Ronald W. Schafer

A digital signal processing approach to interpolation

In many digital signal precessing systems, e.g., vacoders, modulation systems, and digital waveform coding systems, it is necessary to alter the sampling rate of a digital signal Thus it is of considerable interest to examine the problem of interpolation of bandlimited signals from the viewpoint of digital signal processing. A frequency dmnain interpretation of the interpolation process, through which it is clear that interpolation is fundamentally a linear filtering process, is presented, An examination of the relative merits of finite duration impulse response (FIR) and infinite duration impulse response (IIR) digital filters as interpolation filters indicates that FIR filters are generally to be preferred for interpolation. It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter. The use of classical interpolation methods in signal processing applications is illustrated by a discussion of FIR interpolation filters derived from the Lagrange interpolation formula. The limitations of these filters lead us to a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques. Examples are presented to illustrate the significant improvements that are obtained using the optimum filters.

Morphological filters--Part I: Their set-theoretic analysis and relations to linear shift-invariant filters

This paper examines the set-theoretic interpretation of morphological filters in the framework of mathematical morphology and introduces the representation of classical linear filters in terms of morphological correlations, which involve supremum/infimum operations and additions. Binary signals are classified as sets, and multilevel signals as functions. Two set-theoretic representations of signals are reviewed. Filters are classified as set-processing (SP) or function-processing (FP). Conditions are provided for certain FP filters that pass binary signals to commute with signal thresholding because then they can be analyzed and implemented as SP filters. The basic morphological operations of set erosion, dilation, opening, and closing are related to Minkowski set operations and are used to construct FP morphological filters. Emphasis is then given to analytically and geometrically quantifying the similarities and differences between morphological filtering of signals by sets and functions; the latter case allows the definition of morphological convolutions and correlations. Toward this goal, various properties of FP morphological filters are also examined. Linear shift-invariant filters (due to their translation-invariance) are uniquely characterized by their kernel, which is a special collection of input signals. Increasing linear filters are represented as the supremum of erosions by their kernel functions. If the filters are also discrete and have a finite-extent impulse response, they can be represented as the supremum of erosions only by their minimal (with respect to a signal ordering) kernel functions. Stable linear filters can be represented as the sum of (at most) two weighted suprema of erosions. These results demonstrate the power of mathematical morphology as a unifying approach to both linear and nonlinear signal-shaping strategies.

Morphological filters--Part II: Their relations to median, order-statistic, and stack filters

This paper extends the theory of median, order-statistic (OS), and stack filters by using mathematical morphology to analyze them and by relating them to those morphological erosions, dilations, openings, closings, and open-closings that commute with thresholding. The max-min representation of OS filters is introduced by showing that any median or other OS filter is equal to a maximum of erosions (moving local minima) and also to a minimum of dilations (moving local maxima). Thus, OS filters can be computed by a closed formula that involves a max-min on prespecified sets of numbers and no sorting. Stack filters are established as the class of filters that are composed exactly of a finite number of max-min operations. The kernels of median, OS, and stack filters are collections of input signals that uniquely represent these filters due to their translation-invariance. The max-min functional definitions of these nonlinear iliters is shown to be equivalent to a maximum of erosions by minimal (with respect to a signal ordering) kernel elements, and also to a minimum of dilations by minimal kernel elements of dual filters. The representation of stack filters based on their minimal kernel elements is proven to be equivalent to their representation based on irreducible sum-of-products expressions of Boolean functions. It is also shown that median filtering (and its iterations) of any signal by convex 1-D windows is bounded below by openings and above by closings; a signal is a root (fixed point) of the median iff it is a root of both an opening and a closing; the open-closing and clos-opening yield median roots in one pass, suppress impulse noise similarly to the median, can discriminate between positive and negative noise impulses, and are computationally less complex than the median. Some similar results are obtained for 2-D median filtering.

Demosaicking: color filter array interpolation

The author begins by discussing the image formation process. The demosaicking methods are examined in three groups: the first group consists of heuristic approaches. The second group formulates demosaicking as a restoration problem. The third group is a generalization that uses the spectral filtering model given in Wandell.

Constrained iterative restoration algorithms

This paper describes a rather broad class of iterative signal restoration techniques which can be applied to remove the effects of many different types of distortions. These techniques also allow for the incorporation of prior knowledge of the signal in terms of the specification of a constraint operator. Conditions for convergence of the iteration under various combinations of distortions and constraints are explored. Particular attention is given to the use of iterative restoration techniques for constrained deconvolution, when the distortion band-limits the signal and spectral extrapolation must be performed. It is shown that by predistorting the signal (and later removing this predistortion) it is possible to achieve spectral extrapolation, to broaden the class of signals for which these algorithms achieve convergence, and to improve their performance in the presence of broad-band noise.

Morphological skeleton representation and coding of binary images

This paper presents the results of a study on the use of morphological set operations to represent and encode a discrete binary image by parts of its skeleton, a thinned version of the image containing complete information about its shape and size. Using morphological erosions and openings, a finite image can be uniquely decomposed into a finite number of skeleton subsets and then the image can be exactly reconstructed by dilating the skeleton subsets. The morphological skeleton is shown to unify many previous approaches to skeletonization, and some of its theoretical properties are investigated. Fast algorithms that reduce the original quadratic complexity to linear are developed for skeleton decomposition and reconstruction. Partial reconstructions of the image are quantified through the omission of subsets of skeleton points. The concepts of a globally and locally minimal skeleton are introduced and fast algorithms are developed for obtaining minimal skeletons. For images containing blobs and large areas, the skeleton subsets are much thinner than the original image. Therefore, encoding of the skeleton information results in lower information rates than optimum block-Huffman or optimum runlength-Huffman coding of the original image. The highest level of image compression was obtained by using Elias coding of the skeleton.

A regularized iterative image restoration algorithm

The development of the algorithm is based on a set theoretic approach to regularization. Deterministic and/or statistical information about the undistorted image and statistical information about the noise are directly incorporated into the iterative procedure. The restored image is the center of an ellipsoid bounding the intersection of two ellipsoids. The proposed algorithm, which has the constrained least squares algorithm as a special case, is extended into an adaptive iterative restoration algorithm. The spatial adaptivity is introduced to incorporate properties of the human visual system. Convergence of the proposed iterative algorithms is established. For the experimental results which are shown, the adaptively restored images have better quality than the nonadaptively restored ones based on visual observations and on an objective criterion of merit which accounts for the noise masking property of the visual system. >

Morphological systems for multidimensional signal processing

The basic theory and applications of a set-theoretic approach to image analysis called mathematical morphology are reviewed. The goals are to show how the concepts of mathematical morphology geometrical structure in signals to illuminate the ways that morphological systems can enrich the theory and applications of multidimensional signal processing. The topics covered include: applications to nonlinear filtering (morphological and rank-order filters, multiscale smoothing, morphological sampling, and morphological correlation); applications to image analysis (feature extraction, shape representation and description, size distributions, and fractals); and representation theorems, which shows how a large class of nonlinear and linear signal operators can be realized as a combination of simple morphological operations. >

System for Automatic Formant Analysis of Voiced Speech

A system for automatically estimating the lowest three formants and the pitch period of voiced speech is presented. The system is based on a digital computation of the cepstrum (defined as the inverse transform of the log magnitude of the z‐transform). The pitch period estimate and smoothed log magnitude are obtained from the cepstrum. Formants are estimated from the smoothed spectral envelope using constraints on formant frequency ranges and relative levels of spectral peaks at the formant frequencies. These constraints allow the detection of cases where two formants are too close together in frequency to be resolved in the initial spectral envelope. In these cases, a new spectral analysis algorithm (the chirp z‐transform algorithm) allows the efficient computation of a narrow‐band spectrum in which the formant resolution is enhanced. Formant and pitch period data obtained by the analysis system are used to control a digital formant synthesizer. Results, in the form of spectrograms, are presented to illu...

Nonlinear filtering of multiplied and convolved signals

An approach to some nonlinear filtering problems through a generalized notion of superposition has proven useful In this paper this approach is investigated for the nonlinear filtering of signals which can be expressed as products or as convolutions of components. The applications of this approach in audio dynamic range compression and expansion, image enhancement with applications to bandwidth reduction, echo removal, and speech waveform processing are presented.