the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

a new method for varying adaptive bandwidth selection a novel approach is developed to solve a problem of varying bandwidth selection for filtering a signal given with an additive noise the approach is based on the intersection of confidence intervals ici rule and gives the algorithm which is simple to implement and adaptive to unknown smoothness of the signal

image modeling and denoising with orientation adapted gaussian scale mixtures we develop a statistical model to describe the spatially varying behavior of local neighborhoods of coefficients in a multiscale image representation neighborhoods are modeled as samples of a multivariate gaussian density that are modulated and rotated according to the values of two hidden random variables thus allowing the model to adapt to the local amplitude and orientation of the signal a third hidden variable selects between this oriented process and a nonoriented scale mixture of gaussians process thus providing adaptability to the local orientedness of the signal based on this model we develop an optimal bayesian least squares estimator for denoising images and show through simulations that the resulting method exhibits significant improvement over previously published results obtained with gaussian scale mixtures

scale space and edge detection using anisotropic diffusion a new definition of scale space is suggested and a class of algorithms used to realize a diffusion process is introduced the diffusion coefficient is chosen to vary spatially in such a way as to encourage intraregion smoothing rather than interregion smoothing it is shown that the no new maxima should be generated at coarse scales property of conventional scale space is preserved as the region boundaries in the approach remain sharp a high quality edge detector which successfully exploits global information is obtained experimental results are shown on a number of images parallel hardware implementations are made feasible because the algorithm involves elementary local operations replicated over the image

bm3d frames and variational image deblurring a family of the block matching 3 d bm3d algorithms for various imaging problems has been recently proposed within the framework of nonlocal patchwise image modeling in this paper we construct analysis and synthesis frames formalizing bm3d image modeling and use these frames to develop novel iterative deblurring algorithms we consider two different formulations of the deblurring problem i e one given by the minimization of the single objective function and another based on the generalized nash equilibrium gne balance of two objective functions the latter results in the algorithm where deblurring and denoising operations are decoupled the convergence of the developed algorithms is proved simulation experiments show that the decoupled algorithm derived from the gne formulation demonstrates the best numerical and visual results and shows superiority with respect to the state of the art in the field confirming a valuable potential of bm3d frames as an advanced image modeling tool

guided wavelet shrinkage for edge aware smoothing edge aware smoothing has been extensively studied due to its wide range of applications in computer vision and graphics most published works have been focused on formulating the smoothing problem in the spatial domain in this paper we propose a new edge aware smoothing framework called guided wavelet shrinkage gws which is formulated in the wavelet domain as a maximum a posterior estimation problem we impose a number of desirable properties on the statistical models and the associated parameters in order to derive an effective and computationally efficient algorithm we compare the proposed gws with classical image denoising in the wavelet domain we have also compared different wavelet representations and found that the double density dual tree wavelet transform is the best choice we show that the gws can be configured as either self guidance or external guidance it can also be configured to operate in an iterative or non iterative way experimental results and comparison with many state of the algorithms demonstrate that the gws based algorithm can produce competitive results with mathcal o n computational complexity

a statistical approach to texture classification from single images we investigate texture classification from single images obtained under unknown viewpoint and illumination a statistical approach is developed where textures are modelled by the joint probability distribution of filter responses this distribution is represented by the frequency histogram of filter response cluster centres textons recognition proceeds from single uncalibrated images and the novelty here is that rotationally invariant filters are used and the filter response space is low dimensional classification performance is compared with the filter banks and methods of leung and malik ijcv 2001 schmid cvpr 2001 and cula and dana ijcv 2004 and it is demonstrated that superior performance is achieved here classification results are presented for all 61 materials in the columbia utrecht texture database we also discuss the effects of various parameters on our classification algorithm such as the choice of filter bank and rotational invariance the size of the texton dictionary as well as the number of training images used finally we present a method of reliably measuring relative orientation co occurrence statistics in a rotationally invariant manner and discuss whether incorporating such information can enhance the classifier s performance

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

rigorous computation of the global dynamics of integrodifference equations with smooth nonlinearities topological tools such as conley index theory have inspired rigorous computational methods for studying dynamics these methods rely on the construction of an outer approximation a combinatorial representation of the system that incorporates small bounded error in this work we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities chebyshev interpolants and galerkin projections form the basis for the construction while analysis and interval arithmetic are used to incorporate explicit error bounds this represents a significant advance to the approach given by day junge and mischaikow siam j appl dyn syst 3 2004 pp 117 160 extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set as a demonstration of the techniques a morse decomposition of the

image denoising using scale mixtures of gaussians in the wavelet domain we describe a method for removing noise from digital images based on a statistical model of the coefficients of an overcomplete multiscale oriented basis neighborhoods of coefficients at adjacent positions and scales are modeled as the product of two independent random variables a gaussian vector and a hidden positive scalar multiplier the latter modulates the local variance of the coefficients in the neighborhood and is thus able to account for the empirically observed correlation between the coefficient amplitudes under this model the bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable we demonstrate through simulations with images contaminated by additive white gaussian noise that the performance of this method substantially surpasses that of previously published methods both visually and in terms of mean squared error

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

modified nystrom methods for semi discrete hyperbolic differential equations first and second order nystrom type methods are derived for second order differential equations without first derivatives possessing the following properties i the stability interval equals 4m 2 m denoting the number of stages per integration step ii the method is internally stable irrespective of the value of m iii the storage requirements are limited and iv the costs per integration step are one right hand side evaluation one evaluation of the jacobian matrix and m 1 matrix vector multiplications these four properties are of interest in the integration of the usually very large systems of ordinary differential equations resulting from the semi discretization of partial differential equations which are of hyperbolic type and of second order in time

empirical bias bandwidths for local polynomial nonparametric regression and density estimation abstract a data based local bandwidth selector is proposed for nonparametric regression by local fitting of polynomials the estimator called the empirical bias bandwidth selector ebbs is rather simple and easily allows multivariate predictor variables and estimation of any order derivative of the regression function ebbs minimizes an estimate of mean squared error consisting of a squared bias term plus a variance term the variance term used is exact not asymptotic though it involves the conditional variance of the response given the predictors that must be estimated the bias term is estimated empirically not from an asymptotic expression thus ebbs is similar to the double smoothing approach of hardle hall and marron and a local bandwidth selector of schucany but is developed here for a far wider class of estimation problems than what those authors considered ebbs is tested on simulated data and its performance seems quite satisfactory local polynomial smoothing of a histogram is a high

indices of the iterates of r 3 homeomorphisms at lyapunov stable fixed points abstract given any positive sequence c n n n we construct orientation preserving homeomorphisms f r 3 r 3 such that fix f per f 0 0 is lyapunov stable and lim sup i f m 0 c m we will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

video denoising deblocking and enhancement through separable 4 d nonlocal spatiotemporal transforms we propose a powerful video filtering algorithm that exploits temporal and spatial redundancy characterizing natural video sequences the algorithm implements the paradigm of nonlocal grouping and collaborative filtering where a higher dimensional transform domain representation of the observations is leveraged to enforce sparsity and thus regularize the data 3 d spatiotemporal volumes are constructed by tracking blocks along trajectories defined by the motion vectors mutually similar volumes are then grouped together by stacking them along an additional fourth dimension thus producing a 4 d structure termed group where different types of data correlation exist along the different dimensions local correlation along the two dimensions of the blocks temporal correlation along the motion trajectories and nonlocal spatial correlation i e self similarity along the fourth dimension of the group collaborative filtering is then realized by transforming each group through a decorrelating 4 d separable transform and then by shrinkage and inverse transformation in this way the collaborative filtering provides estimates for each volume stacked in the group which are then returned and adaptively aggregated to their original positions in the video the proposed filtering procedure addresses several video processing applications such as denoising deblocking and enhancement of both grayscale and color data experimental results prove the effectiveness of our method in terms of both subjective and objective visual quality and show that it outperforms the state of the art in video denoising

image denoising using total least squares in this paper we present a method for removing noise from digital images corrupted with additive multiplicative and mixed noise an image patch from an ideal image is modeled as a linear combination of image patches from the noisy image we propose to fit this model to the real world image data in the total least square tls sense because the tls formulation allows us to take into account the uncertainties in the measured data we develop a method to reduce the contribution from the irrelevant image patches which will sharpen the edges and reduce edge artifacts at the same time although the proposed algorithm is computationally demanding the image quality of the output image demonstrates the effectiveness of the tls algorithms

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

wavelets ridgelets and curvelets for poisson noise removal in order to denoise poisson count data we introduce a variance stabilizing transform vst applied on a filtered discrete poisson process yielding a near gaussian process with asymptotic constant variance this new transform which can be deemed as an extension of the anscombe transform to filtered data is simple fast and efficient in very low count situations we combine this vst with the filter banks of wavelets ridgelets and curvelets leading to multiscale vsts ms vsts and nonlinear decomposition schemes by doing so the noise contaminated coefficients of these ms vst modified transforms are asymptotically normally distributed with known variances a classical hypothesis testing framework is adopted to detect the significant coefficients and a sparsity driven iterative scheme reconstructs properly the final estimate a range of examples show the power of this ms vst approach for recovering important structures of various morphologies in very low count images these results also demonstrate that the ms vst approach is competitive relative to many existing denoising methods

adaptive weights smoothing with applications to image restoration we propose a new method of nonparametric estimation which is based on locally constant smoothing with an adaptive choice of weights for every pair of data points some theoretical properties of the procedure are investigated then we demonstrate the performance of the method on some simulated univariate and bivariate examples and compare it with other nonparametric methods finally we discuss applications of this procedure to magnetic resonance and satellite imaging

optimal spatial adaptation to inhomogeneous smoothness an approach based on kernel estimates with variable bandwidth selectors a new v ria e 1 s e 0r estimation is proposed the application of this bandwidth selector leads to kernel estimates that achieve optimal rates of convergence over b 3 sses this implies that the procedure adapts to spatially inhomogeneous s noothness in particu lar the estimates share optimality properties with wavelet estimates based on thresholding of empirical wavelet coefficients 1 introduction in nonparametric curve estimation the statistical anal ysis may focus on the inference of the qualitative structure of the analyzed curve often interesting features of the curve are connected with spatially inhomogeneous smoothness in this case curve estimates that are spatially adaptlve ar e appropriaie a variety of such procedures have been proposed in the literature in breiman friedman olshen and stone 1983 piecewise constant least squares estimates are considered with a data adaptive choice of the pieces cart more generally friedman and silverman 1989 friedman 1991 and luo and wahba 1995 use variable knot splines mars knot points are added removed and allocated recursively using cross validation techniques these methods have shown good performance in simulations and real data examples however no asymptotic theory is available mammen and van de geer 1997 discuss penalized least squares curve estimation for spatially inhomogeneous curves they propose penalty terms which allow more spatial inhomogeneity than the usual l2 norms of deriva tives of the curve the estimates turn out to be variable knot splines see also mammen 1991 results on rates of convergence and a pointwise asymptotic distribution theory are given miiller and stadtmiiller 1987 staniswalis 1989 and brockmann gasser and hermann 1993 propose kernel estimation with locally variable band width selectors the calculation of local bandwidths is based on pilot estima tion of local smoothness characteristics an asymptotic analysis is available

efficient approximation of neural filters for removing quantum noise from images in this paper efficient filters are presented that approximate neural filters nfs that are trained to remove quantum noise from images a novel analysis method is proposed for making clear the characteristics of the trained nf in the proposed analysis method an unknown nonlinear deterministic system with plural inputs such as the trained nf can be analyzed by using its outputs when the specific input signals are input to it the experiments on the nfs trained to remove quantum noise from medical and natural images were performed the results have demonstrated that the approximate filters which are realized by using the results of the analysis are sufficient for approximation of the trained nfs and efficient at computational cost

