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

convergence of em image reconstruction algorithms with gibbs smoothing p j green has defined an osl one step late algorithm that retains the e step of the em algorithm for image reconstruction in emission tomography but provides an approximate solution to the m step further modifications of the osl algorithm guarantee convergence to the unique maximum of the log posterior function convergence is proved under a specific set of sufficient conditions several of these conditions concern the potential function of the gibb s prior and a number of candidate potential functions are identified generalization of the osl algorithm to transmission tomography is also considered

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

phase unwrapping via graph cuts phase unwrapping is the inference of absolute phase from modulo 2pi phase this paper introduces a new energy minimization framework for phase unwrapping the considered objective functions are first order markov random fields we provide an exact energy minimization algorithm whenever the corresponding clique potentials are convex namely for the phase unwrapping classical lp norm with pges1 its complexity is kt n 3n where k is the length of the absolute phase domain measured in 2pi units and t n m is the complexity of a max flow computation in a graph with n nodes and m edges for nonconvex clique potentials often used owing to their discontinuity preserving ability we face an np hard problem for which we devise an approximate solution both algorithms solve integer optimization problems by computing a sequence of binary optimizations each one solved by graph cut techniques accordingly we name the two algorithms puma for phase unwrapping max flow min cut a set of experimental results illustrates the effectiveness of the proposed approach and its competitiveness in comparison with state of the art phase unwrapping algorithms

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

tomographic reconstruction of gated data acquisition using dft basis functions in image reconstruction gated acquisition is often used in order to deal with blur caused by organ motion in the resulting images however this is achieved almost inevitably at the expense of reduced signal to noise ratio in the acquired data in this work we propose a reconstruction procedure for gated images based upon use of discrete fourier transform dft basis functions wherein the temporal activity at each spatial location is regulated by a fourier representation the gated images are then reconstructed through determination of the coefficients of the fourier representation we demonstrate this approach in the context of single photon emission computed tomography spect for cardiac imaging which is often hampered by the increased noise due to gating and other degrading factors we explore two different reconstruction algorithms one is a penalized least square approach and the other is a maximum a posteriori approach in our experiments we conducted a quantitative evaluation of the proposed approach using monte carlo simulated spect imaging the results demonstrate that use of dft basis functions in gated imaging can improve the accuracy of the reconstruction as a preliminary demonstration we also tested this approach on a set of clinical acquisition

pet regularization by envelope guided conjugate gradients the authors propose a new way to iteratively solve large scale ill posed problems and in particular the image reconstruction problem in positron emission tomography by exploiting the relation between tikhonov regularization and multiobjective optimization to obtain iteratively approximations to the tikhonov l curve and its corner monitoring the change of the approximate l curves allows the authors to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration so that the desired solution can be reconstructed with a small number of iterations

galerkin method for constrained variational equations and a collage based approach to related inverse problems for a general variational equation with constraints we present both the stability of the corresponding galerkin method and a collage theorem for a related inverse problem in addition we demonstrate the applicability of the results by considering forward and inverse boundary value problems of some linear impulsive differential equations

simulated annealing acceleration techniques and image restoration typically the linear image restoration problem is an ill conditioned underdetermined inverse problem here stabilization is achieved via the introduction of a first order smoothness constraint which allows the preservation of edges and leads to the minimization of a nonconvex functional in order to carry through this optimization task we use stochastic relaxation with annealing we prefer the metropolis dynamics to the popular but computationally much more expensive gibbs sampler still metropolis type annealing algorithms are also widely reported to exhibit a low convergence rate their finite time behavior is outlined and we investigate some inexpensive acceleration techniques that do not alter their theoretical convergence properties namely restriction of the state space to a locally bounded image space and increasing concave transform of the cost functional successful experiments about space variant restoration of simulated synthetic aperture imaging data illustrate the performance of the resulting class of algorithms and show significant benefits in terms of convergence speed

ml parameter estimation for markov random fields with applications to bayesian tomography markov random fields mrfs have been widely used to model images in bayesian frameworks for image reconstruction and restoration typically these mrf models have parameters that allow the prior model to be adjusted for best performance however optimal estimation of these parameters sometimes referred to as hyperparameters is difficult in practice for two reasons i direct parameter estimation for mrfs is known to be mathematically and numerically challenging ii parameters can not be directly estimated because the true image cross section is unavailable we propose a computationally efficient scheme to address both these difficulties for a general class of mrf models and we derive specific methods of parameter estimation for the mrf model known as generalized gaussian mrf ggmrf we derive methods of direct estimation of scale and shape parameters for a general continuously valued mrf for the ggmrf case we show that the ml estimate of the scale parameter spl sigma has a simple closed form solution and we present an efficient scheme for computing the ml estimate of the shape parameter p by an off line numerical computation of the dependence of the partition function on p we present a fast algorithm for computing ml parameter estimates when the true image is unavailable to do this we use the expectation maximization em algorithm we develop a fast simulation method to replace the e step and a method to improve the parameter estimates when the simulations are terminated prior to convergence experimental results indicate that our fast algorithms substantially reduce the computation and result in good scale estimates for real tomographic data sets

iterative maximum likelihood displacement field estimation in quantum limited image sequences we develop an algorithm for obtaining the maximum likelihood ml estimate of the displacement vector field dvp from two consecutive image frames of an image sequence acquired under quantum limited conditions the estimation of the dvf has applications in temporal filtering object tracking stereo matching and frame registration in low light level image sequences as well as low dose clinical x ray image sequences in the latter case a controlled x ray dosage reduction may be utilized to lower the radiation exposure to the patient and the medical staff the quantum limited effect is modeled as an undesirable poisson distributed signal dependent noise artifact a fisher bayesian formulation is used to estimate the dvf and a block component search algorithm is employed in obtaining the solution several experiments involving a phantom sequence and a teleconferencing image sequence with realistic motion demonstrate the effectiveness of this estimator in obtaining the dvf under severe quantum noise conditions 20 25 events pixel

stopping rule for the mle algorithm based on statistical hypothesis testing it is known that when the maximum likelihood estimator mle algorithm passes a certain point it produces images that begin to deteriorate we propose a quantitative criterion with a simple probabilistic interpretation that allows the user to stop the algorithm just before this effect begins the mle algorithm searches for the image that has the maximum probability to generate the projection data the underlying assumption of the algorithm is a poisson distribution of the data therefore the best image according to the mle algorithm is the one that results in projection means which are as close to the data as possible it is shown that this goal conflicts with the assumption that the data are poisson distributed we test a statistical hypothesis whereby the projection data could have been generated by the image produced after each iteration the acceptance or rejection of the hypothesis is based on a parameter that decreases as the images improve and increases as they deteriorate we show that the best mle images which pass the test result in somewhat lower noise in regions of high activity than the filtered back projection results and much improved images in low activity regions the applicability of the proposed stopping rule to other iterative schemes is discussed

a context free and a 1 counter geodesic language for a baumslag solitar group we give a language of unique geodesic normal forms for the baumslag solitar group bs 1 2 that is context free and 1 counter we discuss the classes of context free 1 counter and counter languages and explain how they are inter related

a statistical multiscale framework for poisson inverse problems this paper describes a statistical multiscale modeling and analysis framework for linear inverse problems involving poisson data the framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity a corresponding multiscale factorization of the likelihood induced by this analysis and a choice of prior probability distribution made to match this factorization by modeling the splits in the underlying partition the class of priors used here has the interesting feature that the noninformative member yields the traditional maximum likelihood solution other choices are made to reflect prior belief as to the smoothness of the unknown intensity adopting the expectation maximization em algorithm for use in computing the maximum a posteriori map estimate corresponding to our model we find that our model permits remarkably simple closed form expressions for the em update equations the behavior of our em algorithm is examined and it is shown that convergence to the global map estimate can be guaranteed applications in emission computed tomography and astronomical energy spectral analysis demonstrate the potential of the new approach

maximum likelihood least squares and penalized least squares for pet the em algorithm is the basic approach used to maximize the log likelihood objective function for the reconstruction problem in positron emission tomography pet the em algorithm is a scaled steepest ascent algorithm that elegantly handles the nonnegativity constraints of the problem it is shown that the same scaled steepest descent algorithm can be applied to the least squares merit function and that it can be accelerated using the conjugate gradient approach the experiments suggest that one can cut the computation by about a factor of 3 by using this technique the results are applied to various penalized least squares functions which might be used to produce a smoother image

adaptive langevin sampler for separation of t distribution modelled astrophysical maps we propose to model the image differentials of astrophysical source maps by student s t distribution and to use them in the bayesian source separation method as priors we introduce an efficient markov chain monte carlo mcmc sampling scheme to unmix the astrophysical sources and describe the derivation details in this scheme we use the langevin stochastic equation for transitions which enables parallel drawing of random samples from the posterior and reduces the computation time significantly by two orders of magnitude in addition student s t distribution parameters are updated throughout the iterations the results on astrophysical source separation are assessed with two performance criteria defined in the pixel and the frequency domains

a maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography the expectation maximization method for maximum likelihood image reconstruction in emission tomography based on the poisson distribution of the statistically independent components of the image and measurement vectors is extended to a maximum aposteriori image reconstruction using a multivariate gaussian a priori probability distribution of the image vector the approach is equivalent to a penalized maximum likelihood estimation with a special choice of the penalty function the expectation maximization method is applied to find the a posteriori probability maximizer a simple iterative formula is derived for a penalty function that is a weighted sum of the squared deviations of image vector components from their a priori mean values the method is demonstrated to be superior to pure likelihood maximization in that the penalty function prevents the occurrence of irregular high amplitude patterns in the image with a large number of iterations the so called checkerboard effect or noise artifact

functional inequalities motivated by the lax milgram theorem abstract by adopting some ideas of the theory of functional inequalities we obtain a modification of the lax milgram theorem we drop linearity assumptions by assuming that respective inequalities hold true and we impose only a mild regularity assumption moreover we do not assume the coercivity however as a consequence our assertions are adequately weaker one of our tools is a selection theorem for multifunctions

fully bayesian estimation of gibbs hyperparameters for emission computed tomography data in recent years many investigators have proposed gibbs prior models to regularize images reconstructed from emission computed tomography data unfortunately hyperparameters used to specify gibbs priors can greatly influence the degree of regularity imposed by such priors and as a result numerous procedures have been proposed to estimate hyperparameter values from observed image data many of these procedures attempt to maximize the joint posterior distribution on the image scene to implement these methods approximations to the joint posterior densities are required because the dependence of the gibbs partition function on the hyperparameter values is unknown here the authors use recent results in markov chain monte carlo mcmc sampling to estimate the relative values of gibbs partition functions and using these values sample from joint posterior distributions on image scenes this allows for a fully bayesian procedure which does not fix the hyperparameters at some estimated or specified value but enables uncertainty about these values to be propagated through to the estimated intensities the authors utilize realizations from the posterior distribution for determining credible regions for the intensity of the emission source the authors consider two different markov random field mrf models the power model and a line site model as applications they estimate the posterior distribution of source intensities from computer simulated data as well as data collected from a physical single photon emission computed tomography spect phantom

variational approach for edge preserving regularization using coupled pdes this paper deals with edge preserving regularization for inverse problems in image processing we first present a synthesis of the main results we have obtained in edge preserving regularization by using a variational approach we recall the model involving regularizing functions spl phi and we analyze the geometry driven diffusion process of this model in the three dimensional 3 d case then a half quadratic theorem is used to give a very simple reconstruction algorithm after a critical analysis of this model we propose another functional to minimize for edge preserving reconstruction purposes it results in solving two coupled partial differential equations pdes one processes the intensity the other the edges we study the relationship with similar pde systems in particular with the functional proposed by ambrosio tortorelli 1990 1992 in order to approach the mumford shah 1989 functional developed in the segmentation application experimental results on synthetic and real images are presented

fast maximum entropy approximation in spect using the rbi map algorithm presents a method for approximating constrained maximum entropy me reconstructions of spect data with modifications to a block iterative maximum a posteriori map algorithm maximum likelihood ml based reconstruction algorithms require some form of noise smoothing constrained me provides a more formal method of noise smoothing without requiring the user to select parameters in the context of spect constrained me seeks the minimum information image estimate among those whose projections are a given distance from the noisy measured data with that distance determined by the magnitude of the poisson noise images that meet the distance criterion are referred to as feasible images the authors find that modeling of all principal degrading factors attenuation detector response and scatter in the reconstruction is critical because feasibility is not meaningful unless the projection model is as accurate as possible because the constrained me solution is the same as a map solution for a particular value of the map weighting parameter spl beta the constrained me solution can be found with a map algorithm if the correct value of spl beta is found the authors show that the rbi map algorithm if used with a dynamic scheme for estimating spl beta can approximate constrained me solutions in 20 or fewer iterations the authors compare results for various methods of achieving feasible images on a simulation of tl 201 cardiac spect data results show that the rbi map me approximation provides images and quantitative estimates close to those from a slower algorithm that gives the true me solution also the authors find that the me results have higher spatial resolution and greater high frequency noise content than a feasibility based stopping rule feasibility based low pass filtering and a quadratic gibbs prior with spl beta selected according to the feasibility criterion the authors conclude that fast me approximation is possible using either rbi map with the dynamic procedure or a feasibility based stopping rule and that such reconstructions may be particularly useful in applications where resolution is critical

