shape estimation from shading defocus and correspondence using light field angular coherence light field cameras are quickly becoming commodity items with consumer and industrial applications they capture many nearby views simultaneously using a single image with a micro lens array thereby providing a wealth of cues for depth recovery defocus correspondence and shading in particular apart from conventional image shading one can refocus images after acquisition and shift one s viewpoint within the sub apertures of the main lens effectively obtaining multiple views we present a principled algorithm for dense depth estimation that combines defocus and correspondence metrics we then extend our analysis to the additional cue of shading using it to refine fine details in the shape by exploiting an all in focus image in which pixels are expected to exhibit angular coherence we define an optimization framework that integrates photo consistency depth consistency and shading consistency we show that combining all three sources of information defocus correspondence and shading outperforms state of the art light field depth estimation algorithms in multiple scenarios

consistency and convergence rates of one class svms and related algorithms we determine the asymptotic behaviour of the function computed by support vector machines svm and related algorithms that minimize a regularized empirical convex loss function in the reproducing kernel hilbert space of the gaussian rbf kernel in the situation where the number of examples tends to infinity the bandwidth of the gaussian kernel tends to 0 and the regularization parameter is held fixed non asymptotic convergence bounds to this limit in the l2 sense are provided together with upper bounds on the classification error that is shown to converge to the bayes risk therefore proving the bayes consistency of a variety of methods although the regularization term does not vanish these results are particularly relevant to the one class svm for which the regularization can not vanish by construction and which is shown for the first time to be a consistent density level set estimator

scalable gaussian kernel support vector machines with sublinear training time complexity abstract gaussian kernel support vector machines svms deliver state of the art generalization performance for non linear classification but the time complexity of their training process is at least quadratic w r t the size of training set preventing them from scaling up on large datasets to address this issue we propose a novel approach to large scale kernel svms in sublinear time w r t the size of training set which combines three well known and efficient techniques with theoretical guarantees first we subsample massive samples to reduce the sample size then we use the random fourier feature mapping on the subsamples to construct explicit random feature space where we can train a linear svm to approximate the corresponding gaussian kernel svm finally we use parallel algorithms to make our approach more scalable deriving the upper bounds of kernel matrix approximation error hypothesis error and excess risk w r t the size of training set and the dimension of random feature space we establish the theoretical foundation of our proposed approach in this way we can reduce the time complexity of training kernel svms without sacrificing much accuracy our proposed approach achieves high accuracy and has sublinear training time complexity which exhibits good scalability theoretically and empirically

bounds on colength and maximal multiplicity sequences let a be a characteristic zero p i algebra with cocharacters n a m we obtain upper and lower bounds on max n m and n m and compute the exact rates of growth when a is a prime product algebra

statistical analysis of some multi category large margin classification methods the purpose of this paper is to investigate statistical properties of risk minimization based multi category classification methods these methods can be considered as natural extensions of binary large margin classification we establish conditions that guarantee the consistency of classifiers obtained in the risk minimization framework with respect to the classification error examples are provided for four specific forms of the general formulation which extend a number of known methods using these examples we show that some risk minimization formulations can also be used to obtain conditional probability estimates for the underlying problem such conditional probability information can be useful for statistical inferencing tasks beyond classification

a six generalized squares theorem with applications to polynomial identity algebras the theories of superalgebras and of p i algebras lead to a natural z2 graded extension of the integers for these generalized integers a six generalized squares theorem is proved which can be considered as a z2 graded analogue of the classical four squares theorem for the natural numbers this theorem was conjectured by a berele and a regev exponential growth of some p i algebras br2 and has applications to p i algebras

shape from sharp and motion blurred image pair motion blur due to camera shake is a common occurrence during image capture the apparent motion of a scene point in the image plane varies according to both camera motion and scene structure our objective is to infer the camera motion and the depth map of static scenes using motion blur as a cue to this end we use an unblurred blurred image pair initially we develop a technique to estimate the transformation spread function tsf which symbolizes the camera shake this technique uses blur kernels estimated at different points across the image based on the estimated tsf we recover the complete depth map of the scene within a regularization framework

shape from focus the shape from focus method presented here uses different focus levels to obtain a sequence of object images the sum modified laplacian sml operator is developed to provide local measures of the quality of image focus the operator is applied to the image sequence to determine a set of focus measures at each image point a depth estimation algorithm interpolates a small number of focus measure values to obtain accurate depth estimates a fully automated shape from focus system has been implemented using an optical microscope and tested on a variety of industrial samples experimental results are presented that demonstrate the accuracy and robustness of the proposed method these results suggest shape from focus to be an effective approach for a variety of challenging visual inspection tasks

fast rates for support vector machines using gaussian kernels for binary classification we establish learning rates up to the order of n 1 for support vector machines svms with hinge loss and gaussian rbf kernels these rates are in terms of two assumptions on the considered distributions tsybakov s noise assumption to establish a small estimation error and a new geometric noise condition which is used to bound the approximation error unlike previously proposed concepts for bounding the approximation error the geometric noise assumption does not employ any smoothness assumption 1 introduction in recent years support vector machines svms have been the subject of many theoretical considerations despite this effort their learning performance on restricted classes of distributions is still widely unknown in particular it is unknown under which nontrivial circumstances svms can guarantee fast learning rates the aim of this work is to use concepts like tsybakov s noise assumption and local rademacher averages to establish learning rates up to the order of n 1 for nontrivial distributions in addition to these concepts that are used to deal with the stochastic part of the analysis we also introduce a geometric assumption for distributions that allows us to estimate the approximation properties of gaussian rbf kernels unlike many other concepts introduced for bounding the approximation error our geometric assumption is not in terms of smoothness but describes the concentration and the noisiness of the data generating distribution near the decision boundary let us formally introduce the statistical classification problem to this end let us fix a subset x r d we write y 1 1 given a finite training set

absolute depth estimation from a single defocused image shape from defocus sfd is one of the most popular techniques in monocular 3d vision while most sfd approaches require two or more images of the same scene captured at a fixed view point this paper presents an efficient approach to estimate absolute depth from a single defocused image instead of directly measuring defocus level of each pixel we propose to design a sequence of aperture shape filters to segment a defocused image by defocus level a boundary weighted belief propagation algorithm is employed to obtain a smooth depth map we also give an estimation of depth error extensive experiments show that our approach outperforms the state of the art single image sfd approaches both in precision of the estimated absolute depth and running time

contour detection and hierarchical image segmentation this paper investigates two fundamental problems in computer vision contour detection and image segmentation we present state of the art algorithms for both of these tasks our contour detector combines multiple local cues into a globalization framework based on spectral clustering our segmentation algorithm consists of generic machinery for transforming the output of any contour detector into a hierarchical region tree in this manner we reduce the problem of image segmentation to that of contour detection extensive experimental evaluation demonstrates that both our contour detection and segmentation methods significantly outperform competing algorithms the automatically generated hierarchical segmentations can be interactively refined by user specified annotations computation at multiple image resolutions provides a means of coupling our system to recognition applications

coherence functions with applications in large margin classification methods support vector machines svms naturally embody sparseness due to their use of hinge loss functions however svms can not directly estimate conditional class probabilities in this paper we propose and study a family of coherence functions which are convex and differentiable as surrogates of the hinge function the coherence function is derived by using the maximum entropy principle and is characterized by a temperature parameter it bridges the hinge function and the logit function in logistic regression the limit of the coherence function at zero temperature corresponds to the hinge function and the limit of the minimizer of its expected error is the minimizer of the expected error of the hinge loss we refer to the use of the coherence function in large margin classification as c learning and we present efficient coordinate descent algorithms for the training of regularized c learning models

sparseness of support vector machines support vector machines svms construct decision functions that are linear combinations of kernel evaluations on the training set the samples with non vanishing coefficients are called support vectors in this work we establish lower asymptotical bounds on the number of support vectors on our way we prove several results which are of great importance for the understanding of svms in particular we describe to which limit svm decision functions tend discuss the corresponding notion of convergence and provide some results on the stability of svms using subdifferential calculus in the associated reproducing kernel hilbert space

deblurring saturated night image with function form kernel deblurring saturated night images are a challenging problem because such images have low contrast combined with heavy noise and saturated regions unlike the deblurring schemes that discard saturated regions when estimating blur kernels this paper proposes a novel scheme to deduce blur kernels from saturated regions via a novel kernel representation and advanced algorithms our key technical contribution is the proposed function form representation of blur kernels which regularizes existing matrix form kernels using three functional components 1 trajectory 2 intensity and 3 expansion from automatically detected saturated regions their skeleton brightness and width are fitted into the corresponding three functional components of blur kernels such regularization significantly improves the quality of kernels deduced from saturated regions second we propose an energy minimizing algorithm to select and assign the deduced function form kernels to partitioned image regions as the initialization for non uniform deblurring finally we convert the assigned function form kernels into matrix form for more detailed estimation in a multi scale deconvolution experimental results show that our scheme outperforms existing schemes on challenging real examples

on the optimal parameter choice for spl nu support vector machines we determine the asymptotically optimal choice of the parameter spl nu for classifiers of spl nu support vector machine spl nu svm type which has been introduced by scholkopf et al 2000 it turns out that spl nu should be a close upper estimate of twice the optimal bayes risk provided that the classifier uses a so called universal kernel such as the gaussian rbf kernel moreover several experiments show that this result can be used to implement some modified cross validation procedures which improve standard cross validation for spl nu svms

training a support vector machine in the primal most literature on support vector machines svms concentrates on the dual optimization problem in this letter we point out that the primal problem can also be solved efficiently for both linear and nonlinear svms and that there is no reason for ignoring this possibility on the contrary from the primal point of view new families of algorithms for large scale svm training can be investigated

exponential growth for codimensions of some p i algebras abstract by the giambruno zaicev theorem for associative p i algebras the exponential rate of growth of the codimensions of such a p i algebra is always a positive integer here we calculate that integer for various generic p i algebras which are given by a single identity these include capelli type identities and the various powers of the standard polynomials

applications of belov s theorem to the cocharacter sequence of p i algebras although this can be considereda signi cant generalization of belov s theorem it can be proven with essentially the sameproof there is an excellent exposition of the proof in 2 since the proof with all of itsdetails would be long and since we do not presume that we could improve belov androwen s treatment we merely state the result we need and sketch the changes needed inthe proof

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

non parametric blur map regression for depth of field extension real camera systems have a limited depth of field dof which may cause an image to be degraded due to visible misfocus or too shallow dof in this paper we present a blind deblurring pipeline able to restore such images by slightly extending their dof and recovering sharpness in regions slightly out of focus to address this severely ill posed problem our algorithm relies first on the estimation of the spatially varying defocus blur drawing on local frequency image features a machine learning approach based on the recently introduced regression tree fields is used to train a model able to regress a coherent defocus blur map of the image labeling each pixel by the scale of a defocus point spread function a non blind spatially varying deblurring algorithm is then used to properly extend the dof of the image the good performance of our algorithm is assessed both quantitatively using realistic ground truth data obtained with a novel approach based on a plenoptic camera and qualitatively with real images

richardson lucy deblurring for scenes under a projective motion path this paper addresses how to model and correct image blur that arises when a camera undergoes ego motion while observing a distant scene in particular we discuss how the blurred image can be modeled as an integration of the clear scene under a sequence of planar projective transformations i e homographies that describe the camera s path this projective motion path blur model is more effective at modeling the spatially varying motion blur exhibited by ego motion than conventional methods based on space invariant blur kernels to correct the blurred image we describe how to modify the richardson lucy rl algorithm to incorporate this new blur model in addition we show that our projective motion rl algorithm can incorporate state of the art regularization priors to improve the deblurred results the projective motion path blur model along with the modified rl algorithm is detailed together with experimental results demonstrating its overall effectiveness statistical analysis on the algorithm s convergence properties and robustness to noise is also provided

understanding blind deconvolution algorithms blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown recent algorithms have afforded dramatic progress yet many aspects of the problem remain challenging and hard to understand the goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally we explain the previously reported failure of the naive map approach by demonstrating that it mostly favors no blur explanations we show that using reasonable image priors a naive simulations map estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior on the other hand we show that since the kernel size is often smaller than the image size a map estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur the plethora of recent deconvolution techniques makes an experimental evaluation on ground truth data important as a first step toward this experimental evaluation we have collected blur data with ground truth and compared recent algorithms under equal settings additionally our data demonstrate that the shift invariant blur assumption made by most algorithms is often violated

a sparsity driven kernel machine based on minimizing a generalization error bound a new sparsity driven kernel classifier is presented based on the minimization of a recently derived data dependent generalization error bound the objective function consists of the usual hinge loss function penalizing training errors and a concave penalty function of the expansion coefficients the problem of minimizing the non convex bound is addressed by a successive linearization approach whereby the problem is transformed into a sequence of linear programs the algorithm produced comparable error rates to the standard support vector machine but significantly reduced the number of support vectors and the concomitant classification time

estimating spatially varying defocus blur from a single image estimating the amount of blur in a given image is important for computer vision applications more specifically the spatially varying defocus point spread functions psfs over an image reveal geometric information of the scene and their estimate can also be used to recover an all in focus image a psf for a defocus blur can be specified by a single parameter indicating its scale most existing algorithms can only select an optimal blur from a finite set of candidate psfs for each pixel some of those methods require a coded aperture filter inserted in the camera in this paper we present an algorithm estimating a defocus scale map from a single image which is applicable to conventional cameras this method is capable of measuring the probability of local defocus scale in the continuous domain it also takes smoothness and color edge information into consideration to generate a coherent blur map indicating the amount of blur at each pixel simulated and real data experiments illustrate excellent performance and its successful applications in foreground background segmentation

convexity classification and risk bounds many of the classification algorithms developed in the machine learning literature including the support vector machine and boosting can be viewed as minimum contrast methods that minimize a convex surrogate of the 0 1 loss function the convexity makes these algorithms computationally efficient the use of a surrogate however has statistical consequences that must be balanced against the computational virtues of convexity to study these issues we provide a general quantitative relationship between the risk as assessed using the 0 1 loss and the risk as assessed using any nonnegative surrogate loss function we show that this relationship gives nontrivial upper bounds on excess risk under the weakest possible condition on the loss function that it satisfies a pointwise form of fisher consistency for classification the relationship is based on a simple variational transformation of the loss function that is easy to compute in many applications we also present a refined version of this result in the

colength sequences for matrices over grassmann algebras abstract we prove that the colength sequence of k k matrices over the grassmann algebra is asymptotic to a constant times n k 4

colength sequences for kemer algebras this paper is a sequel to 4 in that paper we calculated the asymptotic behavior of the colength sequence for k k matrices and in this paper we will calculate the asymptotic behavior of the colength sequence for the algebras mk given a p i algebra a the cocharacter sequence n a n 0 is a sequence of sncharacters if we decompose each character into a sum of irreducible sn characters we would get something of the form n a m a let ln a be the length of n a so ln a par n m a the sequence of numbers ln a is the colength sequence of a similar constructions can also be made for other cocharacter sequences such as the trace cocharacter if a is an algebra with trace kemer classified verbally prime p i algebras into three families see 8 one is k k matrices whose colength sequence we investigated in 4 one is k k matrices over an infinite dimensional grassmann algebra we do not know yet how to deal with this case and the third is the algebras mk let e be an infinite dimensional grassmann algebra and let k n then mk is a subalgebra of the matrix algebra mn e in order to define it note that the grassmann algebra has a natural z 2z grading e e0 e1 with respect to which it is anticommutative i e degree 0 elements are central and degree 1 elements anticommute with each other let i and j be between 1 and n we define the ordered pair i j to be of degree 0 if either 1 i j k or k 1 i j n and of degree 1 otherwise then mk will be the set of all matrices aij mn e in which the degree of each aij

an investigation of methods for determining depth from focus the concept of depth from focus involves calculating distances to points in an observed scene by modeling the effect that the camera s focal parameters have on images acquired with a small depth of field this technique is passive and requires only a single camera the most difficult segment of calculating depth from focus is deconvolving the defocus operator from the scene and modeling it most current methods for determining the defocus operator employ inverse filtering the authors reveal some fundamental problems with inverse filtering inaccuracies in finding the frequency domain representation windowing effects and border effects a general matrix based method using regularization is presented which eliminates these problems the new method is confirmed experimentally with the results showing an rms error of 1 3

rational filters for passive depth from defocus a fundamental problem in depth from defocus is the measurement of relative defocus between images the performance of previously proposed focus operators are inevitably sensitive to the frequency spectra of local scene textures as a result focus operators such as the laplacian of gaussian result in poor depth estimates an alternative is to use large filter banks that densely sample the frequency space though this approach can result in better depth accuracy it sacrifices the computational efficiency that depth from defocus offers over stereo and structure from motion we propose a class of broadband operators that when used together provide invariance to scene texture and produce accurate and dense depth maps since the operators are broadband a small number of them are sufficient for depth estimation of scenes with complex textural properties in addition a depth confidence measure is derived that can be computed from the outputs of the operators this confidence measure permits further refinement of computed depth maps experiments are conducted on both synthetic and real scenes to evaluate the performance of the proposed operators the depth detection gain error is less than 1 irrespective of texture frequency depth accuracy is found to be 0 5 1 2 of the distance of the object from the imaging optics

fast approximate energy minimization via graph cuts many tasks in computer vision involve assigning a label such as disparity to every pixel a common constraint is that the labels should vary smoothly almost everywhere while preserving sharp discontinuities that may exist e g at object boundaries these tasks are naturally stated in terms of energy minimization the authors consider a wide class of energies with various smoothness constraints global minimization of these energy functions is np hard even in the simplest discontinuity preserving case therefore our focus is on efficient approximation algorithms we present two algorithms based on graph cuts that efficiently find a local minimum with respect to two types of large moves namely expansion moves and swap moves these moves can simultaneously change the labels of arbitrarily large sets of pixels in contrast many standard algorithms including simulated annealing use small moves where only one pixel changes its label at a time our expansion algorithm finds a labeling within a known factor of the global minimum while our swap algorithm handles more general energy functions both of these algorithms allow important cases of discontinuity preserving energies we experimentally demonstrate the effectiveness of our approach for image restoration stereo and motion on real data with ground truth we achieve 98 percent accuracy

complexities of convex combinations and bounding the generalization error in classification we introduce and study several measures of complexity of functions from the convex hull of a given base class these complexity measures take into account the sparsity of the weights of a convex combination as well as certain clustering properties of the base functions involved in it we prove new upper confidence bounds on the generalization error of ensemble voting classification algorithms that utilize the new complexity measures along with the empirical distributions of classification margins providing a better explanation of generalization performance of large margin classification methods

new support vector algorithms we propose a new class of support vector algorithms for regression and classification in these algorithms a parameter lets one effectively control the number of support vectors while this can be useful in its own right the parameterization has the additional benefit of enabling us to eliminate one of the other free parameters of the algorithm the accuracy parameter epsilon in the regression case and the regularization constant c in the classification case we describe the algorithms give some theoretical results concerning the meaning and the choice of and report experimental results

nonconvex online support vector machines in this paper we propose a nonconvex online support vector machine svm algorithm lasvm nc based on the ramp loss which has the strong ability of suppressing the influence of outliers then again in the online learning setting we propose an outlier filtering mechanism lasvm i based on approximating nonconvex behavior in convex optimization these two algorithms are built upon another novel svm algorithm lasvm g that is capable of generating accurate intermediate models in its iterative steps by leveraging the duality gap we present experimental results that demonstrate the merit of our frameworks in achieving significant robustness to outliers in noisy data classification where mislabeled training instances are in abundance experimental evaluation shows that the proposed approaches yield a more scalable online svm algorithm with sparser models and less computational running time both in the training and recognition phases without sacrificing generalization performance we also point out the relation between nonconvex optimization and min margin active learning

integration of multiresolution image segmentation and neural networks for object depth recovery a novel technique for three dimensional depth recovery based on two coaxial defocused images of an object with added pattern illumination is presented the approach integrates object segmentation with depth estimation firstly segmentation is performed by a multiresolution based approach to isolate object regions from the background given the presence of blur and pattern illumination the segmentation has three sub procedures image pyramid formation linkage adaptation and unsupervised clustering these maximise the object recognition capability while ensuring accurate position information for depth estimation lower resolution information with a strong correlation to depth is fed into a three layered neural network as input feature vectors and processed using a back propagation algorithm the resulting depth model of object recovery is then used with higher resolution data to obtain high accuracy depth measurements experimental results are presented that show low error rates and the robustness of the model with respect to pattern variation and inaccuracy in optical settings

consistency of support vector machines and other regularized kernel classifiers it is shown that various classifiers that are based on minimization of a regularized risk are universally consistent i e they can asymptotically learn in every classification task the role of the loss functions used in these algorithms is considered in detail as an application of our general framework several types of support vector machines svms as well as regularization networks are treated our methods combine techniques from stochastics approximation theory and functional analysis

maximal multiplicities in cocharacter sequences if a is a p i algebra mn a is the maximum multiplicity of the sn irreducible characters in the nth cocharacter of a we prove that if 1 a then mn a is asymptotic to a polynomial and we compute the degree of this polynomial for the verbally prime p i algebras

colength sequences for matrices 1 a berele homogeneous polynomial identities israel j math 42 1982 258 272 2 a berele a regev some remarks on trace cocharacters j algebra 176 1995 1013 1024 3 a berele d j saltman the centers of generic division algebras with involution israel j math 63 1988 98 118 4 v s drensky codimensions of t ideals and hilbert series of relatively free algebras j algebra 91 1984 1 17 5 e formanek invariants and the ring of generic matrices j algebra 89 1984 178 223 6 e formanek a conjecture of regev about the capelli polynomial j algebra 109 1987 93 114 7 i n herstein notes from a ring theory conference cbms vol 9 amer math soc 1971 8 g james a kerber the representation theory of the symmetric group addison wesley reading ma 1981 9 i g macdonald symmetric functions and hall polynomials second ed oxford university press claren don 1995

asymptotics in cocharacters of simple algebras with involution abstract we compute the asymptotics of the colength and maximal multiplicity sequences of simple algebras with involution

asymptotic analysis of penalized likelihood and related estimators a general approach to the first order asymptotic analysis ofpenalized likelihood and related estimators is described the method gives expansions for the systematic and random error asymptotic convergence rates in a family of spectral norms are obtained the theory applies to a broad range of function estimation prob erns including non paxametric dellsity hazard and generalized regression curve estimation some examples are provided ams 1980 subject classifications primary 62 g05 secondary 62j05 41 a35 41 a25 47 a53 45 llo 45 m05

probability estimation for large margin classifiers summary large margin classifiers have proven to be effective in delivering high predictive accuracy par ticularly those focusing on the decision boundaries and bypassing the requirement of estimating the class probability given input for discrimination as a result these classifiers may not directly yield an estimated class probability which is of interest itself to overcome this difficulty this article proposes a novel method for estimating the class probability through sequential classi fications by using features of interval estimation of large margin classifiers the method uses sequential classifications to bracket the class probability to yield an estimate up to the desired level of accuracy the method is implemented for support vector machines and learning in addition to an estimated kullback leibler loss for tuning a solution path of the method is derived for support vector machines to reduce further its computational cost theoretical and numerical analyses indicate that the method is highly competitive against alternatives especially when the dimension of the input greatly exceeds the sample size finally an application to leukaemia data is described

rational filter design for depth from defocus the paper describes a new simple procedure to determine the rational filters that are used in the depth from defocus dfd procedure previously researched by watanabe and nayar 1998 4 their dfd uses two differently defocused images and the filters accurately model the relative defocus in the images and provide a fast calculation of distance this paper presents a simple method to determine the filter coefficients by separating the m p ratio into a linear and a cubic error correction model the method avoids the previous iterative minimisation technique and computes efficiently the model has been verified by comparison with the theoretical m p ratio the proposed filters have been compared with the previous for frequency response closeness of fit to m p rotational symmetry and measurement accuracy experiments were performed for several defocus conditions it was observed that the new filters were largely insensitive to object texture and modelled the blur more precisely than the previous experiments with real planar images demonstrated a maximum rms depth error of 1 18 for the proposed compared to 1 54 for the previous filters complicated objects were also accurately measured

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

image deblurring with an inaccurate blur kernel using a group based low rank image prior abstract we address the problem of restoring an original image from its blurry and noisy observation together with inaccurate information of the blurring process for this purpose we propose an enhanced regularized structured total least squares rstls model that can estimate the latent image and blur kernel simultaneously in the proposed model both the image and the blur kernel are characterized by a group based low rank prior which assumes that a group of vectorized similar data patches can be well approximated by a low rank matrix we develop an alternating minimization algorithm to solve the proposed model efficiently numerical experiments demonstrate the effectiveness of our method in terms of both quantitative measures and visual quality

on robustness properties of convex risk minimization methods for pattern recognition the paper brings together methods from two disciplines machine learning theory and robust statistics we argue that robustness is an important aspect and we show that many existing machine learning methods based on the convex risk minimization principle have besides other good properties also the advantage of being robust robustness properties of machine learning methods based on convex risk minimization are investigated for the problem of pattern recognition assumptions are given for the existence of the influence function of the classifiers and for bounds on the influence function kernel logistic regression support vector machines least squares and the adaboost loss function are treated as special cases some results on the robustness of such methods are also obtained for the sensitivity curve and the maxbias which are two other robustness criteria a sensitivity analysis of the support vector machine is given

bayesian depth from defocus with shading constraints we present a method that enhances the performance of depth from defocus dfd through the use of shading information dfd suffers from important limitations namely coarse shape reconstruction and poor accuracy on textureless surfaces that can be overcome with the help of shading we integrate both forms of data within a bayesian framework that capitalizes on their relative strengths shading data however is challenging to accurately recover from surfaces that contain texture to address this issue we propose an iterative technique that utilizes depth information to improve shading estimation which in turn is used to elevate depth estimation in the presence of textures the shading estimation can be performed in general scenes with unknown illumination using an approximate estimate of scene lighting with this approach we demonstrate improvements over existing dfd techniques as well as effective shape reconstruction of textureless surfaces

shape from defocus via diffusion defocus can be modeled as a diffusion process and represented mathematically using the heat equation where image blur corresponds to the diffusion of heat this analogy can be extended to nonplanar scenes by allowing a space varying diffusion coefficient the inverse problem of reconstructing 3d structure from blurred images corresponds to an inverse diffusion that is notoriously ill posed we show how to bypass this problem by using the notion of relative blur given two images within each neighborhood the amount of diffusion necessary to transform the sharper image into the blurrier one depends on the depth of the scene this can be used to devise a global algorithm to estimate the depth profile of the scene without recovering the deblurred image using only forward diffusion

exact simplification of support vector solutions this paper demonstrates that standard algorithms for training support vector machines generally produce solutions with a greater number of support vectors than are strictly necessary an algorithm is presented that allows unnecessary support vectors to be recognized and eliminated while leaving the solution otherwise unchanged the algorithm is applied to a variety of benchmark data sets for both classification and regression and in most cases the procedure leads to a reduction in the number of support vectors in some cases the reduction is substantial

generalization bounds and complexities based on sparsity and clustering for convex combinations of functions from random classes a unified approach is taken for deriving new generalization data dependent bounds for several classes of algorithms explored in the existing literature by different approaches this unified approach is based on an extension of vapnik s inequality for vc classes of sets to random classes of sets that is classes depending on the random data invariant under permutation of the data and possessing the increasing property generalization bounds are derived for convex combinations of functions from random classes with certain properties algorithms such as svms support vector machines boosting with decision stumps radial basis function networks some hierarchies of kernel machines or convex combinations of indicator functions over sets with finite vc dimension generate classifier functions that fall into the above category we also explore the individual complexities of the classifiers such as sparsity of weights and weighted variance over clusters from the convex combination introduced by koltchinskii and panchenko 2004 and show sparsity type and cluster variance type generalization bounds for random classes

when is there a representer theorem vector versus matrix regularizers we consider a general class of regularization methods which learn a vector of parameters on the basis of linear measurements it is well known that if the regularizer is a nondecreasing function of the l2 norm then the learned vector is a linear combination of the input data this result known as the representer theorem lies at the basis of kernel based methods in machine learning in this paper we prove the necessity of the above condition in the case of differentiable regularizers we further extend our analysis to regularization methods which learn a matrix a problem which is motivated by the application to multi task learning in this context we study a more general representer theorem which holds for a larger class of regularizers we provide a necessary and sufficient condition characterizing this class of matrix regularizers and we highlight some concrete examples of practical importance our analysis uses basic principles from matrix theory especially the useful notion of matrix nondecreasing functions

robust image deblurring with an inaccurate blur kernel most existing nonblind image deblurring methods assume that the blur kernel is free of error however it is often unavoidable in practice that the input blur kernel is erroneous to some extent sometimes the error could be severe e g for images degraded by nonuniform motion blurring when an inaccurate blur kernel is used as the input significant distortions will appear in the image recovered by existing methods in this paper we present a novel convex minimization model that explicitly takes account of error in the blur kernel the resulting minimization problem can be efficiently solved by the so called accelerated proximal gradient method in addition a new boundary extension scheme is incorporated in the proposed model to further improve the results the experiments on both synthesized and real images showed the efficiency and robustness of our algorithm to both the image noise and the model error in the blur kernel

learning from examples as an inverse problem many works related learning from examples to regularization techniques for inverse problems emphasizing the strong algorithmic and conceptual analogy of certain learning algorithms with regularization algorithms in particular it is well known that regularization schemes such as tikhonov regularization can be effectively used in the context of learning and are closely related to algorithms such as support vector machines nevertheless the connection with inverse problem was considered only for the discrete finite sample problem and the probabilistic aspects of learning from examples were not taken into account in this paper we provide a natural extension of such analysis to the continuous population case and study the interplay between the discrete and continuous problems from a theoretical point of view this allows to draw a clear connection between the consistency approach in learning theory and the stability convergence property in ill posed inverse problems the main mathematical result of the paper is a new probabilistic bound for the regularized least squares algorithm by means of standard results on the approximation term the consistency of the algorithm easily follows

on the representer theorem and equivalent degrees of freedom of svr support vector regression svr for discrete data is considered an alternative formulation of the representer theorem is derived this result is based on the newly introduced notion of pseudoresidual and the use of subdifferential calculus the representer theorem is exploited to analyze the sensitivity properties of e insensitive svr and introduce the notion of approximate degrees of freedom the degrees of freedom are shown to play a key role in the evaluation of the optimism that is the difference between the expected in sample error and the expected empirical risk in this way it is possible to define a cp like statistic that can be used for tuning the parameters of svr the proposed tuning procedure is tested on a simulated benchmark problem and on a real world problem boston housing data set

on the influence of the kernel on the consistency of support vector machines in this article we study the generalization abilities of several classifiers of support vector machine svm type using a certain class of kernels that we call universal it is shown that the soft margin algorithms with universal kernels are consistent for a large class of classification problems including some kind of noisy tasks provided that the regularization parameter is chosen well in particular we derive a simple sufficient condition for this parameter in the case of gaussian rbf kernels on the one hand our considerations are based on an investigation of an approximation property the so called universality of the used kernels that ensures that all continuous functions can be approximated by certain kernel expressions this approximation property also gives a new insight into the role of kernels in these and other algorithms on the other hand the results are achieved by a precise study of the underlying optimization problems of the classifiers furthermore we show consistency for the maximal margin classifier as well as for the soft margin svm s in the presence of large margins in this case it turns out that also constant regularization parameters ensure consistency for the soft margin svm s finally we prove that even for simple noise free classification problems svm s with polynomial kernels can behave arbitrarily badly

a geometric approach to shape from defocus we introduce a novel approach to shape from defocus i e the problem of inferring the three dimensional 3d geometry of a scene from a collection of defocused images typically in shape from defocus the task of extracting geometry also requires deblurring the given images a common approach to bypass this task relies on approximating the scene locally by a plane parallel to the image the so called equifocal assumption we show that this approximation is indeed not necessary as one can estimate 3d geometry while avoiding deblurring without strong assumptions on the scene solving the problem of shape from defocus requires modeling how light interacts with the optics before reaching the imaging surface this interaction is described by the so called point spread function psf when the form of the psf is known we propose an optimal method to infer 3d geometry from defocused images that involves computing orthogonal operators which are regularized via functional singular value decomposition when the form of the psf is unknown we propose a simple and efficient method that first learns a set of projection operators from blurred images and then uses these operators to estimate the 3d geometry of the scene from novel blurred images our experiments on both real and synthetic images show that the performance of the algorithm is relatively insensitive to the form of the psf our general approach is to minimize the euclidean norm of the difference between the estimated images and the observed images the method is geometric in that we reduce the minimization to performing projections onto linear subspaces by using inner product structures on both infinite and finite dimensional hilbert spaces both proposed algorithms involve only simple matrix vector multiplications which can be implemented in real time

depth from defocus a spatial domain approach a new method named stm is described for determining distance of objects and rapid autofocusing of camera systems stm uses image defocus information and is based on a new spatial domain convolution deconvolution transform the method requires only two images taken with different camera parameters such as lens position focal length and aperture diameter both images can be arbitrarily blurred and neither of them needs to be a focused image therefore stm is very fast in comparison with depth from focus methods which search for the lens position or focal length of best focus the method involves simple local operations and can be easily implemented in parallel to obtain the depth map of a scene stm has been implemented on an actual camera system named sparcs experiments on the performance of stm and their results on real world planar objects are presented the results indicate that the accuracy of stm compares well with depth from focus methods and is useful in practical applications the utility of the method is demonstrated for rapid autofocusing of electronic cameras

high order total variation based image restoration the total variation tv denoising method is a pde based technique that preserves edges well but has the sometimes undesirable staircase effect namely the transformation of smooth regions ramps into piecewise constant regions stairs in this paper we present an improved model constructed by adding a nonlinear fourth order diffusive term to the euler lagrange equations of the variational tv model our technique substantially reduces the staircase effect while preserving sharp jump discontinuities edges we show numerical evidence of the power of resolution of this novel model with respect to the tv model in some 1d and 2d numerical examples

efficient focus sampling through depth of field calibration due to the limited depth of field dof of conventional digital cameras only the objects within a certain distance range from the camera are in focus objects outside the dof are observed with different amounts of defocus depending on their position focus sampling consists of capturing different images of the same scene by changing the focus configuration of the camera in order to alternately bring objects at different depths into focus focus sampling is an important part of different focus related applications such as autofocus focus stacking and depth estimation this work proposes a calibration procedure for modeling the depth of field of conventional cameras in order to perform an efficient focus sampling the method is simple in terms of repeatability and can be easily implemented in different imaging devices experimental tests are presented in order to illustrate the effectiveness of the proposed approach in autofocus results demonstrate that a significant reduction in the number of frames required to capture during autofocusing can be achieved by means of the proposed methodology

a direct method for building sparse kernel learning algorithms many kernel learning algorithms including support vector machines result in a kernel machine such as a kernel classifier whose key component is a weight vector in a feature space implicitly introduced by a positive definite kernel function this weight vector is usually obtained by solving a convex optimization problem based on this fact we present a direct method to build sparse kernel learning algorithms by adding one more constraint to the original convex optimization problem such that the sparseness of the resulting kernel machine is explicitly controlled while at the same time performance is kept as high as possible a gradient based approach is provided to solve this modified optimization problem applying this method to the support vectom machine results in a concrete algorithm for building sparse large margin classifiers these classifiers essentially find a discriminating subspace that can be spanned by a small number of vectors and in this subspace the different classes of data are linearly well separated experimental results over several classification benchmarks demonstrate the effectiveness of our approach

some properties of regularized kernel methods in regularized kernel methods the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term in this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk first we prove a quantitative version of the representer theorem holding for both regression and classification for both differentiable and non differentiable loss functions and for arbitrary offset terms second we show that the case in which the offset space is non trivial corresponds to solving a standard problem of regularization in a reproducing kernel hilbert space in which the penalty term is given by a seminorm finally we discuss the issues of existence and uniqueness of the solution from the specialization of our analysis to the discrete setting it is immediate to establish a connection between the solution properties of sparsity and coefficient boundedness and some properties of the loss function for the case of support vector machines for classification we also obtain a complete characterization of the whole method in terms of the khun tucker conditions with no need to introduce the dual formulation

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets