an algorithm for redundant binary bit pipelined rational arithmetic the authors introduce a redundant binary representation of the rationals and an associated algorithm for computing the sum difference product quotient and other useful functions of two rational operands using this representation the algorithm extends r w gosper s 1972 partial quotient arithmetic algorithm and allows the design of an online arithmetic unit with computations granularized at the signed bit level each input or output port can be independently set to receive produce operands result in either binary radix or the binary rational representation the authors investigate by simulation the interconnection of several such units for the parallel computation of more complicated expressions in a tree pipelined manner with particular regard to measuring individual and compounded online delays

approximation order of bivariate spline interpolation for arbitrary smoothness by using the algorithm of nurnberger and riessinger 1995 we construct hermite interpolation sets for spaces of bivariate splines sqr 1 of arbitrary smoothness defined on the uniform type triangulations it is shown that our hermite interpolation method yields optimal approximation order for q 3 5r 1 in order to prove this we use the concept of weak interpolation and arguments of birkhoff interpolation

exceptional scatteredness in prime degree abstract let q be an odd prime power and n an integer let l f q n x be a q linearized t scattered polynomial of linearized degree r let d max t r be an odd prime number in this paper we show that under these assumptions it follows that l x our technique involves a galois theoretical characterization of t scattered polynomials combined with the classification of transitive subgroups of the general linear group over the finite field f q

on smooth multivariate spline functions in this paper the dimensions of bivariate spline spaces with simple cross cut grid partitions are determined and expressions of their basis functions are given consequently the closures of these spaces over all partitions of the same type can be determined a somewhat more detailed study on bivariate splines with rectangular grid partitions is included the results in this paper can be applied to problems on interpolation and approximation by bivariate spline functions

construction of local c 1 quartic spline elements for optimal order approximation this paper is concerned with a study of approximation order and construction of locally supported elements for the space s 4 1 of c 1 pp piecewise polynomial functions on an arbitrary triangulation of a connected polygonal domain in r 2 it is well known that even when a is a three directional mesh 1 the order of approximation of s 4 1 1 is only 4 not 5 the objective of this paper is two fold i a local clough tocher refinement procedure of an arbitrary triangulation a is introduced so as to yield the optimal fifth order of approximation where locality means that only a few isolated triangles need refinement and ii locally supported hermite elements are constructed to achieve the optimal order of approximation

hex splines a novel spline family for hexagonal lattices this paper proposes a new family of bivariate nonseparable splines called hex splines especially designed for hexagonal lattices the starting point of the construction is the indicator function of the voronoi cell which is used to define in a natural way the first order hex spline higher order hex splines are obtained by successive convolutions a mathematical analysis of this new bivariate spline family is presented in particular we derive a closed form for a hex spline of arbitrary order we also discuss important properties such as their fourier transform and the fact they form a riesz basis we also highlight the approximation order for conventional rectangular lattices hex splines revert to classical separable tensor product b splines finally some prototypical applications and experimental results demonstrate the usefulness of hex splines for handling hexagonally sampled data

on the dimension of bivariate spline spaces of smoothnessr and degreed 3r 1 we consider the well known spaces of bivariate piecewise polynomials of degreed defined over arbitrary triangulations of a polygonal domain and possessingr continuous derivatives globally to date dimension formulae for such spaces have been established only whend 3r 2 except for the special case wherer 1 andd 4 in this paper we establish dimension formulae for allr 1 andd 3r 1 for almost all triangulations

on the role of exponential splines in image interpolation a sobolev reproducing kernel hilbert space approach to image interpolation is introduced the underlying kernels are exponential functions and are related to stochastic autoregressive image modeling the corresponding image interpolants can be implemented effectively using compactly supported exponential b splines a tight l2 upper bound on the interpolation error is then derived suggesting that the proposed exponential functions are optimal in this regard experimental results indicate that the proposed interpolation approach with properly tuned signal dependent weights outperforms currently available polynomial b spline models of comparable order furthermore a unified approach to image interpolation by ideal and nonideal sampling procedures is derived suggesting that the proposed exponential kernels may have a significant role in image modeling as well our conclusion is that the proposed sobolev based approach could be instrumental and a preferred alternative in many interpolation tasks

