on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

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

a new solution for the nonlinear diffusion convection equation the cauchy problem for a one dimensional nonlinear porous media diffusion convection differential equation is considered a self similar solution is derived that after the riccati transformation is described in terms of the airy functions the asymptotic behavior of the solution for a small amount of species introduced into the porous media is investigated

discretization of the radon transform and of its inverse by spline convolutions we present an explicit formula for b spline convolution kernels these are defined as the convolution of several b splines of variable widths h sub i and degrees n sub i we apply our results to derive spline convolution based algorithms for two closely related problems the computation of the radon transform and of its inverse first we present an efficient discrete implementation of the radon transform that is optimal in the least squares sense we then consider the reverse problem and introduce a new spline convolution version of the filtered back projection algorithm for tomographic reconstruction in both cases our explicit kernel formula allows for the use of high degree splines these offer better approximation performance than the conventional lower degree formulations e g piecewise constant or piecewise linear models we present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity in particular we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate

multiresolution tomographic reconstruction using wavelets shows how the separable two dimensional wavelet representation leads naturally to an efficient multiresolution tomographic reconstruction algorithm this algorithm is similar to the conventional filtered backprojection algorithm except that the filters are now angle dependent and the backprojection gives the wavelet coefficients of the reconstruction which are then used to synthesize the reconstruction at various resolution levels by reconstructing only a small localized region at high resolution the authors show how radiation exposure and computation can be significantly reduced compared to a standard reconstruction

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

wavelet based multiresolution local tomography we develop an algorithm to reconstruct the wavelet coefficients of an image from the radon transform data the proposed method uses the properties of wavelets to localize the radon transform and can be used to reconstruct a local region of the cross section of a body using almost completely local data that significantly reduces the amount of exposure and computations in x ray tomography the property that distinguishes our algorithm from the previous algorithms is based on the observation that for some wavelet bases with sufficiently many vanishing moments the ramp filtered version of the scaling function as well as the wavelet function has extremely rapid decay we show that the variance of the elements of the null space is negligible in the locally reconstructed image also we find an upper bound for the reconstruction error in terms of the amount of data used in the algorithm to reconstruct a local region 16 pixels in radius in a 256 spl times 256 image we require 22 of full exposure data

bilinear approach to delay painleve equations we present the list of all known to date delay painleve equations together with their hirota bilinear forms

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

integrable mappings and soliton equations ii abstract some simple solutions of discrete soliton equations are shown to satisfy 2d mappings we show that these belong to a recently introduced 18 parameter family of integrable reversible mappings of the plane thus lending weight to a previous conjecture we also give an example of an integrable mapping occuring in an exactly solvable model in statistical mechanics finally we discuss the notion of generalized reversibility

painleve analysis and singularity confinement the ultimate conjecture the authors propose a new integrability criterion based on the combination of two powerful integrability detectors the painleve method for continuous systems and the singularity confinement for discrete systems with this new criterion they can treat a large variety of systems that combine discrete and continuous characteristics differential difference systems ordinary and partial differential delay problems and even a class of integro differential equations

a fast digital radon transform an efficient means for evaluating the hough transform a fast digital radon transform based on recursively defined digital straight lines is described which has the sequential complexity of n 2 log n additions for an n x n image this transform can be used to evaluate the hough transform to detect straight lines in a digital image whilst a parallel implementation of the hough transform algorithm is difficult because of global memory access requirements the fast digital radon transform is vectorizable and therefore well suited for parallel computation the structure of the fast algorithm is shown to be quite similar to the fft algorithm for decimation in frequency it is demonstrated that even for sequential computation the fast radon transform is an attractive alternative to the classical hough transform algorithm

discrete painleve equations from singularity patterns the asymmetric trihomographic case we derive the discrete painleve equations associated with the affine weyl group e8 1 that can be represented by an in the quispel roberts thompson sense asymmetric trihomographic system the method used in this paper is based on singularity confinement we start by obtaining all possible singularity patterns for a general asymmetric trihomographic system and discard those patterns that cannot lead to confined singularities working with the remaining ones we implement the confinement conditions and derive the corresponding discrete painleve equations which involve two variables by eliminating either of these variables we obtain a symmetric equation examining all these equations of a single variable we find that they coincide exactly with those derived in previous works of ours thereby establishing the completeness of our results

addendum b spline interpolation in medical image processing analyzes b spline interpolation techniques of degree 2 4 and 5 with respect to all criteria that have been applied to evaluate various interpolation schemes in a recently published survey on image interpolation in medical imaging lehmann et al 1999 it is shown that high degree b spline interpolation has superior fourier properties smallest interpolation error and reasonable computing times therefore high degree b splines are preferable interpolators for numerous applications in medical image processing particularly if high precision is required if no aliasing occurs this result neither depends on the geometric transform applied for the tests nor the actual content of images

integrability criteria for differential difference systems a comparison of singularity confinement and low growth requirements a new approach for the study of the integrability of differential difference systems is introduced given a differential difference system of such a form that it can be iterated without necessitating the integration of a differential equation at any step a two stage strategy is used first the singularity confinement necessary integrability criterion is used in order to limit the possible choices once the system is sufficiently reduced the much stronger requirement of nonexponential growth of the degree of the iterates of some initial condition is implemented this method turns out to be powerful and practical the investigation of a given class of differential difference systems has resulted in some well known integrable systems but also to two promising integrability candidates one of which is reduced to a known integrable case

bandlimited and haar filtered back projection reconstructions a new way to discretize the filtered back projection fbp algorithm is presented the function basis is the haar system 2d product of rectangular windows this scheme allows one to derive the optimal shape of the apodisation window which is angle varying and the oversampling ratio between the pixel and the projection cell size the discrete equivalent filter is also derived the comparison of standard radial band limited and separable haar reconstructions shows that improvements in terms of linearity shift invariance and aliasing can be obtained even for the case of a limited number of views considerations of projection degradations are then analyzed according to a specific imaging device to derive the optimum oversampling ratio

delay differential equations and the painleve transcendents abstract we apply the recently proposed integrability criterion for differential difference systems that blends the classical painleve analysis with singularity confinement for discrete systems to a class of first order differential delay equations our analysis singles out the family of bi riccati equations as integrability candidates among these equations that pass the test some are integrable in a straightforward way usually by reduction to a standard riccati equation for some transformed variable while the remaining ones define new hysterodifferential forms of the painleve transcendental equations

large time solution of a nonlinear diffusion equation in this paper we consider the way in which the solution to a class of initial boundary value problems for the plane nonlinear diffusion equation frac partial partial t frac partial partial x u beta frac partial u partial x t geqslant 0 x geqslant 0 beta 0 approaches the similarity form as t rightarrow infty the problem we solve is chosen for two main reasons firstly the equation above is of widespread use in modelling physical situations and secondly it provides a tractable but significant example of a free boundary problem

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

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

wavelet localization of the radon transform the authors develop an algorithm which significantly reduces radiation exposure in x ray tomography when a local region of the body is to be imaged the algorithm uses the properties of wavelets to essentially localize the radon transform this algorithm differs from previous algorithms for doing local tomography because it recovers an approximation to the original image not the image module the nullspace of the local tomography operator or the lambda transform of the image this is possible because the authors do not truly invert the interior radon transform but rather sample the radon transform sparsely away from the local region of interest much attention in the field has been directed towards localized tomography the authors believe that this technique represents a significant contribution towards this effort

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

survey interpolation methods in medical image processing image interpolation techniques often are required in medical imaging for image generation e g discrete back projection for inverse radon transform and processing such as compression or resampling since the ideal interpolation function spatially is unlimited several interpolation kernels of finite size have been introduced this paper compares 1 truncated and windowed sine 2 nearest neighbor 3 linear 4 quadratic 5 cubic b spline 6 cubic g lagrange and 7 gaussian interpolation and approximation techniques with kernel sizes from 1 spl times 1 up to 8 spl times 8 the comparison is done by 1 spatial and fourier analyses 2 computational complexity as well as runtime evaluations and 3 qualitative and quantitative interpolation error determinations for particular interpolation tasks which were taken from common situations in medical image processing for local and fourier analyses a standardized notation is introduced and fundamental properties of interpolators are derived successful methods should be direct current dc constant and interpolators rather than dc inconstant or approximators each method s parameters are tuned with respect to those properties this results in three novel kernels which are introduced in this paper and proven to be within the best choices for medical image interpolation the 6 spl times 6 blackman harris windowed sinc interpolator and the c2 continuous cubic kernels with n 6 and n 8 supporting points for quantitative error evaluations a set of 50 direct digital x rays was used they have been selected arbitrarily from clinical routine in general large kernel sizes were found to be superior to small interpolation masks except for truncated sine interpolators all kernels with n 6 or larger sizes perform significantly better than n 2 or n 3 point methods p spl lt 0 005 however the differences within the group of large sized kernels were not significant summarizing the results the cubic 6 spl times 6 interpolator with continuous second derivatives as defined in 24 can be recommended for most common interpolation tasks it appears to be the fastest six point kernel to implement computationally it provides eminent local and fourier properties is easy to implement and has only small errors the same characteristics apply to b spline interpolation but the 6 spl times 6 cubic avoids the intrinsic border effects produced by the b spline technique however the goal of this study was not to determine an overall best method but to present a comprehensive catalogue of methods in a uniform terminology to define general properties and requirements of local techniques and to enable the reader to select that method which is optimal for his specific application in medical imaging

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

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

a spline inversion formula for the radon transform an approximate inversion formula is given for the radon transform of a function f in an arbitrary number of dimensions the basic idea is to consider the least squares approximation of f in a finite element space and to transform the normal equations into equations involving radon data the radon transform of finite elements is given in terms of multivariate b splines and error estimates are deduced in an l 2 setting

performance evaluation of an iterative image reconstruction algorithm for positron emission tomography an image reconstruction method motivated by positron emission tomography pet is discussed the measurements tend to be noisy and so the reconstruction method should incorporate the statistical nature of the noise the authors set up a discrete model to represent the physical situation and arrive at a nonlinear maximum a posteriori probability map formulation of the problem an iterative approach which requires the solution of simple quadratic equations is proposed the authors also present a methodology which allows them to experimentally optimize an image reconstruction method for a specific medical task and to evaluate the relative efficacy of two reconstruction methods for a particular task in a manner which meets the high standards set by the methodology of statistical hypothesis testing the new map algorithm is compared to a method which maximizes likelihood and with two variants of the filtered backprojection method

analysis of the direct fourier method for computer tomography the authors develop a direct fourier method dfm for reconstructing a function from its x ray projections they introduce a framework that can be used to get a quantitative comparison between different choices of basis functions in the step of resampling from polar to cartesian coordinates they use the framework to compare polynomial interpolation approximated sinc functions gaussians splines and kaiser bessel functions the resulting algorithm is very fast requiring 12 5 n sup 2 log n 49 n sup 2 flops numerical experiments show it to be efficient

on the direct fourier method for computer tomography considers the direct fourier methods dfm s for reconstructing an image from its given x ray projections the main purpose here is to use concepts from numerical analysis to estimate the errors in the methods the authors also suggest an alternative to the interpolations involved in the dfm s and estimate the number of terms involved realizing that one of the main reasons for the degradation of the quality of the reconstruction is the gibbs phenomenon the authors suggest a way to overcome this difficulty and give numerical evidence of its efficacy

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

on the exactly solvable equation s_t beta s gamma 2 s_x _x alpha beta s gamma 2 s_x occurring in two phase flow in porous media the symmetry approach of fokas j math phys 21 1980 pp 1318 1325 is used to show that the most general equation of the form s t g s s xx f s s x dg ds ne 0 which is exactly solvable is equivalent to the equation s t beta s gamma 2 s x x alpha beta s gamma 2 s x this equation is then mapped to the linear heat equation through a series of backlund transformations the above result is used to solve an initial boundary value problem modeling the process of waterflooding i e the displacement of oil initially in a porous reservoir by continuously injected water

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure