maximum likelihood reconstruction for magnetic resonance fingerprinting this paper introduces a statistical estimation framework for magnetic resonance mr fingerprinting a recently proposed quantitative imaging paradigm within this framework we present a maximum likelihood ml formalism to estimate multiple mr tissue parameter maps directly from highly undersampled noisy bf k space data a novel algorithm based on variable splitting the alternating direction method of multipliers and the variable projection method is developed to solve the resulting optimization problem representative results from both simulations and in vivo experiments demonstrate that the proposed approach yields significantly improved accuracy in parameter estimation compared to the conventional mr fingerprinting reconstruction moreover the proposed framework provides new theoretical insights into the conventional approach we show analytically that the conventional approach is an approximation to the ml reconstruction more precisely it is exactly equivalent to the first iteration of the proposed algorithm for the ml reconstruction provided that a gridding reconstruction is used as an initialization

a theorem of real analysis and its application to free boundary problems in this note we show how comparison theorems for solutions to equations with non smooth coefficients yield in a very simple fashion some free boundary regularity theorems in cases where there is a priori geometrical information for instance when one knows that the solution u is monotone in some given direction

a fast sinc function gridding algorithm for fourier inversion in computer tomography the fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the cartesian grid required for fast numerical fourier inversion although the fourier inversion method is recognized as being computationally faster than the back projection method for parallel ray projection data the artifacts resulting from inaccurate interpolation have generally limited application of the method this paper presents a computationally efficient gridding algorithm which can be used with direct fourier transformation to achieve arbitrarily small artifact levels the method has potential for application to other measurement geometries such as fan beam projections and diffraction tomography and nmr imaging

reactive trajectories and the transition path process we study the trajectories of a solution x t to an ito stochastic differential equation in mathbb r d as the process passes between two disjoint open sets a and b these segments of the trajectory are called transition paths or reactive trajectories and they are of interest in the study of chemical reactions and thermally activated processes in that context the sets a and b represent reactant and product states our main results describe the probability law of these transition paths in terms of a transition path process y t which is a strong solution to an auxiliary sde having a singular drift term we also show that statistics of the transition path process may be recovered by empirical sampling of the original process x t as an application of these ideas we prove various representation formulas for statistics of the transition paths we also identify the density and current of transition paths our results fit into the framework of the transition path theory by weinan and vanden eijnden

smooth bumps a borel theorem and partitions of smooth functions on p c f fractals we provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals one using a probabilistic approach and sub gaussian estimates for the heat operator and the other using the analytic theory for p c f fractals and a fixed point argument the heat semigroup probabilistic method is applicable to a more general class of metric measure spaces with laplacian including certain infinitely ramified fractals however the cutoff technique involves some loss in smoothness from the analytic approach we establish a borel theorem for p c f fractals showing that to any prescribed jet at a junction point there is a smooth function with that jet as a consequence we prove that on p c f fractals smooth functions may be cut off with no loss of smoothness and thus can be smoothly decomposed subordinate to an open cover the latter result provides a replacement for classical partition of unity arguments in the p c f fractal setting

what is not in the domain of the laplacian on sierpinski gasket type fractals abstract we consider the analog of the laplacian on the sierpinski gasket and related fractals constructed by kigami a function f is said to belong to the domain of if f is continuous and f is defined as a continuous function we show that if f is a nonconstant function in the domain of then f2 is not in the domain of we give two proofs of this fact the first is based on the analog of the pointwise identity f2 2f f f 2 where we show that f 2 does not exist as a continuous function in fact the correct interpretation of f2 is as a singular measure a result due to kusuoka we give a new proof of this fact the second is based on a dichotomy for the local behavior of a function in the domain of at a junction point x0 of the fractal in the typical case nonvanishing of the normal derivative we have upper and lower bounds for f x f x0 in terms of d x x0 for a certain value and in the nontypical case vanishing normal derivative we have an upper bound with an exponent greater than 2 this method allows us to show that general nonlinear functions do not operate on the domain of

nonuniform interpolation of noisy signals using support vector machines the problem of signal interpolation has been intensively studied in the information theory literature in conditions such as unlimited band nonuniform sampling and presence of noise during the last decade support vector machines svm have been widely used for approximation problems including function and signal interpolation however the signal structure has not always been taken into account in svm interpolation we propose the statement of two novel svm algorithms for signal interpolation specifically the primal and the dual signal model based algorithms shift invariant mercer s kernels are used as building blocks according to the requirement of bandlimited signal the sine kernel which has received little attention in the svm literature is used for bandlimited reconstruction well known properties of general svm algorithms sparseness of the solution robustness and regularization are explored with simulation examples yielding improved results with respect to standard algorithms and revealing good characteristics in nonuniform interpolation of noisy signals

the structure of the free boundary for lower dimensional obstacle problems we study the regularity of the free surface in boundary obstacle problems we show that near a non degenerate point the free boundary is a c 1 a n 2 dimensional surface in r n 1

behavior near the boundary of positive solutions of second order parabolic equations ii a boundary backward harnack inequality is proved for positive solutions of second order parabolic equations in non divergence form in a bounded cylinder q q x 0 t which vanish on o q oq x 0 t where q is a bounded lipschitz domain in r this inequality is applied to the proof of the holder continuity of the quotient of two positive solutions vanishirlg on a portion of o q

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

a new boundary harnack principle equations with right hand side we introduce a new boundary harnack principle in lipschitz domains for equations with a right hand side our approach which uses comparisons and blow ups will adapt to more general domains as well as other types of operators we prove the principle for divergence form elliptic equations with lower order terms including zero order terms the inclusion of a zero order term appears to be new even in the absence of a right hand side

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

taylor approximations on sierpinski gasket type fractals for a class of fractals that includes the familiar sierpinski gasket there is now a theory involving laplacians dirichlet forms normal derivatives green s functions and the gauss green integration formula analogous to the theory of analysis on manifolds this theory was originally developed as a by product of the construction of stochastic processes analogous to brownian motion but has been given by a direct analytic construction in the work of kigami until now this theory has not provided anything analogous to the gradient of a function or a local taylor approximation in this paper we construct a family of derivatives which includes the known normal derivative at vertex points in the graphs that approximate the fractal and obtain taylor approximations at these points we show that a function in the domain of n can be locally well approximated by an n harmonic function solution of nu 0 one novel feature of this result is that it requires several different estimates to describe the optimal rate of approximation

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

fractal interpolation on the koch curve the koch curve can be obtained as an iterated function system construction self similar interpolation is possible for any function on the sets that are defined recursively we prove that the koch curve kc is an analogue of the fractal interpolation theorem of barnsley also the classical harmonic functions are defined on the kc as the degree 1 polynomials for self similar interpolation

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

a practical guide to the recovery of wavelet coefficients from fourier measurements in a series of recent papers b adcock a c hansen and c poon appl comput harmon anal 36 2014 pp 387 415 b adcock m gataric and a c hansen siam j imaging sci 7 2014 pp 1690 1723 adcock et al siam j math anal 47 2015 pp 1196 1233 it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its fourier transform via the method of generalized sampling while these papers focused on the optimality of generalized sampling in terms of its stability and error bounds the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm in particular we show that generalized sampling has a computational complexity of mathcal o left m n log n right when recovering the first n boundary corrected wavelet coefficients of an unknown compactly supported function from m n fourier samples therefore due to the linear correspondences bet

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

regularity of a free boundary with application to the pompeiu problem in the unit ball b 0 1 let u and a domain in n solve the following overdetermined problem u in b 0 1 02 u jruj 0 inb 0 1 n where denotes the characteristic function and the equation is satisfled in the sense of distributions if the complement of does not develop cusp singularities at the origin then we prove is analytic in some small neighborhood of the origin the result can be modifled to yield for more general divergence form operators as an application of this then we obtain the regularity of the boundary of a domain without the pompeiu property provided its complement has no cusp singularities

the hot spots conjecture for the sierpinski gasket abstract there are many works on the hot spots conjecture for domains in euclidean space since the conjecture was posed by j rauch in 1974 in this paper using spectral decimation we prove that the conjecture holds on the sierpinski gasket i e every eigenfunction of the second smallest eigenvalue of the neumann laplacian introduced by j kigami attains its maximum and minimum on the boundary

regularity of one phase stefan problem near lipschitz initial data in this paper we show that starting from lipschitz initial free boundary with small lipschitz constant the solution of the one phase stefan problem st immediately regularizes and is smooth in space and time for a small positive time

nonlinear approximation by sums of exponentials and translates in this paper we discuss the numerical solution of two nonlinear approximation problems many applications in electrical engineering signal processing and mathematical physics lead to the following problem let h be a linear combination of exponentials with real frequencies determine all frequencies all coefficients and the number of summands if finitely many perturbed uniformly sampled data of h are given we solve this problem by an approximate prony method apm and prove the stability of the solution in the square and uniform norm further an apm for nonuniformly sampled data is proposed too the second approximation problem is related to the first one and reads as follows let varphi be a given 1 periodic window function as defined in section 4 further let f be a linear combination of translates of varphi determine all shift parameters all coefficients and the number of translates if finitely many perturbed uniformly sampled data of f are given using fourier technique this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by apm the stability of the solution is discussed in the square and uniform norm too numerical experiments show the performance of our approximation methods

gradients on fractals in this paper we define and study a gradient on p c f post critically finite or finitely ramified fractals we use dirichlet energy form analysis developed for such fractals by kigami we consider both nondegenerate and degenerate harmonic structures where a nonzero harmonic function can be identically zero on an open set we show that the energy is equal to the integral of a certain seminorm of the gradient if the harmonic structure is weakly nondegenerate this result was proved by kusuoka in a different form we show that for a c 1 function on the sierpin ski gasket the gradient considered here and kusuoka s gradient essentially coincide with a gradient considered by kigami the gradient at a junction point was studied by strichartz in relation to the taylor approximation on fractals he also proved the existence of the gradient almost everywhere with respect to the hausdorff bernoulli measure for a function in the domain of the laplacian in this paper we obtain certain continuity properties of the gradient for a function in the domain of the laplacian as an appendix we prove an estimate of the local energy of harmonic functions which was stated by strichartz as a hypothesis 2000 academic press

caloric functions in lipschitz domains and the regularity of solutions to phase transition problems in some domain d c rn l that vanishes locally on some distinguished part of od the term refers to the fact that in some sense along that part of od u resembles the fundamental solution of the heat equation in d this is particularly the case when the domain under consideration d is the intersection of some n 1 dimensional cube q with q one side of a lipschitz graph i e q fx f x t that is d q n q and the distinguished part of od is precisely q n q this is the main first area of study of this paper relying on work of fabes garofalo salsa fgs on backward harnack type inequalities for such domains lipschitz regularity in time versus lipschitz regularity in space is not of course the natural homogeneity balance for the study of parabolic equations but it is so for the study of phase transition relations of the form

radial imaging with multipolar magnetic encoding fields we present reconstruction methods for radial magnetic resonance imaging mri data which were spatially encoded using a pair of orthogonal multipolar magnetic fields for in plane encoding and parallel imaging it is shown that a direct method exists in addition to iterative reconstruction standard direct projection reconstruction algorithms can be combined with a previously developed direct reconstruction for multipolar encoding fields acquired with cartesian trajectories the algorithm is simplified by recasting the reconstruction problem into polar coordinates in this formulation distortion and aliasing become separate effects distortion occurs only along the radial direction and aliasing along the azimuthal direction moreover aliased points are equidistantly distributed in this representation and consequently cartesian sense is directly applicable with highly effective unfolding properties for radio frequency coils arranged with a radial symmetry the direct and iterative methods are applied to simulated data to analyze basic properties of the algorithms and for the first time also measured in vivo data are presented the results are compared to linear spatial encoding using a radial trajectory and quadrupolar encoding using a cartesian trajectory the direct reconstruction gives good results for fully sampled datasets undersampled datasets however show star shaped artifacts which are significantly reduced with the iterative reconstruction

regularity properties of a free boundary near contact points with the fixed boundary in the upper half of the unit ball b x 0 let u and omega a domain in r n x is an element of r n x 1 0 solve the following overdetermined problem deltau chi omega in b u

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

the sub elliptic obstacle problem c1 a regularity of the free boundary in carnot groups of step two the sub elliptic obstacle problem arises in various branches of the applied sciences e g in mechanical engineering and robotics mathematical finance image reconstruction and neurophysiology in the recent paper donatella danielli nicola garofalo sandro salsa variational inequalities with lack of ellipticity i optimal interior regularity and non degeneracy of the free boundary indiana univ math j 52 2 2003 361 398 mr1976081 2004c 35424 it was proved that weak solutions to the sub elliptic obstacle problem in a carnot group belong to the folland stein optimal lipschitz class the analogue of the well known interior local regularity for the classical obstacle problem however the regularity of the free boundary remained a challenging open problem in this paper we prove that in carnot groups of step r 2 the free boundary is euclidean c1 a near points satisfying a certain thickness condition this constitutes the sub elliptic counterpart of a celebrated result due to caffarelli luis a caffarelli the regularity of free boundaries in higher dimensions acta math 139 3 4 1977 155 184 mr0454350 56 12601

resampling of data between arbitrary grids using convolution interpolation for certain medical applications resampling of data is required in magnetic resonance tomography mrt or computer tomography ct e g data may be sampled on nonrectilinear grids in the fourier domain for the image reconstruction a convolution interpolation algorithm often called gridding can be applied for resampling of the data onto a rectilinear grid resampling of data from a rectilinear onto a nonrectilinear grid are needed e g if projections of a given rectilinear data set are to be obtained in this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another the basic algorithm can be split into two steps first the data are resampled from the arbitrary input grid onto a rectilinear grid and second the rectilinear data is resampled onto the arbitrary output grid furthermore the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm for fast sampling pattern independent determination of the sampling density function the voronoi diagram of the sample distribution is calculated the volume of the voronoi cell around each sample is used as a measure for the sampling density it is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids furthermore it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral radial and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image

fractal interpolation on the sierpinski gasket abstract we prove for the sierpinski gasket sg an analogue of the fractal interpolation theorem of barnsley let v 0 p 1 p 2 p 3 be the set of vertices of sg and u i x 1 2 x p i the three contractions of the plane of which the sg is the attractor fix a number n and consider the iterations u w u w 1 u w 2 u w n for any sequence w w 1 w 2 w n 1 2 3 n the union of the images of v 0 under these iterations is the set of nth stage vertices v n of sg let f v n r be any function given any numbers w w 1 2 3 n with 0 w 1 there exists a unique continuous extension f sg r of f such that f u w x w f x h w x for x sg where h w are harmonic functions on sg for w 1 2 3 n interpreting the harmonic functions as the degree 1 polynomials on sg is thus a self similar interpolation obtained for any start function f v n r

estimates for the resolvent kernel of the laplacian on p c f self similar fractals and blowups one of the main features of analysis on post critically finite self similar pcfss sets is that it is possible to understand the behavior of the laplacian and its inverse the green operator in terms of the self similar structure of the set indeed a major step in the approach to analysis on self similar fractals via dirichlet forms was kigami s proof 8 10 that for a self similar dirichlet form the green kernel can be written explicitly as a series in which each term is a rescaling of a single expression via the self similar structure in 6 this result was extended to show that the resolvent kernel of the laplacian meaning the kernel of z 1 can also be written as a self similar series for suitable values of z c part of the motivation for that work was that it gives a new understanding of functions of the laplacian such as the heat operator et by writing them as integrals of the resolvent the purpose of the present work is to establish estimates that permit the above approach to be carried out we study the functions occurring in the series decomposition from 6 see theorem 3 3 below for this decomposition and give estimates on their decay from this we determine estimates on the resolvent kernel and on kernels of operators defined as integrals of the resolvent kernel in particular we recover the sharp upper estimates for the heat kernel see theorem 10 2 that were proved for pcfss sets by hambly and kumagai 4 by probabilistic methods see also 1 13 3 for earlier results of this type on less general classes of sets it is important to note that the preceding authors were able to prove not just upper estimates but also lower bounds for the heat kernel and therefore were able to prove sharpness of their bounds our methods permit sharp bounds for the resolvent kernel on the positive real axis but we do not know how to obtain these globally in the complex plane or how to obtain lower estimates for the heat kernel from them therefore in this direction our results are not as strong as those obtained in 4 however in other directions we obtain more information than that known from heat kernel estimates and we hope that our approach will complement the existing probabilistic methods in particular we are able to obtain resolvent bounds on any ray in c other than the negative real axis where the spectrum lies while standard calculations from heat kernel bounds only give these estimates in a half plane a further consequence of our approach is that we extend in theorem 9 7 the decomposition from 6 to the case of blowups which are non compact sets with local structure equivalent to that of the underlying self similar sets the blowup of a pcfss set bears the same relation to the original set as the real line bears to the unit interval see 17 for details the structure of the paper is as follows in section 3 we recall some basic features of analysis on pcfss sets as well as the main result of 6 which is the decomposition of the resolvent as a weighted sum of piecewise eigenfunctions section 4 then discusses

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

kaiser bessel basis for particle mesh interpolation in this work we introduce the kaiser bessel interpolation basis for the particle mesh interpolation in the fast ewald method a reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems and is shown to be effective in optimizing the shape parameter of the kaiser bessel basis in terms of accuracy by comparing the optimized kaiser bessel basis with the traditional b spline basis we demonstrate that the former is more accurate than the latter in part of the working parameter space say a relatively small real space cutoff a relatively small reciprocal space mesh and a relatively large truncation of basis in some cases the kaiser bessel basis is found to be more than one order of magnitude more accurate

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

fourier based forward and back projectors in iterative fan beam tomographic image reconstruction fourier based forward and back projection methods can reduce computation in iterative tomographic image reconstruction recently an optimized nonuniform fast fourier transform nufft approach was shown to yield accurate parallel beam projections in this paper we extend the nufft approach to describe an o n sup 2 log n projector backprojector pair for fan beam transmission tomography simulations and experiments with real ct data show that fan beam fourier based forward and back projection methods can reduce computation for iterative reconstruction while still providing accuracy comparable to their o n sup 3 space based counterparts

function spaces on fractals abstract we construct function spaces analogs of holder zygmund besov and sobolev spaces on a class of post critically finite self similar fractals in general and the sierpinski gasket in particular based on the laplacian and effective resistance metric of kigami this theory is unrelated to the usual embeddings of these fractals in euclidean space and so our spaces are distinct from the function spaces of jonsson and wallin although there are some coincidences for small orders of smoothness we show that the laplacian acts as one would expect an elliptic pseudodifferential operator of order d 1 on a space of dimension d to act where d is determined by the growth rate of the measure of metric balls we establish some sobolev embedding theorems and some results on complex interpolation on these spaces

calculus on the sierpinski gasket i polynomials exponentials and power series abstract we study the analog of power series expansions on the sierpinski gasket for analysis based on the kigami laplacian the analog of polynomials are multiharmonic functions which have previously been studied in connection with taylor approximations and splines here the main technical result is an estimate of the size of the monomials analogous to x n n we propose a definition of entire analytic functions as functions represented by power series whose coefficients satisfy exponential growth conditions that are stronger than what is required to guarantee uniform convergence we present a characterization of these functions in terms of exponential growth conditions on powers of the laplacian of the function these entire analytic functions enjoy properties such as rearrangement and unique determination by infinite jets that one would expect however not all exponential functions eigenfunctions of the laplacian are entire analytic and also many other natural candidates such as the heat kernel do not belong to this class nevertheless we are able to use spectral decimation to study exponentials and in particular to create exponentially decaying functions for negative eigenvalues

multiple overlapping k space junctions for investigating translating objects mojito it is a well known property in fourier transform magnetic resonance imaging mri that rigid body translational motion in image space results in linear phase accumulation in k space this work describes multiple overlapping k space junctions for investigating translating objects mojito a correction scheme based on phase differences at trajectory intersections caused by 2 d alterations in the position of an object during mr imaging the algorithm allows both detection and correction of motion artifacts caused by 2 d rigid body translational motion although similar in concept to navigator echoes mojito does not require a repeating path through k space uses k space data from a broader region of k space and uses the repeated data in image reconstruction this provides the potential for a highly efficient self navigating motion correction method here the concept and theoretical basis of mojito is demonstrated using the continuous sampling bowtie trajectory in simulation and mr experiments this particular trajectory is selected since it is well suited for such an algorithm due to numerous trajectory intersections specifically the validity of the technique in the presence of noise and off resonance effects is demonstrated through simulation

a generalization of the two dimensional prolate spheroidal wave function method for nonrectilinear mri data acquisition methods the two dimensional 2 d prolate spheroidal wave function 2 d pswf method was previously introduced as an efficient method for trading off between spatial and temporal resolution in magnetic resonance imaging mri with minimal penalty due to truncation and partial volume effects in the 2 d pswf method the k space sampling area and a matching 2 d pswf filter with optimal signal concentration and minimal truncation artifacts are determined by the shape and size of a given convex region of interest roi the spatial information in the reduced k space data is used to calculate the total image intensity over a nonsquare roi instead of producing a low resolution image this method can be used for tracking dynamic signals from non square rois using a reduced k space sampling area while achieving minimal signal leakage however the previous theory is limited to the case of rectilinear sampling in order to make the 2 d pswf method more suitable for dynamic studies this paper presents a generalized version of the 2 d pswf theory that can be applied to nonrectilinear data acquisition methods the method is applied to an fmri study using a spiral trajectory which illustrates the methods efficiency at tracking hemodynamic signals with high temporal resolution

calculus on the sierpinski gasket ii point singularities eigenfunctions and normal derivatives of the heat kernel this paper continues the study of fundamental properties of elementary functions on the sierpinski gasket sg related to the laplacian defined by kigami harmonic functions multiharmonic functions and eigenfunctions of the laplacian we describe the possible point singularities of such functions and we use the description at certain periodic points to motivate the definition of local derivatives at these points we study the global behavior of eigenfunctions on all generic infinite blow ups of sg and construct eigenfunctions that decay at infinity in certain directions we study the asymptotic behavior of normal derivatives of dirichlet eigenfunctions at boundary points and give experimental evidence for the behavior of the normal derivatives of the heat kernel at boundary points

fast higher order mr image reconstruction using singular vector separation medical resonance imaging mri conventionally relies on spatially linear gradient fields for image encoding however in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level accounting for field perturbations that are neither linear in space nor constant over time i e dynamic higher order fields is particularly challenging it was previously shown to be feasible with conjugate gradient iteration however so far this approach has been relatively slow due to the need to carry out explicit matrix vector multiplications in each cycle in this work it is proposed to accelerate higher order reconstruction by expanding the encoding matrix such that fast fourier transform can be employed for more efficient matrix vector computation the underlying principle is to represent the perturbing terms as sums of separable functions of space and time compact representations with this property are found by singular vector analysis of the perturbing matrix guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis the proposed technique is demonstrated for the case of higher order field perturbations due to eddy currents caused by diffusion weighting in this example image reconstruction was accelerated by two orders of magnitude

fractafolds based on the sierpinski gasket and their spectra we introduce the notion of fractafold which is to a fractal what a manifold is to a euclidean half space we specialize to the case when the fractal is the sierpinski gasket sg we show that each such compact fractafold can be given by a cellular construction based on a finite cell graph g which is 3 regular in the case that the fractafold has no boundary we show explicitly how to obtain the spectrum of the fractafold from the spectrum of the graph using the spectral decimation method of fukushima and shima this enables us to obtain isospectral pairs of nonhomeomorphic fractafolds we also show that although sg is topologically rigid there are fractafolds based on sg that are not topologically rigid