multimodality image registration by maximization of mutual information a new approach to the problem of multimodality medical image registration is proposed using a basic concept from information theory mutual information mi or relative entropy as a new matching criterion the method presented in this paper applies mi to measure the statistical dependence or information redundancy between the image intensities of corresponding voxels in both images which is assumed to be maximal if the images are geometrically aligned maximization of mi is a very general and powerful criterion because no assumptions are made regarding the nature of this dependence and no limiting constraints are imposed on the image content of the modalities involved the accuracy of the mi criterion is validated for rigid body registration of computed tomography ct magnetic resonance mr and photon emission tomography pet images by comparison with the stereotactic registration solution while robustness is evaluated with respect to implementation issues such as interpolation and optimization and image content including partial overlap and image degradation our results demonstrate that subvoxel accuracy with respect to the stereotactic reference solution can be achieved completely automatically and without any prior segmentation feature extraction or other preprocessing steps which makes this method very well suited for clinical applications

local and pointwise error estimates of the local discontinuous galerkin method applied to the stokes problem we prove local and pointwise error estimates for the local discontinuous galerkin method applied to the stokes problem in two and three dimensions by using techniques originally developed by a schatz math comp 67 1998 877 899 to prove pointwise estimates for the laplace equation we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries

least squares solutions as solutions of a perturbation form of the galerkin methods interior pointwise error estimates and pollution effect first order div least squares ls finite element methods for second order elliptic partial differential equations are considered the main idea of this paper is a consideration of the ls solution u h for the primary function u as the solution of a perturbation form of the galerkin methods consequently the results concerning the perturbation form of the galerkin methods can be applied for the ls methods here we use the pointwise error estimates for the perturbation form provided by schatz a h schatz pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part ii interior estimates siam j numer anal 38 2000 1269 1293 to obtain the pointwise estimates for the ls solution the estimates consist of the local truncation error and pollution terms we provide estimates for pollution terms our process for arriving at the estimates involves obtaining the norm of a linear functional which accounts for the perturbation our estimates are valid for both smooth and nonsmooth problems for the smooth problems the pollution is weaker than the local truncation error for the nonsmooth problems we provide a criterion for determining the strength of the pollution in particular we identify superconvergent points for nonconvex polygonal domains

convergence of an iterative method for total variation denoising in total variation denoising one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion several approaches based on this idea have recently been shown to be very effective particularly for denoising functions with discontinuities this paper analyzes the convergence of an iterative method for solving such problems the iterative method involves a lagged diffusivity approach in which a sequence of linear diffusion problems are solved global convergence in a finite dimensional setting is established and local convergence properties including rates and their dependence on various parameters are examined

asymptotic error expansions for the finite element method for second order elliptic problems in r n n geq2 i local interior expansions our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h valid for h sufficiently small such expansions are obtained from the so called asymptotic expansion inequalities valid in mathbf r n for n geq2 studies by schatz in math comp 67 1998 pp 877 899 and siam j numer anal 38 2000 pp 1269 1293

new entire positive solution for the nonlinear schrodinger equation coexistence of fronts and bumps in this paper we construct new kinds of positive solutions of delta u u u p 0 quad rm on bbb r 2 when p 2 these solutions have the following asymptotic behavior u x z sim omega x f z sum i 1 infty omega 0 x z xi i vec e 1 as l rightarrow infty where omega is a unique positive homoclinic solution of omega omega omega p 0 in bbb r omega 0 is the two dimensional positive solution and vec e 1 1 0 and xi j are points such that xi j jl cal o 1 for all j geq 1 this represents a first result on the it coexistence of fronts and bumps geometrically our new solutions correspond to it triunduloid in the theory of cmc surface

deconvolution based partial volume correction in raclopride pet and monte carlo comparison to mr based method abstract in this work we evaluated three iterative deconvolution algorithms and compared their performance to partial volume pv correction based on structural imaging in brain positron emission tomography pet using a database of monte carlo simulated images we limited our interest to quantitative radioligand pet imaging particularly to 11c raclopride and striatal imaging the studied deconvolution methods included richardson lucy reblurred van cittert and reblurred van cittert with the total variation regularization we studied the bias and variance of the regional estimates of binding potential bp values and the accuracy of regional tacs as a function of the applied image processing the resolution noise tradeoff in parametric bp images was addressed as well the regional bp values and tacs obtained by deconvolution were almost as accurate than those by structural imaging based pv correction gtm method when the ideal volumes of interests vois were used to extract tacs from the images for deconvolution methods the ideal vois were slightly eroded from the exact anatomical voi to limit the bias due to tissue fraction effect which is not corrected for by deconvolution based methods for the gtm method the ideal vois were the exact anatomical vois the bp values and tacs by deconvolution were less affected by segmentation and registration errors than those with the gtm based pv correction the bp estimates and tacs with deconvolution based pv correction were more accurate than bps and tacs derived without pv correction the parametric images obtained by the deconvolution based pv correction showed considerably improved resolution with only slightly increased noise level compared to the case with no pv correction the reblurred van cittert method was the best of the studied deconvolution methods we conclude that the deconvolution is an interesting alternative to structural imaging based pv correction as it leads to quantification results of similar accuracy and it is less prone to registration and segmentation errors than structural imaging based pv correction moreover pv corrected parametric images can be readily computed based on deconvolved dynamic images

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

sharply localized pointwise and w 1 estimates for finite element methods for quasilinear problems we establish pointwise and w 1 estimates for finite element methods for a class of second order quasilinear elliptic problems defined on domains in r these estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order our pointwise estimates are similar to and rely on results and analysis techniques of schatz for linear problems we also extend estimates of schatz and wahlbin for pointwise differences e x 1 e x 2 in pointwise errors to quasilinear problems finally we establish estimates for the error in w 1 d where d is a subdomain these negative norm estimates are novel for linear as well as for nonlinear problems our analysis heavily exploits the fact that galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships thus allowing easy application of properly formulated results for linear problems

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

perturbations of forms and error estimates for the finite element method at a point with an application to improved superconvergence error estimates for subspaces that are symmetric with respect to a point we first derive a variety of local error estimates for u uh at a point x0 where uh belongs to a finite element space shr and is an approximation to u satisfying the local equations a u u h varphi f varphi for all varphi in shr with compact support in a neighborhood of x0 here the a cdot cdot are bilinear forms associated with second order elliptic equations and the f are linear functionals in the case that f equiv 0 our results coincide with those of schatz siam j numer anal 38 2000 pp 1269 1293 but are improvements when f neq 0 we apply these results to improve the superconvergence error estimates obtained by schatz sloan and wahlbin siam j numer anal 33 1996 pp 505 521 at points x0 where the subspaces are symmetric with respect to x0

solutions without any symmetry for semilinear elliptic problems abstract we prove the existence of infinitely many solitary waves for the nonlinear klein gordon or schrodinger equation u u u 3 0 in r 2 which have finite energy and whose maximal group of symmetry reduces to the identity

interior maximum norm estimates for finite element methods part ii we consider bilinear forms a connected with second order elliptic problems and assume that for u h in a finite element space s h we have a u u h x f x for x in s h with local compact support we give local estimates for u u h in l and w 1 of the type local best approximation plus weak outside influences plus the local size of f

cluster analysis in kinetic modelling of the brain a noninvasive alternative to arterial sampling in emission tomography quantification of brain tracer uptake metabolism or binding requires knowledge of the cerebral input function traditionally this is achieved with arterial blood sampling we propose a noninvasive alternative via the use of a blood vessel time activity curve tac extracted directly from dynamic positron emission tomography pet scans by cluster analysis five healthy subjects were injected with the 5ht 2a receptor ligand 18 f altanserin and blood samples were subsequently taken from the radial artery and cubital vein eight regions of interest roi tacs were extracted from the pet data set hierarchical k means cluster analysis was performed on the pet time series to extract a cerebral vasculature roi the number of clusters was varied from k 1 to 10 for the second of the two stage method determination of the correct number of clusters was performed by the within variance measure and by 3d visual inspection of the homogeneity of the determined clusters the cluster determined input curve was then used in logan plot analysis and compared with the arterial and venous blood samples and additionally with one of the currently used alternatives to arterial blood sampling the simplified reference tissue model srtm and logan analysis with cerebellar tac as an input there was a good agreement p 0 05 between the values of distribution volume dv obtained from the k means clustered input function and those from the arterial blood samples this work acts as a proof of principle that the use of cluster analysis on a pet data set could obviate the requirement for arterial cannulation when determining the input function for kinetic modelling of ligand binding and that this may be a superior approach as compared to the other noninvasive alternatives

a new approach to richardson extrapolation in the finite element method for second order elliptic problems this paper presents a nonstandard local approach to richardson extrapolation when it is used to increase the accuracy of the standard finite element approximation of solutions of second order elliptic boundary value problems in mathbb r n n ge 2 the main feature of the approach is that it does not rely on a traditional asymptotic error expansion but rather depends on a more easily proved weaker a priori estimate derived in 19 called an asymptotic error expansion inequality in order to use this inequality to verify that the richardson procedure works at a point we require a local condition which links the different subspaces used for extrapolation roughly speaking this condition says that the subspaces are similar about a point i e any one of them can be made to locally coincide with another by a simple scaling of the independent variable about that point examples of finite element subspaces that occur in practice and satisfy this condition are given

existence of quasiperiodic solutions of elliptic equations on rn 1 via center manifold and kam theorems abstract we consider elliptic equations on r n 1 of the form 1 x u u y y g x u 0 x y r n r where g x u is a sufficiently regular function with g 0 0 we give sufficient conditions for the existence of solutions of 1 which are quasiperiodic in y and decaying as x uniformly in y such solutions are found using a center manifold reduction and results from the kam theory we discuss several classes of nonlinearities g to which our results apply

constant mean curvature surfaces with three ends abstract we announce the classification of complete almost embedded surfaces of constant mean curvature with three ends and genus zero they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends

linear and nonlinear image deblurring a documented study nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods this paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging where reconstruction of high frequency detail is of prime interest and where avoidance of a priori smoothness constraints is a major concern the theoretical basis behind each of the following nonlinear procedures is examined the lucy richardson method the maximum likelihood e m algorithm the poisson maximum a posteriori method and the nunez llacer version of the maximum entropy method a linear iterative method vancittert s iteration is also studied it is shown that each of the first three methods as well as vancittert s method lack a necessary ingredient for successful solution of the ill posed deblurring problem while in the maximum entropy method the enforced smoothness may have adverse consequences in medical imaging a direct linear method the slow evolution from the continuation boundary secb method designed specifically for nonsmooth images is also considered that method is stabilized by constraining the blurring operator as well as the solution and does not require smoothness constraints it is shown that useful error estimates can be obtained in the secb method while this is impossible in tikhonov s method without a priori bounds on derivatives of the unknown solution reconstruction experiments on low noise synthetic mri data show that thousands of iterations are necessary to achieve sufficient resolution in the iterative procedures however the secb method provides higher resolution at considerable savings in computer time at high noise levels the iterative algorithms are shown to diverge at these same noise levels the secb method produces reconstructions comparable in quality to those that would be obtained in the iterative methods were one able to terminate the divergent algorithm at that iteration which best approximates the true solution in the l1 norm

pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids part i global estimates this part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in r n in a sense to be discussed below these sharpen known quasi optimal l and w 1 estimates for the error on irregular quasi uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u we note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces as a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities these expansion inequalities are valid for large classes of finite elements defined on irregular grids in r n and have applications to superconvergence and extrapolation and a posteriori estimates part ii of this series will contain local estimates applicable to non smooth problems

holder estimates for green s functions on convex polyhedral domains and their applications to finite element methods a model second order elliptic equation on a general convex polyhedral domain in three dimensions is considered the aim of this paper is twofold first sharp holder estimates for the corresponding green s function are obtained as an applications of these estimates to finite element methods we show the best approximation property of the error in w 1 infty in contrast to previously known results w p 2 regularity for p 3 which does not hold for general convex polyhedral domains is not required furthermore the new green s function estimates allow us to obtain localized error estimates at a point

the topology and geometry of embedded surfaces of constant mean curvature si m est une surface proprement plongee dans r 3 avec une courbure moyenne constante non nulle alors m n est pas homeomorphe a une surface fermee avec un seul point elimine soit m une surface proprement plongee dans r 3 a courbure moyenne constante non nulle si m est homeomorphe a une surface fermee avec 2 points elimines alors m est une distance bornee a partir d une ligne droite soit m une surface proprement plongee de courbure moyenne non nulle qui est homeomorphe a une surface fermee avec 3 points elimines alors m reste une distance bornee a partir d un plan

some error estimates for the lumped mass finite element method for a parabolic problem we study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous dirichlet boundary conditions improving earlier results we show that known optimal order smooth initial data error estimates for the standard galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation we also discuss the application to time discretization by the backward euler and crank nicolson methods

multiple end solutions to the allen cahn equation in r2 we construct a new class of entire solutions for the allen cahn equation u 1 u2 u 0 in r2 c given k 1 we find a family of solutions whose zero level sets are away from a compact set asymptotic to 2k straight lines which we call the ends these solutions have the property that there exist 0 1 2k 0 2 such that limr u rei 1 j uniformly in on compact subsets of j j 1 for j 0 2k 1