The complete list

Using the J Language for Neural Net Experiments

Training Wheels for Encoder Networks

Multidimensional Golden Means

Using Quasi-Random Numbers in Neural Networks

Linear Pixel Shuffling for Image Processing, an Introduction

The Polynomial Method Augmented by Supervised Training for Hand-Printed Character Recognition

Genetic Algorithm Selection of Features for Handwritten Character Identification

Typewriter Keyboards via Simulated Annealing

Ordered Greed

Ordered Greed, II: Graph Coloring

Advances in Ordered Greed

Error Diffusion Using Linear Pixel Shuffling

Linear Pixel Shuffling (I): New Paradigms for New Printers

Advances in Linear Pixel Shuffling

Error Diffusion and Edge Enhancement: Raster Versus Omni-Directional Processing

The Unit RBF Network: Experiments and Preliminary Results

Lectures on Genetic Algorithms

Color Uniformity and Moire in Dispersed Dot Halftone Masks Generated by Linear Pixel Shuffling

J Language Lectures

A Genetic Algorithm Search for Improved Halftone Masks

Optimizing Halftone Masks with Genetic Algorithms and a Printer Model

Good Halftone Masks via Genetic Algorithms

A Genetic Algorithm Search for Improved Halftone Masks

Compressible Halftoning

Combinatorial Proofs of Fermat's, Lucas's, and Wilson's Theorems

Convolutions Combinatorially

December, 1995

We introduce the programming language J and show its applicability for experimenting with neural networks and genetic algorithms. We illustrate the use of J with complete programs for perceptron and back propagation learning.

We develop a new approach to training encoder feed-forward neural networks and apply it to two classes of problems. Our approach is to initially train the network with a related, relatively easy-to-learn problem, and then gradually replace the training set with harder problems, until the network learns the problem we originally intended to solve. The problems we address are modifications of the common N-2-N encoder network problem with N exemplars, the unit vectors, e[k] in N-space.

Our first modification of the problem is to use objects consisting of paired 1's, e[k]+e[k+1], with subscripts taken mod N. This requires an N-2-N net to organize the images of the exemplars in 2-space ordered around a circle.

Our second modification is to use patterns consisting of two objects; each object is a pair of adjacent 1's; the objects must be separated from each other. This problem can be learned by a N-4-N network which must organize the images of the exemplars in 4-space in the form of a mobius strip.

The easy-to-learn problem in both cases involves replacing the two-ones signal e[k]+e[k+1] with a block-signal of length B: e[k]+e[k+1]+...+e[k+B-1].

In several cases, our method allowed us to train networks that otherwise fail to train. In some other cases, our method proved to be ten times faster than otherwise.

We investigate a geometric construction which yields periodic continued fractions and generalize it to higher dimensions. The simplest of these constructions yields a number which we call a two (or higher) dimensional golden mean, since it appears as a limit of ratios of a generalized Fibonacci sequence. Multiples of these golden points, considered "mod 1" (i.e., points on a torus) prove to be good probes for applications such as Monte Carlo integration and image processing.

June, 1993

We investigate a method of ordering pixels (the elements of a rectangular matrix) based on an arithmetic progression with wraparound(modular arithmetic). For appropriate choices of the progression's parameters based on a generalization of Fibonacci numbers and the golden mean, we find equidistributed collections of pixels formed by subintervals of the pixel progression or "shuffle." We illustrate this equidistributivity with a novel approach to progressive rendering of a synthetic image, and we suggest several opportunities for its application to other areas of image processing.

March, 1995

We present a novel training algorithm for a feed forward neural network with a single hidden layer of nodes (i.e., two layers of connection weights). Our algorithm is capable of training networks for hard problems, such as the classic two-spirals problem.

The weights in the first layer are determined using a quasirandom number generator. These weights are frozen---they are never modified during the training process.

The second layer of weights is trained as a simple linear discriminator using methods such as the pseudo-inverse, with possible iterations.

We also study the problem of reducing the hidden layer: pruning low-weight nodes and a genetic algorithm search for good subsets.

Presented at The International Conference on Artificial Neural Networks & Genetic Algorithms, ANNGA 93, Innsbruck, Austria,

April 1993.

We present a pattern recognition algorithm for hand-printed characters, based on a combination of the classical least squares method and a neural-network-type supervised training algorithm. Characters are mapped, nonlinearly, to feature vectors using selected quadratic polynomials of the given pixels. We use a method for extracting an equidistributed subsample of all possible quadratic features.

This method creates pattern classifiers with accuracy competitive to feed-forward systems trained using back propagation; however back propagation training takes longer by a factor of ten to fifty. (This makes our system particularly attractive for experimentation with other forms of feature representation, other character sets, etc.)

The resulting classifier runs much faster in use than the back propagation trained systems, because all arithmetic is done using bit and integer operations.

Genetic Algorithm Selection of Features for Handwritten Character Identification,

Presented at The International Conference on Artificial Neural Networks Genetic Algorithms, ANNGA 93, Innsbruck, Austria, April 1993.

We present a pattern recognition We have constructed a linear discriminator for hand-printed character recognition that uses a (binary) vector of 1,500 features based on an equidistributed collection of products of pixel pairs. This classifier is competitive with other techniques, but faster to train and to run for classification.

However, the 1,500-member feature set clearly contains many redundant (overlapping or useless) members, and a significantly smaller set would be very desirable (e.g., for faster training, a faster and smaller application program, and a smaller system suitable for hardware implementation). A system using the small set of features should also be better at generalization, since fewer features are less likely to allow a system to memorize noise in the training data.

Several approaches to using a genetic algorithm to search for effective small subsets of features have been tried, and we have successfully derived a 300-element set of features and built a classifier whose performance is as good on our training and testing set as the system using the full set.

Typewriter Keyboards via Simulated Annealing,

Printed in

We apply the simulated annealing algorithm to the combinatorial optimization problem of typewriter keyboard design, yielding nearly optimal key-placements using a figure of merit based on English letter pair frequencies and finger travel-times. Our keyboards are demonstrably superior to both the ubiquitous QWERTY keyboard and the less common Dvorak keyboard.

The paper is constructed as follows: first we discuss the historical background of keyboard design; this includes August Dvorak's work, and a figure-of-merit (scalar) metric for keyboards. We discuss a theory of keyboard designs: why keyboard design is a combinatorial problem, how combinatorial problems are typically solved, what is simulated annealing, and why it is especially suitable for the problem at hand. Next we discussed the results, and compare the keyboards produced by simulated annealing to QWERTY and Dvorak's keyboard. Finally, we suggest some future lines of inquiry.

Ordered Greed,

Scheduling problems are among the most challenging and realistic problems application of problem solving heuristics, such as genetic algorithms (GAs). The naive greedy algorithm for scheduling simply assigns, in turn, each item to be scheduled the best yet untaken position for that item. We investigate using a genetic algorithm to search the space of orderings for this greedy algorithm. That is, the GA individuals are permutations that determine the permutations that are the schedules, rather than the GA individuals directly being the schedules.

We have experimented with the classical N Queens problem and a realistic soccer tournament scheduling problem, comparing the GA individual as the assignment with our greedy hybrid algorithm (ordered greed).

Warnsdorff's heuristic is introduced to modify blind greed with excellent results.

We also introduce the use of signatures in our GAs
to represent permutations. Signatures are easy to
create and manipulate in crossover and mutation
operations.

Ordered Greed, II: Graph Coloring

Presented at the ICSC/NAISO Conference, Information Science Innovations (ISI 2001), Dubai, UAE, March, 2001.

A popular application of genetic algorithms (GAs) is to attempt to generate good, rapid, approximate solutions to NP-complete or NP-hard problems.

Previously, we introduced a hybrid algorithm combining a GA with simple greedy algorithms applied to the N-Queens problem and to sports tournament scheduling. The greedy algorithm makes locally optimal assignments (to Queens or matches) in some order. We treat that ordering as the sought after goal, and thus work with a population whose individuals are permutations.

The subject of the present paper is the problem of graph coloring. We focus the present paper on the single benchmark problem of coloring a three-colorable graph that was constructed as a subgraph of the complete 3-partite graph K{p,q,r} in which each edge exists with probability 0.1. (We have applied our method successfully to several other categories of graphs, but present space limitations dictate presenting the results for this special case.)

Error Diffusion Using Linear Pixel Shuffling

Presented at the IS&T Conference, PICS2000, Portland, OR, March, 2000.

Linear pixel shuffling error diffusion is a digital halftoning algorithm that combines the linear pixel shuffling (LPS) order of visiting pixels in an image with diffusion of quantization errors in all directions.

LPS uses a simple linear rule to produce a pixel ordering giving a smooth, uniform probing of the image. This paper elucidates that algorithm.

Like the Floyd-Steinberg algorithm, LPS error
diffusion enhances edges and retains high-frequency
image information. LPS error diffusion avoids some of
the artifacts (worms, tears, and
checkerboarding) often associated with the
Floyd-Steinberg algorithm. LPS error diffusion
requires the entire image be available in memory; the
Floyd-Steinberg algorithm requires storage
proportional only to a single scan line.

Linear Pixel Shuffling (I): New Paradigms for New Printers

Presented at the IS&T Conference, on Non-Impact Printing, Vancouver, 2001

Linear Pixel Shuffling (LPS) is a novel order for image pixel processing which provides opportunities for construction of dot placement algorithms for high-resolution printers through micro-clumping and the formation of pseudo clustered dots. We present the details of LPS, how to program using it, several of its properties and applications, especially for electrophotographic (EP) imaging.

Advances in Linear Pixel Shuffling

Presented at the Fibonacci Association conference, Summer 1992, St. Andrews, Scotland

Given an interval or a higher dimensional block of points, that may be either continuous or discrete, how can we probe that set in a smooth manner, visiting all its regions without slighting some and overprobing others? The method should be easy to program, to understand, and to run efficiently.

We investigate a method of visiting the pixels (the elements of a rectangular matrix) and the points in the real unit cube based on an arithmetic progression with wrap-around (modular arithmetic). For appropriate choices of parameters, choices that generalize Fibonacci numbers and the golden mean, we find equidistributed collections of pixels or points, respectively.

We illustrate this equidistributivity with a
novel approach to progressive rendering of
digital images. We also suggest several
opportunities for its application to other areas
of image processing and computing.

Linear Pixel Shuffling for Image Processing, an Introduction

This article appears in The Journal of Electronic Imaging, April, 1993, pp. 147-154.

We present a method of ordering pixels (the elements of a rectangular matrix) based on an arithmetic progression with wrap-around (modular arithmetic). For appropriate choices of the progression's parameters based on a generalization of Fibonacci numbers and the golden mean, we achieve uniformly distributed collections of pixels formed by intervals of the pixel progression or shuffle.

We illustrate this uniformity with a novel approach to progressive rendering of a synthetic image, and we note several opportunities for applications to other areas of image processing.

Error Diffusion and Edge Enhancement: Raster Versus Omni-Directional Processing

Paper present at the Western NY Conference on Image Processing, Rochester, NY, September 9, 2001.

Error diffusion is a well known technique for generating bi-level images and is often used instead of a halftone screen process in order to minimize the visual impact of quantization error. In addition, an alternative to raster image processing, involving a linear algorithm called linear pixel shuffling, was used to perform error diffusion. This allows the use of symmetrical kernels and the omni-directional diffusion of quantization error. Edge enhancement with omni-directional error diffusion was examined and found to be capable of more directionally uniform enhancement of edges than is possible with raster error diffusion.

URBF, the unit radial basis function network is an RBF neural network with all second layer weights set to +/- 1. The URBF models functions or physical phenomena by sampling their behaviors at various probe points, and correcting the model, more and more delicately (i.e., using Gaussian functions with ever narrower spread), when discrepancies are discovered. The probe points---input space positions to test and adjust the network---are linear pixel shuffling points, used for their highly uniform sampling property. We demonstrate the network's performance on several examples. It shows its power via good extrapolation behavior: for smooth-boundary discriminations, very few new hidden units need to be added for a large number of probe points.

An introduction to emulating the problem solving according to nature's method: via evolution, selective breeding, ``survival of the fittest.''

We will present the fundamental algorithms
and present several examples---especially some problems that are
hard to solve otherwise.

A directory containing several pdf slides.

We investigate a method for combining two dispersed-dot halftoning masks for two colors that generalizes the traditional angular displacement used with clustered-dot halftone screens.

Our masks were two variations of
the Linear Pixel Shuffling algebraic mask
algorithm involving numbers arising from two
different
third order, Fibonacci-like recurrences, As an
alternative to screen rotation, these masks were
repositioned by flipping and rotating, and all
pairwise positional combinations of the
transposed, inverted masks were used two at a
time to generate cyan and magenta images, each at
nominal dot fractions of 0.25. The resulting
blue images were evaluated for color uniformity
and moire using visual evaluations, Fourier
analysis, and a color micro-variation analysis.
This presentation will describe the analytical
techniques used to evaluate color uniformity and
moire.

A Genetic Algorithm Search for Improved Halftone Masks

We present a genetic algorithm that automatically generates halftone masks optimized for use in specific printing systems. The search is guided by a single figure of merit based on a detailed model of the printing process and the human visual system. Our experiments show that genetic algorithms are effective in finding improved halftone masks and that two methods of reducing the search space to particular subsets of possible halftone masks greatly enhance the performance of the GA.

Good Halftone Masks via Genetic Algorithms

Presented at the 2003 Western New York Image Processing Workshop, Rochester, NY, October 17, 2003.

We present a genetic algorithm that automatically generates halftone masks optimized for use in specific printing systems. The search is guided by a single figure of merit based on a model of the printing process and the human visual system.

Our
experiments show that genetic algorithms are effective
in finding improved halftone masks and that two
methods of reducing the search space to particular
subsets of possible halftone masks greatly enhance
the search performance.

A Genetic Algorithm Search for Improved Halftone Masks

Presented at the ANNIE Conference, November 2003, St. Louis, MO.

We present a genetic algorithm that automatically generates halftone masks optimized for use in specific printing systems. The search is guided by a single figure of merit based on a detailed model of the printing process and the human visual system. Our experiments show that genetic algorithms are effective in finding improved halftone masks and that two methods of reducing the search space to particular subsets of possible halftone masks greatly enhance the performance of the GA.

Optimizing Halftone Masks with Genetic Algorithms and a Printer Model

Techniques are described for determining optimum halftone masks for electrophotographic laser printers. Earlier techniques were based primarily on up-shifting the noise power of the halftone pattern in order to minimize visual granularity. This strategy worked well with low addressable printers of the 1980s and early '90s. However, current addressability of EP printers is so high that too much up-shifting introduces printer instabilities that can introduce different kinds of granularity and/or mottle. Thus, the optimum halftone mask for image quality should have noise power concentrated within at frequency band between the edge of human vision and the edge of printer stability. The authors have developed search techniques for examining permutations of halftone masks of size HxW. The techniques are based on a strategy called a genetic algorithm (GA).

J Language Lectures

This is a collection of overheads introducing the J language with many examples and explanations.

Compressible Halftoning

We present a technique for producing an bi-level, halftoned imaged from a gray-scale image in such a way that the halftoned image is compressible.

The technique involves sorting the halftoned image using the halftone mask as a sort key, then using a Huffman code to represent groups of the rearranged pixels.

We get compression ratios of 3.5--10 for halftoned photographs.

This talk was presented at Electronic Imaging,
Santa Clara, January 2003. However, material on multilevel
halftone compression was added later.

Combinatorial Proofs of Fermat's, Lucas's, and Wilson's Theorems

In this note, we observe that many classical theorems from number theory are simple consequences of a simple combinatorial lemma.

To appear in the American Mathematical Monthly

Convoutions Combinatorially

Inspired by Benjamin and Quinn's combinatorial approach to the Fibonacci numbers, etc. (Proofs that Really Count, The Art of Combinatorial Proof), we observe that the Fibonacci convolution sequence has a conceptually simple combinatorial interpretation which leads immediately to a fourth-order recurrence and other properties.

We also find a combinatorial nearly proof-without-words for the convolution formula for the difference between the Pell and Fibonacci sequences as well as differences for other linear recurrences. This uses the observation that the number of ways to tile an n-board with squares and dominoes is a Fibonacci number; in case the squares come in two colors, it's a Pell number.

Finally, a consideration of the placement of the second color in tiling an n-board gives the Pell numbers as another convolution of the Fibonacci numbers.

Presentation at the Midwest Conference
on Combinatorics, Cryptography, and Computing,
Rochester, New York, October 2004

Advances in Ordered Greed

Ordered Greed is a form of genetic algorithm that uses a population of permutations whose fitnesses depend on their use as the orders in which parts of a problem are solved. For example, a permutation may specify the order of the rows in which chess-board queens are placed to try to avoid attacks by other queens, or it may specify the order in which vertices of a graph are colored to avoid adjacent vertices getting the same color. The problems mentioned are surrogates for practical, difficult, real-life problems such as scheduling.

Ordered greed requires its own form of crossover. We have developed and investigate two new crossover operators. The first uses the standard combinatorial mathematics notion of permutations' signatures, which are a list of numbers that specify a permutation, but for which the usual bit-string notions of substring-swapping crossovers apply. The second form of crossover works with permutations directly, merging two parents, as in a riffle shuffle, and extracting two children from the merged list consisting of the lists of first or second instances of each value. We compare the new methods with each other and with the well-known PMX

The application for ordered greed that we have chosen for this investigation is that of coloring Hamiltonian planar graphs (i.e., map graphs), which are well-known to be four colorable.

An abbreviated version of this paper was
accepted by the ANNIE Conference, St. Louis, Nov. 2004.

**For a pdf version of this paper, **click here.

**For the visuals, **click here.

Back to Peter Anderson's Home Page

``Compressible Halftones,'' with Changmeng Liu, Proceedings of The SPIE/IS&T Symposium on Electronic Imaging, January, 2003, Santa Clara, CA.

``A Printer Model that is Independent of the Halftone Algorithm,'' with Jonathan S. Arney and Sunadi Ganawan, submitted to the Journal of Imaging Science and Technology, 2002

``Kubelka-Munk Theory and the MTF of Paper,'' with Jonathan S. Arney, Josh Nauman, and Jim Chauvin, submitted to the Journal of Imaging Science and Technology, 2002

``Using Genetic Algorithms to Design Digital Halftone Masks,'' with Jon Arney, Sam Inverso, and Dan Kunkle, (in process, 2003)

``Error Diffusion and Edge Enhancement Raster Versus Omni-Directional Processing,'' with Jonathan S. Arney and Sunadi Ganawan Accepted for publication in J. Imag. Sci. & Technol., Fall, 2002.

``Reversed Zeckendorff Radix,'' presented at the Fibonacci Association Conference, Flagstaff, AZ, June 2002.

``The Unit RBF Network: Experiments and Preliminary Results,'' Cybernetics and Systems, an International Journal, 2002.

