aspects of total variation regularized l1 function approximation the total variation based image denoising model of rudin osher and fatemi phys d 60 1992 pp 259 268 has been generalized and modified in many ways in the literature one of these modifications is to use the l1 norm as the fidelity term we study the interesting consequences of this modification especially from the point of view of geometric properties of its solutions it turns out to have interesting new implications for data driven scale selection and multiscale image decomposition

bayesian reconstructions from emission tomography data using a modified em algorithm a novel method of reconstruction from single photon emission computerized tomography data is proposed this method builds on the expectation maximization em approach to maximum likelihood reconstruction from emission tomography data but aims instead at maximum posterior probability estimation which takes account of prior belief about smoothness in the isotope concentration a novel modification to the em algorithm yields a practical method the method is illustrated by an application to data from brain scans

on the unification of line processes outlier rejection and robust statistics with applications in early vision the modeling of spatial discontinuities for problems such as surface recovery segmentation image reconstruction and optical flow has been intensely studied in computer vision while line process models of discontinuities have received a great deal of attention there has been recent interest in the use of robust statistical techniques to account for discontinuities this paper unifies the two approaches to achieve this we generalize the notion of a line process to that of an analog outlier process and show how a problem formulated in terms of outlier processes can be viewed in terms of robust statistics we also characterize a class of robust statistical problems for which an equivalent outlier process formulation exists and give a straightforward method for converting a robust estimation problem into an outlier process formulation we show how prior assumptions about the spatial structure of outliers can be expressed as constraints on the recovered analog outlier processes and how traditional continuation methods can be extended to the explicit outlier process formulation these results indicate that the outlier process approach provides a general framework which subsumes the traditional line process approaches as well as a wide class of robust estimation problems examples in surface reconstruction image segmentation and optical flow are presented to illustrate the use of outlier processes and to show how the relationship between outlier processes and robust statistics can be exploited an appendix provides a catalog of common robust error norms and their equivalent outlier process formulations

a variational bayesian method to inverse problems with impulsive noise we propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers the approach is of bayesian type and it exploits a heavy tailed t distribution for data noise to achieve robustness with respect to outliers a hierarchical model with all hyper parameters automatically determined from the given data is described an algorithm of variational type by minimizing the kullback leibler divergence between the true posteriori distribution and a separable approximation is developed the numerical method is illustrated on several one and two dimensional linear and nonlinear inverse problems arising from heat conduction including estimating boundary temperature heat flux and heat transfer coefficient the results show its robustness to outliers and the fast and steady convergence of the algorithm

generalised nonlocal image smoothing we propose a discrete variational approach for image smoothing consisting of nonlocal data and smoothness constraints that penalise general dissimilarity measures defined on image patches one of such dissimilarity measures is the weighted l 2 distance between patches in such a case we derive an iterative neighbourhood filter that induces a new similarity measure in the photometric domain it can be regarded as an extended patch similarity measure that evaluates not only the patch similarity of two chosen pixels but also the similarity of their corresponding neighbours this leads to a more robust smoothing process since the pixels selected for averaging are more coherent with the local image structure by slightly modifying the way the similarities are computed we obtain two related filters the nl means filter of buades et al siam multiscale model simul 4 2 490 530 2005b and the nds filter of mrazek et al geometric properties for incomplete data computational imaging and vision vol 31 pp 335 352 springer dordrecht 2006 in fact the proposed approach can be considered as a generalisation of the latter filter to the space of patches we also provide novel insights into relations of the nds filter with diffusion regularisation methods as well as with some recently proposed graph regularisation techniques we evaluate our method for the task of denoising greyscale and colour images degraded with gaussian and salt and pepper noise demonstrating that it compares very well to other more sophisticated approaches