penalized maximum likelihood image reconstruction using space alternating generalized em algorithms most expectation maximization em type algorithms for penalized maximum likelihood image reconstruction converge slowly particularly when one incorporates additive background effects such as scatter random coincidences dark current or cosmic radiation in addition regularizing smoothness penalties or priors introduce parameter coupling rendering intractable the m steps of most em type algorithms this paper presents space alternating generalized em sage algorithms for image reconstruction which update the parameters sequentially using a sequence of small hidden data spaces rather than simultaneously using one large complete data space the sequential update decouples the m step so the maximization can typically be performed analytically we introduce new hidden data spaces that are less informative than the conventional complete data space for poisson data and that yield significant improvements in convergence rate this acceleration is due to statistical considerations not numerical overrelaxation methods so monotonic increases in the objective function are guaranteed we provide a general global convergence proof for sage methods with nonnegativity constraints

roof edge preserving image smoothing based on mrfs a novel markov random field mrf model is proposed for roof edge as well as step edge preserving image smoothing image surfaces containing roof edges are represented by piecewise continuous polynomial functions governed by a few parameters piecewise smoothness constraint is imposed on these parameters rather than on the surface heights as is in traditional models for step edges in this way roof edges are preserved without the necessity to deal with unstable higher order derivatives

the application of the gibbs bogoliubov feynman inequality in mean field calculations for markov random fields the gibbs bogoliubov feynman gbf inequality of statistical mechanics is adopted with an information theoretic interpretation as a general optimization framework for deriving and examining various mean field approximations for markov random fields mrf s the efficacy of this approach is demonstrated through the compound gauss markov cgm model comparisons between different mean field approximations and experimental results in image restoration

penalized weighted least squares image reconstruction for positron emission tomography presents an image reconstruction method for positron emission tomography pet based on a penalized weighted least squares pwls objective for pet measurements that are precorrected for accidental coincidences the author argues statistically that a least squares objective function is as appropriate if not more so than the popular poisson likelihood objective the author proposes a simple data based method for determining the weights that accounts for attenuation and detector efficiency a nonnegative successive over relaxation sor algorithm converges rapidly to the global minimum of the pwls objective quantitative simulation results demonstrate that the bias variance tradeoff of the pwls sor method is comparable to the maximum likelihood expectation maximization ml em method but with fewer iterations and is improved relative to the conventional filtered backprojection fbp method qualitative results suggest that the streak artifacts common to the fbp method are nearly eliminated by the pwls sor method and indicate that the proposed method for weighting the measurements is a significant factor in the improvement over fbp

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

regularized image reconstruction algorithms for positron emission tomography we develop algorithms for obtaining regularized estimates of emission means in positron emission tomography the first algorithm iteratively minimizes a penalized maximum likelihood pml objective function it is based on standard de coupled surrogate functions for the ml objective function and de coupled surrogate functions for a certain class of penalty functions as desired the pml algorithm guarantees nonnegative estimates and monotonically decreases the pml objective function with increasing iterations the second algorithm is based on an iteration dependent de coupled penalty function that introduces smoothing while preserving edges for the purpose of making comparisons the mlem algorithm and a penalized weighted least squares algorithm were implemented in experiments using synthetic data and real phantom data it was found that for a fixed level of background noise the contrast in the images produced by the proposed algorithms was the most accurate

vectorial additive half quadratic minimization for isotropic regularization we propose the vectorial additive half quadratic vahq algorithm to minimize the isotropic regularized cost function with the general edge preserving potential functions pfs in image restoration by introducing an auxiliary vectorial variable the cost function is changed into an augmented one which can be alternately minimized one minimization is solved with an explicit expression the other is implemented by fast fourier transform fft vahq is shown to globally converge to a stationary point for nonconvex pfs providing all stationary points are isolated and to a unique minimum for convex pfs without any isolation assumption on an extended domain of parameters what is more the linear convergence rate of vahq is proved to be less than 1 the explicit expression of the optimal parameters and the optimal bound of the convergence rate are present for convex pfs image restoration examples show the restoration performance of nonconvex pfs the lower computation cost of our algorithm compared to majorize minimize memory gradient mmmg algorithm with the help of fft and some interesting phenomena confirming our conclusions on the convergence domain and the convergence rate

analysis of resolution and noise properties of nonquadratically regularized image reconstruction methods for pet we present accurate and efficient methods for estimating the spatial resolution and noise properties of nonquadratically regularized image reconstruction for positron emission tomography pet it is well known that quadratic regularization tends to over smooth sharp edges many types of edge preserving nonquadratic penalties have been proposed to overcome this problem however there has been little research on the quantitative analysis of nonquadratic regularization due to its nonlinearity in contrast quadratically regularized estimators are approximately linear and are well understood in terms of resolution and variance properties we derive new approximate expressions for the linearized local perturbation response llpr and variance using the taylor expansion with the remainder term although the expressions are implicit we can use them to accurately predict resolution and variance for nonquadratic regularization where the conventional expressions based on the first order taylor truncation fail they also motivate us to extend the use of a certainty based modified penalty to nonquadratic regularization cases in order to achieve spatially uniform perturbation responses analogous to uniform spatial resolution in quadratic regularization finally we develop computationally efficient methods for predicting resolution and variance of nonquadratically regularized reconstruction and present simulations that illustrate the validity of these methods

minimax emission computed tomography using high resolution anatomical side information and b spline models in this paper a minimax methodology is presented for combining information from two imaging modalities having different intrinsic spatial resolutions the focus application is emission computed tomography ect a low resolution modality for reconstruction of radionuclide tracer density when supplemented by high resolution anatomical boundary information extracted from a magnetic resonance image mri of the same imaging volume the mri boundary within the two dimensional 2 d slice of interest is parameterized by a closed planar curve the cramer rao cr lower bound is used to analyze estimation errors for different boundary shapes under a spatially inhomogeneous gibbs field model for the tracer density a representation for the minimax mri enhanced tracer density estimator is obtained it is shown that the estimator is asymptotically equivalent to a penalized maximum likelihood pml estimator with resolution selective gibbs penalty quantitative comparisons are presented using the iterative space alternating generalized expectation maximization sage fm algorithm to implement the pml estimator with and without minimax weight averaging

denoising for 3 d photon limited imaging data using nonseparable filterbanks in this paper we present a novel frame based denoising algorithm for photon limited 3d images we first construct a new 3d nonseparable filterbank by adding elements to an existing frame in a structurally stable way in contrast with the traditional 3d separable wavelet system the new filterbank is capable of using edge information in multiple directions we then propose a data adaptive hysteresis thresholding algorithm based on this new 3d nonseparable filterbank in addition we develop a new validation strategy for denoising of photon limited images containing sparse structures such as neurons the structure of interest is less than 5 of total volume the validation method based on tubular neighborhoods around the structure is used to determine the optimal threshold of the proposed denoising algorithm we compare our method with other state of the art methods and report very encouraging results on applications utilizing both synthetic and real data

space alternating generalized expectation maximization algorithm the expectation maximization em method can facilitate maximizing likelihood functions that arise in statistical estimation problems in the classical em paradigm one iteratively maximizes the conditional log likelihood of a single unobservable complete data space rather than maximizing the intractable likelihood function for the measured or incomplete data em algorithms update all parameters simultaneously which has two drawbacks 1 slow convergence and 2 difficult maximization steps due to coupling when smoothness penalties are used the paper describes the space alternating generalized em sage method which updates the parameters sequentially by alternating between several small hidden data spaces defined by the algorithm designer the authors prove that the sequence of estimates monotonically increases the penalized likelihood objective derive asymptotic convergence rates and provide sufficient conditions for monotone convergence in norm two signal processing applications illustrate the method estimation of superimposed signals in gaussian noise and image reconstruction from poisson measurements in both applications the sage algorithms easily accommodate smoothness penalties and converge faster than the em algorithms

simulation evaluation of gibbs prior distributions for use in maximum a posteriori spect reconstructions the effects of several of gibbs prior distributions in terms of noise characteristics edge sharpness and overall quantitative accuracy of the final estimates obtained from an iterative maximum a posteriori map procedure applied to data from a realistic chest phantom are demonstrated the effects of the adjustable parameters built into the prior distribution on these properties are examined it is found that these parameter values influence the noise and edge characteristics of the final estimate and can generate reconstructions closer to the actual solution than maximum likelihood ml in addition it is found that the choice of the shape of the prior distribution affects the noise characteristics and edge sharpness in the final estimate

solution of an inverse problem for fractals and other sets abstract the problem of providing succinct approximate descriptions of given bounded subsets of rn can be solved by application of the contraction mapping principle

a primal dual method for large scale image reconstruction in emission tomography in emission tomography images can be reconstructed from a set of measured projections using a maximum likelihood ml criterion in this paper we present a primal dual algorithm for large scale three dimensional image reconstruction the primal dual method is specialized to the ml reconstruction problem the reconstruction problem is extremely large in several of our data sets the hessian of the objective function is the product of a 1 4 million by 63 million matrix and its scaled transpose as such we consider only approaches that are suitable for large scale parallel computation we apply a stabilization technique to the system of equations for computing the primal direction and demonstrate the need for stabilization when approximately solving the system using an early terminated conjugate gradient iteration we demonstrate that the primal dual method for this problem converges faster than the logarithmic barrier method and considerably faster than the expectation maximization algorithm the use of extrapolation in conjunction with the primal dual method further reduces the overall computation required to achieve convergence

iterative image reconstruction algorithms based on cross entropy minimization the related problems of minimizing the functionals f x alpha kl y px 1 alpha kl p x and g x alpha kl px y 1 alpha kl x p respectively over the set of vectors x or 0 are considered kl a b is the cross entropy or kullback leibler distance between two nonnegative vectors a and b iterative algorithms for minimizing both functionals using the method of alternating projections are derived a simultaneous version of the multiplicative algebraic reconstruction technique mart algorithm called smart is introduced and its convergence is proved

restoration of severely blurred high range images using stochastic and deterministic relaxation algorithms in compound gauss markov random fields abstract over the last few years a growing number of researchers from varied disciplines have been utilizing markov random fields mrf models for developing optimal robust algorithms for various problems such as texture analysis image synthesis classification and segmentation surface reconstruction integration of several low level vision modules sensor fusion and image restoration however no much work has been reported on the use of simulated annealing sa and iterative conditional mode icm algorithms for maximum a posteriori estimation in image restoration problems when severe blurring is present in this paper we examine the use of compound gauss markov random fields cgmrf to restore severely blurred high range images for this deblurring problem the convergence of the sa and icm algorithms has not been established we propose two new iterative restoration algorithms which can be considered as extensions of the classical sa and icm approaches and whose convergence is established finally they are tested on real and synthetic images and the results compared with the restorations obtained by other iterative schemes

tomographic image sequence reconstruction by edge preserving interslice map methods we consider a new problem in tomographic imaging in a sequence of transverse images adjacent slices often look similar to each other we investigate the possibility of using this correlation to help reconstruct the sequence with 2 d in plane data the problem is formulated as a maximum a posteriori map estimation problem using an edge preserving markov random field prior model both gaussian and poisson data are considered some improvements over the single slice map reconstructions are observed

convex total variation denoising of poisson fluorescence confocal images with anisotropic filtering fluorescence confocal microscopy fcm is now one of the most important tools in biomedicine research in fact it makes it possible to accurately study the dynamic processes occurring inside the cell and its nucleus by following the motion of fluorescent molecules over time due to the small amount of acquired radiation and the huge optical and electronics amplification the fcm images are usually corrupted by a severe type of poisson noise this noise may be even more damaging when very low intensity incident radiation is used to avoid phototoxicity in this paper a bayesian algorithm is proposed to remove the poisson intensity dependent noise corrupting the fcm image sequences the observations are organized in a 3 d tensor where each plane is one of the images acquired along the time of a cell nucleus using the fluorescence loss in photobleaching flip technique the method removes simultaneously the noise by considering different spatial and temporal correlations this is accomplished by using an anisotropic 3 d filter that may be separately tuned in space and in time dimensions tests using synthetic and real data are described and presented to illustrate the application of the algorithm a comparison with several state of the art algorithms is also presented

tomographic reconstruction of dynamic cardiac image sequences in this paper we propose an approach for the reconstruction of dynamic images from a gated cardiac data acquisition the goal is to obtain an image sequence that can show simultaneously both cardiac motion and time varying image activities to account for the cardiac motion the cardiac cycle is divided into a number of gate intervals and a time varying image function is reconstructed for each gate in addition to cope with the under determined nature of the problem the time evolution at each pixel is modeled by a b spline function the dynamic images for the different gates are then jointly determined using maximum a posteriori estimation in which a motion compensated smoothing prior is introduced to exploit the similarity among the different gates the proposed algorithm is evaluated using a dynamic version of the 4 d gated mathematical cardiac torso phantom simulating a gated single photon emission computed tomography perfusion acquisition with technitium 99m labeled teboroxime we thoroughly evaluated the performance of the proposed algorithm using several quantitative measures including signal to noise ratio analysis bias variance plot and time activity curves our results demonstrate that the proposed joint reconstruction approach can improve significantly the accuracy of the reconstruction

a generalized em algorithm for 3 d bayesian reconstruction from poisson data using gibbs priors a generalized expectation maximization gem algorithm is developed for bayesian reconstruction based on locally correlated markov random field priors in the form of gibbs functions and on the poisson data model for the m step of the algorithm a form of coordinate gradient ascent is derived the algorithm reduces to the em maximum likelihood algorithm as the markov random field prior tends towards a uniform distribution three different gibbs function priors are examined reconstructions of 3 d images obtained from the poisson model of single photon emission computed tomography are presented

application of the em algorithm for the multitarget multisensor tracking problem an important problem in surveillance and reconnaissance systems is the tracking of multiple moving targets in cluttered noise environments using outputs from a number of sensors possessing wide variations in individual characteristics and accuracies a number of approaches have been proposed for this multitarget multisensor tracking problem ranging from reasonably simple though ad hoc schemes to fairly complex but theoretically optimum approaches in this paper we describe an iterative procedure for time recursive multitarget multisensor tracking based on use of the expectation maximization em algorithm more specifically we pose the multitarget multisensor tracking problem as an incomplete data problem with the observable sensor outputs representing the incomplete data whereas the target associated sensor outputs constitute the complete data target updates at each time use an em based approach that calculates the maximum a posteriori map estimate of the target states under the assumption of appropriate motion models based on the outputs of disparate sensors the approach uses a markov random field mrf model of the associations between observations and targets and allows for estimation of joint association probabilities without explicit enumeration the advantage of this em based approach is that it provides a computationally efficient means for approaching the performance offered by theoretically optimum approaches that use explicit enumeration of the joint association probabilities we provide selected results illustrating the performance complexity characteristics of this em based approach compared with competing schemes

a generalized gaussian image model for edge preserving map estimation the authors present a markov random field model which allows realistic edge modeling while providing stable maximum a posterior map solutions the model referred to as a generalized gaussian markov random field ggmrf is named for its similarity to the generalized gaussian distribution used in robust detection and estimation the model satisfies several desirable analytical and computational properties for map estimation including continuous dependence of the estimate on the data invariance of the character of solutions to scaling of data and a solution which lies at the unique global minimum of the a posteriori log likelihood function the ggmrf is demonstrated to be useful for image reconstruction in low dosage transmission tomography

expectation maximization algorithms null spaces and map image restoration a computationally efficient easily implementable algorithm for map restoration of images degraded by blur and additive correlated gaussian noise using gibbs prior density functions is derived this algorithm is valid for a variety of complete data spaces the constraints upon the complete data space arising from the gaussian image formation model are analyzed and a motivation is provided for the choice of the complete data based upon the ease of computation of the resulting em algorithms the overlooked role of the null space of the blur operator in image restoration is introduced an examination of this role reveals an important drawback to the use of the simulated annealing algorithm in maximizing a specific class of functionals an alternative iterative method for computing the nullspace component of a vector is given the ability of a simple gibbs prior density function to enable partial recovery of the component of an image within the nullspace of the blur operator is demonstrated

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

line detection in images through regularized hough transform the problem of determining the location and orientation of straight lines in images is of great importance in the fields of computer vision and image processing traditionally the hough transform a special case of the radon transform has been widely used to solve this problem for binary images in this paper we pose the problem of detecting straight lines in gray scale images as an inverse problem our formulation is based on use of the inverse radon operator which relates the parameters determining the location and orientation of the lines in the image to the noisy input image the advantage of this formulation is that we can then approach the problem of line detection within a regularization framework and enhance the performance of the hough based line detector through the incorporation of prior information in the form of regularization we discuss the type of regularizers that are useful for this problem and derive efficient computational schemes to solve the resulting optimization problems enabling their use in large applications finally we show how our new approach can be alternatively viewed as one of finding an optimal representation of the noisy image in terms of elements chosen from a dictionary of lines this interpretation relates the problem of hough based line finding to the body of work on adaptive signal representation

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

a generalized collage method based upon the lax milgram functional for solving boundary value inverse problems abstract some inverse problems can be cast as approximating a given target function perhaps the interpolation of observational data points by the fixed points of an operator equation the collage method refers to the solution strategy of bounding above the true approximation error by a more readily minimizable quantity in this paper we present a collage method for solving inverse problems for boundary value problems the approach is based upon the lax milgram representation theorem for applications we focus on one and two dimensional steady state reaction diffusion equations

on image reconstruction algorithms for binary electromagnetic geotomography in this paper we discuss the selected image reconstruction methods of binary tomography in the context of their application to geophysical imaging we restrict our considerations to a discrete version of high frequency electromagnetic geotomography which we label as binary electromagnetic geotomography beg basically such an imaging technique may be applied to detect subsurface anomalies air filled voids whose attenuation coefficient is very low nearly zero value and considerably different from that for the background the assumption for a binary representation of the image to be reconstructed substantially relaxes image reconstruction problems related to ill posedness that comes from an intrinsic limitation of an angular range of projections we test two algorithms for binary tomography where the penalty term is based on the markov random field mrf model the mean field reference distribution and mean field annealing are applied to estimate the global maximizer of the gibbs boltzmann distribution associated with the objective function we also apply the projected gradient algorithm that uses a binary steering very efficient implementations of the algorithms are also given the numerical results are presented for noise free noisy and real data

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

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

growth functions for some nonautomatic baumslag solitar groups the growth function of a group is a generating function whose coefficients a are the number of elements in the group whose minimum length as a word in the generators is n in this paper we use finite state automata to investigate the growth function for the baumslag solitar group of the form a b a a lba a2 based on an analysis of its combinatorial and geometric structure in particular we obtain a set of length minimal normal forms for the group which although it does not form the language of a finite state automata is nevertheless built up in a sufficiently coherent way that the growth function can be shown to be rational the rationality of the growth function of this group is particularly interesting as it is known not to be synchronously automatic the results in this paper generalize to the groups a b i a ba am for all positive integers m

a geometric projection space reconstruction algorithm abstract we present a method to reconstruct images from finite sets of noisy projections that may be available only over limited or sparse angles the algorithm calculates the maximum a posteriori map estimate of the full sinogram which is an image of the 2 d radon transform of the object from the available data it is implemented using a primal dual constrained optimization procedure that solves a partial differential equation in the primal phase with an efficient local relaxation algorithm and uses a simple lagrange multiplier update in the dual phase the sinogram prior probability is given by a markov random field mrf that includes information about the mass center of mass and convex hull of the object and about the smoothness fundamental constraints and periodicity of the 2 d radon transform the object is reconstructed using convolution backprojection applied to the estimated sinogram we show several reconstructed objects which are obtained from simulated limited and sparse angle data using the described algorithm and compare these results with images obtained using convolution backprojection directly

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

interior point methodology for 3 d pet reconstruction interior point methods have been successfully applied to a wide variety of linear and nonlinear programming applications this paper presents a class of algorithms based on path following interior point methodology for performing regularized maximum likelihood ml reconstructions on three dimensional 3 d emission tomography data the algorithms solve a sequence of subproblems that converge to the regularized maximum likelihood solution from the interior of the feasible region the nonnegative orthant the authors propose two methods a primal method which updates only the primal image variables and a primal dual method which simultaneously updates the primal variables and the lagrange multipliers a parallel implementation permits the interior point methods to scale to very large reconstruction problems termination is based on well defined convergence measures namely the karush kuhn tucker first order necessary conditions for optimality the authors demonstrate the rapid convergence of the path following interior point methods using both data from a small animal scanner and monte carlo simulated data the proposed methods can readily be applied to solve the regularized weighted least squares reconstruction problem

statistic based map image reconstruction from poisson data using gibbs priors a statistical method for selecting the gibbs parameter in map image restoration from poisson data using gibbs priors is presented the gibbs parameter determines the degree to which the prior influences the restoration the presented method yields a map restored image minimally influenced by the prior for which a statistic falls within an appropriate confidence interval the method assumes that a close approximation to the blurring function is known a simple iterative feedback algorithm is presented to statistically select the parameter as the map image restoration is being performed this algorithm is heuristically based on a model reference control formulation but it requires only a minimal number of iterations for the parameter to settle to its statistically specified value the performance of the statistical method for selecting the prior parameter and that of the iterative feedback algorithm are demonstrated using both 2 d and 3 d images

global two stage filtering of clinical pet parametric maps application to 11 c r pk11195 abstract introduction in positron emission tomography pet quantification of physiological parameters at the voxel level may result in unreliable estimates due to the high noise of voxel time activity curves global two stage gts an estimation technique belonging to the group of population approaches can be used to tackle this problem gts was previously tested on simulated pet data and yielded substantial improvements when compared to standard estimation approaches such as weighted nonlinear least squares wnlls and basis function method bfm in this work gts performance is assessed in a clinical context using the neuroinflammation marker 11 c r pk11195 applied to a cohort of huntington s disease hd patients with and without symptoms materials and methods parametric maps of binding potential bp nd of 12 normal controls nc 9 symptomatic and 9 presymptomatic hd patients were generated by applying a modified reference tissue model that accounts for tracer vascular activity in both reference and target tissues srtmv gts was then applied to srtmv maps and its performance compared with that of srtmv three smoothed versions of srtmv obtained by filtering the original srtmv maps with gaussian filters of 3 mm 5 mm and 7 mm full width half maximum fwhm were also included in the comparison since striatal degeneration is the hallmark of hd sensitivity was assessed for all methods by computing the mean of z scores in caudate putamen and globus pallidus in the voxel by voxel statistical comparison of bp nd between hd and nc results application of gts to parametric maps brought a substantial qualitative improvement to srtmv maps to the extent that anatomical structures often became visible in addition most parameter estimates that were outside the physiological range with srtmv were corrected by gts gts yielded a 2 3 fold increase in sensitivity with respect to srtmv for the symptomatic cohort mean of striatal z scores of 0 76 for srtmv and 1 79 for gts and an even more substantial increase for the presymptomatic cohort mean of striatal z scores of 0 34 for srtmv and 0 96 for gts the sensitivity of gts was similar to the one obtained with a filter of 7 mm fwhm applied to the initial srtmv maps but gts images were not characterized by the notable loss of resolution typical of smoothed maps gts additionally does not require to change define settings according to the tracer and level of noise whereas the choice of the fwhm value of the gaussian filter normally employed in the smoothing procedure is typically arbitrary conclusions gts is a powerful and robust tool for improving the quality of parametric maps in pet the method is particularly appealing in that it can be applied to any tracer and estimation method provided that initial estimates of the parameter vector and of its covariance are available although the benefits of gts are far from being exhaustively assessed the significant improvements obtained both on real and simulated data suggest that it could become an important tool for dynamic pet in the future

parallelizable bayesian tomography algorithms with rapid guaranteed convergence bayesian tomographic reconstruction algorithms generally require the efficient optimization of a functional of many variables in this setting as well as in many other optimization tasks functional substitution fs has been widely applied to simplify each step of the iterative process the function to be minimized is replaced locally by an approximation having a more easily manipulated form e g quadratic but which maintains sufficient similarity to descend the true functional while computing only the substitute we provide two new applications of fs methods in iterative coordinate descent for bayesian tomography the first is a modification of our coordinate descent algorithm with one dimensional 1 d newton raphson approximations to an alternative quadratic which allows convergence to be proven easily in simulations we find essentially no difference in convergence speed between the two techniques we also present a new algorithm which exploits the fs method to allow parallel updates of arbitrary sets of pixels using computations similar to iterative coordinate descent the theoretical potential speed up of parallel implementations is nearly linear with the number of processors if communication costs are neglected

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

image registration under illumination variations using region based confidence weighted m estimators we present an image registration model for image sets with arbitrarily shaped local illumination variations between images any nongeometric variations tend to degrade the geometric registration precision and impact subsequent processing traditional image registration approaches do not typically account for changes and movement of light sources which result in interimage illumination differences with arbitrary shape in addition these approaches typically use a least square estimator that is sensitive to outliers where interimage illumination variations are often large enough to act as outliers in this paper we propose an image registration approach that compensates for arbitrarily shaped interimage illumination variations which are processed using robust m estimators tuned to that region each m estimator for each illumination region has a distinct cost function by which small and large interimage residuals are unevenly penalized since the segmentation of the interimage illumination variations may not be perfect a segmentation confidence weighting is also imposed to reduce the negative effect of mis segmentation around illumination region boundaries the proposed approach is cast in an iterative coarse to fine framework which allows a convergence rate similar to competing intensity based image registration approaches the overall proposed approach is presented in a general framework but experimental results use the bisquare m estimator with region segmentation confidence weighting a nearly tenfold improvement in subpixel registration precision is seen with the proposed technique when convergence is attained as compared with competing techniques using both simulated and real data sets with interimage illumination variations

noniterative map reconstruction using sparse matrix representations we present a method for noniterative maximum a posteriori map tomographic reconstruction which is based on the use of sparse matrix representations our approach is to precompute and store the inverse matrix required for map reconstruction this approach has generally not been used in the past because the inverse matrix is typically large and fully populated i e not sparse in order to overcome this problem we introduce two new ideas the first idea is a novel theory for the lossy source coding of matrix transformations which we refer to as matrix source coding this theory is based on a distortion metric that reflects the distortions produced in the final matrix vector product rather than the distortions in the coded matrix itself the resulting algorithms are shown to require orthonormal transformations of both the measurement data and the matrix rows and columns before quantization and coding the second idea is a method for efficiently storing and computing the required orthonormal transformations which we call a sparse matrix transform smt the smt is a generalization of the classical fft in that it uses butterflies to compute an orthonormal transform but unlike an fft the smt uses the butterflies in an irregular pattern and is numerically designed to best approximate the desired transforms we demonstrate the potential of the noniterative map reconstruction with examples from optical tomography the method requires offline computation to encode the inverse transform however once these offline computations are completed the noniterative map algorithm is shown to reduce both storage and computation by well over two orders of magnitude as compared to a linear iterative reconstruction methods

a stochastic continuation approach to piecewise constant reconstruction we address the problem of reconstructing a piecewise constant 3 d object from a few noisy 2 d line integral projections more generally the theory developed here readily applies to the recovery of an ideal n d signal n ges 1 from indirect measurements corrupted by noise stabilization of this ill conditioned inverse problem is achieved with the potts prior model which leads to a challenging optimization task to overcome this difficulty we introduce a new class of hybrid algorithms that combines simulated annealing with deterministic continuation we call this class of algorithms stochastic continuation sc we first prove that under mild assumptions sc inherits the finite time convergence properties of generalized simulated annealing then we show that sc can be successfully applied to our reconstruction problem in addition we look into the concave distortion acceleration method introduced for standard simulated annealing and we derive an explicit formula for choosing the free parameter of the cost function numerical experiments using both synthetic data and real radiographic testing data show that sc outperforms standard simulated annealing

efficient algorithm for nonconvex minimization and its application to pm regularization in image processing nonconvex regularization has the ability to smooth homogeneous regions and sharpen edges but leads to challenging computation we propose some iterative schemes to minimize the energy function with nonconvex edge preserving potential the schemes are derived from the duality based algorithm proposed by bermudez and moreno and the fixed point iteration the convergence is proved for the convex energy function with nonconvex potential and the linear convergence rate is given applying the proposed schemes to perona and malik s nonconvex regularization we present some efficient algorithms based on our schemes and show the approximate convergence behavior for nonconvex energy function experimental results are presented which show the efficiency of our algorithms including better denoised performance of nonconvex regularization faster convergence speed higher calculation precision lower calculation cost under the same number of iterations and less implementation time under the same peak signal noise ratio level

coupling image restoration and segmentation a generalized linear model bregman perspective we introduce a new class of data fitting energies that couple image segmentation with image restoration these functionals model the image intensity using the statistical framework of generalized linear models by duality we establish an information theoretic interpretation using bregman divergences we demonstrate how this formulation couples in a principled way image restoration tasks such as denoising deblurring deconvolution and inpainting with segmentation we present an alternating minimization algorithm to solve the resulting composite photometric geometric inverse problem we use fisher scoring to solve the photometric problem and to provide asymptotic uncertainty estimates we derive the shape gradient of our data fitting energy and investigate convex relaxation for the geometric problem we introduce a new alternating split bregman strategy to solve the resulting convex problem and present experiments and comparisons on both synthetic and real world images

image classification using spectral and spatial information based on mrf models a new criterion for classifying multispectral remote sensing images or textured images by using spectral and spatial information is proposed the images are modeled with a hierarchical markov random field mrf model that consists of the observed intensity process and the hidden class label process the class labels are estimated according to the maximum a posteriori map criterion but some reasonable approximations are used to reduce the computational load a stepwise classification algorithm is derived and is confirmed by simulation and experimental results

acceleration of maximum likelihood reconstruction using frequency amplification and attenuation compensation algorithms that calculate maximum likelihood ml and maximum a posteriori solutions using expectation maximization have been successfully applied to spect and pet these algorithms are appealing because of their solid theoretical basis and their guaranteed convergence a major drawback is the slow convergence which results in long processing times the authors present 2 new heuristic acceleration methods for maximum likelihood reconstruction of ect images the first method incorporates a frequency dependent amplification in the calculations to compensate for the low pass filtering of the backprojection operation in the second method an amplification factor is incorporated that suppresses the effect of attenuation on the updating factors both methods are compared to the 1 dimensional line search method proposed by lewitt all 3 methods accelerate the ml algorithm on the authors test images lewitt s method produced the strongest acceleration of the three individual methods however the combination of the frequency amplification with the line search method results in a new algorithm with still better performance under certain conditions an effective frequency amplification can be already achieved by skipping some of the calculations required for ml

bayesian image reconstruction in spect using higher order mechanical models as priors while the ml em algorithm for reconstruction for emission tomography is unstable due to the ill posed nature of the problem bayesian reconstruction methods overcome this instability by introducing prior information often in the form of a spatial smoothness regularizer more elaborate forms of smoothness constraints may be used to extend the role of the prior beyond that of a stabilizer in order to capture actual spatial information about the object previously proposed forms of such prior distributions were based on the assumption of a piecewise constant source distribution here the authors propose an extension to a piecewise linear model the weak plate which is more expressive than the piecewise constant model the weak plate prior not only preserves edges but also allows for piecewise ramplike regions in the reconstruction indeed for the authors application in spect such ramplike regions are observed in ground truth source distributions in the form of primate autoradiographs of rcbf radionuclides to incorporate the weak plate prior in a map approach the authors model the prior as a gibbs distribution and use a gem formulation for the optimization they compare quantitative performance of the ml em algorithm a gem algorithm with a prior favoring piecewise constant regions and a gem algorithm with their weak plate prior pointwise and regional bias and variance of ensemble image reconstructions are used as indications of image quality the authors results show that the weak plate and membrane priors exhibit improved bias and variance relative to ml em techniques

the rationality of sol manifolds abstract let be the fundamental group of a manifold modeled on 3 dimensional sol geometry we prove that has a finite index subgroup g which has a rational growth series with respect to a natural generating set we do this by enumerating g by a regular language however in contrast to most earlier proofs of this sort our regular language is not a language of words in the generating set but rather reflects a different geometric structure in g

a variational model for image classification and restoration we present a variational model devoted to image classification coupled with an edge preserving regularization process the discrete nature of classification i e to attribute a label to each pixel has led to the development of many probabilistic image classification models but rarely to variational ones in the last decade the variational approach has proven its efficiency in the field of edge preserving restoration we add a classification capability which contributes to provide images composed of homogeneous regions with regularized boundaries a region being defined as a set of pixels belonging to the same class the soundness of our model is based on the works developed on the phase transition theory in mechanics the proposed algorithm is fast easy to implement and efficient we compare our results on both synthetic and satellite images with the ones obtained by a stochastic model using a potts regularization

image relaxation restoration and feature extraction the techniques of a posteriori image restoration and iterative image feature extraction are described and compared image feature extraction methods known as graduated nonconvexity gnc variable conductance diffusion vcd anisotropic diffusion and biased anisotropic diffusion bad which extract edges from noisy images are compared with a restoration feature extraction method known as mean field annealing mfa all are shown to be performing the same basic operation image relaxation this equivalence shows the relationship between energy minimization methods and spatial analysis methods and between their respective parameters of temperature and scale as a result of the equivalence vcd is demonstrated to minimize a cost function and that cost is specified explicitly furthermore operations over scale space are shown to be a method of avoiding local minima

bayesian image processing in two dimensions a bayesian image processing bip formalism which incorporates a priori amplitude and spatial probability density information was applied to two dimensional source fields for valid moderately restrictive a priori information strikingly improved results for ideal and experimental radioisotope phantom imaging data compared to a standard non bayesian formalism maximum likelihood ml were obtained the applicability of a fast fourier transform technique for convolution calculations a reduced region restriction for the initial deconvolution calculations and a relaxation parameter for accelerating convergence are considered

the application of mean field theory to image motion estimation previously markov random field mrf model based techniques have been proposed for image motion estimation since motion estimation is usually an ill posed problem various constraints are needed to obtain a unique and stable solution the main advantage of the mrf approach is its capacity to incorporate such constraints for instance motion continuity within an object and motion discontinuity at the boundaries between objects in the mrf approach motion estimation is often formulated as an optimization problem and two frequently used optimization methods are simulated annealing sa and iterative conditional mode icm although the sa is theoretically optimal in the sense of finding the global optimum it usually takes many iterations to converge the icm on the other hand converges quickly but its results are often unsatisfactory due to its hard decision nature previously the authors have applied the mean field theory to image segmentation and image restoration problems it provides results nearly as good as sa but with much faster convergence the present paper shows how the mean field theory can be applied to mrf model based motion estimation this approach is demonstrated on both synthetic and real world images where it produced good motion estimates

multiscale poisson intensity and density estimation the nonparametric poisson intensity and density estimation methods studied in this paper offer near minimax convergence rates for broad classes of densities and intensities with arbitrary levels of smoothness the methods and theory presented here share many of the desirable features associated with wavelet based estimators computational speed spatial adaptivity and the capability of detecting discontinuities and singularities with high resolution unlike traditional wavelet based approaches which impose an upper bound on the degree of smoothness to which they can adapt the estimators studied here guarantee nonnegativity and do not require any a priori knowledge of the underlying signal s smoothness to guarantee near optimal performance at the heart of these methods lie multiscale decompositions based on free knot free degree piecewise polynomial functions and penalized likelihood estimation the degrees as well as the locations of the polynomial pieces can be adapted to the observed data resulting in near minimax optimal convergence rates for piecewise analytic signals in particular the error of this estimator converges at nearly the parametric rate these methods can be further refined in two dimensions and it is demonstrated that platelet based estimators in two dimensions exhibit similar near optimal error convergence rates for images consisting of smooth surfaces separated by smooth boundaries

linear quadratic noise smoothing filters for quantum limited images we consider the use of nonlinear estimators for the noise smoothing of images obtained under quantum limited imaging conditions a volterra expansion is investigated from which a set of linear quadratic filters is derived using higher order statistics the filters are applicable for single frame and multiple frames of a single scene imaged under low light levels

a unified approach to statistical tomography using coordinate descent optimization over the past years there has been considerable interest in statistically optimal reconstruction of cross sectional images from tomographic data in particular a variety of such algorithms have been proposed for maximum a posteriori map reconstruction from emission tomographic data while map estimation requires the solution of an optimization problem most existing reconstruction algorithms take an indirect approach based on the expectation maximization em algorithm we propose a new approach to statistically optimal image reconstruction based on direct optimization of the map criterion the key to this direct optimization approach is greedy pixel wise computations known as iterative coordinate decent icd we propose a novel method for computing the icd updates which we call icd newton raphson we show that icd newton raphson requires approximately the same amount of computation per iteration as em based approaches but the new method converges much more rapidly in our experiments typically five to ten iterations other advantages of the icd newton raphson method are that it is easily applied to map estimation of transmission tomograms and typical convex constraints such as positivity are easily incorporated

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

galerkin schemes and inverse boundary value problems in reflexive banach spaces we develop the galerkin method for a recent version of the lax milgram theorem the generation of the corresponding finite dimensional subspaces for concrete boundary value problems leads us to consider certain biorthogonal systems in the reflexive banach spaces in question in addition we present an application to the numerical solution of inverse problems involving certain elliptic boundary value problems