The output image of a digital camera is subject to a severe degradation due to noise in the image sensor. This paper proposes a novel technique to combine demosaicing and denoising procedures systematically into a single operation by exploiting their obvious similarities. We first design a filter as if we are optimally estimating a pixel value from a noisy single-color (sensor) image. With additional constraints, we show that the same filter coefficients are appropriate for color filter array interpolation (demosaicing) given noisy sensor data. The proposed technique can combine many existing denoising algorithms with the demosaicing operation. In this paper, a total least squares denoising method is used to demonstrate the concept. The algorithm is tested on color images with pseudorandom noise and on raw sensor data from a real CMOS digital camera that we calibrated. The experimental results confirm that the proposed method suppresses noise (CMOS/CCD image sensor noise model) while effectively interpolating the missing pixel components, demonstrating a significant improvement in image quality when compared to treating demosaicing and denoising problems independently.

In this paper, we present a method for removing noise from digital images corrupted with additive, multiplicative, and mixed noise. An image patch from an ideal image is modeled as a linear combination of image patches from the noisy image. We propose to fit this model to the real-world image data in the total least square (TLS) sense, because the TLS formulation allows us to take into account the uncertainties in the measured data. We develop a method to reduce the contribution from the irrelevant image patches, which will sharpen the edges and reduce edge artifacts at the same time. Although the proposed algorithm is computationally demanding, the image quality of the output image demonstrates the effectiveness of the TLS algorithms.

In this paper, we present a method for removing noise from digital images corrupted with additive, multiplicative, and mixed noise. An image patch from an ideal image is modeled as a linear combination of image patches from the noisy image. We propose to fit this image model to the real-world image data in the total least square (TLS) sense, because the TLS formulation allows us to take into account the uncertainties in the measured data. We develop a method to reduce the contribution from the irrelevant image patches, which will sharpen the edges and reduce edge artifacts at the same time. Although the proposed algorithm is computationally demanding, the image quality of the output image demonstrates the effectiveness of the TLS algorithms.

The output image of a digital camera is subject to a severe degradation due to noise in the image sensor. This paper proposes a novel technique to combine demosaicing and denoising procedures systematically into a single operation by exploiting their obvious similarities. We first design a filter as if we are optimally estimating a pixel value from a noisy single-color image. With additional constraints, we show that the same filter coefficients are appropriate for CFA interpolation (demosaicing) given noisy sensor data. The proposed technique can combine many existing denoising algorithms with the demosaicing operation. In this paper, a total least squares denoising method is used to demonstrate the concept. The algorithm is tested on color images with pseudo-random noise and on raw sensor data from a real CMOS digital camera that we calibrated. The experimental results confirm that the proposed method suppresses noise (CMOS image sensor noise model) while effectively interpolating the missing pixel components, demonstrating a significant improvement in image quality when compared to treating demosaicing and denoising problems independently.

The problem of adjusting the color such that the output image from a digital camera, viewed under a standard condition, matches the scene observed by the photographer's eye is called white-balance. While most white-balance algorithms approach the problem using the coefficient law (von Kries), the coefficient law has been shown inaccurate. In this paper, we instead formulate the white-balance problem using Jameson and Hurvich's induced opponent response chromatic adaptation theory. The solution to this white-balance problem reduces to a single matrix multiplication. The experimental results using existing illuminant estimation methods verify that the induced opponent response approach to solving the white-balance problem yields more neutral colors in the white panels of the Macbeth color chart than the traditional methods. The computational cost of the proposed method is virtually zero.

A cost-effective digital camera uses a single-image sensor, applying alternating patterns of red, green, and blue color filters to each pixel location. A way to reconstruct a full three-color representation of color images by estimating the missing pixel components in each color plane is called a demosaicing algorithm. This paper presents three inherent problems often associated with demosaicing algorithms that incorporate two-dimensional (2-D) directional interpolation: misguidance color artifacts, interpolation color artifacts, and aliasing. The level of misguidance color artifacts present in two images can be compared using metric neighborhood modeling. The proposed demosaicing algorithm estimates missing pixels by interpolating in the direction with fewer color artifacts. The aliasing problem is addressed by applying filterbank techniques to 2-D directional interpolation. The interpolation artifacts are reduced using a nonlinear iterative procedure. Experimental results using digital images confirm the effectiveness of this approach.

Color images in single-chip digital cameras are obtained by interpolating mosaiced color samples. These samples are encoded in a single-chip charge-coupled device by sampling the light after it passes through a color filter array (CFA) that contains different color filters (i.e., red, green, and blue) placed in some pattern. The resulting sparsely sampled images of the three-color planes are interpolated to obtain the complete color image. Interpolation usually introduces color artifacts due to the phase-shifted, aliased signals introduced by the sparse sampling of the CFAs. This paper introduces a nonlinear interpolation scheme based on edge information that produces high-quality visual results. The new method is especially good at reconstructing the image around edges, a place where the visual human system is most sensitive.

This paper begins with a description of the curve tracing problem and compares it with curve extraction, a fundamentally different problem. Issues concerning curve tracing are addressed. We discuss a good curve tracing algorithm proposed by Carsten Steger and improve this method to suit the goals of curve tracing. Results show that the improved algorithm emulates human perception by using both global and local features.

Image interpolation is a key aspect of digital image processing. This paper presents a novel interpolation method based on optimal recovery and adaptively determining the quadratic signal class from the local image behavior. The advantages of the new interpolation method are the ability to interpolate directly by any factor and to model properties of the data acquisition system into the algorithm itself. Through comparisons with other algorithms it is shown that the new interpolation is not only mathematically optimal with respect to the underlying image model, but visually it is very efficient at reducing jagged edges, a place where most other interpolation algorithms fail.

Most cost-effective digital camera uses a single image sensor, applying alternating patterns of red, green, and blue color filters to each pixel location. Demosaicing algorithm reconstructs a full three-color representation of color images from this sensor data. This paper identifies three inherent problems often associated with directional interpolation approach to demosaicing algorithms: misguidance color artifacts, interpolation color artifacts, and aliasing. The level of misguidance color artifacts present in two images can be compared using metric neighborhood modeling. The proposed demosaicing algorithm estimates missing pixels by interpolating in the direction with fewer color artifacts. The aliasing problem is addressed by applying filterbank techniques to directional interpolation. The interpolation artifacts are reduced using a nonlinear iterative procedure. Experimental results using digital images confirm the effectiveness of this approach.

This paper presents a novel approach to image denoising using adaptive principal components. Our assumptions are that the image is corrupted by additive white Gaussian noise. The new denoising technique performs well in terms of image visual fidelity, and in terms of PSNR values, the new technique compares very well against some of the most recently published denoising algorithms.

In previous work Muresan and Parks (see ICIP 2001, Greece, 2001) considered the problem of image interpolation from an adaptive optimal recovery point of view. They showed how a training set S determines a quadratic signal class and how to use this signal class to perform image interpolation. In that work the training set S was taken from the low resolution version of the image they were interpolating. In this paper we continue our discussion of the method presented previously by looking more closely at the training set S. In particular, we show how a training set of high resolution images can give very good interpolation results through the use of the method.

Hidden Markov trees in the wavelet domain are capable of accurately modeling the statistical behavior of real world signals by exploiting relationships between coefficients in different scales. The model is used to interpolate images by predicting coefficients at finer scales. Various optimizations and post-processing steps are also investigated to determine their effect on the performance of the interpolation. The interpolation algorithm was found to produce noticeably sharper images with PSNR values which outperform many other interpolation techniques on a variety of images.

We consider the problem of image interpolation using adaptive optimal recovery. We adaptively estimate the local quadratic signal class of our image pixels. We then use optimal recovery to estimate the missing local samples based on this quadratic signal class. This approach tends to preserve edges, interpolating along edges and not across them.

We consider the problem of image interpolation from an adaptive optimal recovery point of view. Many different standard interpolation approaches may be viewed through the prism of optimal recovery. We review some standard image interpolation methods and how they relate to optimal recovery as well as introduce a broader, more general and systematic approach to image interpolation using adaptive optimal recovery.

The detection and estimation of multi-periodic signals of unknown periods in white Gaussian noise is investigated. New estimates for the sub-signals (signals making up the received signal) and their periods are derived using an orthogonal subspace decomposition approach.

In the problem of image interpolation, most of the difficulties arise in areas around edges and sharp changes. Around edges, many interpolation methods tend to smooth and blur image detail. Fortunately, most of the signal information is often carried around edges and areas of sharp changes and can be used to predict these missing details from a sampled image. A method for adding image detail based on the cone of influence, the evolution of the wavelet coefficients across scales, is presented.

An analysis of a discrete time-frequency distribution yields a new periodic wide band probing signal for use in unknown system identification. The derivation is based on the mathematical properties of the discrete Wigner distribution. Like the continuous distribution, the discrete version also satisfies the covariance property, meaning transformations in the time-frequency plane are equivalent to transformations in the time domain. By utilizing this property, the linearly swept frequency measurement is extended to discrete, periodic signals. The resulting probing signal possesses favorable characteristics such as a short illumination time requirement and good resistance to noise. The performance of the proposed probing method is compared with m-sequence methods and chirp signal methods.

A reduction in error is obtained for estimates of range computed in a multipath environment. A good estimation technique for this problem typically involves exploiting the multipath interference, that is, analyzing each received signal reflection to improve the estimate. Under certain conditions, however, the errors in multipath estimates can be quite large. We compute the Cramer-Rao bounds for an example satisfying these conditions to demonstrate this phenomenon. We propose a modification to the original signal model which better represents the received signals. While the modified model does introduce a bias error, we provide a suitable estimate of the bias error so that a complete error analysis of the modified model is possible. The total error (noise error plus bias error) for the modified model is computed for an example and compares favorably with the error obtained via the original signal model.

In this correspondence, we address the problem of translation sensitivity of conventional wavelet transforms for two-dimensional (2-D) signals. We propose wavelet transform algorithms that achieve the following desirable properties simultaneously: (i) translation invariance, (ii) reduced edge effects, and (iii) size-limitedness. We apply this translation invariant biorthogonal wavelet transform with symmetric extensions to image coding applications with good results.

Locally most powerful tests and invariant tests are combined in a new detector to test for the presence of a signal in additive white Gaussian noise. Such tests, locally most powerful invariant (LMPI) tests, are defined using a number of orthogonal linear subspaces, and the second-order signal statistics in each subspace.

The design of constrained multirate systems using a relative l/sup 2/ error criterion is considered. A general algorithm is proposed to solve the problem. One application of the algorithm is the design of a new class of multirate filters for signal decomposition-projection filters. These multirate systems are projection operators that optimally approximate linear time-invariant filters in the l/sup 2/ norm. A second application of constrained multirate filter design is also presented-optimal design of multistage multirate systems. Examples illustrate the new design method and its advantages over design methods intended for linear time-invariant systems.

The concept of rotations in continuous-time, continuous-frequency is extended to discrete-time, discrete-frequency as it applies to the Wigner distribution. As in the continuous domain, discrete rotations are defined to be elements of the special orthogonal group over the appropriate (discrete) field. Use of this definition ensures that discrete rotations will share many of the same mathematical properties as continuous ones. A formula is given for the number of possible rotations of a prime-length signal, and an example is provided to illustrate what such rotations look like. In addition, by studying a 90 degree rotation, we formulate an algorithm to compute a prime-length discrete Fourier transform (DFT) based on convolutions and multiplications of discrete, periodic chirps. This algorithm provides a further connection between the DFT and the discrete Wigner distribution based on group theory.

Nonsmooth optimization is applied to the design of transmitter and receiver filters for data transmission. This optimization method allows tradeoffs between several performance measures-ISI, transmitter filter average power output, and receiver filter power gain. An FCC mask constraint on the transmitter filter is imposed. An example illustrates the design method.

The use of cone classes for signal detection and estimation is relatively new. Cone classes and the cone detector have shown potential as a viable alternative to subspace detection. The addition of new quadratic forms that model a wider range of signal characteristics, and the extension to using intersections of single cones, now makes it possible to apply cone classes to a wider range of practical problems. These extensions have resulted in a number of different types of intersection classes. Each type requires a different algorithm to implement the generalized likelihood ratio test (GLRT) used for signal detection. The authors classify these types and outline the possible solutions to implement the GLRT. New examples of signal classes modeled using cones are also given.

The design of rate-changing multirate systems using a maximum relative l/sup 2/-error criterion is analyzed. Using multirate techniques, the criterion is simplified to a matrix-response approximation problem. An algorithm using convex optimization is proposed to solve the problem. An example illustrates the use of the algorithm and effectiveness over methods intended for LTI system design. This approach is useful if little is known about the signals to be processed.

A discrete-time, discrete-frequency Wigner distribution is derived using a group-theoretic approach. It is based upon a study of the Heisenberg group generated by the integers mod N, which represents the group of discrete-time and discrete-frequency shifts. The resulting Wigner distribution satisfies several desired properties. An example demonstrates that it is a full-band time-frequency representation, and, as such, does not require special sampling techniques to suppress aliasing. It also exhibits some interesting and unexpected interference properties. The new distribution is compared with other discrete-time, discrete-frequency Wigner distributions proposed in the literature.

A symmetric wideband ambiguity function is proposed, by recognizing the relationship between ambiguity functions and the theory of group representations. By slightly modifying the 2D Fourier transform of the symmetric wideband ambiguity function, we recover the affine Wigner distribution, a Mellin-frequency/time-scale phase space function with properties that parallel those of the Wigner distribution. The affine Wigner distribution was first proposed by the Bertrands: they defined a class of time-frequency distributions satisfying covariance with respect to time-shifts and dilations, and singled out this function by imposing certain axiomatic requirements, just as the usual Wigner distribution uniquely satisfies certain axioms among the class of time-frequency distributions that are covariant with respect to time-shifts and frequency-shifts. Our derivation, which is different, is based on recognizing that the duality between the narrowband ambiguity function and the usual Wigner distribution is an example of a general framework that has been studied in the subject of Fourier analysis on groups. Our description of this group-theoretic framework is new in the signal processing literature. The classical Fourier and Mellin transforms, Wigner distribution, and affine Wigner distribution result from applications of this theory to different groups, and it therefore provides a unifying perspective. Vorgeschlagen wird eine symmetrische Breitband-Ambiguity-Funktion, wobei von der Beziehung zwischen Ambiguity-Funktionen und der Theorie der Gruppendarstellungen ausgegangen wird. Durch eine geringfügige Modifikation der zweidimensionalen Fourier-Transformation der symmetrischen Breitband-Ambiguity-Funktion finden wir die affine Wigner-Distribution, eine Mellin-Frequenz-/Zeitskalen-/Phasenraum-Funktion mit Eigenschaften, die Parallelen zu denen der Wignerverteilung aufweist. Die affine Wigner-distribution wurde zuerst von den Bertrands vorgeschlagen: Sie definierten eine Klassse von Zeit-/Frequenz-Verteilungen, welche Kovarianzbedingungen einhalten bezüglich Zeitverschiebungen und Dilatationen. Sie schälten diese spezielle Funktion heraus, indem sie bestimmte axiomatische Anforderungen aufstellten, so, wie auch die normale Wignerverteilung bestimmten Axiomen innerhalb der Klasse der Zeit-/Frequenz-Verteilungen mit Kovarianzbeziehungen bezüglich Zeit- und Frequenzverschiebungen in einzigartiger Weise gehorcht. Unsere Herleitung folgt einem anderen Weg. Sie beruht auf der Erkenntnis, daβ die Dualität der Schmallband-Ambiguity-Funktion und der gewöhnlichen Wignerverteilung nur ein Beispiel eines allgemeinen Zusammenhangs ist, welcher im Bereich der Fourieranalyse von Gruppen untersucht worden ist. Unsere Beschriebung dieser gruppentheoretischen Gesamtschau ist neuartig in der Signalverarbeitungs-Literatur. Die klassischen Fourier- und Meilin-Transformationen, die Wignerverteilung und die affine Wigner-Distribution erhält man aus der Anwendung dieser Theorie auf verschiedene Gruppen. Die Theorie liefert damit eine vereinheitlichte Perspektive. Nous proposons une fonction d'ambiguïté large-bande symétrique, basée sur la relation existant entre les fonctions d'ambiguïté et la théorie des représentations de groupe. En modifiant légèrement la transformation de Fourier 2D de la fonction d'ambiguïté large-bande symétrique, nous recouvrons la distribution de Wigner affine, une fonction de l'espace de phase de type fréquence-Mellin/temps-échelle dont les propriétés sont parallèlles à celles de la distribution de Wigner. La distribution de Wigner affine fut tout d'abord proposée par les Bertrands: ils définirent une classe de distributions temps-fréquence satisfaisant les propriétés de covariance vis-á-vis des décalages temporels et des dilatations, et isolérent cette fonction en imposant certaines exigences axiomatiques, de même que la distribution de Wigner usuelle satisfait certains axiomes de manière unique dans la classe des distributions temps-fréquence qui sont covariantes vis-á-vis des décalages temporels et fréquentiels. Notre dérivation, qui est différente, s'appuie sur la reconnaissance du fait que la dualité entre la fonction d'ambiguïté bande-étroite et la distribution de Wigner usuelle est un exemple d'un concept général qui a été, étudié dans le cadre de l'analyse de Fourier sur les groupes. Notre description de ce contexte de théorie des groupes est nouvelle dans la littérature sur le traitement des signaux. Les transformations classiques de Fourier et de Mellin, la distribution de Wigner, et la distribution de Wigner affine résultent d'applications de cette théorie à différents groupes, et elle donne de ce fait une perspective unificatrice.

Reviews the equiripple and least-squares filter design methods. The trade-off between the two methods is illustrated with numerical examples, where ideal filters are approximated by both equiripple and least-squares FIR filters. The authors explain the motivation behind the classical filter design methods, and then extend this reasoning to develop new design criteria for certain multirate filter design problems.>

Addresses the problem of the sensitivity of wavelet representations to translations for two-dimensional signals. The authors describe a fast algorithm to calculate the two-dimensional wavelet transforms for all the circular translates of an input image. They select the optimal translate for the decomposition using a quadtree search algorithm. The resulted wavelet representation is invariant under translations measured by an additive cost criterion. The complexity of the whole algorithm is O(N/sup 2/ log N) for a N/spl times/N input block. They apply this translation invariant wavelet transform to data compression. The results show that by taking into account the effect of translations, additional compression can be achieved beyond that achieved by a standard wavelet transform.>

A new algorithm is presented for the design of two-dimensional linear phase FIR filters that best approximate an arbitrary magnitude response in the Chebyshev sense. The algorithm is an extension of one proposed recently by the authors (see Proc. IEEE ICASSP, vol. 3, 1993) for the design of one dimensional filters. It consists of an iterative sequence of linear programs solved by a variation of the simplex algorithm. Convergence is stable. The true optimal design is found, to within an arbitrarily small error tolerance, in contrast with several of the recently published approximate techniques. Sparse and/or non-causal impulse coefficient designs can be done in a straightforward manner. Examples are given demonstrating the performance of this new algorithm over previously published techniques with respect to speed, stability of convergence, and accuracy. The new method has been used to design filters larger than can be accommodated by other methods known to the authors. A 41/spl times/41 design example is given.>

The authors analyze a cochlear model data representation for use in classifying acoustic transient signals. Comparisons of the cochlear model, short time Fourier transform, and wavelet transform from a filter bank perspective are made. The cochleagram and the spectrogram are used to classify a set of bowhead whales and ice sounds from the Arctic.>

A symmetrized version of the wideband ambiguity function is proposed, by observing the relationship between ambiguity functions and the theory of certain groups. Its 2-D Fourier transform has properties similar to the Wigner distribution, and it is called the affine Wigner distribution, by analogy. This approach is compared with earlier work by R.A. Altes (1989) and by O. Rioul and P. Flandrin (1990).>

The authors consider the problem of interpolation using a set of training signals. The training set is assumed to contain signals representative of the type of the waveform to be interpolated. The authors discuss two procedures for using the training data directly to estimate the missing samples. The interpolated signal is chosen to satisfy the given samples, and to be close to the signals in the training set. The authors discuss the difference between the two methods, and present and example of interpolation of biological transients (sperm whales).>

The problem of interpolating and resampling signals known to result from the application of a linear operator to signals whose minimum and maximum values are known is addressed. These limits are physically meaningful in many applications, and modeling the signals with the limits is preferable to modeling with the unrealistic energy bounds that are sometimes used. It is shown that these more realistic bounds (called amplitude bounds) induce estimation techniques and estimates which have different properties than the ones induced by energy bounds. This allows, potentially, more accurate signal processing and more accurate design of signal processing systems, when the signals are well-described by these bounds. The energy-bound estimation techniques and estimates are reviewed. It is shown that amplitude-bound estimation may be implemented with a modified linear programming technique.>

The notion of concentration is discussed, and concentration with respect to an operator is defined. This is related to the idea of concentration based on second moments in time and frequency. A class of signals B/sub w/( mu ) concentrated with respect to an operator W is introduced. A best n-dimensional approximating subspace for this class is shown to be the space spanned by the n signals most concentrated with respect to the operator defining the class. The case of approximating B/sub W/( mu ) with a subspace which is not maximally concentrated with respect to W, but which consists of signals which are maximally concentrated with respect to some other operator, is examined. This is motivated from time-frequency analysis where useful physical insight may be gained by decomposing a signal into elementary locally concentrated components. The tradeoff between error and concentration is discussed.>

Two signal components are considered resolved in a time-frequency representation when two distinct peaks can be observed. The time-frequency resolution limit of two Gaussian components, alike except for their time and frequency centers, are determined for the Wigner distribution and smoothed variants of the Wigner distribution (the pseudo-Wigner distribution, the smoothed Wigner distribution, and the squared magnitude of the short-time Fourier transform). The matched-window short-time Fourier transform has the highest resolution in time-frequency; it is about 24% greater than that of the Wigner distribution. An explanation of this rather surprising result is presented. The pseudo-Wigner distribution provides more resolution in one direction than the other representations, if substantial resolution in the other direction can be sacrificed.>

The authors begin with a review of estimation using signal classes. They then show that certain spectral estimation procedures can be viewed as methods of estimating a signal class from data. Given a signal class, it is then possible to estimate signal parameters other than the magnitude of the spectrum, such as phase or missing data points.>

The IIR (infinite-impulse response) filter design problem in the complex domain is formulated such that the existence of a best approximation is guaranteed. A modified Ellacott-Williams algorithm (1976) is proposed to solve the design problem. The proposed algorithm uses a complex linear approximation utilized for FIR (finite-impulse response) filter design and it usually converges in several iterations. Design examples are included.>

In many practical applications, it is necessary to interpolate between known samples of a signal which is produced by, or viewed through, an averaging operation. Such a signal is not bandlimited and the application of a bandlimited interpolator may not be the best thing to do. It is pointed out that by formulating the interpolation problem in terms of optimal signal recovery, a best interpolator may be designed for the averaged data and that in certain special cases, the optimal interpolation is simply a piecewise-linear interpolation.>

An algorithm for evaluating the Discrete Fourier Transform (DFT) at particular output frequency is derived using a technique called summation by parts (SBP). This technique is shown to reduce the number of multiplications and the number of bits per multiplicative coefficient needed to implement the DFT. For many transform lengths, only two one-bit multiplications or simple memory shifts are needed to implement the DFT. When the DFT length is prime, a SBP algorithm designed for a fixed output frequency index can be used to evaluate the DFT at any other non-zero output frequency index simply by appropriately changing the order of the input sequence.

We present a data-adaptive time-frequency representation that obtains high resolution of signal components in time-frequency. This representation overcomes the often poor resolution of the traditional short-time Fourier transform, while avoiding the nonlinearities that make the Wigner distribution and other bilinear representations difficult to interpret and use. The new method uses adaptive Gaussian windows, with the window parameters varying at different time-frequency locations to maximize the local signal concentration in time-frequency. Two methods for selecting the Gaussian parameters are presented: a parameter estimation approach, and a method that maximizes a measure of local signal concentration.

Large errors may be produced by filtering finite-length records extracted from a discrete signal instead of filtering the signal itself. The knowledge that the signal is bandlimited in some particular way may be used to reduce these errors, particularly around the ends of the record. An optimal finite-record filtering technique that uses this information is presented here that performs the filtering operation by a single matrix multiplication. An improvement to the basic algorithm is also presented that permits the bandwidth of the signal to be estimated from knowledge of average-power characteristics of the original signal.

The frequency response of a finite-duration impulse response (FIR) filter is degraded by unknown variations in the values of the filter coefficients. If the filter coefficients can only be specified to within a certain percentage accuracy, the solution of a nested optimization problem gives a filter design such that over all possible coefficient variations, the worst frequency response is as good as possible in the Chebyshev sense. The size of the nested optimization problem grows exponentially with the length of the filter. However a tractable optimization problem can be solved whose solution gives a suboptimal filter design that is likely to be nearly as good as the globally optimal solution.

An algorithm is presented for extrapolation and spectral estimation of discrete-time signals from a set of samples. First, we assume the given samples are those of a length N sequence that is a member of a signal class consisting of the collection of outputs of a known filter with limited-energy inputs. Then we estimate by finding the signal that goes through the samples and which gives the best estimate of an arbitrary time or frequency sample using a minmax criterion. This estimate is linear and optimal over all the signals in the class and over all possible data. Next we obtain a series of signal estimates, each of which assumes the signal belongs to a different class. We make these classes depend on the data recursively by re-defining them using the previous signal estimate. Convergence of the algorithm is discussed and an example is provided to illustrate the technique.

The separation of transient plane-waves as a function of their velocity and time-of-arrival can be performed by a time domain beamformer. Modifications in the beamformer output domain, such as masking out coherent interfering signals while taking into account both velocity and time information, are more straightforward than with traditional (k-ω) velocity filtering methods. We describe three methods of inverting the beamformer output to reconstruct the space-time samples. Of the three, one method, a block matrix inversion technique, is new. It is exact when operating on the original data, but fails when the beamformer output is modified. This problem can be alleviated by reducing the dimensionality of the inversion matrix using a singular value decomposition.

The Short-time Fourier transform(STFT), the Ambiguity Function(AF), and the Wigner distribution (WD) are mixed time-frequency signal representations that use Fourier transform techniques to map a one-dimensional function of time into a two-dimensional function of time and frequency. These mixed time-frequency mappings have been used to analyze the local frequency characteristics of a variety of signals and systems. Although much work has also been done to develop STFT and AF synthesis algorithms that can be used to implement a variety of time-varying signal processing operations, no such synthesis techniques have thus far been developed for the WD. In this paper, signal synthesis algorithms that work directly with the real-valued high-resolution WD will be derived. Examples of how this WD synthesis procedure can be used to perform time-varying filtering operations or signal separation will be given.

Time-domain beamforming with an array of space-time samples is equivalent to performing a discrete Radon transform (DRT). The result is a function of slowness (inverse velocity) and time intercept. As a function of slowness, beamforming decimates each channel in proportion to its distance from the beamformer reference channel. This viewpoint allows the slowness bandwidth to be expressed as a function of the array spatial locations and the normalized temporal bandwidth of the received waveforms. These relationships determine the required temporal sample rate increase at each channel when interpolation beamforming is used to avoid slowness aliasing. Examples are presented showing the slowness aliasing that results if either the temporal bandwidths or the array spatial aperture are too large. The beamformer output energy averaged over a time window is shown to require twice the slowness sampling rate as the beamformer output itself.

The Wigner Distribution(WD) is a bilinear signal transformation possessing several properties that are useful in time-frequency signal analysis. Fast Fourier transfom(FFT) techniques have been used to approximate the WD. However, the signal must be sampled at twice the Nyquist rate in order to avoid aliasing errors. This paper demonstrates that implicit spline interpolation of a continuous time signal or an undersampled discrete time sequence can be used to reduce aliasing errors when approximating the WD. The method is said to he implicit since the interpolated samples are never actually computed. An efficient implicit interpolation algorithm that takes advantage of the special structure and symmetry of the WD is proposed.

A technique for spectral estimation given a finite number of measurements of a two-dimensional band-limited signal of time t and space x is used on signals which are transient propagating waves, The theory of optimal estimation for deterministic signals is presented in two-dimensions for signals which are assumed to be contained in a wedge-shaped region of the two-dimensional frequency plane. We also consider samples which contain errors. An example with exact signal samples shows a spectral estimate with all the basic features of our test signal. After adding noise to this signal, we encounter more difficulties in recovering the original spectrum.

This paper addresses the problem of estimating the velocity of a propagating wave in the presence of other waves traveling at different velocities. Signals from a linear array of receivers are first windowed to attenuate interfering signals. Velocities are calculated from spatial frequency estimates obtained for each temporal frequency using a variation of Prony's method. This procedure is applied to the sonic well logging problem where shear wave velocity is estimated from data which contain interfering compressional and guided fluid waves.

This paper discusses the problem of reducing distortion introduced by a known distorting filter G. The distorting filter has a symmetric unit pulse response so that no phase distortion is introduced. However the example presented in this paper involves a filter with several unit circle zeroes. These frequency domain zeroes rule out inverse filtering techniques which amount to choosing a compensation filter H(ω) with H(ω) = 1/G(ω). The problem described here arises in well logging where the system performance is evaluated in terms of step inputs. Chebychev approximation techniques are applied to find an FIR compensator so that the compensated step response has the minimum possible overshoot for a given rise time.

A procedure for designing digital filters with good frequency response and few multiplies is described. The filters are designed by using dynamic programming to reduce the number of non-zero coefficients in a high order numerator. In the designs described here a fixed, low order denominator is assumed add the numerator is optimized.

The equivalent problems of extrapolation and spectral estimation are formulated as deterministic estimation problems for the class of bandlimited signals which are concentrated in a known time interval. Estimators are derived which minimize the maximum magnitude of the error over the class of bandlimited, time-concentrated signals. The dependence of this minimum error on bandwidth, degree of time-concentration, and measurement errors (noise) is discussed. Practical examples are presented using discrete prolate spheroidal sequences to compute spectral estimation and extrapolation operators.

This paper presents methods for optimally estimating a desired linear functional of a bandlimited signal from a given set of measurements. The error measure used is the maximum magnitude of the difference between the desired linear functional of the signal and its estimate normalized by the norm of the signal. The estimate is a linear combination of the signal measurements. These techniques have been used on a finite number of time samples to solve the equivalent problems of extrapolation of the time function and spectral estimation of frequency points.

A class of equiripple IIR (Infinite Impulse Response) digital filters, with all zeros on the unit circle is discussed. These new filters bring together, to some extent, the advantages of FIR and elliptic designs, by having only powers of zDin the denominator (D is the decimation ratio). Only every Dth output is computed with these recursive filters. Comparisons are made with FIR and elliptic designs. It is shown that these new filters require significantly fewer multiplications per second. Coefficient storage requirements of these three designs are also compared. Some interesting properties of filters with only powers of zDin the denominator are briefly discussed.

This paper investigates the error in reconstructions of a signal based on a given finite set of linear measurements, and presents two schemes that, if there is available a priori knowledge of the class of signals of which the measured signal is a member, can achieve a reduction of this error beyond the best that could be done without such knowledge. The error measure used is the supremum over the class of the\mathcal{L}_2distance between a signal and its reconstruction. The essence of the proposed reconstruction techniques is a coordinate transformation from the sampling subspace to a new reconstruction subspace known to be efficient for representation of signals of the given class. This study applies the theory of extremal subspaces andnwidths of signal classes originated by Kolmogorov. Results are applied to the much studied class of time-concentrated band-limited signals. The measurement process is here assumed to be the convenient one of Nyquist rate time sampling. For this problem, plots of the error bounds and of several test functions and their reconstructions are presented, both for the proposed reconstructions, and for conventional cardinal-sampling-theorem reconstructions.