high quality image resizing using oblique projection operators the standard interpolation approach to image resizing is to fit the original picture with a continuous model and resample the function at the desired rate however one can obtain more accurate results if one applies a filter prior to sampling a fact well known from sampling theory the optimal solution corresponds to an orthogonal projection onto the underlying continuous signal space unfortunately the optimal projection prefilter is difficult to implement when sine or high order spline functions are used we propose to resize the image using an oblique rather than an orthogonal projection operator in order to make use of faster simpler and more general algorithms we show that we can achieve almost the same result as with the orthogonal projection provided that we use the same approximation space the main advantage is that it becomes perfectly feasible to use higher order models e g splines of degree n spl ges 3 we develop the theoretical background and present a simple and practical implementation procedure using b splines our experiments show that the proposed algorithm consistently outperforms the standard interpolation methods and that it provides essentially the same performance as the optimal procedure least squares solution with considerably fewer computations the method works for arbitrary scaling factors and is applicable to both image enlargement and reduction

wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

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

h sub infinity optimal control with state feedback a h sub infinity optimal control problem in which the measured outputs are the states of the plant is considered the main result shows that the infimum of the norm of the closed loop transfer function using linear static state feedback equals the infimum of the norm of the closed loop transfer function over all stabilizing dynamic even nonlinear time varying state feedback controllers

on the oscillation of some linear difference equations with periodic coefficients abstract some linear difference equations with periodic coefficients not necessarily nonnegative are considered necessary conditions and sufficient conditions for the oscillation of the solutions are established conditions under which all nonoscillatory solutions tend to zero at are also presented the results obtained are the discrete analogues of the oscillation results for some linear delay differential equations with periodic coefficients which were given earlier by the second author oscillations of some delay differential equations with periodic coefficients j math anal appl 162 1991 452 475

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

curvature interpolation method for image zooming we introduce a novel image zooming algorithm called the curvature interpolation method cim which is partial differential equation pde based and easy to implement in order to minimize artifacts arising in image interpolation such as image blur and the checkerboard effect the cim first evaluates the curvature of the low resolution image after interpolating the curvature to the high resolution image domain the cim constructs the high resolution image by solving a linearized curvature equation incorporating the interpolated curvature as an explicit driving force it has been numerically verified that the new zooming method can produce clear images of sharp edges which are already denoised and superior to those obtained from linear methods and pde based methods of no curvature information various results are given to prove effectiveness and reliability of the new method

existence and comparison theorems for algebraic riccati equations for continuous and discrete time systems abstract we discuss two comparison theorems for algebraic riccati equations of the form xbr 1 b x x a br 1 c a br 1 c x q c r 1 c 0 simultaneously we give sufficient conditions to obtain the existence of the maximal hermitian solution of a riccati equation from the existence of a hermitian solution of a second riccati equation further similar results are given for discrete algebraic riccati equations

asymptotic error expansion of wavelet approximations of smooth functions ii we generalize earlier results concerning an asymptotic error expansion of wavelet approximations the properties of the monowavelets which are the building blocks for the error expansion are studied in more detail and connections between spline wavelets and euler and bernoulli polynomials are pointed out the expansion is used to compare the error for different wavelet families we prove that the leading terms of the expansion only depend on the multiresolution subspaces v j and not on how the complementary subspaces w j are chosen consequently for a fixed set of subspaces v j the leading terms do not depend on the fact whether the wavelets are orthogonal or not we also show that daubechies orthogonal wavelets need in general one level more than spline wavelets to obtain an approximation with a prescribed accuracy these results are illustrated with numerical examples

h guaranteed cost control for uncertain continuous time linear systems abstract this paper deals with the h guaranteed cost control problems for continuous time uncertain systems it consist of the determination of a stabilizing state feedback gain which imposes on all possible closed loop models an h norm upper bound 0 assuming that the uncertain domain is convex bounded and the uncertain system is quadratic stabilizable with disturbance attenuation it is shown how to determine by means of a convex programming problem the global minimum of as a particular and important case for precisely known linear systems the last problem reduces to the classical h optimal control problem the results follow from the definition of a special parameter space on which the above mentioned problems are convex

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

on the optimality of ideal filters for pyramid and wavelet signal approximation the reconstructed lowpass component in a quadrature mirror filter qmf bank provides a coarser resolution approximation of the input signal since the outputs of the two qmf branches are orthogonal the transformation that provides the maximum energy compaction in the lowpass channel is also the one that results in the minimum approximation error this property is used as a common strategy for the optimization of qmf banks orthogonal wavelet transforms and least squares pyramids a general solution is derived for the qmf bank that provides the optimal decomposition of an arbitrary wide sense stationary process this approach is extended to the continuous case to obtain the minimum error approximation of a signal at a given sampling rate in particular it is shown that the sine wavelet transform is optimal for the representation at all scales of signals with non increasing spectral density

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

optimized least square nonuniform fast fourier transform the main focus of this paper is to derive a memory efficient approximation to the nonuniform fourier transform of a support limited sequence we show that the standard nonuniform fast fourier transform nufft scheme is a shift invariant approximation of the exact fourier transform based on the theory of shift invariant representations we derive an exact expression for the worst case mean square approximation error using this metric we evaluate the optimal scale factors and the interpolator that provides the least approximation error we also derive the upper bound for the error component due to the lookup tablebased evaluation of the interpolator we use this metric to ensure that this component is not the dominant one theoretical and experimental comparisons with standard nufft schemes clearly demonstrate the significant improvement in accuracy over conventional schemes especially when the size of the uniform fast fourier transform fft is small since the memory requirement of the algorithm is dependent on the size of the uniform fft the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands

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

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

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

viewing distance aware super resolution for high definition display in this paper we propose a novel algorithm for high definition displays to enlarge low resolution images while maintaining perceptual constancy i e the same field of view perceptual blur radius and the retinal image size in viewer s eyes we model the relationship between a viewer and a display by considering two main aspects of visual perception i e scaling factor and perceptual blur radius as long as we enlarge an image while adjust its image blur levels on the display we can maintain viewer s perceptual constancy we show that the scaling factor should be set in proportion to the viewing distance and the blur levels on the display should be adjusted according to the focal length of a viewer toward this we first refer to edge directions to interpolate a low resolution image with the increasing of viewing distance and the scaling factor after images are interpolated we utilize a local contrast to estimate the spatially varying image blur levels of the interpolated image we then further adjust the image blur levels using a parametric deblurring method which combines l1 as well as l2 reconstruction errors and tikhonov with total variation regularization terms by taking these factors into account high resolution images adaptive to viewing distance on a display can be generated experimental results on both natural image metric and user subjective studies across image scales demonstrate that the proposed super resolution algorithm for high definition displays performs favorably against the state of the art methods

h control by state feedback an iterative algorithm and characterization of high gain occurrence we present an iterative algorithm to compute the achievable h norm by state feedbak for a standard regular problem and give a characterization when high gain feedback is necessary to approach the optimal value if there is no need for high gain components to approximate the optimal value we can reduce the problem to the computation of the h norm of a certain stable transfer matrix

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

h optimal actuator location in many control applications there is freedom to place the actuators the actuator location should be chosen to optimize certain performance objectives in this paper h performance with state feedback is considered that is both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output a framework for calculating h optimal actuator locations is developed conditions for well posedness of the h optimal actuator location problem are presented many optimal actuator problems involve systems modelled by partial differential equations and conditions under which approximations yield reliable results are given a derivative free optimization algorithm to calculate h optimal actuator locations is described the results are illustrated using several examples

all fixed order h sub spl infin controllers observer based structure and covariance bounds this note obtains a parameterization of the set of all stabilizing controllers of order less than or equal to the plant which yields for the closed loop transfer matrix a specified h sub spl infin norm bound the algebraic results of covariance control are applied to the h sub spl infin control problem to yield a parameterization in terms of the lyapunov matrix which carries many system properties such as h sub 2 performance covariance bounds system entropy at infinity etc all low order h sub spl infin controllers are shown to have observer based structure for reduced order models of the plant and are characterized by two riccati equations with a coupling condition

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

shifting interpolation kernel toward orthogonal projection orthogonal projection offers the optimal solution for many sampling reconstruction problems in terms of the least square error in the standard interpolation setting where the sampling is assumed to be ideal however the projection is impossible unless the interpolation kernel is related to the sinc function and the input is bandlimited in this paper we propose a notion of shifting kernel toward the orthogonal projection for a given interpolation kernel we formulate optimization problems whose solutions lead to shifted interpolations that while still being interpolatory are closest to the orthogonal projection in the sense of the minimax regret the quality of interpolation is evaluated in terms of the average approximation error over input shift for the standard linear interpolation we obtain several values of optimal shift dependent on a priori information on input signals for evaluation we apply the new shifted linear interpolations to a gaussian signal an ecg signal a speech signal a two dimensional signal and three natural images significant improvements are observed over the standard and the 0 21 shifted linear interpolation proposed early

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

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

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

decentralized control through parameter space optimization the decentralized control problem for linear dynamic systems is revisited using a parameter space approach which enables the definition of the decentralized feedbacks from the existence of non empty parameter convex sets the convexity property enables the derivation of efficient numerical algorithms based on standard approaches in convex programming the continuous time and discrete time cases are investigated and the decentralized control design is also treated to meet other important assignments such as optimal h2 performance index absolute stability h prescribed attenuation and robustness against actuator failures some numerical experiments illustrate the potential of this new control design

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

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

h control with regional stability constraints abstract the problem of static state feedback h control with various regional stability constraints for the closed loop system is considered a general framework for solving this problem is established using the lagrange multiplier method in conjunction with generalised lyapunov theory and riccati equation based h optimization characterization necessary conditions are given for the existence of the desired optimal feedback gains furthermore as the simplest regional stability case the stability degree problem is studied with two different approaches one an indirect overbounding method that extends the standard h problem formulation with an shift and another that uses the direct optimization of the h performance measure with stability degree as a constraint under the general framework the conservativeness of the former approach and the generality of the latter approach are clearly demonstrated finally the theory is illustrated with examples and the computational issues are discussed it is shown that the successive approximation technique of liu and of liu and yedavalli can be used to partially solve the problem

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

high quality parallel ray x ray ct back projection using optimized interpolation we propose a new cost efficient method for computing back projections in parallel ray x ray ct forward and back projections are the basis of almost all x ray ct reconstruction methods but computing these accurately is costly in the special case of parallel ray geometry it turns out that reconstruction requires back projection only one approach to accelerate the back projection is through interpolation fit a continuous representation to samples of the desired signal then sample it at the required locations instead we propose applying a prefilter that has the effect of orthogonally projecting the underlying signal onto the space spanned by the interpolator which can significantly improve the quality of the interpolation we then build on this idea by using oblique projection which simplifies the computation while giving effectively the same improvement in quality our experiments on analytical phantoms show that this refinement can improve the reconstruction quality for both filtered back projection and iterative reconstruction in the high quality regime i e with low noise and many measurements

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed