probabilistic algorithms for deciding equivalence of straight line programs let q be any algebraic structure and the set of all total programs over q using the instruction set z 1 z x y z x y z x y z x y a program is total if no division by zero occurs during any computation let the equivalence problem for be the problem of deciding for two given programs in whether or not they compute the same funcuon the following results are proved 1 if q is an inftmte field e g the rauonal numbers or the complex numbers then the equwalence problem for is probabilistlcally decidable in polynomml time the result also holds for programs with no dwlslon instructions and q an infimte integral domain e g the integers 2 if q is a finite field or if q is a fimte set of integers of cardmahty 2 then the equivalence problem is np hard the case when the field q is finite but its cardinality is a funcuon of the size of the instance to the eqmvalence problem is also considered an example is shown for which a sharp boundary between the classes np hard and probabihsticauy decidable exists provided they are not identical classes

the hermite transform applications it is demonstrated how the hermite transform can be used for image coding and analysis hierarchical coding structures based on increasingly specified basic patterns i e general 2 d patterns general 1 d patterns and specific 1 d patterns such as edges and corners are presented in the image coding application the relation with existing pyramid coders is described a new coding scheme based on local one dimensional image approximations is introduced in the image analysis application the relation between the hermite transform and existing line edge detection schemes is described it is shown that by concentrating on more specific patterns the coding efficiency can be increased since fewer coefficients have to be coded meanwhile sufficient descriptive power can be maintained for approximating the most interesting features in natural images

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

computation of the para pseudoinverse for oversampled filter banks forward and backward greville formulas frames and oversampled filter banks have been extensively studied over the past few years due to their increased design freedom and improved error resilience in frame expansions the least square signal reconstruction operator is called the dual frame which can be obtained by choosing the synthesis filter bank as the para pseudoinverse of the analysis bank in this paper we study the computation of the dual frame by exploiting the greville formula which was originally derived in 1960 to compute the pseudoinverse of a matrix when a new row is appended here we first develop the backward greville formula to handle the case of row deletion based on the forward greville formula we then study the computation of para pseudoinverse for extended filter banks and laplacian pyramids through the backward greville formula we investigate the frame based error resilient transmission over erasure channels the necessary and sufficient condition for an oversampled filter bank to be robust to one erasure channel is derived a postfiltering structure is also presented to implement the para pseudoinverse when the transform coefficients in one subband are completely lost

computing optical flow via variational techniques defined as the apparent motion in a sequence of images the optical flow is very important in the computer vision community where its accurate estimation is necessary for many applications it is o

subpixel edge localization and the interpolation of still images in this paper we present a nonlinear interpolation scheme for still image resolution enhancement the algorithm is based on a source model emphasizing the visual integrity of detected edges and incorporates a novel edge fitting operator that has been developed for this application a small neighborhood about each pixel in the low resolution image is first mapped to a best fit continuous space step edge the bilevel approximation serves as a local template on which the higher resolution sampling grid can then be superimposed where disputed values in regions of local window overlap are averaged to smooth errors the result is an image of increased resolution with noticeably sharper edges and in all tried cases lower mean squared reconstruction error than that produced by linear techniques

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

color image coding using morphological pyramid decomposition presents a new algorithm that utilizes mathematical morphology for pyramidal coding of color images the authors obtain lossy color image compression by using block truncation coding at the pyramid levels to attain reduced data rates the pyramid approach is attractive due to low computational complexity simple parallel implementation and the ability to produce acceptable color images at moderate data rates in many applications the progressive transmission capability of the algorithm is very useful the authors show experimental results for color images at data rates of 1 89 bits pixel

divergence measure fields and hyperbolic conservation laws we analyze a class of l infty vector fields called divergence measure fields we establish the gauss green formula the normal traces over subsets of lipschitz boundaries and the product rule for this class of l infty fields then we apply this theory to analyze l infty entropy solutions of initial boundary value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data the examples of conservation laws include multidimensional scalar equations the system of nonlinear elasticity and a class of m x m systems with affine characteristic hypersurfaces the analysis in l infty also extends to l p

block median pyramidal transform analysis and denoising applications a nonlinear multiscale pyramidal transform based on nonoverlapping block decompositions using the median operation and a polynomial approximation is considered it is shown that this structure can be useful for denoising of oneand two dimensional 1 d and 2 d signals various denoising techniques are analyzed including methods based on spatially adaptive thresholding and partial cycle spinning algorithms an analytical method for deriving the distribution function of the transform coefficients is also presented this in turn can be used for the selection of thresholds for denoising applications

a new family of nonredundant transforms using hybrid wavelets and directional filter banks we propose a new family of nonredundant geometrical image transforms that are based on wavelets and directional filter banks we convert the wavelet basis functions in the finest scales to a flexible and rich set of directional basis elements by employing directional filter banks where we form a nonredundant transform family which exhibits both directional and nondirectional basis functions we demonstrate the potential of the proposed transforms using nonlinear approximation in addition we employ the proposed family in two key image processing applications image coding and denoising and show its efficiency for these applications

q warping direct computation of quadratic reference surfaces we consider the problem of wrapping around an object of which two views are available a reference surface and recovering the resulting parametric flow using direct computations via spatio temporal derivatives the well known examples are affine flow models and eight parameter flow models both describing a flow field of a planar reference surface we extend those classic flow models to deal with a quadric reference surface and work out the explicit parametric form of the flow field as a result we derive a simple warping algorithm that maps between two views and leaves a residual flow proportional to the 3d deviation of the surface from a virtual quadric surface the applications include image morphing model building image stabilization and disparate view correspondence