error free solution to a toeplitz system of equations a numerical method which allows error free computation of the solution to a toeplitz system of equations is developed the method requires that the system of equations have rational entries to avoid error that is inherent in floating point arithmetic multiple modulus residue arithmetic is applied to a modified version of the levinson algorithm and to an algorithm presented by s y kung and y h hu 1983 the error free method has a highly parallel structure and can be implemented with existing software and hardware the exact method presented here serves as an outline for the general approach to the development of error free solution methods

orthogonal and biorthogonal fir hexagonal filter banks with sixfold symmetry recently hexagonal image processing has attracted attention the hexagonal lattice has several advantages in comparison with the rectangular lattice the conventionally used lattice for image sampling and processing for example a hexagonal lattice needs less sampling points it has better consistent connectivity it has higher symmetry and its structure is plausible to human vision systems the multiresolution analysis method has been used for hexagonal image processing since the hexagonal lattice has high degree of symmetry it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry which is pertinent to the symmetry structure of the hexagonal lattice the orthogonal or prefect reconstruction pr hexagonal filter banks that are available in the literature have only threefold symmetry in this paper we investigate the construction of orthogonal and pr finite impulse response fir hexagonal filter banks with sixfold symmetry we obtain block structures of 7 size refinement seven channel two dimensional orthogonal and pr fir hexagonal filter banks with sixfold rotational symmetry radic7 refinement orthogonal and biorthogonal wavelets based on these block structures are constructed in this paper we also consider fir hexagonal filter banks with axial line symmetry and we present a block structure of fir hexagonal filter banks with pseudo sixfold axial symmetry

sub hexagonal phase correlation for motion estimation we present a novel frequency domain motion estimation technique which operates on hexagonal images and employs the hexagonal fourier transform our method involves image sampling on a hexagonal lattice followed by a normalised hexagonal cross correlation in the frequency domain the term subpixel or subcell is defined on a hexagonal grid in order to achieve floating point registration experiments using both artificially induced motion and actual motion demonstrate that the proposed method outperforms the state of the art in frequency domain motion estimation operating on a square lattice in the shape of phase correlation in terms of subpixel accuracy for a range of test material and motion scenarios

swapping edges of arbitrary triangulations to achieve the optimal order of approximation in the representation of scattered data by smooth pp piecewise polynomial functions perhaps the most important problem is to find an optimal triangulation of the given sample sites called vertices of course the notion of optimality depends on the desirable properties in the approximation or modeling problems in this paper we are concerned with optimal approximation order with respect to the given order r of smoothness and degree k of the polynomial pieces of the smooth pp functions we will only consider c1 pp approximation with r 1 and k 4 the main result in this paper is an efficient method for triangulating any finitely many arbitrarily scattered sample sites such that these sample sites are the only vertices of the triangulation and that for any discrete data given at these sample sites there is a c1 piecewise quartic polynomial on this triangulation that interpolates the given data with the fifth order of approximation

quantitative error analysis for the reconstruction of derivatives we present a general fourier based method which provides an accurate prediction of the approximation error when the derivative of a signal s t is continuously reconstructed from uniform point samples or generalized measurements on s this formalism applies to a wide class of convolution based techniques it provides a key tool the frequency error kernel for designing computationally efficient reconstruction schemes which are near optimal in the least squares sense

approximation in normal linear spaces abstract a historical account is given of the development of methods for solving approximation problems set in normed linear spaces approximation of both real functions and real data is considered with particular reference to lp or lp and chebyshev norms as well as coverage of methods for the usual linear problems an account is given of the development of methods for approximation by functions which are nonlinear in the free parameters and special attention is paid to some particular nonlinear approximating families

fingerprint classification using a hexagonal fast fourier transform in this paper a new method for the analysis of fingerprint images is presented a hexagonal fourier transform is applied that will classify fingerprints into whorls loops and arches the hexagonal fourier transform allows the utilization of hexagonally sampled data and the extention of output data in a rectangular scheme which is more convenient for treatment and interpretation results for fingerprint classification are given

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

a p p bit fraction model of binary floating point division and extremal rounding cases we introduce the ordered series fp p of irreducible p p bit fractions as a model of p bit precision binary floating point division we employ and extend results from the number theoretic literature on farey fractions and continued fractions to provide a foundation for generation and analysis of the series fp p an algorithm for ordered on the fly enumeration of a consecutive subsequence of fp p over a selected interval is introduced which requires only a couple of integer additions and or subtractions per p p bit fraction enumerated we characterize two extremal rounding boundary sets rnp respectively rdp of irreducible p p bit fractions over the standard binade 1 2 whose 2p o 1 member fractions have rational values that are each comparably close to a boundary for rounding to a normalized p bit floating point number by round to nearest respectively by a directed rounding a transformation is shown allowing either set rnp rdp to be simply computed from the other we determine properties of these extremal rounding boundary sets rnp rdp and describe their use in the testing of floating point division implementations

quasi interpolating spline models for hexagonally sampled data the reconstruction of a continuous domain representation from sampled data is an essential element of many image processing tasks in particular image resampling until today most image data have been available on cartesian lattices despite the many theoretical advantages of hexagonal sampling in this paper we propose new reconstruction methods for hexagonally sampled data that use the intrinsically 2 d nature of the lattice and that at the same time remain practical and efficient to that aim we deploy box spline and hex spline models which are notably well adapted to hexagonal lattices we also rely on the quasi interpolation paradigm to design compelling prefilters that is the optimal filter for a prescribed design is found using recent results from approximation theory the feasibility and efficiency of the proposed methods are illustrated and compared for a hexagonal to cartesian grid conversion problem

developments in bivariate spline interpolation the aim of this survey is to describe developments in the field of interpolation by bivariate splines we summarize results on the dimension and the approximation order of bivariate spline spaces and describe interpolation methods for these spaces moreover numerical examples are given

optimal biorthogonal wavelet decomposition of wire frame meshes using box splines and its application to the hierarchical coding of 3 d surfaces optimal mechanisms are determined for the hierarchical decomposition of wire frame surfaces generated by box splines a family of box splines with compact support suitable for the approximation of wire frames is first defined generated by arbitrary sampling matrices with integer eigenvalues for each such box spline the optimal positioning of the wire frame nodes is determined for each level of the hierarchical wire frame decomposition criterion of optimality is the minimization of the variance of the error difference between the original surface and its representation at each resolution level this is needed so as to ensure that the wire mesh produces at each resolution as close a replica of the original surface as possible several such combinations of box spline generated meshes and the corresponding optimal node lattice sequences are examined in detail with a view to practical application their specific application to the hierarchical coding of three dimensional 3 d wire meshes is experimentally evaluated

orthogonal and biorthogonal sqrt 3 refinement wavelets for hexagonal data processing the hexagonal lattice was proposed as an alternative method for image sampling the hexagonal sampling has certain advantages over the conventionally used square sampling hence the hexagonal lattice has been used in many areas a hexagonal lattice allows radic3 dyadic and radic7 refinements which makes it possible to use the multiresolution multiscale analysis method to process hexagonally sampled data the radic3 refinement is the most appealing refinement for multiresolution data processing due to the fact that it has the slowest progression through scale and hence it provides more resolution levels from which one can choose this fact is the main motivation for the study of radic3 refinement surface subdivision and it is also the main reason for the recommendation to use the radic3 refinement for discrete global grid systems however there is little work on compactly supported radic3 refinement wavelets in this paper we study the construction of compactly supported orthogonal and biorthogonal radic3 refinement wavelets in particular we present a block structure of orthogonal fir filter banks with twofold symmetry and construct the associated orthogonal radic3 refinement wavelets we study the sixfold axial symmetry of perfect reconstruction biorthogonal fir filter banks in addition we obtain a block structure of sixfold symmetric radic3 refinement filter banks and construct the associated biorthogonal wavelets

stability of optimal order approximation by bivariate splines over arbitrary triangulations let a be a triangulation of some polygonal domain in r2 and skr a the space of all bivariate cr piecewise polynomials of total degree 3r 2 that can be used to provide the highest order of approximation with the property that the approximation constant of this order is independent of the geometry of a with the exception of the smallest angle in the partition this result is obtained by means of a careful choice of locally supported basis functions which however require a very technical proof to justify their stability in optimal order approximation a new formulation of smoothness conditions for piecewise polynomials in terms of their b net representations is derived for this purpose

finite precision rational arithmetic an arithmetic unit the foundations of an arithmetic unit performing the add subtract multiply and divide operations on rational operands are developed the unit uses the classical euclidean algorithm as one unified algorithm for all the arithmetic operations including rounding binary implementations are discussed based on techniques known from srt division and utilizing ripple free borrow save and carry save addition average time behavior is investigated

increasing the approximation order of spline quasi interpolants abstract in this paper we show how by a very simple modification of bivariate spline discrete quasi interpolants we can construct a new class of quasi interpolants which have remarkable properties such as high order of regularity and polynomial reproduction more precisely given a spline discrete quasi interpolation operator q d which is exact on the space p m of polynomials of total degree at most m we first propose a general method to determine a new differential quasi interpolation operator q r d which is exact on p m r q r d uses the values of the function to be approximated at the points involved in the linear functional defining q d as well as the partial derivatives up to the order r at the same points from this result we then construct and study a first order differential quasi interpolant based on the c 1 cubic b spline on the equilateral triangulation with a hexagonal support when the derivatives are not available or extremely expensive to compute we approximate them by appropriate finite differences to derive new discrete quasi interpolants q d we estimate with small constants the quasi interpolation errors f q r d f and f q d f in the infinity norm finally numerical examples are used to analyze the performance of the method

finite precision rational arithmetic slash number systems fraction number systems characterized by fixed slash and floating slash formats are specified the structure of arithmetic over such systems is prescribed by the rounding obtained from best rational approximation multitiered precision hierarchies of both the fixed slash and floating slash type are described and analyzed with regards to their support of both exact rational and approximate real computation

hexagonal fast fourier transform with rectangular output hexagonal sampling is the most efficient sampling pattern for a two dimensional circularly bandlimited function a separable fast discrete fourier transform dft algorithm for hexagonally sampled data that directly computes output points on a rectangular lattice is reported no interpolation is required the algorithm has computational complexity comparable to that of standard two dimensional fast fourier transforms

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

nonideal sampling and interpolation from noisy observations in shift invariant spaces digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous time signal from a sequence of corrupted discrete time samples in this paper a general formulation of this problem is developed that treats the interpolation problem from ideal noisy samples and the deconvolution problem in which the signal is filtered prior to sampling in a unified way the signal reconstruction is performed in a shift invariant subspace spanned by the integer shifts of a generating function where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter several alternative approaches to designing the correction filter are suggested which differ in their assumptions on the signal and noise the classical deconvolution solutions least squares tikhonov and wiener are adapted to our particular situation and new methods that are optimal in a minimax sense are also proposed the solutions often have a similar structure and can be computed simply and efficiently by digital filtering some concrete examples of reconstruction filters are presented as well as simple guidelines for selecting the free parameters e g regularization of the various algorithms

upper and lower bounds on the dimension of multivariate spline spaces we give upper and lower bounds on the dimensions of multivariate spline spaces defined on triangu lations in mathbb r k the bounds are optimal in a certain sense

constructions of locally recoverable codes which are optimal let q be a prime power and mathbb f q be the finite field of size q in this paper we provide a galois theoretical framework that allows to produce good polynomials for the tamo and barg construction of optimal locally recoverable codes lrc using our approach we construct new good polynomials and therefore optimal lrcs with new parameters the existing theory of good polynomials fits entirely in our new framework the key advantage of our method is that we do not need to rely on arithmetic properties of the pair q r where r is the locality of the code

fir filter banks for hexagonal data processing images are conventionally sampled on a rectangular lattice thus traditional image processing is carried out on the rectangular lattice the hexagonal lattice was proposed more than four decades ago as an alternative method for sampling compared with the rectangular lattice the hexagonal lattice has certain advantages which include that it needs less sampling points it has better consistent connectivity and higher symmetry the hexagonal structure is also pertinent to the vision process in this paper we investigate the construction of symmetric fir hexagonal filter banks for multiresolution hexagonal image processing we obtain block structures of fir hexagonal filter banks with 3 fold rotational symmetry and 3 fold axial symmetry these block structures yield families of orthogonal and biorthogonal fir hexagonal filter banks with 3 fold rotational symmetry and 3 fold axial symmetry in this paper we also discuss the construction of orthogonal and biorthogonal fir filter banks with scaling functions and wavelets having optimal smoothness in addition we present a few of such orthogonal and biorthogonal fir filters banks

an explicit basis for c 1 quartic by various bivariate splines we establish the dimension of the space of c 1 bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain our approach is to construct a minimal determining set and an associated explicit basis for the space for general triangulations the minimal determining set must be defined globally

multivariate spline spaces in principle all of the properties and applications of multivariate spline functions can be derived from the basic theories in chapter 1 the expressions of the multivariate spline functions in 2 of chapter 1 are inconvenient to be applied directly because the expressions also rely on the solutions of the global conformality conditions so as to determine all the smooth cofactors

fast rounding in multiprecision floating slash arithmetic a computational algorithm is described which quickly rounds large fractions into a fixed length multiprecision floating slash representation using single precision operations where possible an easily calculated criterion for terminating the rounding process is given

oversampling in shift invariant spaces with a rational sampling period it is well known that under appropriate hypotheses a sampling formula allows us to recover any function in a principal shift invariant space from its samples taken with sampling period one whenever the generator of the shift invariant space satisfies the strang fix conditions of order r this formula also provides an approximation scheme of order r valid for smooth functions in this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one with the use of this oversampling technique there is not one but an infinite number of sampling formulas whenever the generator has compact support among these formulas it is possible to find one whose associated reconstruction functions have also compact support

quadrilateral finite elements of fvs type and class c rho let cal p be some partition of a planar polygonal domain omega into quadrilaterals given a smooth function u we construct piecewise polynomial functions upsilon in c rho omega of degree n 3 rho for rho odd and n 3 rho 1 for rho even on a subtriangulation tau 4 of cal p the latter is obtained by drawing diagonals in each q in cal p and upsilon vert q is a composite quadrilateral finite element generalizing the classical c 1 cubic fraeijs de veubeke and sander or fvs quadrilateral the function upsilon interpolates the derivatives of u up to order rho rho 2 at the vertices of cal p polynomial degrees obtained in this way are minimal in the family of interpolation schemes based on finite elements

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

mathematical model and implementation of rational processing precision in computations is a considerable challenge to adequately addressing many current scientific or engineering problems the way in which the numbers are represented constitutes the first step to compute them and determines the validity of the results the aim of this research is to provide a formal framework and a set of computational primitives to address high precision problems of mathematical calculation in engineering and numerical simulation the main contribution of this research is a mathematical model to build an exact arithmetical unit able to represent without error rational numbers in positional notation system the functions under consideration are addition and multiplication because they form an algebraic commutative ring which contains a multiplicative inverse for every non zero element this paper reviews other specialized arithmetic units based on existing formats to show ways to make high precision computing it is proposed a formal framework of the whole arithmetic architecture in which the operators are based then the design of the addition operation is detailed and its hardware implementation is described finally extensive evaluation of this operator is performed to prove its ability for exact processing it is introduced a formal framework for processing rational numbers a representation system based on positional notation system is described a method for calculating the addition function is detailed experiments and application example have been made to validate the model

interpolation by c 1 splines of degree q 4 on triangulations let be an arbitrary regular triangulation of a simply connected compact polygonal domain ir and let s1q denote the space of bivariate polynomial splines of degree q and smoothness 1 with respect to we develop an algorithm for constructing point sets admissible for lagrange interpolation by s1q if q 4 in the case q

voronoi splines we introduce a framework for construction of non separable multivariate splines that are geometrically tailored for general sampling lattices voronoi splines are b spline like elements that inherit the geometry of a sampling lattice from its voronoi cell and generate a lattice shift invariant spline space for approximation in rd the spline spaces associated with voronoi splines have guaranteed approximation order and degree of continuity by exploiting the geometric properties of voronoi polytopes and zonotopes we establish the relationship between voronoi splines and box splines which are used for a closed form characterization of the former for cartesian lattices voronoi splines coincide with tensor product b splines and for the 2 d hexagonal lattice the proposed approach offers a reformulation of hex splines in terms of multi box splines while the construction is for general multidimensional lattices we particularly characterize bivariate and trivariate voronoi splines for all 2 d and 3 d lattices and specifically study them for body centered cubic and face centered cubic lattices

edge detection in fingerprints abstract a suitable method for ridge detection in fingerprint is presented in this paper this proposed method blends some of the results developed in image enhancement and edge detection literature to minimize the effect of noise

exceptional scattered polynomials abstract let f be an f q linear function over f q n if the f q subspace u x q t f x x f q n defines a maximum scattered linear set then we call f a scattered polynomial of index t as these polynomials appear to be very rare it is natural to look for some classification of them we say a function f is an exceptional scattered polynomial of index t if the subspace u associated with f defines a maximum scattered linear set in pg 1 q m n for infinitely many m our main results are the classifications of exceptional scattered monic polynomials of index 0 for q 5 and of index 1 the strategy applied here is to convert the original question into a special type of algebraic curves and then to use the intersection theory and the hasse weil theorem to derive contradictions

quasi interpolation by means of filter banks we consider the problem of approximating a regular function f t from its samples f nt taken in a uniform grid quasi interpolation schemes approximate f t with a dilated version of a linear combination of shifted versions of a kernel t specifically fapprox t t af n t t n in a way that the polynomials of degree at most l 1 are recovered exactly these approximation schemes give order l i e the error is o tl where t is the sampling period recently quasi interpolation schemes using a discrete prefiltering of the samples f nt to obtain the coefficients af n have been proposed they provide tight approximation with a low computational cost in this work we generalize considering rational filter banks to prefilter the samples instead of a simple filter this generalization provides a greater flexibility in the design of the approximation scheme the upsampling and downsampling ratio r of the rational filter bank plays a significant role when r 1 the scheme has similar characteristics to those related to a simple filter approximation schemes corresponding to smaller ratios give less approximation quality but in return they have less computational cost and involve less storage load in the system

approximation order from certain spaces of smooth bivariate splines on a three direction mesh determination de l ordre exact d approximation de certains espaces de fonctions splines regulieres a deux variables

a comparison between the sampling kantorovich algorithm for digital image processing with some interpolation and quasi interpolation methods abstract in this paper we study the performance of the sampling kantorovich s k algorithm for image processing with other well known interpolation and quasi interpolation methods the s k algorithm has been implemented with three different families of kernels central b splines jackson type and bochner riesz the above method is compared in term of psnr peak signal to noise ratio and cpu time with the bilinear and bicubic interpolation the quasi fir finite impulse response and quasi iir infinite impulse response approximation experimental results show better performance of s k algorithm than the considered other ones

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

wavelet based fingerprint image retrieval this paper presents a novel approach for personal identification based on a wavelet based fingerprint retrieval system which encompasses three image retrieval tasks namely feature extraction similarity measurement and feature indexing we propose the use of different types of wavelets for representing and describing the textural information presented in fingerprint images in a compact way for that purpose the feature vectors used to characterize the fingerprints are obtained by computing the mean and the standard deviation of the decomposed images in the wavelet domain these feature vectors are used both to retrieve the most similar fingerprints given a query image and their indexation is used to reduce the search spaces of candidate images the different types of wavelets used in our study include gabor wavelets tree structured wavelet decomposition using both orthogonal and bi orthogonal filter banks as well as the steerable wavelets to evaluate the retrieval accuracy of the proposed approach a total number of eight different data sets were considered we also took into account different combinations of the above wavelets with six similarity measures the results show that the gabor wavelets combined with the square chord similarity measure achieves the best retrieval effectiveness

on super splines and finite elements super splines are spaces of multivariate splines consisting of piecewise polynomials defined on triangulations where the degree of smoothness enforced at the vertices is larger than the degree of smoothness enforced across the edges dimension formulae for super spline spaces are established and the construction of explicit bases of locally supported splines is presented in addition it is shown that many classical finite element spaces can be interpreted as super spline spaces providing a link between spline theory and the highly applicable finite element theory

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

reversible fast and high quality grid conversions a new grid conversion method is proposed to resample between two 2 d periodic lattices with the same sampling density the main feature of our approach is the symmetric reversibility which means that when using the same algorithm for the converse operation then the initial data is recovered exactly to that purpose we decompose the lattice conversion process into at most three successive shear operations the translations along the shear directions are implemented by 1 d fractional delay operators which revert to simple 1 d convolutions with appropriate filters that yield the property of symmetric reversibility we show that the method is fast and provides high quality resampled images applications of our approach can be found in various settings such as grid conversion between the hexagonal and the cartesian lattice or fast implementation of affine transformations such as rotations

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

instability in the dimension of spaces of bivariate piecewise polynomials of degree 2 r and smoothness order r the spaces of bivariate splines on a particular triangulation are considered the dimension of the space of splines with degree 2 and with smoothness order 1 on this triangulation has long been known to depend on the geometry of the triangulation this result is extended to the space of splines of degree 2r and smoothness order r the dimension of the spline space is shown to have a similar dependence on geometry for all values of r geqq 1

monotonicity preserving surface interpolation c 1 monotone quadratic splines are analyzed this motivates an algorithm for interpolating monotone data given on a rectangular grid with a c 1 monotone quadratic spline surface error estimates an operation count and numerical examples are given