propagation separation approach for local likelihood estimation the paper presents a unified approach to local likelihood estimation for a broad class of nonparametric models including e g the regression density poisson and binary response model the method extends the adaptive weights smoothing aws procedure introduced in polzehl and spokoiny 2000 in context of image denoising the main idea of the method is to describe a greatest possible local neighborhood of every design point x i in which the local parametric assumption is justified by the data the method is especially powerful for model functions having large homogeneous regions and sharp discontinuities the performance of the proposed procedure is illustrated by numerical examples for density estimation and classification we also establish some remarkable theoretical nonasymptotic results on properties of the new algorithm this includes the propagation property which particularly yields the root n consistency of the resulting estimate in the homogeneous case we also state an oracle result which implies rate optimality of the estimate under usual smoothness conditions and a separation result which explains the sensitivity of the method to structural changes

nonlocal hierarchical dictionary learning using wavelets for image denoising exploiting the sparsity within representation models for images is critical for image denoising the best currently available denoising methods take advantage of the sparsity from image self similarity pre learned and fixed representations most of these methods however still have difficulties in tackling high noise levels or noise models other than gaussian in this paper the multiresolution structure and sparsity of wavelets are employed by nonlocal dictionary learning in each decomposition level of the wavelets experimental results show that our proposed method outperforms two state of the art image denoising algorithms on higher noise levels furthermore our approach is more adaptive to the less extensively researched uniform noise

optimal inversion of the anscombe transformation in low count poisson image denoising the removal of poisson noise is often performed through the following three step procedure first the noise variance is stabilized by applying the anscombe root transformation to the data producing a signal in which the noise can be treated as additive gaussian with unitary variance second the noise is removed using a conventional denoising algorithm for additive white gaussian noise third an inverse transformation is applied to the denoised signal obtaining the estimate of the signal of interest the choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied we introduce optimal inverses for the anscombe transformation in particular the exact unbiased inverse a maximum likelihood ml inverse and a more sophisticated minimum mean square error mmse inverse we then present an experimental analysis using a few state of the art denoising algorithms and show that the estimation can be consistently improved by applying the exact unbiased inverse particularly at the low count regime this results in a very efficient filtering solution that is competitive with some of the best existing methods for poisson image denoising

image denoising via sparse and redundant representations over learned dictionaries we address the image denoising problem where zero mean white and homogeneous gaussian additive noise is to be removed from a given image the approach taken is based on sparse and redundant representations over trained dictionaries using the k svd algorithm we obtain a dictionary that describes the image content effectively two training options are considered using the corrupted image itself or training on a corpus of high quality image database since the k svd is limited in handling small image patches we extend its deployment to arbitrary image sizes by defining a global image prior that forces sparsity over patches in every location in the image we show how such bayesian treatment leads to a simple and effective denoising algorithm this leads to a state of the art denoising performance equivalent and sometimes surpassing recently published leading alternative denoising methods

the design and use of steerable filters the authors present an efficient architecture to synthesize filters of arbitrary orientations from linear combinations of basis filters allowing one to adaptively steer a filter to any orientation and to determine analytically the filter output as a function of orientation steerable filters may be designed in quadrature pairs to allow adaptive control over phase as well as orientation the authors show how to design and steer the filters and present examples of their use in the analysis of orientation and phase angularly adaptive filtering edge detection and shape from shading one can also build a self similar steerable pyramid representation the same concepts can be generalized to the design of 3 d steerable filters

two stage image denoising by principal component analysis with local pixel grouping this paper presents an efficient image denoising scheme by using principal component analysis pca with local pixel grouping lpg for a better preservation of image local structures a pixel and its nearest neighbors are modeled as a vector variable whose training samples are selected from the local window by using block matching based lpg such an lpg procedure guarantees that only the sample blocks with similar contents are used in the local statistics calculation for pca transform estimation so that the image local features can be well preserved after coefficient shrinkage in the pca domain to remove the noise the lpg pca denoising procedure is iterated one more time to further improve the denoising performance and the noise level is adaptively adjusted in the second stage experimental results on benchmark test images demonstrate that the lpg pca method achieves very competitive denoising performance especially in image fine structure preservation compared with state of the art denoising algorithms

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

weighted curvature preserving pde image filtering method the tensor driven curvature preserving partial differential equation is an outstanding anisotropic diffusion filtering model to effectively preserve image edge well a weighted curvature preserving pde based filtering method is proposed which employ local image directional information to design weight coefficient for different vector fields experimental results indicate that new approach shows superior performance on preserving image edge and curvature geometric structure

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

shiftable multiscale transforms one of the major drawbacks of orthogonal wavelet transforms is their lack of translation invariance the content of wavelet subbands is unstable under translations of the input signal wavelet transforms are also unstable with respect to dilations of the input signal and in two dimensions rotations of the input signal the authors formalize these problems by defining a type of translation invariance called shiftability in the spatial domain shiftability corresponds to a lack of aliasing thus the conditions under which the property holds are specified by the sampling theorem shiftability may also be applied in the context of other domains particularly orientation and scale jointly shiftable transforms that are simultaneously shiftable in more than one domain are explored two examples of jointly shiftable transforms are designed and implemented a 1 d transform that is jointly shiftable in position and scale and a 2 d transform that is jointly shiftable in position and orientation the usefulness of these image representations for scale space analysis stereo disparity measurement and image enhancement is demonstrated

image denoising pointwise adaptive approach a new method of pointwise adaptation has been proposed and studied in spokoiny 1998 ann statist 26 1356 1378 in the context of estimation of piecewise smooth univariate functions the present paper extends that method to estimation of bivariate grey scale images composed of large homogeneous regions with smooth edges and observed with noise on a gridded design the proposed estimator hat f x at a point x is simply the average of observations over a window widehat u x selected in a data driven way the theoretical properties of the procedure are studied for the case of piecewise constant images we present a nonasymptotic bound for the accuracy of estimation at a specific grid point x as a function of the number of pixels n of the distance from the point of estimation to the closest boundary and of smoothness properties and orientation of this boundary it is also shown that the proposed method provides a near optimal rate of estimation near edges and inside homogeneous regions we briefly discuss algorithmic aspects and the complexity of the procedure the numerical examples demonstrate a reasonable performance of the method and they are in agreement with the theoretical issues an example from satellite sar imaging illustrates the applicability of the method

image denoising with edge preserving and segmentation based on mask nha in this paper we propose a zero mean white gaussian noise removal method using a high resolution frequency analysis it is difficult to separate an original image component from a noise component when using discrete fourier transform or discrete cosine transform for analysis because sidelobes occur in the results the 2d non harmonic analysis 2d nha is a high resolution frequency analysis technique that improves noise removal accuracy because of its sidelobe reduction feature however spectra generated by nha are distorted because of which the signal of the image is non stationary in this paper we analyze each region with a homogeneous texture in the noisy image non uniform regions that occur due to segmentation are analyzed by an extended 2d nha method called mask nha we conducted an experiment using a simulation image and found that mask nha denoising attains a higher peak signal to noise ratio psnr value than the state of the art methods if a suitable segmentation result can be obtained from the input image even though parameter optimization was incomplete this experimental result exhibits the upper limit on the value of psnr in our mask nha denoising method the performance of mask nha denoising is expected to approach the limit of psnr by improving the segmentation method

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

multiresolution moment filters theory and applications we introduce local weighted geometric moments that are computed from an image within a sliding window at multiple scales when the window function satisfies a two scale relation we prove that lower order moments can be computed efficiently at dyadic scales by using a multiresolution wavelet like algorithm we show that b splines are well suited window functions because in addition to being refinable they are positive symmetric separable and very nearly isotropic gaussian shape we present three applications of these multiscale local moments the first is a feature extraction method for detecting and characterizing elongated structures in images the second is a noise reduction method which can be viewed as a multiscale extension of savitzky golay filtering the third is a multiscale optical flow algorithm that uses a local affine model for the motion field extending the lucas kanade optical flow method the results obtained in all cases are promising

llsure local linear sure based edge preserving image filtering in this paper we propose a novel approach for performing high quality edge preserving image filtering based on a local linear model and using the principle of stein s unbiased risk estimate as an estimator for the mean squared error from the noisy image only we derive a simple explicit image filter which can filter out noise while preserving edges and fine scale details moreover this filter has a fast and exact linear time algorithm whose computational complexity is independent of the filtering kernel size thus it can be applied to real time image processing tasks the experimental results demonstrate the effectiveness of the new filter for various computer vision applications including noise reduction detail smoothing and enhancement high dynamic range compression and flash no flash denoising

structure based low rank model with graph nuclear norm regularization for noise removal nonlocal image representation methods including group based sparse coding and block matching 3 d filtering have shown their great performance in application to low level tasks the nonlocal prior is extracted from each group consisting of patches with similar intensities grouping patches based on intensity similarity however gives rise to disturbance and inaccuracy in estimation of the true images to address this problem we propose a structure based low rank model with graph nuclear norm regularization we exploit the local manifold structure inside a patch and group the patches by the distance metric of manifold structure with the manifold structure information a graph nuclear norm regularization is established and incorporated into a low rank approximation model we then prove that the graph based regularization is equivalent to a weighted nuclear norm and the proposed model can be solved by a weighted singular value thresholding algorithm extensive experiments on additive white gaussian noise removal and mixed noise removal demonstrate that the proposed method achieves a better performance than several state of the art algorithms

a universal procedure for aggregating estimators in this paper we study the aggregation problem that can be formulated as follows assume that we have a family of estimators f built on the basis of available observations the goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family we propose a general aggregation scheme that is universal in the following sense it applies for families of arbitrary estimators and a wide variety of models and global risk measures the procedure is based on comparison of empirical estimates of certain linear functionals with estimates induced by the family f we derive oracle inequalities and show that they are unimprovable in some sense numerical results demonstrate good practical behavior of the procedure

a discriminative approach for wavelet denoising this paper suggests a discriminative approach for wavelet denoising where a set of mapping functions mfs are applied to the transform coefficients in an attempt to produce a noise free image as opposed to the descriptive approaches modeling image or noise priors is not required here and the mfs are learned directly from an ensemble of example images using least squares fitting the suggested scheme generates a novel set of mfs that are essentially different from the traditional soft hard thresholding in the over complete case these mfs are demonstrated to obtain comparable performance to the state of the art denoising approaches additionally this framework enables a seamless customization of the shrinkage operation to a new set of restoration problems that were not addressed previously with shrinkage techniques such as deblurring jpeg artifact removal and various types of additive noise that are not necessarily gaussian white noise

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

a weighted dictionary learning model for denoising images corrupted by mixed noise this paper proposes a general weighted l2 l0 norms energy minimization model to remove mixed noise such as gaussian gaussian mixture impulse noise and gaussian impulse noise from the images the approach is built upon maximum likelihood estimation framework and sparse representations over a trained dictionary rather than optimizing the likelihood functional derived from a mixture distribution we present a new weighting data fidelity function which has the same minimizer as the original likelihood functional but is much easier to optimize the weighting function in the model can be determined by the algorithm itself and it plays a role of noise detection in terms of the different estimated noise parameters by incorporating the sparse regularization of small image patches the proposed method can efficiently remove a variety of mixed or single noise while preserving the image textures well in addition a modified k svd algorithm is designed to address the weighted rank one approximation the experimental results demonstrate its better performance compared with some existing methods

gradient histogram estimation and preservation for texture enhanced image denoising natural image statistics plays an important role in image denoising and various natural image priors including gradient based sparse representation based and nonlocal self similarity based ones have been widely studied and exploited for noise removal in spite of the great success of many denoising algorithms they tend to smooth the fine scale image textures when removing noise degrading the image visual quality to address this problem in this paper we propose a texture enhanced image denoising method by enforcing the gradient histogram of the denoised image to be close to a reference gradient histogram of the original image given the reference gradient histogram a novel gradient histogram preservation ghp algorithm is developed to enhance the texture structures while removing noise two region based variants of ghp are proposed for the denoising of images consisting of regions with different textures an algorithm is also developed to effectively estimate the reference gradient histogram from the noisy observation of the unknown image our experimental results demonstrate that the proposed ghp algorithm can well preserve the texture appearance in the denoised images making them look more natural

image denoising using svm classification in nonsubsampled contourlet transform domain for image denoising the main challenge is how to preserve the information bearing structures such as edges and textures to get satisfactory visual quality when improving the signal to noise ratio snr edge preserving image denoising has become a very intensive research topic in this paper we propose an image denoising using support vector machine svm classification in nonsubsampled contourlet transform nsct domain firstly the noisy image is decomposed into different subbands of frequency and orientation responses using the nsct secondly the feature vector for a pixel in a noisy image is formed by the spatial regularity in nsct domain and the least squares support vector machine ls m model is obtained by training then the nsct detail coefficients are divided into two classes edge related coefficients and noise related ones by ls svm training model finally the detail subbands of nsct coefficients are denoised by using shrink method in which the adaptive bayesian threshold is utilized extensive experimental results demonstrate that our method can obtain better performances in terms of both subjective and objective evaluations than those state of the art denoising techniques especially the proposed method can preserve edges very well while removing noise

weighted averaging for denoising with overcomplete dictionaries we consider the scenario where additive independent and identically distributed i i d noise in an image is removed using an overcomplete set of linear transforms and thresholding rather than the standard approach where one obtains the denoised signal by ad hoc averaging of the denoised estimates provided by denoising with each of the transforms we formulate the optimal combination as a conditional linear estimation problem and solve it for optimal estimates our approach is independent of the utilized transforms and the thresholding scheme and as we illustrate using oracle based denoisers it extends established work by exploiting a separate degree of freedom that is in general not reachable using previous techniques our derivation of the optimal estimates specifically relies on the assumption that the utilized transforms provide sparse decompositions at the same time our work is robust as it does not require any assumptions about image statistics beyond sparsity unlike existing work which tries to devise ever more sophisticated transforms and thresholding algorithms to deal with the myriad types of image singularities our work uses basic tools to obtain very high performance on singularities by taking better advantage of the sparsity that surrounds them with well established transforms we obtain results that are competitive with state of the art methods

layered graph matching with composite cluster sampling this paper presents a framework of layered graph matching for integrating graph partition and matching the objective is to find an unknown number of corresponding graph structures in two images we extract discriminative local primitives from both images and construct a candidacy graph whose vertices are matching candidates i e a pair of primitives and whose edges are either negative for mutual exclusion or positive for mutual consistence then we pose layered graph matching as a multicoloring problem on the candidacy graph and solve it using a composite cluster sampling algorithm this algorithm assigns some vertices into a number of colors each being a matched layer and turns off all the remaining candidates the algorithm iterates two steps 1 sampling the positive and negative edges probabilistically to form a composite cluster which consists of a few mutually conflicting connected components ccps in different colors and 2 assigning new colors to these ccps with consistence and exclusion relations maintained and the assignments are accepted by the markov chain monte carlo mcmc mechanism to preserve detailed balance this framework demonstrates state of the art performance on several applications such as multi object matching with large motion shape matching and retrieval and object localization in cluttered background

infinite series in cohomology attractors and conley index in this paper we study the cohomological conley index of arbitrary isolated invariant continua for continuous maps f u subseteq mathbb r d rightarrow mathbb r d by analyzing the topological structure of their unstable manifold we provide a simple dynamical interpretation for the first cohomological conley index describing it completely and relate it to the cohomological conley index in higher degrees a number of consequences are derived including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3 our approach exploits a certain attractor repeller decomposition of the unstable manifold reducing the study of the cohomological conley index to the relation between the cohomology of an attractor and its basin of attraction this is a classical problem that in the present case is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated to address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology if z is a metric space and k subseteq z is a compact global attractor for a continuous map g z rightarrow z we show how to interpret series of the form sum j ge 0 a j g j as endomorphisms of the cohomology group of the pair z k

handbook of the normal distribution genesis an historical background basic properties expansions and algorithms characterizations sampling distributions limit theorems and expansions normal approximations to distributions order statistics from normal samples the bivariate normal distribution bivariate normal sampling distributions point estimation statistical intervals

image denoising via bandwise adaptive modeling and regularization exploiting nonlocal similarity this paper proposes a new image denoising algorithm based on adaptive signal modeling and regularization it improves the quality of images by regularizing each image patch using bandwise distribution modeling in transform domain instead of using a global model for all the patches in an image it employs content dependent adaptive models to address the non stationarity of image signals and also the diversity among different transform bands the distribution model is adaptively estimated for each patch individually it varies from one patch location to another and also varies for different bands in particular we consider the estimated distribution to have non zero expectation to estimate the expectation and variance parameters for every band of a particular patch we exploit the nonlocal correlation in image to collect a set of highly similar patches as the data samples to form the distribution irrelevant patches are excluded so that such adaptively learned model is more accurate than a global one the image is ultimately restored via bandwise adaptive soft thresholding based on a laplacian approximation of the distribution of similar patch group transform coefficients experimental results demonstrate that the proposed scheme outperforms several state of the art denoising methods in both the objective and the perceptual qualities

low rank prior in single patches for nonpointwise impulse noise removal this paper introduces a low rank prior in small oriented noise free image patches considering an oriented patch as a matrix a low rank matrix approximation is enough to preserve the texture details in the optimally oriented patch based on this prior we propose a single patch method within a generalized joint low rank and sparse matrix recovery framework to simultaneously detect and remove nonpointwise random valued impulse noise e g very small blobs a weighting matrix is incorporated in the framework to encode an initial estimate of the spatial noise distribution an accelerated proximal gradient method is adapted to estimate the optimal noise free image patches experiments show the effectiveness of our framework in detecting and removing nonpointwise random valued impulse noise

estimation of a function with discontinuities via local polynomial fit with an adaptive window choice we propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error at the same time the estimator is shown to be very sensitive to discontinuities or change points of the underlying function f or its derivatives for instance in the case of a jump of a regression function beyond the intervals of length in order n 1 log n around change points the quality of estimation is essentially the same as if locations of jumps were known the method is fully adaptive and no assumptions are imposed on the design number and size of jumps the results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size

fields of experts we develop a framework for learning generic expressive image priors that capture the statistics of natural scenes and can be used for a variety of machine vision tasks the approach provides a practical method for learning high order markov random field mrf models with potential functions that extend over large pixel neighborhoods these clique potentials are modeled using the product of experts framework that uses non linear functions of many linear filter responses in contrast to previous mrf approaches all parameters including the linear filters themselves are learned from training data we demonstrate the capabilities of this field of experts model with two example applications image denoising and image inpainting which are implemented using a simple approximate inference scheme while the model is trained on a generic image database and is not tuned toward a specific application we obtain results that compete with specialized techniques

order reduction and how to avoid it when explicit runge kutta nystrom methods are used to solve linear partial differential equations in this paper we study the order reduction which turns up when explicit runge kutta nystrom methods are used to discretize linear second order hyperbolic equations by means of the method of lines the order observed in practice including its fractional part is obtained it is also proved that the order reduction can be completely avoided taking the boundary values of the intermediate stages of the time semidiscretization the numerical experiments confirm that the optimal order can be reached

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

a universal denoising framework with a new impulse detector and nonlocal means impulse noise detection is a critical issue when removing impulse noise and impulse gaussian mixed noise in this paper we propose a new detection mechanism for universal noise and a universal noise filtering framework based on the nonlocal means nl means the operation is carried out in two stages i e detection followed by filtering for detection first we propose the robust outlyingness ratio ror for measuring how impulselike each pixel is and then all the pixels are divided into four clusters according to the ror values second different decision rules are used to detect the impulse noise based on the absolute deviation to the median in each cluster in order to make the detection results more accurate and more robust the from coarse to fine strategy and the iterative framework are used in addition the detection procedure consists of two stages i e the coarse and fine detection stages for filtering the nl means are extended to the impulse noise by introducing a reference image then a universal denoising framework is proposed by combining the new detection mechanism with the nl means ror nlm finally extensive simulation results show that the proposed noise detector is superior to most existing detectors and the ror nlm produces excellent results and outperforms most existing filters for different noise models unlike most of the other impulse noise filters the proposed ror nlm also achieves high peak signal to noise ratio and great image quality by efficiently removing impulse gaussian mixed noise

image restoration by matching gradient distributions the restoration of a blurry or noisy image is commonly performed with a map estimator which maximizes a posterior probability to reconstruct a clean image from a degraded image a map estimator when used with a sparse gradient image prior reconstructs piecewise smooth images and typically removes textures that are important for visual realism we present an alternative deconvolution method called iterative distribution reweighting idr which imposes a global constraint on gradients so that a reconstructed image should have a gradient distribution similar to a reference distribution in natural images a reference distribution not only varies from one image to another but also within an image depending on texture we estimate a reference distribution directly from an input image for each texture segment our algorithm is able to restore rich mid frequency textures a large scale user study supports the conclusion that our algorithm improves the visual realism of reconstructed images compared to those of map estimators

pointwise shape adaptive dct for high quality denoising and deblocking of grayscale and color images the shape adaptive discrete cosine transform sa dct transform can be computed on a support of arbitrary shape but retains a computational complexity comparable to that of the usual separable block dct b dct despite the near optimal decorrelation and energy compaction properties application of the sa dct has been rather limited targeted nearly exclusively to video compression in this paper we present a novel approach to image filtering based on the sa dct we use the sa dct in conjunction with the anisotropic local polynomial approximation intersection of confidence intervals technique which defines the shape of the transform s support in a pointwise adaptive manner the thresholded or attenuated sa dct coefficients are used to reconstruct a local estimate of the signal within the adaptive shape support since supports corresponding to different points are in general overlapping the local estimates are averaged together using adaptive weights that depend on the region s statistics this approach can be used for various image processing tasks in this paper we consider in particular image denoising and image deblocking and deringing from block dct compression a special structural constraint in luminance chrominance space is also proposed to enable an accurate filtering of color images simulation experiments show a state of the art quality of the final estimate both in terms of objective criteria and visual appearance thanks to the adaptive support reconstructed edges are clean and no unpleasant ringing artifacts are introduced by the fitted transform

susan a new approach to low level image processing this paper describes a new approach to low level image processing in particular edge and corner detection and structure preserving noise reduction non linear filtering is used to define which parts of the image are closely related to each individual pixel each pixel has associated with it a local image region which is of similar brightness to that pixel the new feature detectors are based on the minimization of this local image region and the noise reduction method uses this region as the smoothing neighbourhood the resulting methods are accurate noise resistant and fast details of the new feature detectors and of the new noise reduction method are described along with test results

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

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

fundamental relationship between bilateral filtering adaptive smoothing and the nonlinear diffusion equation in this paper the relationship between bilateral filtering and anisotropic diffusion is examined the bilateral filtering approach represents a large class of nonlinear digital image filters we first explore the connection between anisotropic diffusion and adaptive smoothing and then the connection between adaptive smoothing and bilateral filtering previously adaptive smoothing was considered to be an inconsistent approximation to the nonlinear diffusion equation we extend adaptive smoothing to make it consistent thus enabling a unified viewpoint that relates nonlinear digital image filters and the nonlinear diffusion equation

from local kernel to nonlocal multiple model image denoising we review the evolution of the nonparametric regression modeling in imaging from the local nadaraya watson kernel estimate to the nonlocal means and further to transform domain filtering based on nonlocal block matching the considered methods are classified mainly according to two main features local nonlocal and pointwise multipoint here nonlocal is an alternative to local and multipoint is an alternative to pointwise these alternatives though obvious simplifications allow to impose a fruitful and transparent classification of the basic ideas in the advanced techniques within this framework we introduce a novel single and multiple model transform domain nonlocal approach the block matching and 3 d filtering bm3d algorithm which is currently one of the best performing denoising algorithms is treated as a special case of the latter approach

why simple shrinkage is still relevant for redundant representations shrinkage is a well known and appealing denoising technique introduced originally by donoho and johnstone in 1994 the use of shrinkage for denoising is known to be optimal for gaussian white noise provided that the sparsity on the signal s representation is enforced using a unitary transform still shrinkage is also practiced with nonunitary and even redundant representations typically leading to very satisfactory results in this correspondence we shed some light on this behavior the main argument in this work is that such simple shrinkage could be interpreted as the first iteration of an algorithm that solves the basis pursuit denoising bpdn problem while the desired solution of bpdn is hard to obtain in general we develop a simple iterative procedure for the bpdn minimization that amounts to stepwise shrinkage we demonstrate how the simple shrinkage emerges as the first iteration of this novel algorithm furthermore we show how shrinkage can be iterated turning into an effective algorithm that minimizes the bpdn via simple shrinkage steps in order to further strengthen the denoising effect

nonlocal image and movie denoising neighborhood filters are nonlocal image and movie filters which reduce the noise by averaging similar pixels the first object of the paper is to present a unified theory of these filters and reliable criteria to compare them to other filter classes a ccd noise model will be presented justifying the involvement of neighborhood filters a classification of neighborhood filters will be proposed including classical image and movie denoising methods and discussing further a recently introduced neighborhood filter nl means in order to compare denoising methods three principles will be discussed the first principle method noise specifies that only noise must be removed from an image a second principle will be introduced noise to noise according to which a denoising method must transform a white noise into a white noise contrarily to method noise this principle which characterizes artifact free methods eliminates any subjectivity and can be checked by mathematical arguments and fourier analysis noise to noise will be proven to rule out most denoising methods with the exception of neighborhood filters this is why a third and new comparison principle the statistical optimality is needed and will be introduced to compare the performance of all neighborhood filters the three principles will be applied to compare ten different image and movie denoising methods it will be first shown that only wavelet thresholding methods and nl means give an acceptable method noise second that neighborhood filters are the only ones to satisfy the noise to noise principle third that among them nl means is closest to statistical optimality a particular attention will be paid to the application of the statistical optimality criterion for movie denoising methods it will be pointed out that current movie denoising methods are motion compensated neighborhood filters this amounts to say that they are neighborhood filters and that the ideal neighborhood of a pixel is its trajectory unfortunately the aperture problem makes it impossible to estimate ground true trajectories it will be demonstrated that computing trajectories and restricting the neighborhood to them is harmful for denoising purposes and that space time nl means preserves more movie details

how to compare noisy patches patch similarity beyond gaussian noise many tasks in computer vision require to match image parts while higher level methods consider image features such as edges or robust descriptors low level approaches so called image based compare groups of pixels patches and provide dense matching patch similarity is a key ingredient to many techniques for image registration stereo vision change detection or denoising recent progress in natural image modeling also makes intensive use of patch comparison a fundamental difficulty when comparing two patches from real data is to decide whether the differences should be ascribed to noise or intrinsic dissimilarity gaussian noise assumption leads to the classical definition of patch similarity based on the squared differences of intensities for the case where noise departs from the gaussian distribution several similarity criteria have been proposed in the literature of image processing detection theory and machine learning by expressing patch dis similarity as a detection test under a given noise model we introduce these criteria with a new one and discuss their properties we then assess their performance for different tasks patch discrimination image denoising stereo matching and motion tracking under gamma and poisson noises the proposed criterion based on the generalized likelihood ratio is shown to be both easy to derive and powerful in these diverse applications

lapb locally adaptive patch based wavelet domain edge preserving image denoising abstract image denoising is one of the most diversified research areas in the field of image processing and computer vision it is highly desirable for a denoising technique to preserve important image features such as edges corners and other sharp structures of an image after denoising wavelet transforms show excellent proficiency in providing efficient edge preserving image denoising due to their capability of suppressing noisy signals from an image this paper presents a novel edge preserving image denoising technique based on wavelet transforms the multi level decomposition of the noisy image is carried out to transform the data into the wavelet domain a locally adaptive patch based lapb thresholding scheme is used to effectively reduce noise while preserving relevant features of the original image experimental results on benchmark test images demonstrate that the proposed method achieves competitive denoising performance in comparison to various state of the art algorithms

object categorization with sketch representation and generalized samples in this paper we present a framework for object categorization via sketch graphs that incorporate shape and structure information in this framework we integrate the learnable and or graph model a hierarchical structure that combines the reconfigurability of a stochastic context free grammar scfg with the constraints of a markov random field mrf considering the computation efficiency we generalize instances from the and or graph models and perform a set of sequential tests for cascaded object categorization rather than directly inferring with the and or graph models we study 33 categories each consisting of a small data set of 30 instances and 30 additional templates with varied appearance are generalized from the learned and or graph model these samples better span the appearance space and form an augmented training set w t of 1980 60x33 training templates to perform recognition on a testing image we use a set of sequential tests to project w t into different representation spaces to narrow the number of candidate matches in w t we use graphlets structural elements as our local features and model w t at each stage using histograms of graphlets over categories histograms of graphlets over object instances histograms of pairs of graphlets over objects and shape context each test is increasingly computationally expensive and by the end of the cascade we have a small candidate set remaining to use with our most powerful test a top down graph matching algorithm we apply the proposed approach on the challenging public dataset including 33 object categories and achieve state of the art performance

