We derive an estimator for the standard deviation of a Gaussian random variable that uses the maximum of two observations on the random variables. The process is repeated until the variance of the estimator or the degree of confidence reaches a predetermined level. The estimator is unbiased and consistent, and its variance is only marginally larger than the standard square root of the sum of the squares estimator. Moreover the computation of an estimate requires only a sequence of comparisons of two numbers followed by an addition.

A new technique for designing digital all-pass IIR filters is proposed. The approach is based on the vector space projection method. Constraint sets, and their associated projectors, that capture the properties of the desired group delay are given. Examples that demonstrate the advantages and flexibility of this method as well as comparisons with a well-known method (Nguyen et al., 1994) are furnished.

A new technique for designing finite impulse response (FIR) quadrature mirror filters (QMFs) is proposed. The approach is based on the vector space projection method (VSPM). Convex constraint sets and their projections that capture the properties of the desired QMFs are given. The proposed approach produces equiripple filters. This method is compared with the classical Johnston's (1980) method and is shown to have certain advantages.

A new method of rejecting narrowband interference in direct-sequence spread spectrum (DS/SS) communications is presented. A typical approach is to reject the interference using a filter with large attenuation at the interference frequencies before despreading. The interference rejection method presented incorporates vector space projection techniques to suppress the correlated interference. Several signal characteristics are formulated which lead to constraint surfaces in the vector space of possible solutions. These constraint surfaces describe interference rejection solutions which introduce minimal distortion to the spread spectrum signal and simultaneously remove the interference. The constraint surfaces essentially correspond to spread spectrum signal estimates which, after interference rejection, conform to known characteristics of the transmitted spread spectrum signal. The formulation of the surfaces relies on prior knowledge about the spread spectrum signal correlation and spectral properties.

We present a gradient-projection algorithm for solving the classical blind deconvolution problem. In our approach all known a priori information about both the unknown source and blurring functions is expressed via constraint sets. In computer simulations, the algorithm performed well even when the prior information was not accurate. In this study the algorithm is also compared with a conjugate gradient algorithm proposed by Lane (see J. Opt. Soc. Am. A, vol.9, no.9, p.1508-1514, 1992).>

For the image recovery process the authors use the convex projections method, also known as the method of projections onto convex sets (POCS). Several incomplete-data geometries, including those associated with limited source travel and beam-blocking internal opacities, are considered. To enable the recovery several prior-knowledge constraints including one associated with the directivity of the image vector are used. The overall recovery algorithm can be practically implemented by exploiting the Toeplitz structure of key operators.

The method of generalized projections is applied to the multilayer feedforward neural net problem to derive a learning algorithm. This learning rule is called the projection-method learning rule (PMLR). The authors apply the PMLR to a well-known pattern recognition problem, which cannot be solved by a linear discriminant scheme. The PMLR is compared with the error backpropagation learning rule (BPLR) and is shown to converge faster than the latter for the problems considered. As the degree of nonlinearity of the neuron activation function increases, the PMLR becomes even more superior to the BPLR.>

Projection method formulations of the perceptron and Hopfield associative content-addressable memory (ACAM) neural nets are presented. It is shown that the well-known single-layer-perceptron learning algorithm can be formulated using the method of projections onto convex sets (POCS) and that its performance can be improved using this method. The operation of a modified, binary-valued Hopfield ACAM is shown to be equivalent to the method of generalized projections. A direct extension of the binary-valued ACAM to the continuous-valued case lends itself to a POCS formulation.>

The authors propose a method for reconstructing partial data slices using the theory of projections onto convex sets (POCS). The problem involves the reconstruction of a complex function, which consists of the desired magnitude image and an undesired phase. For that purpose the authors introduce, among other features, a convex constraint set based on a priori information about the phase. The authors show selected results of an extensive experimental study, using head images, taken from an actual MRI session. It is concluded that it is possible to obtain fair quality reconstructions using the proposed procedure, even from a relatively small amount of data.>

The authors apply the method of projections onto convex sets (POCS) to reconstruct an image in computer tomography. Although POCS has been used before to recover missing portions of the data set, this is the first time that it is directly used to invert raysum data. Since POCS is geometry-free, it can be directly applied to cases of incomplete data. The strength of POCS compared to other geometry-free methods is its systematic use of a priori information in a natural and consistent way along with the raysum data.>

The problem of reconstructing a two-dimensional function from a set of spiral samples is addressed. This problem is of practical interest in magnetic resonance imaging (MRI). A theorem that enables exact interpolation from linear spiral samples to a Cartesian lattice is presented, dealing with nonlinear, expanding spirals. These kinds of spirals have a potential advantage in a situation for which the data collection time is limited. In general, there is no exact reconstruction procedure from nonlinear spiral samples. Two practical methods are furnished to reconstruct an image from a set of spiral samples of its Fourier transform and examples are shown of reconstructions obtained through both methods.>

An algorithm is discussed that estimates the smallest convex polygon that circumscribes an opacity in a medical image. This is called the minimum polygon image (MPI). The algorithm requires only O(n) operations to produce the MPI. Thus it offers the following advantages over conventional image reconstruction: (1) it is computationally fast; (2) it submits the patient to much lower X-ray doses; and (3) it is much less likely to ionize a radiation-sensitive substance during inspection. An analysis confirms the high performance of the algorithm but it shows that it depends strongly on an appropriate choice of system parameters. These parameters, the number of views N/sub theta / and the detector spacing Delta s are shown to be linked, and the choice of Delta s strongly constraints the choice of N/sub theta /. It is shown how an optimum N/sub theta / can be determined for a fixed, practical value of Delta s. A series of simulation experiments that verify the theory is reported.>

The a priori information on generalized momentsm_{r}, r = 0, 1, ..., N, (m_{r} \Deltaeq \int_{\Omega} dx g_{r}(x) f(x))is shown to restrict the function f to lie in closed convex sets Cr, r = 0, 1, ..., N, in a Hilbert space setting. Therefore the a priori moment information can also be incorporated into the method of projections onto convex sets (POCS). Given the partial information (including both measurements and a priori constraints), the method of POCS reconstructs or synthesizes a feasible solution. The a priori knowledge of moments up to a finite order is assumed to synthesize a nonnegative function, consistent with a well-defined Rayleigh distribution, via the method of POCS.

Image restoration problems which involve nonconvex constrains such as the restoration from magnitude problem (RFM) cannot be solved by the method of projections onto convex sets (POCS). An algorithm known as generalized projections is derived and its properties are discussed. We introduce a performance criterion which can be used to measure the performance of the algorithm at any iteration. A method of optimizing (via control of relaxation parameters) the algorithm is described. Comparisons with other RFM methods are furnished.

We apply the method of alternating projections onto convex sets to the problem of restoring a signal from the phase of its Fourier transform. A method of improving convergence by adaptively varying a set relaxation parameters in the restoration algorithm is described. The advantages of using the method of convex projections are discussed.

In many foreseeable applications of Computerized tomography (CT) it may not be possible (e.g., because of obstructions) to obtain 360 degree view-data and reconstruction from partial view data may be unsatisfactory. In this paper we describe the method of projections onto convex sets (POCS)[1] by which the missing data can be restored from partial viewdata and a priori known properties of the original image. POCS performs significantly better than the conventional Gerchberg-Papoulis (GP) method.

In this paper we extend the generalized image restoration theory developed by Youla (1) to arbitrary L2(i.e., square-integrable) images or signals. Specifically we furnish recursive algorithms to restore images from (1) extended segments and low-pass spectra and (2) short segments and high-pass spectra. We also assert that it is not possible to restore an arbitrary image from only its low-pass spectrum and a short segment.

To avoid some of the problems that arise in the pattern recognition of waveforms using conventional feature extractors, we derive a feature extraction technique based on autoregressive (AR) modelling. The technique generates a feature vector whose components are the coefficients of the "best" AR fit to the data. Levinson's algorithm and Akaike's Information Criterion (AIC) are used to help furnish the best coefficients and best AR order respectively.

An optical-digital computer (ODC) that enables parallel processing of images is discussed. A number of optical and analog electronic preprocessing operations are described. These include the use of the texture variance to separate objects in an image on the basis of texture, application of autocorrelation techniques to find hidden periodicities in images, masking on the basis of color, Fourier analysis, and planimetric analysis of color scenes. A particular design of an ODC is discussed. The computer consists of an optical system, an image analyzer, a bandwidth compressor and digitizer, and a digital computer. A number of applications are furnished. Emphasis is given to processing aerial and radiographic imagery.