complete parameterization of piecewise polynomial interpolation kernels every now and then a new design of an interpolation kernel appears in the literature while interesting results have emerged the traditional design methodology proves laborious and is riddled with very large systems of linear equations that must be solved analytically we propose to ease this burden by providing an explicit formula that can generate every possible piecewise polynomial kernel given its degree its support its regularity and its order of approximation this formula contains a set of coefficients that can be chosen freely and do not interfere with the four main design parameters it is thus easy to tune the design to achieve any additional constraints that the designer may care for

a new two step p stable hybrid obrechkoff method for the numerical integration of second order ivps in this paper we introduce a class of new p stable hybrid obrechkoff methods for the numerical solution of second order initial value problems we generate a two step symmetric p stable hybrid obrechkoff method of order 8 with two off step symmetric points and phase lag of order at least 2 and for special cases this order can be increased up to 6 the numerical results obtained by the new method for some problems show its superiority in efficiency accuracy and stability

invariant subspaces dilation theory and the structure of the predual of a dual algebra i on etudie la structure des preduaux d algebres d operateurs on obtient un nouveau theoreme de sous espace invariant on trouve des proprietes concernant le treillis sous espace invariant de l operateur decalage de bergman

a four step trigonometric fitted p stable obrechkoff method for periodic initial value problems in this paper we present a new p stable obrechkoff four step method which greatly improves the performance of our previous obrechkoff four step method and extends its application range by trigonometric fitting we extend the interval of periodicity of the previous four step method from about h2 16 to infinity and at the same time we keep all its advantage in the accuracy and efficiency we have tested the new method by four well known problems 1 the test equation 2 stiefel and bettis problem 3 duffing equation without damping and 4 bessel equation the numerical results show that the new method is more accurate than any previous method it also has great advantage in stability and efficiency

operators similar to their restrictions to invariant subspaces abstract an operator t on a separable infinite dimensional banach space x is called crystal like if for each nonzero invariant subspace m of t there exists a positive number r such that t m the restriction of t to m is similar to rt in this paper we investigate some elementary properties of crystal like operators such as the spectrum cyclicity strong irreducibility essential similarity reflectivity and the commutativity of the commutant

color image resolution conversion in this paper we look at the problem of spatially scaling color images we focus on an approach that takes advantage of the human visual system s color spatial frequency sensitivity the algorithm performs an efficient least squares ls resolution conversion for the luminance channel and a low complexity pixel replication reduction in the chrominance channels the performance of the algorithm is compared to a ls method in srgb and cielab color spaces as well as standard bilinear interpolation in srgb space the comparisons are made in terms of computational cost and color error in scielab

comparison of interpolating methods for image resampling when resampling an image to a new set of coordinates for example when rotating an image there is often a noticeable loss in image quality to preserve image quality the interpolating function used for the resampling should be an ideal low pass filter to determine which limited extent convolving functions would provide the best interpolation five functions were compared a nearest neighbor b linear c cubic b spline d high resolution cubic spline with edge enhancement a 1 and e high resolution cubic spline a 0 5 the functions which extend over four picture elements c d e were shown to have a better frequency response than those which extend over one a or two b pixels the nearest neighbor function shifted the image up to one half a pixel linear and cubic b spline interpolation tended to smooth the image the best response was obtained with the high resolution cubic spline functions the location of the resampled points with respect to the initial coordinate system has a dramatic effect on the response of the sampled interpolating function the data are exactly reproduced when the points are aligned and the response has the most smoothing when the resampled points are equidistant from the original coordinate points thus at the expense of some increase in computing time image quality can be improved by resampled using the high resolution cubic spline function as compared to the nearest neighbor linear or cubic b spline functions

moms maximal order interpolation of minimal support we consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly supported basis function spl phi z we first give the expression for the cases of spl phi that have minimal support for a given accuracy also known as approximation order this class of functions which we call maximal order minimal support functions moms is made of linear combinations of the b spline of the same order and of its derivatives we provide an explicit form of the moms that maximizes the approximation accuracy when the step size is small enough we compute the sampling gain obtained by using these optimal basis functions over the splines of the same order we show that it is already substantial for small orders and that it further increases with the approximation order l when l is large this sampling gain becomes linear more specifically its exact asymptotic expression is 2 spl pi e l since the optimal functions are continuous but not differentiable for even orders and even only piecewise continuous for odd orders our result implies that regularity has little to do with approximating performance these theoretical findings are corroborated by experimental evidence that involves compounded rotations of images

transformation between continuous and discrete representations of images a perceptual approach the transformation of image signals between the continuous domain of the real world and the discrete domain of modern data processing can have a significant effect on quality and efficiency analysis based on perceptual rather than mathematical considerations has been carried out a series of experiments based on the analysis has shown that substantial improvement over usual techniques can be achieved by the use of a cascade of a presharpening filter combined with gaussian presampling and interpolation filters the resulting sharpened gaussian filter although not exactly circularly symmetrical gives a high degree of isotropy each element in the cascade is separable so that computational efficiency is high a favorable tradeoff is achieved among sharpness smoothness and the effects of aliasing subjective testing in comparison with other commonly used filters has shown the clear superiority of this filter

recursive regularization filters design properties and applications least squares approximation problems that are regularized with specified highpass stabilizing kernels are discussed for each problem there is a family of discrete regularization filters r filters which allow an efficient determination of the solutions these operators are stable symmetric lowpass filters with an adjustable scale factor two decomposition theorems for the z transform of such systems are presented one facilitates the determination of their impulse response while the other allows an efficient implementation through successive causal and anticausal recursive filtering a case of special interest is the design of r filters for the first and second order difference operators these results are extended for two dimensional signals and for illustration purposes are applied to the problem of edge detection this leads to a very efficient implementation 8 multiplies 10 adds per pixel of the optimal canny edge detector based on the use of a separable second order r filter

pgscm a family of p stable boundary value methods for second order initial value problems in this paper we introduce a family of linear multistep methods used as boundary value methods for the numerical solution of initial value problems for second order ordinary differential equations of special type we rigorously prove that these schemes are p stable in a generalized sense of arbitrarily high order this overcomes the barrier that lambert and watson established in lambert and watson 1976 1 on linear multistep methods used in the classic way that is as initial value methods we call the new methods pgscms an acronym for p n stable generalized stormer cowell methods numerical illustrations which confirm the theoretical results of the paper are finally given

on the comparison of interpolation methods a study of different cubic interpolation kernels in the frequency domain is presented that reveals novel aspects of both cubic spline and cubic convolution interpolation the kernel used in cubic convolution is of finite support and depends on a parameter to be chosen at will at the nyquist frequency the spectrum attains a value that is independent of this parameter exactly the same value is found at the nyquist frequency in the cubic spline interpolation if a strictly positive interpolation kernel is of importance in applications cubic convolution with the parameter value zero is recommended

a p stable complete in phase obrechkoff trigonometric fitted method for periodic initial value problems in this paper we derive a p stable trigonometric fitted obrechkoff method with phase lag frequency distortion infinity it is easy to see from numerical results presented that the new method is much more accurate than previous methods

quadratic interpolation for image resampling nearest neighbor linear and various cubic interpolation functions are frequently used in image resampling quadratic functions have been disregarded largely because they have been thought to introduce phase distortions this is shown not to be the case and a family of quadratic functions is derived the interpolating member of this family has visual quality close to that of the catmull rom cubic yet requires only 60 of the computation time

p stable symmetric super implicitmethods for periodic initial value problems the idea of super implicit methods requiring not just past and present but also future values was suggested by fukushima recently here we construct p stable super implicit methods for the solution of second order initial value problems the benefit of such methods is realized when using vector or parallel computers

quasitriangular small compact strongly irreducible let t be a bounded linear operator acting on a separable infinite dimensional hilbert space let be a positive number in this article we prove that the perturbation of t by a compact operator k with k 0 be given is there a compact operator k with k such that t k is strongly irreducible

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

fast b spline transforms for continuous image representation and interpolation efficient algorithms for the continuous representation of a discrete signal in terms of b splines direct b spline transform and for interpolative signal reconstruction indirect b spline transform with an expansion factor m are described expressions for the z transforms of the sampled b spline functions are determined and a convolution property of these kernels is established it is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering fast computational algorithms based on the recursive implementations of these filters are proposed a b spline interpolator can also be characterized in terms of its transfer function and its global impulse response cardinal spline of order n the case of the cubic spline is treated in greater detail the present approach is compared with previous methods that are reexamined from a critical point of view it is concluded that b spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner

p stable high order super implicit and obrechkoff methods for periodic initial value problems this paper discusses the numerical solution of periodic initial value problems two classes of methods are discussed super implicit and obrechkoff the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in cases when the right hand side is very complex we may prefer super implicit methods we develop a super implicit p stable method of order 12 and obrechkoff method of order 18

an improved trigonometrically fitted p stable obrechkoff method for periodic initial value problems in this paper we present an improved p stable trigonometrically fitted obrechkoff method with phase lag frequency distortion infinity compared with the previous p stable trigonometrically fitted obrechkoff method developed by simos our new method is simpler in structure and more stable in computation we have also improved the accuracy of the first order derivative formula from the numerical illustration presented we can show that the new method is much more accurate than the previous methods

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

obrechkoff versus super implicit methods for the solution of first and second order initial value problems abstract this paper discusses the numerical solution of first order initial value problems and a special class of second order ones those not containing first derivative two classes of methods are discussed super implicit and obrechkoff we will show equivalence of super implicit and obrechkoff schemes the advantage of obrechkoff methods is that they are high order one step methods and thus will not require additional starting values on the other hand they will require higher derivatives of the right hand side in case the right hand side is complex we may prefer super implicit methods the disadvantage of super implicit methods is that they in general have a larger error constant to get the same error constant we require one or more extra future values we can use these extra values to increase the order of the method instead of decreasing the error constant one numerical example shows that the super implicit methods are more accurate than the obrechkoff schemes of the same order

an optimal rotator for iterative reconstruction for implementations of iterative reconstruction algorithms that rotate the image matrix the characteristics of the rotator may affect the reconstruction quality desirable qualities for the rotator include 1 preservation of global and local image counts 2 accurate count positioning 3 a uniform and predictable amount of blurring due to the rotation a new rotation method for iterative reconstruction is proposed which employs gaussian interpolation this method was compared to standard rotation techniques and is shown to be superior to standard techniques when measured by these qualities the computational cost was demonstrated to be only slightly more than bilinear interpolation

obrechkoff methods having additional parameters for general second order differential equations abstract a class of two step implicit methods involving higher order derivatives of y for initial value problems of the form y f t y y is developed the methods involve arbitrary parameters p and q which are determined so that the methods become absolutely stable when applied to the test equation y y y 0 numerical results for bessel s and general second order differential equations are presented to illustrate that the methods are absolutely stable and are of order o h 4 o h 6 and o h 8

some new implicit two step multiderivative methods for solving special second order ivp s abstract the aim of this paper is to select from the huge family of implicit two step multiderivative methods those which have better characteristics as regards the error order and error constant stability and easy of use with most of the coefficients equal to zero a few of these multiderivative methods that are a 0 stable and a stable have been obtained this feature makes them useful for solving stiff problems the stability intervals or stability regions for the methods chosen are presented some numerical tests are included the numerical test concerning the wave equation shows the good performance of the methods of lower order but a 0 stable in comparison with the poor performance of higher order methods