees operation on the convex hull of the operational r d curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity applications for this coding technique include r coefman et al s yale univ 1990 generalized multiresolution wavelet packet decomposition iterative subband coders and quadtree structures applications to image processing involving wavelet packets as well as discrete cosine transform dct quadtrees are presented

optimal n term approximation by linear splines over anisotropic delaunay triangulations anisotropic triangulations provide efficient geometrical methods for sparse representations of bivariate functions from discrete data in particular from image data in previous work we have proposed a locally adaptive method for efficient image approximation called adaptive thinning which relies on linear splines over anisotropic delaunay triangulations in this paper we prove asymptotically optimal n term approximation rates for linear splines over anisotropic delaunay triangulations where our analysis applies to relevant classes of target functions a piecewise linear horizon functions across holder smooth boundaries b functions of w p regularity where 2 p 1 c piecewise regular horizon functions of w 2 regularity where 1

planelets a piecewise linear fractional model for preserving scene geometry in intra coding of indoor depth images geometrical wavelets have already proved their strength in approximation compression and denoising of piecewise constant and piecewise linear images in this paper we extend this family by introducing planelets toward an effective representation of indoor depth images it uses a linear fractional model to capture non linearity of depth values in the planar regions of the output images of kinect like sensors a block based compression framework based on planelet approximation is then presented which uses quadtree decomposition along with spatial predictions as an effective intra coding scheme compared with both classical geometric wavelets and some state of the art image coding algorithms our method provides desirable quality by explicitly representing edges and planar patches

adaptive multivariate approximation using binary space partitions and geometric wavelets the binary space partition bsp technique is a simple and efficient method to adaptively partition an initial given domain to match the geometry of a given input function as such the bsp technique has been widely used by practitioners but up until now no rigorous mathematical justification for it has been offered here we attempt to put the technique on sound mathematical foundations and we offer an enhancement of the bsp algorithm in the spirit of what we are going to call geometric wavelets this new approach to sparse geometric representation is based on recent developments in the theory of multivariate nonlinear piecewise polynomial approximation we provide numerical examples of n term geometric wavelet approximations of known test images and compare them with dyadic wavelet approximation we also discuss applications to image denoising and compression

brief box jenkins continuous time modeling this paper treats the identification of continuous time models using arbitrary band limited excitation signals a modeling approach is presented that has the following two advantages 1 asymptotically the amount of data tends to infinity there is no approximation error over the complete frequency band from dc to nyquist 2 it allows to identify general parametric noise models the key idea is to combine a continuous time plant model with a discrete time noise model hybrid box jenkins model structure

a new hybrid method for image approximation using the easy path wavelet transform the easy path wavelet transform epwt has recently been proposed by one of the authors as a tool for sparse representations of bivariate functions from discrete data in particular from image data the epwt is a locally adaptive wavelet transform it works along pathways through the array of function values and exploits the local correlations of the given data in a simple appropriate manner however the epwt suffers from its adaptivity costs that arise from the storage of path vectors in this paper we propose a new hybrid method for image approximation that exploits the advantages of the usual tensor product wavelet transform for the representation of smooth images and uses the epwt for an efficient representation of edges and texture numerical results show the efficiency of this procedure

low bit rate image coding using adaptive geometric piecewise polynomial approximation we present a new image coding algorithm the geometric piecewise polynomials gpp method that draws on recent developments in the theory of adaptive multivariate piecewise polynomials approximation the algorithm relies on a segmentation stage whose goal is to minimize a functional that is conceptually similar to the mumford shah functional except that it measures the smoothness of the segmentation instead of the length the initial segmentation is ldquoprunedrdquo and the remaining curve portions are lossy encoded the image is then further partitioned and approximated by low order polynomials on the subdomains we show examples where our algorithm outperforms state of the art wavelet coding in the low bit rate range the gpp algorithm significantly outperforms wavelet based coding methods on graphic and cartoon images also at the bit rate 0 05 bits per pixel the gpp algorithm achieves on the test image cameraman which has a geometric structure a psnr of 21 5 db while the jpeg2000 kakadu software obtains psnr of 20 db for the test image lena the gpp algorithm obtains the same psnr as jpeg2000 but with better visual quality at 0 03 bpp

tiling and adaptive image compression we investigate the task of compressing an image by using different probability models for compressing different regions of the image in this task using a larger number of regions would result in better compression but would also require more bits for describing the regions and the probability models used in the regions we discuss using quadtree methods for performing the compression we introduce a class of probability models for images the k rectangular tilings of an image that is formed by partitioning the image into k rectangular regions and generating the coefficients within each region by using a probability model selected from a finite class of n probability models for an image of size n spl times n we give a sequential probability assignment algorithm that codes the image with a code length which is within o k log nn k of the code length produced by the best probability model in the class the algorithm has a computational complexity of o nn sup 3 an interesting subclass of the class of k rectangular tilings is the class of tilings using rectangles whose widths are powers of two this class is far more flexible than quadtrees and yet has a sequential probability assignment algorithm that produces a code length that is within o k log nn k of the best model in the class with a computational complexity of o nn sup 2 logn similar to the computational complexity of sequential probability assignment using quadtrees we also consider progressive transmission of the coefficients of the image

space frequency quantization for image compression with directionlets the standard separable 2 d wavelet transform wt has recently achieved a great success in image processing because it provides a sparse representation of smooth images however it fails to efficiently capture 1 d discontinuities like edges or contours these features being elongated and characterized by geometrical regularity along different directions intersect and generate many large magnitude wavelet coefficients since contours are very important elements in the visual perception of images to provide a good visual quality of compressed images it is fundamental to preserve good reconstruction of these directional features in our previous work we proposed a construction of critically sampled perfect reconstruction transforms with directional vanishing moments imposed in the corresponding basis functions along different directions called directionlets in this paper we show how to design and implement a novel efficient space frequency quantization sfq compression algorithm using directionlets our new compression method outperforms the standard sfq in a rate distortion sense both in terms of mean square error and visual quality especially in the low rate compression regime we also show that our compression method does not increase the order of computational complexity as compared to the standard sfq algorithm

image coding with geometric wavelets this paper describes a new and efficient method for low bit rate image coding which is based on recent development in the theory of multivariate nonlinear piecewise polynomial approximation it combines a binary space partition scheme with geometric wavelet gw tree approximation so as to efficiently capture curve singularities and provide a sparse representation of the image the gw method successfully competes with state of the art wavelet methods such as the ezw spiht and ebcot algorithms we report a gain of about 0 4 db over the spiht and ebcot algorithms at the bit rate 0 0625 bits per pixels bpp it also outperforms other recent methods that are based on sparse geometric representation for example we report a gain of 0 27 db over the bandelets algorithm at 0 1 bpp although the algorithm is computationally intensive its time complexity can be significantely reduced by collecting a global gw n term approximation to the image from a collection of gw trees each constructed separately over tiles of the image

wavelet footprints theory algorithms and applications wavelet based algorithms have been successful in different signal processing tasks the wavelet transform is a powerful tool because it manages to represent both transient and stationary behaviors of a signal with few transform coefficients discontinuities often carry relevant signal information and therefore they represent a critical part to analyze we study the dependency across scales of the wavelet coefficients generated by discontinuities we start by showing that any piecewise smooth signal can be expressed as a sum of a piecewise polynomial signal and a uniformly smooth residual theorem 1 we then introduce the notion of footprints which are scale space vectors that model discontinuities in piecewise polynomial signals exactly we show that footprints form an overcomplete dictionary and develop efficient and robust algorithms to find the exact representation of a piecewise polynomial function in terms of footprints this also leads to efficient approximation of piecewise smooth functions finally we focus on applications and show that algorithms based on footprints outperform standard wavelet methods in different applications such as denoising compression and nonblind deconvolution in the case of compression we also prove that at high rates footprint based algorithms attain optimal performance theorem 3

all pole estimation in spectral domain autoregressive ar modeling is a popular spectral analysis method commonly resolved in the time domain this paper presents a novel ar analysis framework dealing with the estimation of poles directly from spectral samples the basis of the method lies on a minimizing functional built with a certain mapping of the spectral residue the optimization mechanism is based on the multivariate newton raphson algorithm two different mappings are considered namely linear and logarithmic the linear case results in a nonquadratic convex functional whose global minimum is equivalent to that of the time domain autocorrelation method the logarithmic case under the maximum likelihood criterion turns out equivalent to the whittle likelihood proven here to be suitable for frequency selective estimation the statistical and convergence performance of the method is demonstrated with simulations on stochastic and deterministic harmonic signals

rate distortion analysis of directional wavelets the inefficiency of separable wavelets in representing smooth edges has led to a great interest in the study of new 2 d transformations the most popular criterion for analyzing these transformations is the approximation power transformations with near optimal approximation power are useful in many applications such as denoising and enhancement however they are not necessarily good for compression therefore most of the nearly optimal transformations such as curvelets and contourlets have not found any application in image compression yet one of the most promising schemes for image compression is the elegant idea of directional wavelets diws while these algorithms outperform the state of the art image coders in practice our theoretical understanding of them is very limited in this paper we adopt the notion of rate distortion and calculate the performance of the diw on a class of edge like images our theoretical analysis shows that if the edges are not sharp the diw will compress them more efficiently than the separable wavelets it also demonstrates the inefficiency of the quadtree partitioning that is often used with the diw to solve this issue we propose a new partitioning scheme called megaquad partitioning our simulation results on real world images confirm the benefits of the proposed partitioning algorithm promised by our theoretical analysis