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

adaptive bandwidth choice for kernel regression abstract a data based procedure is introduced for local bandwidth selection for kernel estimation of a regression function at a point the estimated bandwidth is shown to be consistent and asymptotically normal as an estimator of the asymptotic optimal value for minimum mean square estimation simulation studies indicate satisfactory behavior of the new bandwidth estimator in finite samples the findings are improvements over a global bandwidth estimator the same methodology works for local linear regression and extends easily to weighted local polynomial fits

adapting to unknown smoothness via wavelet shrinkage abstract we attempt to recover a function of unknown smoothness from noisy sampled data we introduce a procedure sureshrink that suppresses noise by thresholding the empirical wavelet coefficients the thresholding is adaptive a threshold level is assigned to each dyadic resolution level by the principle of minimizing the stein unbiased estimate of risk sure for threshold estimates the computational effort of the overall procedure is order n log n as a function of the sample size n sureshrink is smoothness adaptive if the unknown function contains jumps then the reconstruction essentially does also if the unknown function has a smooth piece then the reconstruction is essentially as smooth as the mother wavelet will allow the procedure is in a sense optimally smoothness adaptive it is near minimax simultaneously over a whole interval of the besov scale the size of this interval depends on the choice of mother wavelet we know from a previous paper by the authors that traditional smoot

realization of all dold s congruences with stability the main goal of this paper is to prove that for each n 2 every sequence of integers satisfying dold s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving r n homeomorphism at an isolated stable fixed point

local adaptivity to variable smoothness for exemplar based image regularization and representation a novel adaptive and exemplar based approach is proposed for image restoration denoising and representation the method is based on a pointwise selection of similar image patches of fixed size in the variable neighborhood of each pixel the main idea is to associate with each pixel the weighted sum of data points within an adaptive neighborhood we use small image patches e g 7 7 or 9 9 patches to compute these weights since they are able to capture local geometric patterns and texels seen in images in this paper we mainly focus on the problem of adaptive neighborhood selection in a manner that balances the accuracy of approximation and the stochastic error at each spatial position the proposed pointwise estimator is then iterative and automatically adapts to the degree of underlying smoothness with minimal a priori assumptions on the function to be recovered the method is applied to artificially corrupted real images and the performance is very close and in some cases even surpasses to that of the already published denoising methods the proposed algorithm is demonstrated on real images corrupted by non gaussian noise and is used for applications in bio imaging

index filtrations and the homology index braid for partially ordered morse decompositions abstract on a morse decomposition of an invariant set in a flow there are partial orderings defined by the flow these are called admissable orderings of the morse decomposition the index filtrations for a total ordering of a morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a morse decomposition it is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration the homology index braid of the corresponding admissable ordering of the morse decomposition is obtained by passing to homology in the chain complex braid keywords morse decomposition conley index index filtration

spatially adaptive estimation via fitted local likelihood techniques this paper offers a new technique for spatially adaptive estimation the local likelihood is exploited for nonparametric modeling of observations and estimated signals the approach is based on the assumption of a local homogeneity of the signal for every point there exists a neighborhood in which the signal can be well approximated by a constant the fitted local likelihood statistics are used for selection of an adaptive size and shape of this neighborhood the algorithm is developed for a quite general class of observations subject to the exponential distribution the estimated signal can be uni and multivariable we demonstrate a good performance of the new algorithm for image denoising and compare the new method versus the intersection of confidence interval ici technique that also exploits a selection of an adaptive neighborhood for estimation

patch based nonlocal functional for denoising fluorescence microscopy image sequences we present a nonparametric regression method for denoising 3 d image sequences acquired via fluorescence microscopy the proposed method exploits the redundancy of the 3 d time information to improve the signal to noise ratio of images corrupted by poisson gaussian noise a variance stabilization transform is first applied to the image data to remove the dependence between the mean and variance of intensity values this preprocessing requires the knowledge of parameters related to the acquisition system also estimated in our approach in a second step we propose an original statistical patch based framework for noise reduction and preservation of space time discontinuities in our study discontinuities are related to small moving spots with high velocity observed in fluorescence video microscopy the idea is to minimize an objective nonlocal energy functional involving spatio temporal image patches the minimizer has a simple form and is defined as the weighted average of input data taken in spatially varying neighborhoods the size of each neighborhood is optimized to improve the performance of the pointwise estimator the performance of the algorithm which requires no motion estimation is then evaluated on both synthetic and real image sequences using qualitative and quantitative criteria

numerical methods for nonlinear second order hyperbolic partial differential equations ii rothe s techniques for 1 d problems abstract two families of rothe s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second order hyperbolic equations with damping and nonlinear source terms are presented the first family is based on time integration and results in a volterra integro differential equation which upon approximating certain time integrals can be written as an ordinary differential equation in space upon discretizing the spatial derivatives by means of finite difference formula this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step and its accuracy is shown to be lower than that of an explicit second order accurate discretization in both space and time of second order hyperbolic equations the second family of methods is based on the discretization of the time derivatives and time linearization and results in linear ordinary differential equations in space upon freezing the coefficients of these equations one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space and three point finite difference equations which depend exponentially on the time step spatial grid size and the diffusion relaxation damping and reaction times the finite difference equations are shown to result in non diagonally dominant matrices unless the time step is smaller than the characteristic relaxation diffusion damping and reaction times to avoid this problem two rothe s techniques that do not account for the jacobian of the reaction terms in the ordinary differential operator are developed and it is shown that the rothe s techniques belonging to the second family are as accurate as the linearly implicit methods presented in part i provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space when the relaxation time is greater than the critical one rothe s methods have been found to be less accurate than the finite difference techniques of part i the rothe s techniques belonging to the second family are generalized to systems of nonlinear second order hyperbolic equations and mixed systems of parabolic and second order hyperbolic partial differential equations