fuzzy peer groups for reducing mixed gaussian impulse noise from color images the peer group of an image pixel is a pixel similarity based concept which has been successfully used to devise image denoising methods however since it is difficult to define the pixel similarity in a crisp way we propose to represent this similarity in fuzzy terms in this paper we introduce the fuzzy peer group concept which extends the peer group concept in the fuzzy setting a fuzzy peer group will be defined as a fuzzy set that takes a peer group as support set and where the membership degree of each peer group member will be given by its fuzzy similarity with respect to the pixel under processing the fuzzy peer group of each image pixel will be determined by means of a novel fuzzy logic based procedure we use the fuzzy peer group concept to design a two step color image filter cascading a fuzzy rule based switching impulse noise filter by a fuzzy average filtering over the fuzzy peer group both steps use the same fuzzy peer group which leads to computational savings the proposed filter is able to efficiently suppress both gaussian noise and impulse noise as well as mixed gaussian impulse noise experimental results are provided to show that the proposed filter achieves a promising performance

a nonconvex l1 l1 l2 model for image restoration with impulse noise abstract in this paper we propose a model for image restoration with blur and impulse noise it is composed of l 1 data fitting term and a nonconvex regularization term which is the weighted difference of l 1 norm and l 2 norm based on wavelet frame the combined model is difficult to solve by the corresponding euler lagrangian equations here we solve it by alternating direction method of multipliers admm we describe the detailed process of the algorithm and establish the convergence of the algorithm the experimental outcomes on different blurs and different impulse noises demonstrate that the proposed approach is better than those existing methods in terms of standard signal to noise ratio psnr relative error reerr and visual quality

a new impulse detection and filtering method for removal of wide range impulse noises a new impulse detection and filtering algorithm is proposed for restoration of images that are highly corrupted by impulse noise it is based on the minimum absolute value of four convolutions obtained by one dimensional laplacian operators the proposed algorithm can effectively remove the impulse noise with a wide range of noise density and produce better results in terms of the qualitative and quantitative measures of the images even at noise density as high as 90 extensive simulations show that the proposed algorithm provides better performance than many of the existing switching median filters in terms of noise suppression and detail preservation

finite element techniques for removing the mixture of gaussian and impulsive noise the finite element method has become a very powerful and popular tool to solve boundary value problems coming from science and engineering here we consider a scattered data fitting method based on the finite element method and apply the method to remove the mixture of gaussian and impulsive noise from an image numerical results show the performance of the approach

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

minimizers of cost functions involving nonsmooth data fidelity terms application to the processing of outliers we present a theoretical study of the recovery of an unknown vector x in mathbb r p such as a signal or an image from noisy data y in mathbb r q by minimizing with respect to x a regularized cost function cal f x y psi x y alpha phi x where psi is a data fidelity term phi is a smooth regularization term and alpha 0 is a parameter typically psi x y ax y 2 where a is a linear operator the data fidelity terms psi involved in regularized cost functions are generally smooth functions only a few papers make an exception to this and they consider restricted situations nonsmooth data fidelity terms are avoided in image processing in spite of this we consider both smooth and nonsmooth data fidelity terms our goal is to capture essential features exhibited by the local minimizers of regularized cost functions in relation to the smoothness of the data fidelity term in order to fix the context of our study we consider psi x y sum i psi a i tx y i where a i t are the rows of a and psi is cal c m on mathbb r setminus 0 we show that if psi 1 0 psi 1 0 then typical data y give rise to local minimizers hat x of cal f y which fit exactly a certain number of the data entries there is a possibly large set hat h of indexes such that a i t hat x y i for every i in hat h in contrast if psi is smooth on mathbb r for almost every y the local minimizers of cal f y do not fit any entry of y thus the possibility that a local minimizer fits some data entries is due to the nonsmoothness of the data fidelity term this is a strong mathematical property which is useful in practice by way of application we construct a cost function allowing aberrant data outliers to be detected and to be selectively smoothed our numerical experiments advocate the use of nonsmooth data fidelity terms in regularized cost functions for special purposes in image and signal processing

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

on discontinuity adaptive smoothness priors in computer vision a variety of analytic and probabilistic models in connection with markov random fields mrfs have been proposed in the last decade for solving low level vision problems involving discontinuities this paper presents a systematic study of these models and defines a general discontinuity adaptive da mrf model by analyzing the euler equation associated with the energy minimization it shows that the fundamental difference between different models lies in the behavior of interaction between neighboring points which is determined by the a priori smoothness constraint encoded into the energy function an important necessary condition is derived for the interaction to be adaptive to discontinuities to avoid oversmoothing this forms the basis on which a class of adaptive interaction functions aifs is defined the da model is defined in terms of the euler equation constrained by this class of aifs its solution is c sup 1 continuous and allows arbitrarily large but bounded slopes in dealing with discontinuities because of the continuous nature it is stable to changes in parameters and data a good property for regularizing ill posed problems experimental results are shown

deblurring of color images corrupted by impulsive noise we consider the problem of restoring a multichannel image corrupted by blur and impulsive noise e g salt and pepper noise using the variational framework we consider the l1 fidelity term and several possible regularizers in particular we use generalizations of the mumford shah ms functional to color images and gamma convergence approximations to unify deblurring and denoising experimental comparisons show that the ms stabilizer yields better results with respect to beltrami and total variation regularizers color edge detection is a beneficial by product of our methods

salt and pepper noise removal by median type noise detectors and detail preserving regularization this paper proposes a two phase scheme for removing salt and pepper impulse noise in the first phase an adaptive median filter is used to identify pixels which are likely to be contaminated by noise noise candidates in the second phase the image is restored using a specialized regularization method that applies only to those selected noise candidates in terms of edge preservation and noise suppression our restored images show a significant improvement compared to those restored by using just nonlinear filters or regularization methods only our scheme can remove salt and pepper noise with a noise level as high as 90

the idude framework for grayscale image denoising we present an extension of the discrete universal denoiser dude specialized for the denoising of grayscale images the original dude is a low complexity algorithm aimed at recovering discrete sequences corrupted by discrete memoryless noise of known statistical characteristics it is universal in the sense of asymptotically achieving without access to any information on the statistics of the clean sequence the same performance as the best denoiser that does have access to such information the dude however is not effective on grayscale images of practical size the difficulty lies in the fact that one of the dude s key components is the determination of conditional empirical probability distributions of image samples given the sample values in their neighborhood when the alphabet is relatively large as is the case with grayscale images even for a small sized neighborhood the required distributions would be estimated from a large collection of sparse statistics resulting in poor estimates that would not enable effective denoising the present work enhances the basic dude scheme by incorporating statistical modeling tools that have proven successful in addressing similar issues in lossless image compression instantiations of the enhanced framework which is referred to as idude are described for examples of additive and nonadditive noise the resulting denoisers significantly surpass the state of the art in the case of salt and pepper s p and ary symmetric noise and perform well for gaussian noise

deterministic edge preserving regularization in computed imaging many image processing problems are ill posed and must be regularized usually a roughness penalty is imposed on the solution the difficulty is to avoid the smoothing of edges which are very important attributes of the image in this paper we first give conditions for the design of such an edge preserving regularization under these conditions we show that it is possible to introduce an auxiliary variable whose role is twofold first it marks the discontinuities and ensures their preservation from smoothing second it makes the criterion half quadratic the optimization is then easier we propose a deterministic strategy based on alternate minimizations on the image and the auxiliary variable this leads to the definition of an original reconstruction algorithm called artur some theoretical properties of artur are discussed experimental results illustrate the behavior of the algorithm these results are shown in the field of 2d single photon emission tomography but this method can be applied in a large number of applications in image processing

fast multilevel algorithm for a minimization problem in impulse noise removal an effective 2 phase method for removing impulse noise was recently proposed its phase 1 identifies noisy pixel candidates by using median type filters then in phase 2 it restores only the noisy pixel candidates by some variational methods the resulting method can handle salt and pepper noise and random valued impulse noise at a level as high as 90 and 60 respectively the aim of this paper is to generalize a fast multilevel method for gaussian denoising to solving the minimization problem arising in phase 2 of the 2 phase method the multilevel algorithm gives better images than standard optimization methods such as the newton method or the conjugate gradient method also it can handle more general regularization functionals than the smooth ones previously considered supporting numerical experiments on two dimensional gray scale images are presented

adaptive variational method for restoring color images with high density impulse noise in this paper a new variational framework of restoring color images with impulse noise is presented the novelty of this work is the introduction of an adaptively weighting data fidelity term in the cost functional the fidelity term is derived from statistical methods and contains two weighting functions as well as some statistical control parameters of noise this method is based on the fact that impulse noise can be approximated as an additive noise with probability density function pdf being the finite mixture model a bayesian framework is then formulated in which likelihood functions are given by the mixture model inspired by the expectation maximization em algorithm we present two models with variational framework in this study the superiority of the proposed models is that the weighting functions can effectively detect the noise in the image with the noise information the proposed algorithm can automatically balance the regularity of the restored image and the fidelity term by updating the weighting functions and the control parameters these two steps ensure that one can obtain a good restoration even though the degraded color image is contaminated by impulse noise with large ration 90 or more in addition the numerical implementation of this algorithm is very fast by using a split algorithm some numerical experimental results and comparisons with other methods are provided to show the significant effectiveness of our approach

a multiplicative iterative algorithm for box constrained penalized likelihood image restoration image restoration is a computationally intensive problem as a large number of pixel values have to be determined since the pixel values of digital images can attain only a finite number of values e g 8 bit images can have only 256 gray levels one would like to recover an image within some dynamic range this leads to the imposition of box constraints on the pixel values the traditional gradient projection methods for constrained optimization can be used to impose box constraints but they may suffer from either slow convergence or repeated searching for active sets in each iteration in this paper we develop a new box constrained multiplicative iterative bcmi algorithm for box constrained image restoration the bcmi algorithm just requires pixelwise updates in each iteration and there is no need to invert any matrices we give the convergence proof of this algorithm and apply it to total variation image restoration problems where the observed blurry images contain poisson gaussian or salt and pepper noises

noise adaptive soft switching median filter existing state of the art switching based median filters are commonly found to be nonadaptive to noise density variations and prone to misclassifying pixel characteristics at high noise density interference this reveals the critical need of having a sophisticated switching scheme and an adaptive weighted median filter we propose a novel switching based median filter with incorporation of fuzzy set concept called the noise adaptive soft switching median nasm filter to achieve much improved filtering performance in terms of effectiveness in removing impulse noise while preserving signal details and robustness in combating noise density variations the proposed nasm filter consists of two stages a soft switching noise detection scheme is developed to classify each pixel to be uncorrupted pixel isolated impulse noise nonisolated impulse noise or image object s edge pixel no filtering or identity filter standard median sm filter or our developed fuzzy weighted median fwm filter will then be employed according to the respective characteristic type identified experimental results show that our nasm filter impressively outperforms other techniques by achieving fairly close performance to that of ideal switching median filter across a wide range of noise densities ranging from 10 to 70

a detection statistic for random valued impulse noise this paper proposes an image statistic for detecting random valued impulse noise by this statistic we can identify most of the noisy pixels in the corrupted images combining it with an edge preserving regularization we obtain a powerful two stage method for denoising random valued impulse noise even for noise levels as high as 60 simulation results show that our method is significantly better than a number of existing techniques in terms of image restoration and noise detection

a robust hidden markov gauss mixture vector quantizer for a noisy source noise is ubiquitous in real life and changes image acquisition communication and processing characteristics in an uncontrolled manner gaussian noise and salt and pepper noise in particular are prevalent in noisy communication channels camera and scanner sensors and medical mri images it is not unusual for highly sophisticated image processing algorithms developed for clean images to malfunction when used on noisy images for example hidden markov gauss mixture models hmgmm have been shown to perform well in image segmentation applications but they are quite sensitive to image noise we propose a modified hmgmm procedure specifically designed to improve performance in the presence of noise the key feature of the proposed procedure is the adjustment of covariance matrices in gauss mixture vector quantizer codebooks to minimize an overall minimum discrimination information distortion mdi in adjusting covariance matrices we expand or shrink their elements based on the noisy image while most results reported in the literature assume a particular noise type we propose a framework without assuming particular noise characteristics without denoising the corrupted source we apply our method directly to the segmentation of noisy sources we apply the proposed procedure to the segmentation of aerial images with salt and pepper noise and with independent gaussian noise and we compare our results with those of the median filter restoration method and the blind deconvolution based method respectively we show that our procedure has better performance than image restoration based techniques and closely matches to the performance of hmgmm for clean images in terms of both visual segmentation results and error rate

switching bilateral filter with a texture noise detector for universal noise removal in this paper we propose a switching bilateral filter sbf with a texture and noise detector for universal noise removal operation was carried out in two stages detection followed by filtering for detection we propose the sorted quadrant median vector sqmv scheme which includes important features such as edge or texture information this information is utilized to allocate a reference median from sqmv which is in turn compared with a current pixel to classify it as impulse noise gaussian noise or noise free the sbf removes both gaussian and impulse noise without adding another weighting function the range filter inside the bilateral filter switches between the gaussian and impulse modes depending upon the noise classification result simulation results show that our noise detector has a high noise detection rate as well as a high classification rate for salt and pepper uniform impulse noise and mixed impulse noise unlike most other impulse noise filters the proposed sbf achieves high peak signal to noise ratio and great image quality by efficiently removing both types of mixed noise salt and pepper with uniform noise and salt and pepper with gaussian noise in addition the computational complexity of sbf is significantly less than that of other mixed noise filters

geometric features based filtering for suppression of impulse noise in color images a geometric features based filtering technique named as the adaptive geometric features based filtering technique agff is presented for removal of impulse noise in corrupted color images in contrast with the traditional noise detection techniques where only 1d statistical information is used for noise detection and estimation a novel noise detection method is proposed based on geometric characteristics and features i e the 2 d information of the corrupted pixel or the pixel region leading to effective and efficient noise detection and estimation outcomes a progressive restoration mechanism is devised using multipass nonlinear operations which adapt to the intensity and the types of the noise extensive experiments conducted using a wide range of test color images have shown that the agff is superior to a number of existing well known benchmark techniques in terms of standard image restoration performance criteria including objective measurements the visual image quality and the computational complexity

self tuning to the hopf bifurcation in fluctuating systems the problem of self tuning a system to the hopf bifurcation in the presence of noise and periodic external forcing is discussed we find that the response of the system has a non monotonic dependence on the noise strength and displays an amplified response which is more pronounced for weaker signals the observed effect is to be distinguished from stochastic resonance for the feedback we have studied the unforced self tuned hopf oscillator in the presence of fluctuations exhibits sharp peaks in its spectrum the implications of our general results are briefly discussed in the context of sound detection by the inner ear

selective removal of impulse noise based on homogeneity level information we propose a decision based signal adaptive median filtering algorithm for removal of impulse noise our algorithm achieves accurate noise detection and high snr measures without smearing the fine details and edges in the image the notion of homogeneity level is defined for pixel values based on their global and local statistical properties the cooccurrence matrix technique is used to represent the correlations between a pixel and its neighbors and to derive the upper and lower bound of the homogeneity level noise detection is performed at two stages noise candidates are first selected using the homogeneity level and then a refining process follows to eliminate false detections the noise detection scheme does not use a quantitative decision measure but uses qualitative structural information and it is not subject to burdensome computations for optimization of the threshold values empirical results indicate that our scheme performs significantly better than other median filters in terms of noise suppression and detail preservation

probabilistic tensor canonical polyadic decomposition with orthogonal factors tensor canonical polyadic decomposition cpd which recovers the latent factor matrices from multidimensional data is an important tool in signal processing in many applications some of the factor matrices are known to have orthogonality structure and this information can be exploited to improve the accuracy of latent factors recovery however existing methods for cpd with orthogonal factors all require the knowledge of tensor rank which is difficult to acquire and have no mechanism to handle outliers in measurements to overcome these disadvantages in this paper a novel tensor cpd algorithm based on the probabilistic inference framework is devised in particular the problem of tensor cpd with orthogonal factors is interpreted using a probabilistic model based on which an inference algorithm is proposed that alternatively estimates the factor matrices recovers the tensor rank and mitigates the outliers simulation results using synthetic data and real world applications are presented to illustrate the excellent performance of the proposed algorithm in terms of accuracy and robustness

variational image segmentation model coupled with image restoration achievements image segmentation and image restoration are two important topics in image processing with a number of important applications in this paper we propose a new multiphase segmentation model by combining image restoration and image segmentation models utilizing aspects of image restoration the proposed segmentation model can effectively and robustly tackle images with a high level of noise or blurriness missing pixels or vector values in particular one of the most important segmentation models the piecewise constant mumford shah model can be extended easily in this way to segment gray and vector valued images corrupted for example by noise blur or information loss after coupling a new data fidelity term which borrowed from the field of image restoration it can be solved efficiently using the alternating minimization algorithm and we prove the convergence of this algorithm with three variables under mild conditions experiments on many synthetic and real world images demonstrate that our method gives better segmentation results in terms of quality and quantity in comparison to other state of the art segmentation models especially for blurry images and those with information loss highlightsprovide a new methodology and a new model for multiphase image segmentation extend the piecewise constant mumford shah model to handle blurry case easily has the ability to segment images with gaussian poisson and impulsive noises effective in segmenting vector valued image and image observed with information loss the convergence of the am algorithm with three variables adopted is proved

image deblurring in the presence of impulsive noise consider the problem of image deblurring in the presence of impulsive noise standard image deconvolution methods rely on the gaussian noise model and do not perform well with impulsive noise the main challenge is to deblur the image recover its discontinuities and at the same time remove the impulse noise median based approaches are inadequate because at high noise levels they induce nonlinear distortion that hampers the deblurring process distinguishing outliers from edge elements is difficult in current gradient based edge preserving restoration methods the suggested approach integrates and extends the robust statistics line process half quadratic and anisotropic diffusion points of view we present a unified variational approach to image deblurring and impulse noise removal the objective functional consists of a fidelity term and a regularizer data fidelity is quantified using the robust modified l 1 norm and elements from the mumford shah functional are used for regularization we show that the mumford shah regularizer can be viewed as an extended line process it reflects spatial organization properties of the image edges that do not appear in the common line process or anisotropic diffusion this allows to distinguish outliers from edges and leads to superior experimental results

cognition and removal of impulse noise with uncertainty uncertainties are the major inherent feature of impulse noise this fact makes image denoising a difficult task understanding the uncertainties can improve the performance of image denoising this paper presents a novel adaptive detail preserving filter based on the cloud model cm to remove impulse noise it is called the cm filter first an uncertainty based detector identifies the pixels corrupted by impulse noise then a weighted fuzzy mean filter is applied to remove the noise candidates the experimental results show that compared with the traditional switching filters the cm filter makes a great improvement in image denoising even at a noise level as high as 95 the cm filter still can restore the image with good detail preservation

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

an lp lq minimization method with cross validation for the restoration of impulse noise contaminated images abstract discrete ill posed problems arise in many areas of science and engineering their solutions if they exist are very sensitive to perturbations in the data regularization aims to reduce this sensitivity many regularization methods replace the original problem by a minimization problem with a fidelity term and a regularization term recently the use of a p norm to measure the fidelity term and a q norm to measure the regularization term has received considerable attention the relative importance of these terms is determined by a regularization parameter when the perturbation in the available data is made up of impulse noise and a sparse solution is desired it often is beneficial to let 0 p q 1 then the p and q norms are not norms the choice of a suitable regularization parameter is crucial for the quality of the computed solution it therefore is important to develop methods for determining this parameter automatically without user interaction however the latter has so far not received much attention when the data is contaminated by impulse noise this paper discusses two approaches based on cross validation for determining the regularization parameter in this situation computed examples that illustrate the performance of these approaches when applied to the restoration of impulse noise contaminated images are presented

discriminatively regularized least squares classification over the past decades regularization theory is widely applied in various areas of machine learning to derive a large family of novel algorithms traditionally regularization focuses on smoothing only and does not fully utilize the underlying discriminative knowledge which is vital for classification in this paper we propose a novel regularization algorithm in the least squares sense called discriminatively regularized least squares classification drlsc method which is specifically designed for classification inspired by several new geometrically motivated methods drlsc directly embeds the discriminative information as well as the local geometry of the samples into the regularization term so that it can explore as much underlying knowledge inside the samples as possible and aim to maximize the margins between the samples of different classes in each local area furthermore by embedding equality type constraints in the formulation the solutions of drlsc can follow from solving a set of linear equations and the framework naturally contains multi class problems experiments on both toy and real world problems demonstrate that drlsc is often superior in classification performance to the classical regularization algorithms including regularization networks support vector machines and some of the recent studied manifold regularization techniques