a spectral ct method to directly estimate basis material maps from experimental photon counting data the proposed spectral ct method solves the constrained one step spectral ct reconstruction cosscir optimization problem to estimate basis material maps while modeling the nonlinear x ray detection process and enforcing convex constraints on the basis map images in order to apply the optimization based reconstruction approach to experimental data the presented method empirically estimates the effective energy window spectra using a calibration procedure the amplitudes of the estimated spectra were further optimized as part of the reconstruction process to reduce ring artifacts a validation approach was developed to select constraint parameters the proposed spectral ct method was evaluated through simulations and experiments with a photon counting detector basis material map images were successfully reconstructed using the presented empirical spectral modeling and cosscir optimization approach in simulations the cosscir approach accurately reconstructed the basis map images 1 error in experiments the proposed method estimated the low density polyethylene region of the basis maps with 0 5 error in the pmma image and 4 error in the aluminum image for the teflon region the experimental results demonstrated 8 and 31 error in the pmma and aluminum basis material maps respectively compared with 24 and 126 error without estimation of the effective energy window spectra with residual errors likely due to insufficient modeling of detector effects the cosscir algorithm estimated the material decomposition angle to within 1 3 degree error where for reference the difference in angle between pmma and muscle tissue is 2 1 degrees the joint estimation of spectral response scaling coefficients and basis material maps was found to reduce ring artifacts in both a phantom and tissue specimen the presented validation procedure demonstrated feasibility for the automated determination of algorithm constraint parameters

hybrid pre log and post log image reconstruction for computed tomography tomographic image reconstruction for low dose computed tomography ct is increasingly challenging as dose continues to reduce in clinical applications pre log domain methods and post log domain methods have been proposed individually and each method has its own disadvantage while having the potential to improve image quality for low dose data by using an accurate imaging model pre log domain methods suffer slow convergence in practice due to the nonlinear transformation from the image to measurements in contrast post log domain methods have fast convergence speed but the resulting image quality is suboptimal for low dose ct data because the log transformation is extremely unreliable for low count measurements and undefined for negative values this paper proposes a hybrid method that integrates the pre log model and post log model together to overcome the disadvantages of individual pre log and post log methods we divide a set of ct data into high count and low count regions the post log weighted least squares model is used for measurements in the high count region and the pre log shifted poisson model for measurements in the low count region the hybrid likelihood function can be optimized using an existing iterative algorithm computer simulations and phantom experiments show that the proposed hybrid method can achieve faster early convergence than the pre log shifted poisson likelihood method and better signal to noise performance than the post log weighted least squares method

a parametric level set approach to simultaneous object identification and background reconstruction for dual energy computed tomography dual energy computerized tomography has gained great interest because of its ability to characterize the chemical composition of a material rather than simply providing relative attenuation images as in conventional tomography the purpose of this paper is to introduce a novel polychromatic dual energy processing algorithm with an emphasis on detection and characterization of piecewise constant objects embedded in an unknown cluttered background physical properties of the objects particularly the compton scattering and photoelectric absorption coefficients are assumed to be known with some level of uncertainty our approach is based on a level set representation of the characteristic function of the object and encompasses a number of regularization techniques for addressing both the prior information we have concerning the physical properties of the object and the fundamental physics based limitations associated with our ability to jointly recover the compton scattering and photoelectric absorption properties of the scene in the absence of an object with appropriate physical properties our approach returns a null characteristic function and thus can be viewed as simultaneously solving the detection and characterization problems unlike the vast majority of methods that define the level set function nonparametrically i e as a dense set of pixel values we define our level set parametrically via radial basis functions and employ a gauss newton type algorithm for cost minimization numerical results show that the algorithm successfully detects objects of interest finds their shape and location and gives an adequate reconstruction of the background

accelerating ordered subsets image reconstruction for x ray ct using spatially nonuniform optimization transfer statistical image reconstruction algorithms in x ray computed tomography ct provide improved image quality for reduced dose levels but require substantial computation time iterative algorithms that converge in few iterations and that are amenable to massive parallelization are favorable in multiprocessor implementations the separable quadratic surrogate sqs algorithm is desirable as it is simple and updates all voxels simultaneously however the standard sqs algorithm requires many iterations to converge this paper proposes an extension of the sqs algorithm that leads to spatially nonuniform updates the nonuniform nu sqs encourages larger step sizes for the voxels that are expected to change more between the current and the final image accelerating convergence while the derivation of nu sqs guarantees monotonic descent ordered subsets os algorithms can also accelerate sqs provided suitable subset balance conditions hold these conditions can fail in 3 d helical cone beam ct due to incomplete sampling outside the axial region of interest roi this paper proposes a modified os algorithm that is more stable outside the roi in helical ct we use ct scans to demonstrate that the proposed nu os sqs algorithm handles the helical geometry better than the conventional os methods and converges in less than half the time of ordinary os sqs

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

a splitting based iterative algorithm for accelerated statistical x ray ct reconstruction statistical image reconstruction using penalized weighted least squares pwls criteria can improve image quality in x ray computed tomography ct however the huge dynamic range of the statistical weights leads to a highly shift variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly we propose to alleviate the problem by using a variable splitting scheme that separates the shift variant and nearly invariant components of the statistical data model and also decouples the regularization term this leads to an equivalent constrained problem that we tackle using the classical method of multipliers framework with alternating minimization the specific form of our splitting yields an alternating direction method of multipliers admm algorithm with an inner step involving a nearly shift invariant linear system that is suitable for fft based preconditioning using cone type filters the proposed method can efficiently handle a variety of convex regularization criteria including smooth edge preserving regularizers and non smooth sparsity promoting ones based on the l1 norm and total variation numerical experiments with synthetic and real in vivo human data illustrate that cone filter preconditioners accelerate the proposed admm resulting in fast convergence of admm compared to conventional nonlinear conjugate gradient ordered subsets and state of the art mfista split bregman algorithms that are applicable for ct

3d forward and back projection for x ray ct using separable footprints iterative methods for 3d image reconstruction have the potential to improve image quality over conventional filtered back projection fbp in x ray computed tomography ct however the computation burden of 3d cone beam forward and back projectors is one of the greatest challenges facing practical adoption of iterative methods for x ray ct moreover projector accuracy is also important for iterative methods this paper describes two new separable footprint sf projector methods that approximate the voxel footprint functions as 2d separable functions because of the separability of these footprint functions calculating their integrals over a detector cell is greatly simplified and can be implemented efficiently the sf tr projector uses trapezoid functions in the transaxial direction and rectangular functions in the axial direction whereas the sf tt projector uses trapezoid functions in both directions simulations and experiments showed that both sf projector methods are more accurate than the distance driven dd projector which is a current state of the art method in the field the sf tt projector is more accurate than the sf tr projector for rays associated with large cone angles the sf tr projector has similar computation speed with the dd projector and the sf tt projector is about two times slower

the importance of the privilege for appearance of inverse power solutions in ito equations abstract it is shown that the true cause of inverse power distributions in the ito equation is some kind of privilege which is hidden in the course of evolution of the system connections between ito equations with additive noise or and multiplicative noise with additive processes multiplicative processes multiplication of probabilities and return to the origin problem are found on the basis of two toy models the appearance of particular functions for deterministic and stochastic forces in the ito equation is explained the paper stands as the next contribution confirming the hypothesis that the adequate privilege is the cause for the origin of inverse power distributions in many phenomena

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

combining ordered subsets and momentum for accelerated x ray ct image reconstruction statistical x ray computed tomography ct reconstruction can improve image quality from reduced dose scans but requires very long computation time ordered subsets os methods have been widely used for research in x ray ct statistical image reconstruction and are used in clinical pet and spect reconstruction in particular os methods based on separable quadratic surrogates os sqs are massively parallelizable and are well suited to modern computing architectures but the number of iterations required for convergence should be reduced for better practical use this paper introduces os sqs momentum algorithms that combine nesterov s momentum techniques with os sqs methods greatly improving convergence speed in early iterations if the number of subsets is too large the os sqs momentum methods can be unstable so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior experiments with simulated and real 3d ct scan data illustrate the performance of the proposed algorithms

globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms we present two types of globally convergent relaxed ordered subsets os algorithms for penalized likelihood image reconstruction in emission tomography modified block sequential regularized expectation maximization bsrem and relaxed os separable paraboloidal surrogates os sps the global convergence proof of the existing bsrem de pierro and yamagishi 2001 required a few a posteriori assumptions by modifying the scaling functions of bsrem we are able to prove the convergence of the modified bsrem under realistic assumptions our modification also makes stepsize selection more convenient in addition we introduce relaxation into the os sps algorithm erdogan and fessler 1999 that otherwise would converge to a limit cycle we prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed os sps is a special case main results of the proofs are from nedic and bertsekas 2001 and correa and lemarechal 1993 simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms

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

the importance of the privilege in resource redistribution models for appearance of inverse power solutions abstract a polemics with the opinion that power laws generically appear in systems with finite resources is carried on examples of simple models of redistribution of resources which lead to short tail distributions are given it is shown that the true cause of inverse power distributions is some kind of privilege which is hidden at a specific method of redistribution it is explained why some multiplicative processes lead to logexponential i e inverse power distribution functions rather than to lognormal ones the paper stands as the next contribution confirming the hypothesis that the adequate privilege is the cause of origin of inverse power distributions in many phenomena

grouped coordinate ascent algorithms for penalized likelihood transmission image reconstruction presents a new class of algorithms for penalized likelihood reconstruction of attenuation maps from low count transmission scans we derive the algorithms by applying to the transmission log likelihood a version of the convexity technique developed by de pierro for emission tomography the new class includes the single coordinate ascent sca algorithm and lange s convex algorithm for transmission tomography as special cases the new grouped coordinate ascent gca algorithms in the class overcome several limitations associated with previous algorithms 1 fewer exponentiations are required than in the transmission maximum likelihood expectation maximization ml em algorithm or in the sca algorithm 2 the algorithms intrinsically accommodate nonnegativity constraints unlike many gradient based methods 3 the algorithms are easily parallelizable unlike the sca algorithm and perhaps line search algorithms we show that the gca algorithms converge faster than the sca algorithm even on conventional workstations an example from a low count positron emission tomography pet transmission scan illustrates the method

penalized likelihood reconstruction with high fidelity measurement models for high resolution cone beam imaging we present a novel reconstruction algorithm based on a general cone beam ct forward model which is capable of incorporating the blur and noise correlations that are exhibited in flat panel cbct measurement data specifically the proposed model may include scintillator blur focal spot blur and noise correlations due to light spread in the scintillator the proposed algorithm gpl bc uses a gaussian penalized likelihood objective function which incorporates models of blur and correlated noise in a simulation study gpl bc was able to achieve lower bias as compared with deblurring followed by fdk as well as a model based reconstruction method without integration of measurement blur in the same study gpl bc was able to achieve better line pair reconstructions in terms of segmented image accuracy as compared with deblurring followed by fdk a model based method without blur and a model based method with blur but not noise correlations a prototype extremities quantitative cone beam ct test bench was used to image a physical sample of human trabecular bone these data were used to compare reconstructions using the proposed method and model based methods without blur and or correlation to a registered mu ct image of the same bone sample the gpl bc reconstructions resulted in more accurate trabecular bone segmentation multiple trabecular bone metrics including trabecular thickness tb th were computed for each reconstruction approach as well as the mu ct volume the gpl bc reconstruction provided the most accurate tb th measurement 0 255 mm as compared with the mu ct derived value of 0 193 mm followed by the gpl b reconstruction the gpl i reconstruction and then the fdk reconstruction 0 271 mm 0 309 mm and 0 335 mm respectively

power law distributions resulting from finite resources abstract an elementary stochastic model termed the normalization model is put forward which does demonstrate that power laws generically occur in systems with finite resources the model is capable to exhibit power law distributions with arbitrary power law exponents nevertheless for a large fraction of the parameter space power law exponents near unity are obtained as an application of the normalization mechanism we consider a network growth saturation model this model extends the scale free network model sf to include the fact of finite resources in the network growth saturation model the scale free property holds only for the growth period within the stationary regime we obtain power law distributions of the weight of the edges among the vertices we conjecture that this pattern will be found in the internet if it reaches the steady state

three level description of the domino cellular automaton motivated by the approach of kinetic theory of gases a three level description microscopic mesoscopic and macroscopic of cellular automaton is presented to provide an analytical treatment a simple domino cellular automaton with avalanches is constructed formulas concerning exact relations for density clusters avalanches and other parameters in a stationary state are derived some relations approximately valid for deviations from the stationary state are found and the adequate ito equation is constructed it provides the time evolution description of the density on the macroscopic level the results support an idea of application of the ito equation to some natural time series

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

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

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

penalized weighted least squares image reconstruction for dual energy x ray transmission tomography presents a dual energy de transmission computed tomography ct reconstruction method it is statistically motivated and features nonnegativity constraints in the density domain a penalized weighted least squares pwls objective function has been chosen to handle the non poisson noise added by amorphous silicon asi h detectors a gauss seidel algorithm has been used to minimize the objective function the behavior of the method in terms of bias standard deviation tradeoff has been compared to that of a de method that is based on filtered back projection fbp the advantages of the de pwls method are largest for high noise and or low flux cases qualitative results suggest this as well also the reconstructed images of an object with opaque regions are presented possible applications of the method are attenuation correction for positron emission tomography pet images various quantitative computed tomography qct methods such as bone mineral densitometry bmd and the removal of metal streak artifacts

sparse view spectral ct reconstruction using spectral patch based low rank penalty spectral computed tomography ct is a promising technique with the potential for improving lesion detection tissue characterization and material decomposition in this paper we are interested in kvp switching based spectral ct that alternates distinct kvp x ray transmissions during gantry rotation this system can acquire multiple x ray energy transmissions without additional radiation dose however only sparse views are generated for each spectral measurement and the spectra themselves are limited in number to address these limitations we propose a penalized maximum likelihood method using spectral patch based low rank penalty which exploits the self similarity of patches that are collected at the same position in spectral images the main advantage is that the relatively small number of materials within each patch allows us to employ the low rank penalty that is less sensitive to intensity changes while preserving edge directions in our optimization formulation the cost function consists of the poisson log likelihood for x ray transmission and the nonconvex patch based low rank penalty since the original cost function is difficult to minimize directly we propose an optimization method using separable quadratic surrogate and concave convex procedure algorithms for the log likelihood and penalty terms which results in an alternating minimization that provides a computational advantage because each subproblem can be solved independently we performed computer simulations and a real experiment using a kvp switching based spectral ct with sparse view measurements and compared the proposed method with conventional algorithms we confirmed that the proposed method improves spectral images both qualitatively and quantitatively furthermore our gpu implementation significantly reduces the computational cost

a new inexact alternating directions method for monotone variational inequalities the alternating directions method adm is an effective method for solving a class of variational inequalities vi when the proximal and penalty parameters in sub vi problems are properly selected in this paper we propose a new adm method which needs to solve two strongly monotone sub vi problems in each iteration approximately and allows the parameters to vary from iteration to iteration the convergence of the proposed adm method is proved under quite mild assumptions and flexible parameter conditions

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

multiplicative processes and power laws takayasu sato and takayasu phys rev lett 79 966 1997 revisited the question of stochastic processes with multiplicative noise which have been studied in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we briefly explain the physical mechanism leading to a power law probability distribution function and provide a list of references for these results dating back from a quarter of century we explain how the formulation in terms of the characteristic function developed by takayasu sato and takayasu can be extended to exponents ensuremath mu g2 which explains the reason for the lucky coincidence the multidimensional generalization of the results of takayasu sato and takayasu and the present status of the problem are briefly summarized the discovery of stretched exponential tails in the presence of the cutoff introduced by takayasu sato and takayasu is explained theoretically we end by briefly listing applications

ticmr total image constrained material reconstruction via nonlocal total variation regularization for spectral ct this work develops a material reconstruction method for spectral ct namely total image constrained material reconstruction ticmr to maximize the utility of projection data in terms of both spectral information and high signal to noise ratio snr this is motivated by the following fact when viewed as a spectrally integrated measurement the projection data can be used to reconstruct a total image without spectral information which however has a relatively high snr when viewed as a spectrally resolved measurement the projection data can be utilized to reconstruct the material composition which however has a relatively low snr the material reconstruction synergizes material decomposition and image reconstruction i e the direct reconstruction of material compositions instead of a two step procedure that first reconstructs images and then decomposes images for material reconstruction with high snr we propose ticmr with nonlocal total variation nltv regularization that is first we reconstruct a total image using spectrally integrated measurement without spectral binning and build the nltv weights from this image that characterize nonlocal image features then the nltv weights are incorporated into a nltv based iterative material reconstruction scheme using spectrally binned projection data so that these weights serve as a high snr reference to regularize material reconstruction note that the nonlocal property of nltv is essential for material reconstruction since material compositions may have significant local intensity variations although their structural information is often similar in terms of solution algorithm ticmr is formulated as an iterative reconstruction method with the nltv regularization in which the nonlocal divergence is utilized based on the adjoint relationship the alternating direction method of multipliers is developed to solve this sparsity optimization problem the proposed ticmr method was validated using both simulated and experimental data in comparison with fbp and total variation based iterative method ticmr had improved image quality e g contrast to noise ratio and spatial resolution

fast model based x ray ct reconstruction using spatially nonhomogeneous icd optimization recent applications of model based iterative reconstruction mbir algorithms to multislice helical ct reconstructions have shown that mbir can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts however high computational cost and long reconstruction times remain as a barrier to the use of mbir in practical applications among the various iterative methods that have been studied for mbir iterative coordinate descent icd has been found to have relatively low overall computational requirements due to its fast convergence this paper presents a fast model based iterative reconstruction algorithm using spatially nonhomogeneous icd nh icd optimization the nh icd algorithm speeds up convergence by focusing computation where it is most needed the nh icd algorithm has a mechanism that adaptively selects voxels for update first a voxel selection criterion vsc determines the voxels in greatest need of update then a voxel selection algorithm vsa selects the order of successive voxel updates based upon the need for repeated updates of some locations while retaining characteristics for global convergence in order to speed up each voxel update we also propose a fast 1 d optimization algorithm that uses a quadratic substitute function to upper bound the local 1 d objective function so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm we examine the performance of the proposed algorithm using several clinical data sets of various anatomy the experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3 d multislice geometries

an expanded theoretical treatment of iteration dependent majorize minimize algorithms the majorize minimize mm optimization technique has received considerable attention in signal and image processing applications as well as in statistics literature at each iteration of an mm algorithm one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate the next iterate is obtained by minimizing this tangent majorant function resulting in a sequence of iterates that reduces the cost function monotonically a well known special case of mm methods are expectation maximization algorithms in this paper we expand on previous analyses of mm due to fessler and hero that allowed the tangent majorants to be constructed in iteration dependent ways also this paper overcomes an error in one of those earlier analyses there are three main aspects in which our analysis builds upon previous work first our treatment relaxes many assumptions related to the structure of the cost function feasible set and tangent majorants for example the cost function can be nonconvex and the feasible set for the problem can be any convex set second we propose convergence conditions based on upper curvature bounds that can be easier to verify than more standard continuity conditions furthermore these conditions allow for considerable design freedom in the iteration dependent behavior of the algorithm finally we give an original characterization of the local region of convergence of mm algorithms based on connected e g convex tangent majorants for such algorithms cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods this expanded treatment widens the scope of the mm algorithm designs that can be considered for signal and image processing applications allows us to verify the convergent behavior of previously published algorithms and gives a fuller understanding overall of how these algorithms behave

joint statistical iterative material image reconstruction for spectral computed tomography using a semi empirical forward model by acquiring tomographic measurements with several distinct photon energy spectra spectral computed tomography spectral ct is able to provide additional material specific information compared with conventional ct this information enables the generation of material selective images which have found various applications in medical imaging however material decomposition typically leads to noise amplification and a degradation of the signal to noise ratio this is still a fundamental problem of spectral ct especially for low dose medical applications inspired by the success for low dose conventional ct several statistical iterative reconstruction algorithms for spectral ct have been developed these algorithms typically rely on detailed knowledge about the spectrum and the detector response obtaining this knowledge is often difficult in practice especially if photon counting detectors are used to acquire the energy specific information in this paper a new algorithm for joint statistical iterative material image reconstruction is presented it relies on a semi empirical forward model which is tuned by calibration measurements this strategy allows to model spatially varying properties of the imaging system without requiring detailed prior knowledge of the system parameters we employ an efficient optimization algorithm based on separable surrogate functions to accelerate convergence and reduce the reconstruction time numerical as well as real experiments show that our new algorithm leads to reduced statistical bias and improved image quality compared with projection based material decomposition followed by analytical or iterative image reconstruction

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

least squares restoration of multichannel images multichannel restoration using both within and between channel deterministic information is considered a multichannel image is a set of image planes that exhibit cross plane similarity existing optimal restoration filters for single plane images yield suboptimal results when applied to multichannel images since between channel information is not utilized multichannel least squares restoration filters are developed using the set theoretic and the constrained optimization approaches a geometric interpretation of the estimates of both filters is given color images three channel imagery with red green and blue components are considered constraints that capture the within and between channel properties of color images are developed issues associated with the computation of the two estimates are addressed a spatially adaptive multichannel least squares filter that utilizes local within and between channel image properties is proposed experiments using color images are described

a convergent incremental gradient method with a constant step size an incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented the method requires a single gradient evaluation per iteration and uses a constant step size for the case that the gradient is bounded and lipschitz continuous we show that the method visits infinitely often regions in which the gradient is small under certain unimodality assumptions global convergence is established in the quadratic case a global linear rate of convergence is shown the method is applied to distributed optimization problems arising in wireless sensor networks and numerical experiments compare the new method with other incremental gradient methods

line integral alternating minimization algorithm for dual energy x ray ct image reconstruction we propose a new algorithm called line integral alternating minimization liam for dual energy x ray ct image reconstruction instead of obtaining component images by minimizing the discrepancy between the data and the mean estimates liam allows for a tunable discrepancy between the basis material projections and the basis sinograms a parameter is introduced that controls the size of this discrepancy and with this parameter the new algorithm can continuously go from a two step approach to the joint estimation approach liam alternates between iteratively updating the line integrals of the component images and reconstruction of the component images using an image iterative deblurring algorithm an edge preserving penalty function can be incorporated in the iterative deblurring step to decrease the roughness in component images images from both simulated and experimentally acquired sinograms from a clinical scanner were reconstructed by liam while varying the regularization parameters to identify good choices the results from the dual energy alternating minimization algorithm applied to the same data were used for comparison using a small fraction of the computation time of dual energy alternating minimization liam achieves better accuracy of the component images in the presence of poisson noise for simulated data reconstruction and achieves the same level of accuracy for real data reconstruction

a model based iterative reconstruction algorithm for high angle annular dark field scanning transmission electron microscope haadf stem tomography high angle annular dark field haadf scanning transmission electron microscope stem data is increasingly being used in the physical sciences to research materials in 3d because it reduces the effects of bragg diffraction seen in bright field tem data typically tomographic reconstructions are performed by directly applying either filtered back projection fbp or the simultaneous iterative reconstruction technique sirt to the data since haadf stem tomography is a limited angle tomography modality with low signal to noise ratio these methods can result in significant artifacts in the reconstructed volume in this paper we develop a model based iterative reconstruction algorithm for haadf stem tomography we combine a model for image formation in haadf stem tomography along with a prior model to formulate the tomographic reconstruction as a maximum a posteriori probability map estimation problem our formulation also accounts for certain missing measurements by treating them as nuisance parameters in the map estimation framework we adapt the iterative coordinate descent algorithm to develop an efficient method to minimize the corresponding map cost function reconstructions of simulated as well as experimental data sets show results that are superior to fbp and sirt reconstructions significantly suppressing artifacts and enhancing contrast

time evolution of stochastic processes with correlations in the variance stability in power law tails of distributions abstract we model the time series of the s p500 index by a combined process the ar garch process where ar denotes the autoregressive process which we use to account for the short range correlations in the index changes and garch denotes the generalized autoregressive conditional heteroskedastic process which takes into account the long range correlations in the variance we study the ar garch process with an initial distribution of truncated levy form we find that this process generates a new probability distribution with a crossover from a levy stable power law to a power law with an exponent outside the levy range beyond the truncation cutoff we analyze the sum of n variables of the ar garch process and find that due to the correlations the ar garch process generates a probability distribution which exhibits stable behavior in the tails for a broad range of values n a feature which is observed in the probability distribution of the s p500 index we find that this power law stability depends on the characteristic scale in the correlations we also find that inclusion of short range correlations through the ar process is needed to obtain convergence to a limiting gaussian distribution for large n as observed in the data

modeling statistics and kinetics of the natural aggregation structures and processes with the solution of generalized logistic equation abstract the generalized logistic equation is proposed to model kinetics and statistics of natural processes such as earthquakes forest fires floods landslides and many others this equation has the form d n a d a s 1 n a n a q a q 0 and a 0 is the size of an element of a structure and 0 the equation contains two exponents and q taking into account two important properties of elements of a system their fractal geometry and their ability to interact either to enhance or to damp the process of aggregation the function n a can be understood as an approximation to the number of elements the size of which is less than a the function d n a d a where n a is the general solution of this equation for q 1 is a product of an increasing bounded function and power law function with stretched exponential cut off the relation with tsallis non extensive statistics is demonstrated by solving the generalized logistic equation for q 0 in the case 0 q 1 the equation models super additive and the case q 1 it models sub additive structures the gutenberg richter g r formula results from interpretation of empirical data as a straight line in the area of stretched exponent with small the solution is applied for modeling distribution of foreshocks and aftershocks in the regions of napa valley 2014 and sumatra 2004 earthquakes fitting the observed data well both qualitatively and quantitatively

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

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

pwls ultra an efficient clustering and learning based approach for low dose 3d ct image reconstruction the development of computed tomography ct image reconstruction methods that significantly reduce patient radiation exposure while maintaining high image quality is an important area of research in low dose ct imaging we propose a new penalized weighted least squares pwls reconstruction method that exploits regularization based on an efficient union of learned transforms pwls ultra the union of square transforms is pre learned from numerous image patches extracted from a dataset of ct images or volumes the proposed pwls based cost function is optimized by alternating between a ct image reconstruction step and a sparse coding and clustering step the ct image reconstruction step is accelerated by a relaxed linearized augmented lagrangian method with ordered subsets that reduces the number of forward and back projections simulations with 2 d and 3 d axial ct scans of the extended cardiac torso phantom and 3 d helical chest and abdomen scans show that for both normal dose and low dose levels the proposed method significantly improves the quality of reconstructed images compared to pwls reconstruction with a nonadaptive edge preserving regularizer pwls with regularization based on a union of learned transforms leads to better image reconstructions than using a single learned square transform we also incorporate patch based weights in pwls ultra that enhance image quality and help improve image resolution uniformity the proposed approach achieves comparable or better image quality compared to learned overcomplete synthesis dictionaries but importantly is much faster computationally more efficient

a least squares method for the inverse reflector problem in arbitrary orthogonal coordinates abstract in this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far field of the reflector by use of a least squares method we derive the monge ampere equation expressing conservation of energy while assuming an arbitrary coordinate system we generalize a cartesian coordinate least squares method presented earlier by prins et al 13 to arbitrary orthogonal coordinate systems this generalized least squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source this results in significantly increased numerical accuracy decrease of errors by factors up to 104 is reported we present the generalized least squares method and compare its numerical results with the cartesian version for a disk shaped light source

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

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

proximal admm for multi channel image reconstruction in spectral x ray ct the development of spectral x ray computed tomography ct using binned photon counting detectors has received great attention in recent years and has enabled selective imaging of contrast agents loaded with k edge materials a practical issue in implementing this technique is the mitigation of the high noise levels often present in material decomposed sinogram data in this work the spectral x ray ct reconstruction problem is formulated within a multi channel mc framework in which statistical correlations between the decomposed material sinograms can be exploited to improve image quality specifically a mc penalized weighted least squares pwls estimator is formulated in which the data fidelity term is weighted by the mc covariance matrix and sparsity promoting penalties are employed this allows the use of any number of basis materials and is therefore applicable to photon counting systems and k edge imaging to overcome numerical challenges associated with use of the full covariance matrix as a data fidelity weight a proximal variant of the alternating direction method of multipliers is employed to minimize the mc pwls objective function computer simulation and experimental phantom studies are conducted to quantitatively evaluate the proposed reconstruction method

spatial resolution properties of penalized likelihood image reconstruction space invariant tomographs this paper examines the spatial resolution properties of penalized likelihood image reconstruction methods by analyzing the local impulse response the analysis shows that standard regularization penalties induce space variant local impulse response functions even for space invariant tomographic systems paradoxically for emission image reconstruction the local resolution is generally poorest in high count regions we show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections this analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution the modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized likelihood methods

scale free networks with exponent one a majority of studied models for scale free networks have degree distributions with exponents greater than two real networks however can demonstrate essentially more heavy tailed degree distributions we explore two models of scale free equilibrium networks that have the degree distribution exponent ensuremath gamma 1 p q ensuremath sim q ensuremath ensuremath gamma such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high our models exploit a rewiring mechanism they are local in the sense that no knowledge of the network structure apart from the immediate neighborhood of the vertices is required these models generate uncorrelated networks in the infinite size limit where they are solved explicitly we investigate finite size effects by the use of simulations we find that both models exhibit disassortative degree degree correlations for finite network sizes in addition we observe a markedly degree dependent clustering in the finite networks we indicate a real world network with a similar degree distribution

an incremental gradient projection method with momentum term and adaptive stepsize rule we consider an incremental gradient method with momentum term for minimizing the sum of continuously differentiable functions this method uses a new adaptive stepsize rule that decreases the stepsize whenever sufficient progress is not made we show that if the gradients of the functions are bounded and lipschitz continuous over a certain level set then every cluster point of the iterates generated by the method is a stationary point in addition if the gradient of the functions have a certain growth property then the method is either linearly convergent in some sense or the stepsizes are bounded away from zero the new stepsize rule is much in the spirit of heuristic learning rules used in practice for training neural networks via backpropagation as such the new stepsize rule may suggest improvements on existing learning rules finally extension of the method and the convergence results to constrained minimization is discussed as are some implementation issues and numerical experience

regularized image reconstruction algorithms for dual isotope myocardial perfusion spect mps imaging using a cross tracer prior in simultaneous dual isotope myocardial perfusion spect mps imaging data are simultaneously acquired to determine the distributions of two radioactive isotopes the goal of this work was to develop penalized maximum likelihood pml algorithms for a novel cross tracer prior that exploits the fact that the two images reconstructed from simultaneous dual isotope mps projection data are perfectly registered in space we first formulated the simultaneous dual isotope mps reconstruction problem as a joint estimation problem a cross tracer prior that couples voxel values on both images was then proposed we developed an iterative algorithm to reconstruct the mps images that converges to the maximum a posteriori solution for this prior based on separable surrogate functions to accelerate the convergence we developed a fast algorithm for the cross tracer prior based on the complete data os em cosem framework the proposed algorithm was compared qualitatively and quantitatively to a single tracer version of the prior that did not include the cross tracer term quantitative evaluations included comparisons of mean and standard deviation images as well as assessment of image fidelity using the mean square error we also evaluated the cross tracer prior using a three class observer study with respect to the three class mps diagnostic task i e classifying patients as having either no defect reversible defect or fixed defects for this study a comparison with conventional ordered subsets expectation maximization os em reconstruction with postfiltering was performed the comparisons to the single tracer prior demonstrated similar resolution for areas of the image with large intensity changes and reduced noise in uniform regions the cross tracer prior was also superior to the single tracer version in terms of restoring image fidelity results of the three class observer study showed that the proposed cross tracer prior and the convergent algorithms improved the image quality of dual isotope mps images compared to os em

designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport we study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position shape and intensity such interfaces can be refracting surfaces of two different lenses or of one lens in geometrical optics approximation the analytic formulation of this problem in both cases requires construction of maps with controlled jacobian though this jacobian can be expressed as a second order partial differential equation of monge ampere type for a scalar function defining one of the refracting surfaces its analysis is not straightforward in this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions existence and uniqueness of weak solutions in lipschitz classes for both cases are established by variational methods our results show in particular that two types of interfaces exist in each case for the same data one of these types always consists of two interfaces one of which is concave or convex and the second convex or concave while the interfaces of the second type may be neither convex nor concave the availability of a design with convex concave lenses is particularly important for fabrication the truly geometric nature of this problem permits its statement and investigation in mathbb r n 1 n geqq 1

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

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

penalized likelihood estimators and noise analysis for randoms precorrected pet transmission scans this paper analyzes and compares image reconstruction methods based on practical approximations to the exact log likelihood of randoms precorrected positron emission tomography pet measurements the methods apply to both emission and transmission tomography however in this paper the authors focus on transmission tomography the results of experimental pet transmission scans and variance approximations demonstrate that the shifted poisson sp method avoids the systematic bias of the conventional data weighted least squares wls method and leads to significantly lower variance than conventional statistical methods based on the log likelihood of the ordinary poisson of model the authors develop covariance approximations to analyze the propagation of noise from attenuation maps into emission images via the attenuation correction factors acf s empirical pixel and region variances from real transmission data agree closely with the analytical predictions both the approximations and the empirical results show that the performance differences between the op model and sp model are even larger when considering noise propagation from the transmission images into the final emission images than the differences in the attenuation maps themselves

model based iterative reconstruction for dual energy x ray ct using a joint quadratic likelihood model dual energy x ray ct dect has the potential to improve contrast and reduce artifacts as compared to traditional ct moreover by applying model based iterative reconstruction mbir to dual energy data one might also expect to reduce noise and improve resolution however the direct implementation of dual energy mbir requires the use of a nonlinear forward model which increases both complexity and computation alternatively simplified forward models have been used which treat the material decomposed channels separately but these approaches do not fully account for the statistical dependencies in the channels in this paper we present a method for joint dual energy mbir jde mbir which simplifies the forward model while still accounting for the complete statistical dependency in the material decomposed sinogram components the jde mbir approach works by using a quadratic approximation to the polychromatic log likelihood and a simple but exact nonnegativity constraint in the image domain we demonstrate that our method is particularly effective when the dect system uses fast kvp switching since in this case the model accounts for the inaccuracy of interpolated sinogram entries both phantom and clinical results show that the proposed model produces images that compare favorably in quality to previous decomposition based methods including fbp and other statistical iterative approaches

multi material decomposition using statistical image reconstruction for spectral ct spectral computed tomography ct provides information on material characterization and quantification because of its ability to separate different basis materials dual energy de ct provides two sets of measurements at two different source energies in principle two materials can be accurately decomposed from dect measurements however many clinical and industrial applications require three or more material images for triple material decomposition a third constraint such as volume conservation mass conservation or both is required to solve three sets of unknowns from two sets of measurements the recently proposed flexible image domain id multi material decomposition method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel we propose a penalized likelihood pl method with edge preserving regularizers for each material to reconstruct multi material images using a similar constraint from sinogram data we develop an optimization transfer method with a series of pixel wise separable quadratic surrogate pwsqs functions to monotonically decrease the complicated pl cost function the pwsqs algorithm separates pixels to allow simultaneous update of all pixels but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material and pixel wise sqs algorithms comparing with the id method using 2 d fan beam simulations the pl method greatly reduced noise streak and cross talk artifacts in the reconstructed basis component images and achieved much smaller root mean square errors

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

alternating minimization algorithms for transmission tomography a family of alternating minimization algorithms for finding maximum likelihood estimates of attenuation functions in transmission x ray tomography is described the model from which the algorithms are derived includes polyenergetic photon spectra background events and nonideal point spread functions the maximum likelihood image reconstruction problem is reformulated as a double minimization of the i divergence a novel application of the convex decomposition lemma results in an alternating minimization algorithm that monotonically decreases the objective function each step of the minimization is in closed form the family of algorithms includes variations that use ordered subset techniques for increasing the speed of convergence simulations demonstrate the ability to correct the cupping artifact due to beam hardening and the ability to reduce streaking artifacts that arise from beam hardening and background events

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices