regularity preserving image interpolation assumptions about image continuity lead to oversmoothed edges in common image interpolation algorithms a wavelet based interpolation method that imposes no continuity constraints is introduced the algorithm estimates the regularity of edges by measuring the decay of wavelet transform coefficients across scales and preserves the underlying regularity by extrapolating a new subband to be used in image resynthesis the algorithm produces visibly sharper edges than traditional techniques and exhibits an average peak signal to noise ratio psnr improvement of 2 5 db over bilinear and bicubic techniques

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

convolution based interpolation for fast high quality rotation of images this paper focuses on the design of fast algorithms for rotating images and preserving high quality the basis for the approach is a decomposition of a rotation into a sequence of one dimensional translations as the accuracy of these operations is critical we introduce a general theoretical framework that addresses their design and performance we also investigate the issue of optimality and present an improved least square formulation of the problem this approach leads to a separable three pass implementation of a rotation using one dimensional convolutions only we provide explicit filter formulas for several continuous signal models including spline and bandlimited representations finally we present rotation experiments and compare the currently standard techniques with the various versions of our algorithm our results indicate that the present algorithm in its higher order versions outperforms all standard high accuracy methods of which we are aware both in terms of speed and quality its computational complexity increases linearly with the order of accuracy the best quality results are obtained with the sine based algorithm which can be implemented using simple one dimensional ffts

linear interpolation revitalized we present a simple original method to improve piecewise linear interpolation with uniform knots we shift the sampling knots by a fixed amount while enforcing the interpolation property we determine the theoretical optimal shift that maximizes the quality of our shifted linear interpolation surprisingly enough this optimal value is nonzero and close to 1 5 we confirm our theoretical findings by performing several experiments a cumulative rotation experiment and a zoom experiment both show a significant increase of the quality of the shifted method with respect to the standard one we also observe that in these results we get a quality that is similar to that of the computationally more costly high quality cubic convolution

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

least squares image resizing using finite differences we present an optimal spline based algorithm for the enlargement or reduction of digital images with arbitrary noninteger scaling factors this projection based approach can be realized thanks to a new finite difference method that allows the computation of inner products with analysis functions that are b splines of any degree n a noteworthy property of the algorithm is that the computational complexity per pixel does not depend on the scaling factor a for a given choice of basis functions the results of our method are consistently better than those of the standard interpolation procedure the present scheme achieves a reduction of artifacts such as aliasing and blocking and a significant improvement of the signal to noise ratio the method can be generalized to include other classes of piecewise polynomial functions expressed as linear combinations of b splines and their derivatives

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

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

construction of image reduction operators using averaging aggregation functions in this work we present an image reduction algorithm based on averaging aggregation functions we axiomatically define the concepts of image reduction operator and local reduction operator we study the construction of the latter by means of averaging functions and we propose an image reduction algorithm image reduction operator we analyze the properties of several averaging functions and their effect on the image reduction algorithm finally we present experimental results where we apply our algorithm in two different applications analyzing the best operators for each concrete application

an efficient method for fitting large data sets using t splines data fitting is a fundamental tool in scientific research and engineering applications generally there are two ingredients in solving data fitting problems one is the fitting representation and the other is the fitting method nowadays the fitting of larger and larger quantities of data sets requires more compact fitting representation and faster fitting methods the t spline is a recently invented spline representation whose control mesh t mesh allows a row of control points to terminate thus reducing the number of superfluous control points in the b spline representation significantly this property makes t splines more compact than b splines in fitting large data sets however the adaptivity of the t spline causes the coefficient matrix of a least squares fitting linear system to lose its block structure thus when fitting large data sets with t splines by iterative methods only point iterative methods can be used and the iteration speeds of typically employed point iterative methods are r

interpolation by the fft revisited an experimental investigation a numerical investigation into the accuracy of interpolation by fast fourier transform fft using a sinusoidal test signal is described the method is precisely defined including a previously unnoticed detail which makes a significant difference to the accuracy of the result the experiments show that with no input windowing the accuracy of interpolation is almost independent of sinusoidal wavelength very close to the nyquist limit the resulting rms error is inversely proportional to input sequence length and is very low for sequence lengths likely to be encountered in practice as wavelength passes through the nyquist limit there is a sudden increase in error as is expected from sampling theory if the sequence ends are windowed by short cosine half bells accuracy is further improved at longer wavelengths in comparison small kernal convolution methods such as linear interpolation and cubic convolution perform badly at wavelengths anywhere near the nyquist limit

enlargement or reduction of digital images with minimum loss of information the purpose of this paper is to derive optimal spline algorithms for the enlargement or reduction of digital images by arbitrary noninteger scaling factors in our formulation the original and rescaled signals are each represented by an interpolating polynomial spline of degree n with step size one and spl delta respectively the change of scale is achieved by determining the spline with step size spl delta that provides the closest approximation of the original signal in the l sub 2 norm we show that this approximation can be computed in three steps i a digital prefilter that provides the b spline coefficients of the input signal ii a resampling using an expansion formula with a modified sampling kernel that depends explicitly on spl delta and iii a digital postfilter that maps the result back into the signal domain we provide explicit formulas for n 0 1 and 3 and propose solutions for the efficient implementation of these algorithms we consider image processing examples and show that the present method compares favorably with standard interpolation techniques finally we discuss some properties of this approach and its connection with the classical technique of bandlimiting a signal which provides the asymptotic limit of our algorithm as the order of the spline tends to infinity

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

quasi exact solvability resonances and trivial monodromy in ordinary differential equations a correspondence between the sextic anharmonic oscillator and a pair of third order ordinary differential equations is used to investigate the phenomenon of quasi exact solvability for eigenvalue problems involving differential operators with order greater than 2 in particular links with bender dunne polynomials and resonances between independent solutions are observed for certain second order cases and extended to the higher order problems this article is part of a special issue of journal of physics a mathematical and theoretical devoted to quantum physics with non hermitian operators

sampling for spline reconstruction the use of a prefilter prior to sampling or resampling can lead to reduced mean square error after reconstruction this means that the samples should be obtained from the original data by convolution with an optimal local weighting function whose form depends upon the reconstruction method to be used we display and discuss these weighting functions for the most common reconstruction methods namely nearest neighbor linear and cubic spline interpolation

an edge guided image interpolation algorithm via directional filtering and data fusion preserving edge structures is a challenge to image interpolation algorithms that reconstruct a high resolution image from a low resolution counterpart we propose a new edge guided nonlinear interpolation technique through directional filtering and data fusion for a pixel to be interpolated two observation sets are defined in two orthogonal directions and each set produces an estimate of the pixel value these directional estimates modeled as different noisy measurements of the missing pixel are fused by the linear minimum mean square error estimation lmmse technique into a more robust estimate using the statistics of the two observation sets we also present a simplified version of the lmmse based interpolation algorithm to reduce computational cost without sacrificing much the interpolation performance experiments show that the new interpolation techniques can preserve edge sharpness and reduce ringing artifacts

polynomial spline signal approximations filter design and asymptotic equivalence with shannon s sampling theorem the least squares polynomial spline approximation of a signal g t in l sub 2 r is obtained by projecting g t on s sup n r the space of polynomial splines of order n it is shown that this process can be linked to the classical problem of cardinal spline interpolation by first convolving g t with a b spline of order n more specifically the coefficients of the b spline interpolation of order 2n 1 of the sampled filtered sequence are identical to the coefficients of the least squares approximation of g t of order n it is shown that this approximation can be obtained from a succession of three basic operations prefiltering sampling and postfiltering which confirms the parallel with the classical sampling reconstruction procedure for bandlimited signals the frequency responses of these filters are determined for three equivalent spline representations using alternative sets of shift invariant basis functions of s sup n r the standard expansion in terms of b spline coefficients a representation in terms of sampled signal values and a representation using orthogonal basis functions

image data compression using cubic convolution spline interpolation a new cubic convolution spline interpolation ccsi for both one dimensional 1 d and two dimensional 2 d signals is developed in order to subsample signal and image compression data the ccsi yields a very accurate algorithm for smoothing it is also shown that this new and fast smoothing filter for ccsi can be used with the jpeg standard to design an improved jpeg encoder decoder for a high compression ratio

warped distance for space variant linear image interpolation the problem of image interpolation using linear techniques is dealt with in this paper conventional space invariant methods are revisited and changed into space variant ones by introducing the concept of the warped distance among the pixels of an image a better perceptual rendition of the image details is obtained in this way this effect is proved both via the evaluation of the response to an idealized sigmoidal edge model and with experiments on real world images the computational costs of the proposed approach are very small when compared to those of state of the art nonlinear interpolation operators

rectification for cone beam projection and backprojection the purpose of this paper is to derive a technique for accelerating the computation of cone beam forward and backward projections that are the basic steps of tomographic reconstruction the cone beam geometry of c arm systems is commonly described with projection matrices such matrices provide a continuous framework for analyzing the flow of operations needed to compute backprojection for analytical reconstruction as well as the combination of forward and backward projections for iterative reconstruction the proposed rectification technique resamples the original data to planes that are aligned with two of the reconstructed volume main axes so that the original cone beam geometry can be replaced by a simpler geometry where succession of plane magnifications are involved only rectification generalizes previous independent results to the cone beam backprojection of preprocessed data as well as to cone beam iterative reconstruction the memory access pattern of simple magnifications provides superior predictability and is therefore easier to optimize independently of the choice of the interpolation technique rectification is also shown to provide control over interpolation errors through oversampling allowing tradeoffs between computation speed and precision to be set experimental results are provided for linear and nearest neighbor interpolations based on simulations as well as phantom and patient data acquired on a digital c arm system

a characterisation of oscillations in the discrete two dimensional convection diffusion equation it is well known that discrete solutions to the convection diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation however except for one dimensional problems there is little analysis of this phenomenon in this paper we present an analysis of the two dimensional problem with constant flow aligned with the grid based on a fourier decomposition of the discrete solution for galerkin bilinear finite element discretisations we derive closed form expressions for the fourier coefficients showing them to be weighted sums of certain functions which are oscillatory when the mesh peclet number is large the oscillatory functions are determined as solutions to a set of three term recurrences and the weights are determined by the boundary conditions these expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh peclet number and boundary conditions of the problem

edge forming methods for color image zooming this paper introduces edge forming schemes for image zooming of color images by general magnification factors in order to remove reduce artifacts arising in image interpolation such as image blur and the checkerboard effect an edge forming method is suggested to be applied as a postprocess of standard interpolation methods the method is based on nonconvex nonlinear partial differential equations the equations are carefully discretized incorporating numerical schemes of anisotropic diffusion to be able to form reliable edges satisfactorily the alternating direction implicit adi method is employed for an efficient simulation of the model it has been numerically verified that the resulting algorithm can form clear edges in 2 to 3 adi iterations various results are given to show th effectiveness and reliability of the algorithm