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

solving structured linear systems with large displacement rank linear systems with structures such as toeplitz vandermonde or cauchy likeness can be solved in email protected a 2n operations where n is the matrix size a is its displacement rank and email protected denotes the omission of logarithmic factors we show that for such matrices this cost can be reduced to email protected a w 1n where w is a feasible exponent for matrix multiplication over the base field the best known estimate for w is w 2 38 resulting in costs of order email protected a 1 3 8n we present consequences for hermite pade approximation and bivariate interpolation

parallel evaluation of arithmetic circuits abstract in this paper a generic algorithm designed for the parallel evaluation of arithmetic circuits is given this algorithm can be used in the domain of vlsi design in order to get tight upper bounds on the computing time of a circuit it can also be used in automatic parallelization of numerical programs as a guide for the detection of some predefinite schemes such as dot products or reductions more generally the theoretical algorithm presented in section 2 evaluates very quickly arithmetic straight line programs and its evaluation time serves as a good upper bound this algorithm generalizes miller ramachandran and kaltofen s algorithm 1988 in the sense it deals with a great variety of algebraic structures semi rings rings or lattices our contribution resides on the one hand in a new bound for the evaluation of circuits over lattices which improves previous results miller and teng 1987 and on the other hand in the unified formulation for the evaluation algorithm this algorithm runs in o min log n log d log n ha log n log n parallel time d being the algebraic degree in an extended sense of the circuit and ha the maximal number of alternances of and on a path of the circuit if the and operations define a lattice with m n processors where m n is the number of processors necessary for the multiplication of two n n matrices in the structure in o log n parallel time after presenting this algorithm its efficiency is shown on particular cases taking as input a simple and sequential algorithm it can be used as a compiler to produce a sorting circuit as fast as cole s circuit with logarithmic depth or an adder equivalent to brent and kung s adder in terms of size and depth these academic examples confirm the relevance of the algorithm presented here in the area of conception of fast vlsi arithmetic operators

relative affine structure canonical model for 3d from 2d geometry and applications we propose an affine framework for perspective views captured by a single extremely simple equation based on a viewer centered invariant we call relative affine structure via a number of corollaries of our main results we show that our framework unifies previous work including euclidean projective and affine in a natural and simple way and introduces new extremely simple algorithms for the tasks of reconstruction from multiple views recognition by alignment and certain image coding applications

interpolation of polynomials given by straight line programs we give an algorithm for the interpolation of a polynomial a given by a straight line program its complexity is polynomial in t log d l n where t is an input bound on the number of terms in a d is a bound on its partial degree in all variables l is the length of the given straight line program and n is the number of variables

ranging through gabor logons a consistent hierarchical approach in this work the correspondence problem in stereo vision is handled by matching two sets of dense feature vectors inspired by biological evidence these feature vectors are generated by a correlation between a bank of gabor sensors and the intensity image the sensors consist of two dimensional gabor filters at various scales spatial frequencies and orientations which bear close resemblance to the receptive field profiles of simple v1 cells in visual cortex a hierarchical stochastic relaxation method is then used to obtain the dense stereo disparities unlike traditional hierarchical methods for stereo feature based hierarchical processing yields consistent disparities to avoid false matchings due to static occlusion a dual matching based on the imaging geometry is used

image enhancement based on a nonlinear multiscale method an image enhancement method that reduces speckle noise and preserves edges is introduced the method is based on a new nonlinear multiscale reconstruction scheme that is obtained by successively combining each coarser scale image with the corresponding modified interscale image simulation results are included to demonstrate the performance of the proposed method

detail enhanced exposure fusion in a typical processing chain of image enhancement an exposure fusion scheme can be used to synthesize a more detailed low dynamic range ldr image directly from a set of differently exposed ldr images without generation of an intermediate high dynamic range image in this brief we introduce a new quadratic optimization based method to extract fine details from a vector field the new method extracts fine details from a set of differently exposed ldr images simultaneously the extracted fine details are then added to an intermediate ldr image which is fused by simply using an existing exposure fusion scheme with this the proposed scheme can enhance fine details to produce sharper images

the binomial qmf wavelet transform for multiresolution signal decomposition perfect reconstruction quadrature mirror filters pr qmf s have been proposed as structures suitable for hierarchical subband coding ll 4 and also for multiresolution signal decomposition as might be used in image pyramid coding 5 more recently multiresolution signal decomposition methods are being examined from the standpoint of the discrete wavelet transform for continuous time signals 6 8 in this paper we describe a class of orthogonal binomial filters that provide basis functions for a perfect reconstruction bank of finite impulse response qmf s the orthonormal wavelet filters derived by daubechies 171 from a discrete wavelet transform approach are shown to be the same as the solutions inherent in the binomial based filters the energy compaction performance of the binomial qmf decomposition is computed and shown to be better than the dct for the markov source models as well as real world images considered the proposed binomial structure is efficient simple to implement on vlsi and suitable for multiresolution signal decomposition and coding applications

a fast search algorithm for vector quantization using mean pyramids of codewords one of the most serious problems for vector quantization especially for high dimensional vectors is the high computational complexity of searching for the closest codeword in the codebook design and encoding phases although quantizing high dimensional vectors rather than low dimensional vectors results in better performance the computation time needed for vector quantization grows exponentially with the vector dimension this makes high dimensional vectors unsuitable for vector quantization to overcome this problem a fast search algorithm under the assumption that the distortion is measured by the squared euclidean distance is proposed using the mean pyramids of codewords the algorithm ran reject many codewords that are impossible matches and hence save a great deal of computation time the algorithm is efficient for high dimensional codeword searches experimental results confirm the effectiveness of the proposed method

directional multiscale modeling of images using the contourlet transform the contourlet transform is a new two dimensional extension of the wavelet transform using multiscale and directional filter banks the contourlet expansion is composed of basis images oriented at various directions in multiple scales with flexible aspect ratios given this rich set of basis images the contourlet transform effectively captures smooth contours that are the dominant feature in natural images we begin with a detailed study on the statistics of the contourlet coefficients of natural images using histograms to estimate the marginal and joint distributions and mutual information to measure the dependencies between coefficients this study reveals the highly non gaussian marginal statistics and strong interlocation interscale and interdirection dependencies of contourlet coefficients we also find that conditioned on the magnitudes of their generalized neighborhood coefficients contourlet coefficients can be approximately modeled as gaussian random variables based on these findings we model contourlet coefficients using a hidden markov tree hmt model with gaussian mixtures that can capture all interscale interdirection and interlocation dependencies we present experimental results using this model in image denoising and texture retrieval applications in denoising the contourlet hmt outperforms other wavelet methods in terms of visual quality especially around edges in texture retrieval it shows improvements in performance for various oriented textures

on the equivalence of soft wavelet shrinkage total variation diffusion total variation regularization and sides soft wavelet shrinkage total variation tv diffusion tv regularization and a dynamical system called sides are four useful techniques for discontinuity preserving denoising of signals and images in this paper we investigate under which circumstances these methods are equivalent in the one dimensional case first we prove that haar wavelet shrinkage on a single scale is equivalent to a single step of space discrete tv diffusion or regularization of two pixel pairs in the translationally invariant case we show that applying cycle spinning to haar wavelet shrinkage on a single scale can be regarded as an absolutely stable explicit discretization of tv diffusion we prove that space discrete tv diffusion and tv regularization are identical and that they are also equivalent to the sides system when a specific force function is chosen afterwards we show that wavelet shrinkage on multiple scales can be regarded as a single step diffusion filtering or regularization of the laplacian pyramid of the signal we analyze possibilities to avoid gibbs like artifacts for multiscale haar wavelet shrinkage by scaling the thresholds finally we present experiments where hybrid methods are designed that combine the advantages of wavelets and pde variational approaches these methods are based on iterated shift invariant wavelet shrinkage at multiple scales with scaled thresholds

wavelet transforms associated with finite cyclic groups multiresolution analysis via decomposition on wavelet bases has emerged as an important tool in the analysis of signals and images when these objects are viewed as sequences of complex or real numbers an important class of multiresolution decompositions are the laplacian pyramid schemes in which the resolution is successively halved by recursively low pass filtering the signal under analysis and decimating it by a factor of two in general the principal framework within which multiresolution techniques have been studied and applied is the same as that used in the discrete time fourier analysis of sequences of complex numbers an analogous framework is developed for the multiresolution analysis of finite length sequences of elements from arbitrary fields attention is restricted to sequences of length 2 sup n for n a positive integer so that the resolution may be recursively halved to completion as in finite length fourier analysis a cyclic group structure of the index set of such sequences is exploited to characterize the transforms of interest for the particular cases of complex and finite fields

an analysis of physics based optical flow a variational solution of the physics based optical flow equation is studied for extraction of high resolution velocity fields from flow visualization images the solution to preserve discontinuities in a fluid velocity field is mathematically justified the uniqueness of the solution and the convergence of a successive approximation sequence are proven a numerical algorithm is developed and examined through numerical experiments

a theoretical framework for convex regularizers in pde based computation of image motion many differential methods for the recovery of the optic flow field from an image sequence can be expressed in terms of a variational problem where the optic flow minimizes some energy typically these energy functionals consist of two terms a data term which requires e g that a brightness constancy assumption holds and a regularizer that encourages global or piecewise smoothness of the flow field in this paper we present a systematic classification of rotation invariant convex regularizers by exploring their connection to diffusion filters for multichannel images this taxonomy provides a unifying framework for data driven and flow driven isotropic and anisotropic as well as spatial and spatio temporal regularizers while some of these techniques are classic methods from the literature others are derived here for the first time we prove that all these methods are well posed they posses a unique solution that depends in a continuous way on the initial data an interesting structural relation between isotropic and anisotropic flow driven regularizers is identified and a design criterion is proposed for constructing anisotropic flow driven regularizers in a simple and direct way from isotropic ones its use is illustrated by several examples

scale based detection of corners of planar curves a technique for detecting and localizing corners of planar curves is proposed the technique is based on gaussian scale space which consists of the maxima of absolute curvature of the boundary function presented at all scales the scale space of isolated simple and double corners is first analyzed to investigate the behavior of scale space due to smoothing and interactions between two adjacent corners the analysis shows that the resulting scale space contains line patterns that either persist terminate or merge with a neighboring line next the scale space is transformed into a tree that provides simple but concise representation of corners at multiple scales finally a multiple scale corner detection scheme is developed using a coarse to fine tree parsing technique the parsing scheme is based on a stability criterion that states that the presence of a corner must concur with a curvature maximum observable at a majority of scales experiments were performed to show that the scale space corner detector is reliable for objects with multiple size features and noisy boundaries and compares favorably with other corner detectors tested

image representations using multiscale differential operators differential operators have been widely used for multiscale geometric descriptions of images the efficient computation of these differential operators is always desirable moreover it has not been clear whether such representations are invertible for certain applications it is usually required that such representations should be invertible so that one can facilitate the processing of information in the transform domain and then recover it in this paper such problems are studied we consider multiscale differential representations of images using different types of operators such as the directional derivative operators and laplacian operators in particular we provide a general approach to represent images by their multiscale and multidirectional derivative components for practical implementation efficient pyramid like algorithms are derived using the spline technique for both the the composition and reconstruction of images it is shown that using these representations various meaningful geometric information of images can be extracted at multiple scales therefore these representations can be used for edge based image processing purposes furthermore the intrinsic relationships of the proposed representations with the compact wavelet models and some classical multiscale approaches are also elucidated in the paper

lossless image compression by quantization feedback in a content driven enhanced laplacian pyramid in this paper the effects of quantization noise feedback on the entropy of laplacian pyramids are investigated this technique makes it possible for the maximum absolute reconstruction error to be easily and strongly upper bounded near lossless coding and therefore allows reversible compression the entropy minimizing optimum quantizer is obtained by modeling the first order distributions of the differential signals as laplacian densities and by deriving a model for the equivalent memoryless entropy a novel approach based on an enhanced laplacian pyramid is proposed for the compression either lossless or lossy of gray scale images major details are prioritized through a content driven decision rule embedded in a uniform threshold quantizer with noise feedback lossless coding shows improvements over reversible joint photographers expert group jpeg and the reduced difference pyramid schemes while lossy coding outperforms jpeg with a significant peak signal to noise ratio psnr gain also subjective quality is higher even at very low bit rates due to the absence of the annoying impairments typical of jpeg moreover image versions having resolution and snr that are both progressively increasing are made available at the receiving end from the earliest retrieval stage on as intermediate steps of the decoding procedure without any additional cost

tests for permutation polynomials if mathbb f q is a finite field and f in mathbb f q x then f is called a permutation polynomial if the mapping mathbb f q to mathbb f q induced by f is bijective this property can be tested by a probabilistic algorithm whose number of operations is polynomial in fact essentially linear in the input size i e in deg f cdot log q this is extended to almost permutation polynomials whose value set consists of almost all elements of mathbb f q

progressive image transmission using vector quantization on images in pyramid form a progressive image transmission scheme in which vector quantization is applied to images represented by pyramids is proposed a mean pyramid representation of an image is first built up by forming a sequence of reduced size images by averaging over blocks of 2 2 pixels a difference pyramid is then built up by taking the differences between successive levels in the mean pyramid progressive transmission is achieved by sending all the nodes in the difference pyramid starting from the top level and ending at the bottom level the kth approximate image can be formed by adding the information of level k to the previously reproduced k 1 st approximation to gain efficiency vector quantization is applied to the difference pyramid of the image on a level by level basis if the errors due to quantization at level k are properly delivered and included in the next level k 1 then it is demonstrated that the original image can be reconstructed an entropy coder is used to encode the final residual error image losslessly thus ensuring perfect reproduction of the original image the experiments demonstrate that it is possible to achieve simultaneously lossless and progressive transmission with compression at the intermediate level the use of vector quantization results in a coding gain over that obtained using only a huffman coder excellent reproduction is achieved at a bit rate of only 0 06 bits pixel

orientation sensitive interpolative pyramids for lossless and progressive image coding this paper presents a modified jpeg coder that is applied to the compression of mixed documents containing text natural images and graphics for printing purposes the modified jpeg coder proposed in this paper takes advantage of the distinct perceptually significant regions in these documents to achieve higher perceptual quality than the standard jpeg coder the region adaptivity is performed via classified thresholding being totally compliant with the baseline standard a computationally efficient classification algorithm is presented and the improved performance of the classified jpeg coder is verified

dense estimation of fluid flows in this paper we address the problem of estimating and analyzing the motion of fluids in image sequences due to the great deal of spatial and temporal distortions that intensity patterns exhibit in images of fluids the standard techniques from computer vision originally designed for quasi rigid motions with stable salient features are not well adapted in this context we thus investigate a dedicated minimization based motion estimator the cost function to be minimized includes a novel data term relying on an integrated version of the continuity equation of fluid mechanics which is compatible with large displacements this term is associated with an original second order div curl regularization which prevents the washing out of the salient vorticity and divergence structures the performance of the resulting fluid flow estimator is demonstrated on meteorological satellite images in addition we show how the sequences of dense motion fields we estimate can be reliably used to reconstruct trajectories and to extract the regions of high vorticity and divergence

on two channel filter banks with directional vanishing moments the contourlet transform was proposed to address the limited directional resolution of the separable wavelet transform one way to guarantee good approximation behavior is to let the directional filters in the contourlet filter bank have sharp frequency response this requires filters with large support size we seek to isolate the key filter property that ensures good approximation in this direction we propose filters with directional vanishing moments dvm these filters we show annihilate information along a given direction we study two channel filter banks with dvm filters we provide conditions under which the design of dvm filter banks is possible a complete characterization of the product filter is thus obtained we propose a design framework that avoids 2 d factorization using the mapping technique the filters designed when used in the contourlet transform exhibit nonlinear approximation comparable to the conventional filters while being shorter and therefore providing better visual quality with less ringing artifacts furthermore experiments show that the proposed filters outperform the conventional ones in image approximation and denoising

a voting based computational framework for visual motion analysis and interpretation most approaches for motion analysis and interpretation rely on restrictive parametric models and involve iterative methods which depend heavily on initial conditions and are subject to instability further difficulties are encountered in image regions where motion is not smooth typically around motion boundaries this work addresses the problem of visual motion analysis and interpretation by formulating it as an inference of motion layers from a noisy and possibly sparse point set in a 4d space the core of the method is based on a layered 4d representation of data and a voting scheme for affinity propagation the inherent problem caused by the ambiguity of 2d to 3d interpretation is usually handled by adding additional constraints such as rigidity however enforcing such a global constraint has been problematic in the combined presence of noise and multiple independent motions by decoupling the processes of matching outlier rejection segmentation and interpretation we extract accurate motion layers based on the smoothness of image motion and then locally enforce rigidity for each layer in order to infer its 3d structure and motion the proposed framework is noniterative and consistently handles both smooth moving regions and motion discontinuities without using any prior knowledge of the motion model

multiscale hybrid linear models for lossy image representation in this paper we introduce a simple and efficient representation for natural images we view an image in either the spatial domain or the wavelet domain as a collection of vectors in a high dimensional space we then fit a piece wise linear model i e a union of affine subspaces to the vectors at each downsampling scale we call this a multiscale hybrid linear model for the image the model can be effectively estimated via a new algebraic method known as generalized principal component analysis gpca the hybrid and hierarchical structure of this model allows us to effectively extract and exploit multimodal correlations among the imagery data at different scales it conceptually and computationally remedies limitations of many existing image representation methods that are based on either a fixed linear transformation e g dct wavelets or an adaptive uni modal linear transformation e g pca or a multimodal model that uses only cluster means e g vq we will justify both quantitatively and experimentally why and how such a simple multiscale hybrid model is able to reduce simultaneously the model complexity and computational cost despite a small overhead of the model our careful and extensive experimental results show that this new model gives more compact representations for a wide variety of natural images under a wide range of signal to noise ratios than many existing methods including wavelets we also briefly address how the same hybrid linear modeling paradigm can be extended to be potentially useful for other applications such as image segmentation

efficient synthesis of gaussian filters by cascaded uniform filters gaussian filtering is an important tool in image processing and computer vision in this paper we discuss the background of gaussian filtering and look at some methods for implementing it consideration of the central limit theorem suggests using a cascade of simple filters as a means of computing gaussian filters among simple filters uniform coefficient finite impulse response digital filters are especially economical to implement the idea of cascaded uniform filters has been around for a while 13 16 we show that this method is economical to implement has good filtering characteristics and is appropriate for hardware implementation we point out an equivalence to one of burt s methods 1 3 under certain circumstances as an extension we describe an approach to implementing a gaussian pyramid which requires approximately two addition operations per pixel per level per dimension we examine tradeoffs in choosing an algorithm for gaussian filtering and finally discuss an implementation

optimal pyramidal and subband decompositions for hierarchical coding of noisy and quantized images optimal hierarchical coding is sought for progressive or scalable image transmission by minimizing the variance of the error difference between the original image and its lower resolution renditions the optimal according to the above criterion pyramidal and subband image coders are determined for images subject to corruption by quantization or transmission noise given arbitrary analysis filters and assuming adequate knowledge of the noise statistics optimal synthesis filters are found the optimal analysis filters are subsequently determined leading to formulas for globally optimal structures for pyramidal and subband image decompositions experimental results illustrate the implementation and performance of the optimal coders

greatest common divisors of polynomials given by straight line programs algorithms on multivariate polynomials represented by straight line programs are developed first it is shown that most algebraic algorithms can be probabilistically applied to data that are given by a straight line computation testing such rational numeric data for zero for instance is facilitated by random evaluations modulo random prime numbers then auxiliary algorithms that determine the coefficients of a multivariate polynomial in a single variable are constructed the first main result is an algorithm that produces the greatest common divisor of the input polynomials all in straight line representation the second result shows how to find a straight line program for the reduced numerator and denominator from one for the corresponding rational function both the algorithm for that construction and the greatest common divisor algorithm are in random polynomial time for the usual coefficient fields and output a straight line program which with controllably high probability correctly determines the requested answer the running times are polynomial functions in the binary input size the input degrees as unary numbers and the logarithm of the inverse of the failure probability the algorithm for straight line programs for the numerators and denominators of rational functions implies that every degree bounded rational function can be computed fast in parallel that is in polynomial size and polylogarithmic depth

b spline snakes a flexible tool for parametric contour detection we present a novel formulation for b spline snakes that can be used as a tool for fast and intuitive contour outlining we start with a theoretical argument in favor of splines in the traditional formulation by showing that the optimal curvature constrained snake is a cubic spline irrespective of the form of the external energy field unfortunately such regularized snakes suffer from slow convergence speed because of a large number of control points as well as from difficulties in determining the weight factors associated to the internal energies of the curve we therefore propose an alternative formulation in which the intrinsic scale of the spline model is adjusted a priori this leads to a reduction of the number of parameters to be optimized and eliminates the need for internal energies i e the regularization term in other words we are now controlling the elasticity of the spline implicitly and rather intuitively by varying the spacing between the spline knots the theory is embedded into a multiresolution formulation demonstrating improved stability in noisy image environments validation results are presented comparing the traditional snake using internal energies and the proposed approach without internal energies showing the similar performance of the latter several biomedical examples of applications are included to illustrate the versatility of the method

image enhancement by nonlinear extrapolation in frequency space a technique for enhancing the perceptual sharpness of an image is described the enhancement algorithm augments the frequency content of the image using shape invariant properties of edges across scale by using a nonlinearity that generates phase coherent higher harmonics the procedure utilizes the laplacian transform and the laplacian pyramid image representation results are presented depicting the power spectra augmentation and the visual enhancement of several images simplicity of computations and ease of implementation allow for real time applications such as high definition television hdtv

optimal generating kernels for image pyramids by piecewise fitting a novel class of generating kernels for image pyramids is introduced when these kernels are convolved with intensity functions of images continuous piecewise surfaces composed of polynomial tensor products are fitted to the intensity functions the fittings are optimal in the sense that the mean square error between them and the original intensity functions is minimized two members of the class are introduced and symmetry normalization unimodality and equal contribution properties are proved these kernels possess attractive properties such as small window size fast inverse transformation and minimum error experiments show that they compare favorably with existing ones in terms of mean square error

direct methods for recovering motion we have developed direct methods for recovering the motion of an observer in a static environment in the case of pure rotation pure translation and arbitrary motion when the rotation is known some of these methods are based on the minimization of the difference between the observed time derivative of brightness and that predicted from the spatial brightness gradient given the estimated motion we minimize the square of the integral of this difference taken over the image region of interest other methods presented here exploit the fact that surfaces have to be in front of the observer in order to be seen

stereo correspondence by surface reconstruction an algorithm that solves the computational stereo correspondence and the surface reconstruction is presented the algorithm integrates the reconstruction process in the correspondence analysis by means of multipass attribute matching and disparity refinement in the matching process the requirement of attribute similarity is relaxed with the pass number while the requirement for agreement between the predicted and the measured disparity is tightened disparity discontinuities and occluded areas are detected by analyzing the partial derivatives of the reconstructed disparity surface results on synthetic and on real stereo image pairs are reported

on convergence of the horn and schunck optical flow estimation method the purpose of this study is to prove convergence results for the horn and schunck optical flow estimation method horn and schunck stated optical flow estimation as the minimization of a functional when discretized the corresponding euler lagrange equations form a linear system of equations we write explicitly this system and order the equations in such a way that its matrix is symmetric positive definite this property implies the convergence gauss seidel iterative resolution method but does not afford a conclusion on the convergence of the jacobi method however we prove directly that this method also converges we also show that the matrix of the linear system is block tridiagonal the blockwise iterations corresponding to this block tridiagonal structure converge for both the jacobi and the gauss seidel methods and the gauss seidel method is faster than the sequential jacobi method

the complexity of two problems on arithmetic circuits by using arithmetic circuits encoding multivariate polynomials may be drastically more efficient than writing down the list of monomials via the study of two examples we show however that such an encoding can be hard to handle with a turing machine even if the degree of the polynomial is low namely we show that deciding whether the coefficient of a given monomial is zero is hard for p p under strong nondeterministic turing reductions as a result this problem does not belong to the polynomial hierarchy unless this hierarchy collapses for polynomials over fields of characteristic k 0 this problem is mod kp complete this gives a conp m o d k p algorithm for deciding an upper bound on the degree of a polynomial given by a circuit in fields of characteristic k 0

significance linked connected component analysis for wavelet image coding the success in wavelet image coding is mainly attributed to a recognition of the importance of data organization and representation there have been several very competitive wavelet coders developed namely shapiro s 1993 embedded zerotree wavelets ezw servetto et al s 1995 morphological representation of wavelet data mrwd and said and pearlman s see ieee trans circuits syst video technol vol 6 p 245 50 1996 set partitioning in hierarchical trees spiht we develop a novel wavelet image coder called significance linked connected component analysis slcca of wavelet coefficients that extends mrwd by exploiting both within subband clustering of significant coefficients and cross subband dependency in significant fields extensive computer experiments on both natural and texture images show convincingly that the proposed slcca outperforms ezw mrwd and spiht for example for the barbara image at 0 25 b pixel slcca outperforms ezw mrwd and spiht by 1 41 db 0 32 db and 0 60 db in psnr respectively it is also observed that slcca works extremely well for images with a large portion of texture for eight typical 256 spl times 256 grayscale texture images compressed at 0 40 b pixel slcca outperforms spiht by 0 16 db 0 63 db in psnr this performance is achieved without using any optimal bit allocation procedure thus both the encoding and decoding procedures are fast

multiscale local polynomial smoothing in a lifted pyramid for non equispaced data this paper integrates burt adelson s laplacian pyramids with lifting schemes for the construction of slightly redundant decompositions these decompositions implement multiscale smoothing on possibly non equidistant point sets thanks to the slight redundancy and to the smoothing operations in the lifting scheme the proposed construction unifies sparsity of the analysis smoothness of the reconstruction and stability of the transforms the decomposition is of linear computational complexity with just a slightly larger constant than the fast lifted wavelet transform this paper also discusses several alternatives in the design of non stationary finite impulse response filters for a stable multiresolution smoothing system these filters are adapted to each other and to the locations of the observations

on polynomials with symmetric galois group which are easy to compute abstract we construct a sequence of univariate polynomials over an arbitrary hilbertian field which are easy to compute and have symmetric galois group as a consequence we obtain that in any complexity class almost all polynomials have symmetric galois group all factors of such polynomials with suitable degree and coefficients from the decomposition field are hard to compute

polyhedral object localization in an image by referencing to a single model view identifying a three dimensional 3d object in an image is traditionally dealt with by referencing to a 3d model of the object in the last few years there has been a growing interest of using not a 3d shape but multiple views of the object as the reference this paper attempts a further step in the direction using not multiple views but a single clean view as the reference model the key issue is how to establish correspondences from the model view where the boundary of the object is explicitly available to the scene view where the object can be surrounded by various distracting entities and its boundary disturbed by noise we propose a solution to the problem which is based upon a mechanism of predicting correspondences from just four particular initial point correspondences the object is required to be polyhedral or near polyhedral the correspondence mechanism has a computational complexity linear with respect to the total number of visible corners of the object in the model view the limitation of the mechanism is also analyzed thoroughly in this paper experimental results over real images are presented to illustrate the performance of the proposed solution

the laplacian pyramid as a compact image code we describe a technique for image encoding in which local operators of many scales but identical shape serve as the basis functions the representation differs from established techniques in that the code elements are localized in spatial frequency as well as in space pixel to pixel correlations are first removed by subtracting a lowpass filtered copy of the image from the image itself the result is a net data compression since the difference or error image has low variance and entropy and the low pass filtered image may represented at reduced sample density further data compression is achieved by quantizing the difference image these steps are then repeated to compress the low pass image iteration of the process at appropriately expanded scales generates a pyramid data structure the encoding process is equivalent to sampling the image with laplacian operators of many scales thus the code tends to enhance salient image features a further advantage of the present code is that it is well suited for many image analysis tasks as well as for image compression fast algorithms are described for coding and decoding

a unified approach to boundary perception edges textures and illusory contours a model consisting of a multistage system which extracts and groups salient features in the image at different spatial scales or frequencies is used in the first stage a gabor wavelet decomposition provides a representation of the image which is orientation selective and has optimal localization properties in space and frequency this decomposition is useful in detecting significant features such as step and line edges at different scales and orientations in the image following the wavelet transformation local competitive interactions are introduced to reduce the effects of noise and changes in illumination interscale interactions help in localizing the line ends and corners and play a crucial role in boundary perception the final stage groups similar features aiding in boundary completion the different stages can be identified with processing by simple complex and hypercomplex cells in the visual cortex of mammals experimental results demonstrate the performance of this model in detecting boundaries both real and illusory in real and synthetic images

image filtering using multiresolution representations it is shown how multiresolution representations can be used for filter design and implementation these representations provide a coarse frequency decomposition of the image which forms the basis for two filtering techniques the first method based on image pyramids is used for approximating the convolution of an image with a given mask in this technique a filter is designed using a least squares procedure based on filters synthesized from the basic pyramid equivalent filters the second method is an adaptive noise reduction algorithm an optimally filtered image is synthesized from the multiresolution levels which in this case are maintained at the original sampling density individual pixels of the image representation are linearly combined under a minimum mean square error criterion this uses a local signal to noise ratio estimate to provide the best compromise between noise removal and resolution loss

on photometric issues in 3d visual recognition from asingle 2d image we describe the problem of recognition under changing illumination conditions and changing viewing positions from a computational and human vision perspective on the computational side we focus on the mathematical problems of creating an equivalence class for images of the same 3d object undergoing certain groups of transformations mostly those due to changing illumination and briefly discuss those due to changing viewing positions the computational treatment culminates in proposing a simple scheme for recognizing via alignment an image of a familiar object taken from a novel viewing position and a novel illumination condition on the human vision aspect the paper is motivated by empirical evidence inspired by mooney images of faces that suggest a relatively high level of visual processing is involved in compensating for photometric sources of variability and furthermore that certain limitations on the admissible representations of image information may exist the psychophysical observations and the computational results that follow agree in several important respects such as the same apparent limitations on image representations

sparse interpolation of multivariate rational functions consider the black box interpolation of a sparse n variate rational function f where is the maximum number of terms in either numerator or denominator when numerator and denominator are at most of degree d then the number of possible terms in f is o dn and explodes exponentially as the number of variables increases the complexity of our sparse rational interpolation algorithm does not depend exponentially on n anymore it still depends on d because we densely interpolate univariate auxiliary rational functions of the same degree we remove the exponent n and introduce the sparsity in the complexity by reconstructing the auxiliary function s coefficients via sparse multivariate interpolation the approach is new and builds on the normalization of the rational function s representation our method can be combined with probabilistic and deterministic components from sparse polynomial black box interpolation to suit either an exact or a finite precision computational environment the latter is illustrated with several examples running from exact finite field arithmetic to noisy floating point evaluations in general the performance of our sparse rational black box interpolation depends on the choice of the employed sparse polynomial black box interpolation if the early termination ben or tiwari algorithm is used our method achieves rational interpolation in o d black box evaluations and thus is sensitive to the sparsity of the multivariate f

computing optical flow with physical models of brightness variation although most optical flow techniques presume brightness constancy it is well known that this constraint is often violated producing poor estimates of image motion this paper describes a generalized formulation of optical flow estimation based on models of brightness variations that are caused by time dependent physical processes these include changing surface orientation with respect to a directional illuminant motion of the illuminant and physical models of heat transport in infrared images with these models we simultaneously estimate the 2d image motion and the relevant physical parameters of the brightness change model the estimation problem is formulated using total least squares with confidence bounds on the parameters experiments in four domains with both synthetic and natural inputs show how this formulation produces superior estimates of the 2d image motion

statistical modeling of 3 d natural scenes with application to bayesian stereopsis we studied the empirical distributions of luminance range and disparity wavelet coefficients using a coregistered database of luminance and range images the marginal distributions of range and disparity are observed to have high peaks and heavy tails similar to the well known properties of luminance wavelet coefficients however we found that the kurtosis of range and disparity coefficients is significantly larger than that of luminance coefficients we used generalized gaussian models to fit the empirical marginal distributions we found that the marginal distribution of luminance coefficients have a shape parameter p between 0 6 and 0 8 while range and disparity coefficients have much smaller parameters p 0 32 corresponding to a much higher peak we also examined the conditional distributions of luminance range and disparity coefficients the magnitudes of luminance and range disparity coefficients show a clear positive correlation which means at a location with larger luminance variation there is a higher probability of a larger range disparity variation we also used generalized gaussians to model the conditional distributions of luminance and range disparity coefficients the values of the two shape parameters p s reflect the observed luminance range disparity dependency as an example of the usefulness of luminance statistics conditioned on range statistics we modified a well known bayesian stereo ranging algorithm using our natural scene statistics models which improved its performance

wrap around effect removal finite ridgelet transform for multiscale image denoising denoising in wavelet domain is an attractive research field recently the research focused on finding some algorithms that can detect line singularities of image while smoothing noise based on the finite radon transform a finite ridgelet transform was derived for this purpose but the wrap around effect of the finite radon transform limits its power greatly in this paper a way is pointed out to remove the wrap around effect of the finite radon transform by embedding it into a moving window pyramid a multiscale image denoising algorithm is developed

reconstruction based recognition of scenes with translationally repeated quadrics this paper addresses the problem of invariant based recognition of quadric configurations from a single image these configurations consist of a pair of rigidly connected translationally repeated quadric surfaces this problem is approached via a reconstruction framework a new mathematical framework using relative affine structure on the lines of luong and vieville 1996 has been proposed using this mathematical framework translationally repeated objects have been projectively reconstructed from a single image with four image point correspondences of the distinguished points on the object and its translate this has been used to obtain a reconstruction of a pair of translationally repeated quadrics we have proposed joint projective invariants of a pair of proper quadrics for the purpose of recognition of quadric configurations we compute these invariants for the pair of reconstructed quadrics experimental results on synthetic and real images establish the discriminatory power and stability of the proposed invariant based recognition strategy as a specific example we have applied this technique for discriminating images of monuments which are characterized by translationally repeated domes modeled as quadrics

efficient image segmentation using partial differential equations and morphology the goal of this paper is to investigate segmentation methods that combine fast preprocessing algorithms using partial differential equations pdes with a watershed transformation with region merging we consider two well founded pde methods a nonlinear isotropic diffusion filter that permits edge enhancement and a convex nonquadratic variational image restoration method which gives good denoising for the diffusion filter an efficient algorithm is applied using an additive operator splitting aos that leads to recursive and separable filters for the variational restoration method a novel algorithm is developed that uses aos schemes within a gaussian pyramid decomposition examples demonstrate that preprocessing by these pde techniques significantly improves the watershed segmentation and that the resulting segmentation method gives better results than some traditional techniques the algorithm has linear complexity and it can be used for arbitrary dimensional data sets the typical cpu time for segmenting a 256 image on a modern pc is far below one second

projective structure from uncalibrated images structure from motion and recognition address the problem of reconstructing 3 d space in a projective framework from two or more views and the problem of artificially generating novel views of the scene from two given views reprojection the author describes an invariance relation that provides a new description of structure which the author calls projective depth that is captured by a single equation relating image point correspondences across two or more views and the homographics of two arbitrary virtual planes the framework is based on knowledge of correspondence of features across views is linear and extremely simple and the computations of structure readily extend to overdetermination using multiple views experimental results demonstrate a high degree of accuracy in both tasks reconstruction and reprojection

texture features for classification of ultrasonic liver images the classification of ultrasonic liver images is studied making use of the spatial gray level dependence matrices the fourier power spectrum the gray level difference statistics and the laws texture energy measures features of these types are used to classify three sets of ultrasonic liver images normal liver hepatoma and cirrhosis 30 samples each the bayes classifier and the hotelling trace criterion are employed to evaluate the performance of these features from the viewpoint of speed and accuracy of classification it is found that these features do not perform well enough hence a new texture feature set multiresolution fractal features based on multiple resolution imagery and the fractional brownian motion model is proposed to detect diffuse liver diseases quickly and accurately fractal dimensions estimated at various resolutions of the image are gathered to form the feature vector texture information contained in the proposed feature vector is discussed a real time implementation of the algorithm produces about 90 correct classification for the three sets of ultrasonic liver images

mode kn factor analysis for image ensembles in this corespondence we study the extra factor estimation problem with the assumption that the training image ensemble is expressed as an nth order tensor with the nth dimension characterizing all features for an image and other dimensions for different extra factors such as illuminations poses and identities to overcome the local minimum issue of conventional algorithms designed for this problem we present a novel statistical learning framework called mode kn factor analysis for obtaining a closed form solution to estimating the extra factors of any test image in the learning stage for the kth k ne n dimension of the data tensor the mode kn patterns are constructed by concatenating the feature dimension and the kth extra factor dimension and then a mode kn factor analysis model is learnt based on the mode kn patterns unfolded from the original data tensor in the inference stage for a test image the mode classification of the kth dimension is performed within a probabilistic framework the advantages of mode kn factor analysis over conventional tensor analysis algorithms are twofold 1 a closed form solution instead of iterative sub optimal solution as conventionally is derived for estimating the extra factor mode of any test image and 2 the classification capability is enhanced by interacting with the process of synthesizing data of all other modes in the k th dimension experiments on the pointing 04 and cmu pie databases for pose and illumination estimation both validate the superiority of the proposed algorithm over conventional algorithms for extra factor estimation

the quadric reference surface theory and applications the conceptual component of this work is about reference surfaces which are the analogous to reference frames often used for shape representation purposes the theoretical component of this work involves the question of whether one can find a unique and simple mapping that aligns two arbitrary perspective views of an opaque textured quadric surface in 3d given i few corresponding points in the two views or ii the outline conic of the surface in one view only and few corresponding points in the two views the practical component of this work is concerned with applying the theoretical results as tools for the task of achieving full correspondence between views of arbitrary objects

fast parallel computation of polynomials using few processors it is shown that any multivariate polynomial of degree d that can be computed sequentially in c steps can be computed in parallel in o log d log c log d steps using only cd o 1 processors

on bounds of shift variance in two channel multirate filter banks critically sampled multirate fir filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage we assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged we express these changes by a so called commutator we then derive a sharp upper bound for shift variance via the operator norm of the commutator which is independent of the input signal its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator leading to a matrix norm which depends on frequency this bound can be regarded as a worst case instance holding for all input signals for two channel fir filter banks with perfect reconstruction pr we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used moreover the framework allows to identify the signals for which the upper bound is almost reached as so called near maximizers of the frequency dependent matrix norm for unitary pr filter banks these near maximizers are shown to be narrow band signals to complement this worst case bound we derive an additional bound on shift variance for input signals with given amplitude spectra where we use wide band model spectra instead of narrow band signals like the operator norm this additional bound is based on the above frequency dependent matrix norm we provide results for various critically sampled two channel filter banks such as quadrature mirror filters pr conjugated quadrature filters wavelets and biorthogonal filters banks

nonlinear image operators for the evaluation of local intrinsic dimensionality local intrinsic dimensionality is shown to be an elementary structural property of multidimensional signals that cannot be evaluated using linear filters we derive a class of polynomial operators for the detection of intrinsically 2 d image features like curved edges and lines junctions line ends etc although it is a deterministic concept intrinsic dimensionality is closely related to signal redundancy since it measures how many of the degrees of freedom provided by a signal domain are in fact used by an actual signal furthermore there is an intimate connection to multidimensional surface geometry and to the concept of gaussian curvature nonlinear operators are inevitably required for the processing of intrinsic dimensionality since linear operators are by the superposition principle restricted to or combinations of their intrinsically 1 d eigenfunctions the essential new feature provided by polynomial operators is their potential to act on multiplicative relations between frequency components therefore such operators can provide the and combination of complex exponentials which is required for the exploitation of intrinsic dimensionality using frequency design methods we obtain a generalized class of quadratic volterra operators that are selective to intrinsically 2 d signals these operators can be adapted to the requirements of the signal processing task for example one can control the curvature tuning by adjusting the width of the stopband for intrinsically 1 d signals or the operators can be provided in isotropic and in orientation selective versions we first derive the quadratic volterra kernel involved in the computation of gaussian curvature and then present examples of operators with other arrangements of stop and passbands some of the resulting operators show a close relationship to the end stopped and dot responsive neurons of the mammalian visual cortex

multiple view geometry of general algebraic curves we introduce a number of new results in the context of multi view geometry from general algebraic curves we start with the recovery of camera geometry from matching curves we first show how one can compute without any knowledge on the camera the homography induced by a single planar curve then we continue with the derivation of the extended kruppa s equations which are responsible for describing the epipolar constraint of two projections of a general algebraic curve as part of the derivation of those constraints we address the issue of dimension analysis and as a result establish the minimal number of algebraic curves required for a solution of the epipolar geometry as a function of their degree and genus we then establish new results on the reconstruction of general algebraic curves from multiple views we address three different representations of curves i the regular point representation in which we show that the reconstruction from two views of a curve of degree d admits two solutions one of degree d and the other of degree d d 1 moreover using this representation we address the problem of homography recovery for planar curves ii dual space representation tangents for which we derive a lower bound for the number of views necessary for reconstruction as a function of the curve degree and genus and iii a new representation to computer vision based on the set of lines meeting the curve which does not require any curve fitting in image space for which we also derive lower bounds for the number of views necessary for reconstruction as a function of curve degree alone