Minimum norm reconstruction of band-limited function from regularly or irregularly spaced samples

A technique is described for reconstructing a band-limited function from samples given at arbitrary locations on a finite interval. The method coincides with the Nyquist-Shannon interpolation formula when the samples are regularly spaced. Improvements over alternative techniques are illustrated. It is shown that the method gives the best possible reconstruction in the least-squares sense.<>     

On the transform property of a band-limited function and its samples

Sampling and reconstruction of band-limited functions are fundamental problems in communications and signal processing. This correspondence introduces a new transform pair for any well behaved but arbitrary function f(t) and, thereafter, establishes two transform relationships between a band-limited function and its sample values. This work can be viewed as yet another comment and extension on the sampling theorem.     

Correction to "On the transform property of a band-limited function and its samples"

The linear detection characteristics derived by G. H. Robertson have been condensed from 14 multicurve graphs to a single function whose accuracy is within 0.2 dB over a wide range of parameters.     

Representation of electromagnetic fields over arbitrary surfaces bya finite and nonredundant number of samples

It is shown that the electromagnetic (EM) field, radiated or scattered by bounded sources, can be accurately represented over a substantially arbitrary surface by a finite number of samples even when the observation domain is unbounded. The number of required samples is nonredundant and essentially coincident with the number of degrees of freedom of the field. This result relies on the extraction of a proper phase factor from the field expression and on the use of appropriate coordinates to parameterize the domain. It is demonstrated that the number of degrees of freedom is independent of the observation domain and depends only on the source geometry. The case of spheroidal sources and observation domains with rotational symmetry is analyzed in detail and the particular cases of spherical and planar sources are explicitly considered. For these geometries, precise and fast sampling algorithms of central type are presented, which allow an efficient recovery of EM fields from a nonredundant finite number of samples. Such algorithms are stable with respect to random errors affecting the data     

A Hybrid system for the reconstruction of a smooth function from its samples

The long-standing problem of reconstructing a function from its samples is considered again. Assuming a sequence of oversampled values, a set of appropriate idealized reconstruction filters can be defined, which do not suffer from instability or slow convergence. The realization — a cascade of a nonrecursive digital filter, D/A-converter, and a fixed/analog smoothing filter — demands the design of the digital filter for the increase of the sampling rate. The design of this nonrecursive filter is the purpose of this paper. Approximations in the frequency as well as in the time domain are presented.     

On the prediction of a band-limited signal from past samples

An alternate proof is given of a recent result due to A. Papoulis on predicting the current value of a wide-sense stationary band-limited random process in terms of its past samples. The possibility of such prediction is shown to be equivalent to the completeness of a certain set of complex exponentials on an interval, a property that also allows extension to the prediction of deterministic band-limited signals.     

Estimation of the mean radar reflectivity from a finite number ofcorrelated samples

The authors compare estimators of the mean radar reflectivity on images with different spectral properties. By working on complex data rather than detected images, they can take the speckle correlation into account and thus obtain more accurate estimates. A robust and computationally efficient approximation of the optimal estimator is proposed     

Linear prediction from samples of a function and its derivatives

Some formulas for the prediction of the values of a band-limited function based on its samples from the past are generalized by including past samples of its first derivative. The new sums, developed by an approach based on Newton series, make it possible to double the distance between the sample points. The resulting formulas are shown to apply to the prediction problem for a large class of entire functions of exponential type. In addition, a related prediction formula which uses past samples of successively higher derivatives is shown to behave similarly to the Taylor series approximation, again for a class of functions that includes the band-limited functions.     

The undecimated wavelet decomposition and its reconstruction.

This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in wavelet-based denoising methods. A range of examples illustrates the results.     

The ill-posedness of restoring lost samples and regularized restoration for band-limited signals

In this paper the ill-posedness of restoring lost samples is discussed. The restoration algorithm by Shannon's Sampling Theorem is analyzed. A regularized restoring algorithm for band-limited signals is presented. The convergence of the regularized restoring algorithm is studied and compared with the restoration algorithm by Shannon's sampling theorem.     

Exact image reconstruction from a limited number of projections

A new method for the exact reconstruction of any gray-scale image from its projections is proposed. The original image is projected into several view angles and the projection samples are stored in an accumulator array. In order to reconstruct the image, the accumulator array is considered as an accumulation of sinusoidal contributions each one corresponding to a certain pixel of the original image. The proposed method defines conditions for the necessary number of projections and the density of ray samples on the projection axis. These conditions insure that, for each pixel, there is at least one sample in the accumulator array where only this particular pixel contributes. This characteristic projection sample is used during the reconstruction phase to determine the coordinates and the gray-scale value of the corresponding image pixel. A variation of the method is also proposed where the reconstruction is performed using a limited number of projection samples in certain view angles. Specifically, the number of necessary samples equals at most the overall number of pixels in the original image. This approach leads to a significant reduction of memory and processing time requirements since it provides exact image reconstruction using one projection sample per pixel.     

Tomographic image reconstruction from a limited number of projections

This paper addresses the reconstruction of a function from its rectangularly sampled Radon transform. After reviewing the Radon transform sampling requirements and describing a convolution/backprojection reconstruction algorithm, two basic issues are addressed. First it is shown that even when the Radon transform of an image is adequately sampled (in the Nyquist sense), the standard convolution/backprojection algorithm may lead to a poor reconstruction unless the Radon transform is first reconstructed to a required set of projections. Secondly, when the Radon transform is undersampled in projections, reconstruction approaches are illustrated which span the range from those which achieve a high uniform-resolution reconstruction with large amounts of aliasing artifacts to those which achieve a low nonuniform-resolution reconstruction with virtually no aliasing artifacts. Reconstruction filters which seek a compromise between those two extremes are defined.     

Calculation of Fourier coefficients of a function given at a set of arbitrary points

A numerical method is presented for the calculation of Fourier coefficients of a function which is given at a discrete set of arbitrary points. The function is approximated by a sum of Cheby?shev polynomials. This is performed by Clenshaw's method of curve fitting, which is a least-squares method. The Cheby?shev coefficients are then used to construct linear combinations of Bessel functions, which are very good approximations of the Fourier coefficients.     

Induced current in and scattered field from a finite cylinder with arbitrary conductivity and permittivity

A finite cylinder with arbitrary conductivity and permittivity is illuminated by an EM wave. The induced current in and the scattered field by the cylinder are determined based on the numerical solution of a tensor integral equation. An experiment has been conducted to verify the theory. Biological and metallic cylinders are emphasized in the study.     

Model-based reconstruction of multiple circular and ellipticalobjects from a limited number of projections

We consider tomographic image reconstruction from a limited number of noisy projections. An efficient algorithm based on maximum likelihood estimation (MLE) is developed to reconstruct images of multiple discs with unknown locations and radii. The algorithm is successfully applied to images with signal-to-noise ratio (SNR) as low as 0 dB, using as few as 16 projections, and containing as many as twelve discs with widely varying radii. Experimental results show that our approach significantly outperforms conventional convolution back projection. The algorithm is successfully extended to the multiple ellipse case.     

General theory of doubly periodic arrays over an arbitrary finite field and its applications

A general theory of doubly periodic (DP) arrays over an arbitrary finite field GF(q)is presented. First the basic properties of DP arrays are examined. Next modules of linear recurring (LR) arrays are defined and their algebraic properties discussed in connection with ideals in an extension ringtilde{R}of the ringRof bivariate polynomials with coefficients in GF(q). A finitetilde{R}-module of DP arrays is shown to coincide with thetilde{R}-module of LR arrays dermed by a zero-dimensional ideal intilde{R}. Equivalence relations between DP arrays are explored, i.e., rearrangements of arrays by means of unimodular transformations. Decimation and interleaving of arrays are defined in a two-dimensional sense. The general theory is followed by application to irreducible LR arrays. Among irreducible arrays,M-arrays are a two-dimensional analog ofM-sequences and may be constructed fromM-sequences by means of unimodular transformations. The results of this paper are also important in studying properties of Abelian codes.     

The reconstruction of analog signals from the sign of their noisy samples

The reconstruction of a continuous-time signals(t)from the sign of its (deliberately) contaminated samples is considered. Sequential, generally nonlinear estimates ofs(t)are established and their performance is studied; error bounds and convergence rates are derived. The signals(t)need not be band-limited. The convergence rates obtained here are faster than those obtained previously for nonsequential estimates. The degradation in the reconstruction of the signal, due to transmission over a noisy channel, is also investigated, and bounds on the additional error are obtained.     

基于图像结构信息复数表示与奇异值分解的灰度图像质量评价方法

为了进一步突出图像结构中人眼敏感的重要特征,采用复数矩阵表示图像结构,将图像的局部方差和像素灰度值分别作为复数的实部和虚部。进而对复数矩阵进行分块奇异值分解,分析了传统奇异值分解图像质量评价方法的特点,将复数矩阵每一分块奇异值分布的标准差作为分块图像结构的表征,分别计算参考图像与待测图像对应图像分块奇异值标准差,从而得到了图像结构失真映射图谱,通过计算图谱中的数据分布特征得到最终的量化评价结果。采用LIVE数据库中包含5种失真类型的779幅测试图像验证所提的算法。试验结果表明,本文方法采用复数矩阵描述图像结构信息,平衡了对各种失真类型的敏感程度,与人眼视觉感知(HVS)的一致性优于传统方法。     

A note on exact image reconstruction from a limited number of projections

In a recent paper in this journal by Kesidis and Papamarkos “A new method for the exact reconstruction of any gray-scale image from its projections is proposed.” In this note we point out that this method is a special case of a well-known approach (peeling) and that it can produce exact reconstructions only under assumptions that are not realistic for practical methods of data collection. Further, we point out that some statements made in the paper regarding disadvantages of the algebraic reconstruction techniques (ART) as compared to the method of the paper are false.