``Error Diffusion and Edge Enhancement Raster Versus Omni-Directional Processing,'' with Jonathan S. Arney and Sunadi Ganawan Presented at the Fall 2001 IEEE Signal Processing Society Western NY Symposium on Image Processing.

``The MTF of a Printing System,'' with Jonathan Arney, Preshant Mehta, and Kevin Ayer, Presented at the IS+T Conference, NIP2000, Vancouver, BC, Canada, October, 2000.

``Linear Pixel Shuffling (I): New Paradigms for New Printers,'' with Jonathan Arney, Preshant Mehta, and Kevin Ayer, Presented at the IS+T Conference, NIP2000, Vancouver, BC, Canada, October, 2000.

``Linear Pixel Shuffling (II): An Experimental Analysis of Tone and Spatial Characteristics,'' with Jonathan Arney, Preshant Mehta, and Kevin Ayer, Presented at the IS+T Conference, NIP2000, Vancouver, BC, Canada, October, 2000.

``Ordered Greed, II: Graph Coloring,'' Presented at the ICSC/NAISO Conference, Information Science Innovations (ISI 2001), Dubai, UAE, March, 2001.

``On the a-density of the Fibonacci sequence,'' with Paul Bruckman, accepted for publication in Notes on Number Theory and Discrete Mathematics, 1999.

``Linear pixel shuffling error diffusion,'' Proceedings of the Imaging Science and Technology conference, PICS 2000, Portland, Oregon, March, 2000.

``Linear Pixel Shuffling (I): New Paradigms for New Printers,'' with Jonathan Arney and Kevin Ayer, Proceedings of the Imaging Science and Technology conference, NIP 2000, Vancouver, British Columbia, October, 2000.

``An expanded Neugebauer model for printer color formation,'' with Huanzhao Zeng, IS&T/SPIE Conference on Color Imaging.

``Ordered greed,'' Proceedings of the ICSC Conference SOCO'99 (soft computing), Genoa, Italy, June, 1999.

``Digital Halftoning Using Error Diffusion and Linear Pixel Shuffling,'' with John Szybist. Eighth International Conference on Fibonacci Numbers and their Applications, Summer, 1998. Published in: Applications of Fibonacci Numbers, Vol. 8, G. Bergum, N. A. Philippou, A. F. Horodam, ed., Kluwer, 1999.

``The Unit RBF Network: Experiments and Preliminary Results,'' Neural Computing '98, Vienna, September, 1998.

``Conjectures on the Z-Densities of the Fibonacci Sequence,'' with Paul S. Bruckman. Accepted for publication by the Fibonacci Quarterly.

``On the a-Densities of the Fibonacci Sequence,'' with Paul S. Bruckman. Submitted to the Fibonacci Quarterly.

``Alpha-Radix Codes and Alpha-Shuffle Trees,'' with Wai-Fong Chuan, Mathematics Department, Chung-yuan Christian University, Chung Li, Taiwan, Republic of China. Submitted to the Fibonacci Quarterly.

.ul 1 United States Patent Number 5,555,317: ``Supervised training augmented polynomial method and apparatus for character recognition'' September 10, 1996.

.ul 1 United States Patent Number 5,555,103: ``Half-tone conversion mask for digital images,'' September 10, 1996.

``A Genetic Algorithm Approach to FET Modeling,'' with Neil Craig and P. R. Mukund: in ((unknown conference))

.ul 3 Proceedings of the International ICSC Symposium on Intelligent Industrial Automation (IIA'96) and Soft Computing (SOCO'96), Edited with Kevin Warwick, March 26-28, Reading, U.K., ICSC Academic Press, ISBN 3-906454-01-0.

``Using Quasi-Random Numbers in Neural Networks,'' invited lecture at the Congress of Non-linear Analysis, Athens, Greece, July, 1996.

``The Fibonacci Shuffle Tree,'' presented to the International Conference on Fibonacci Numbers and their Applications, Graz, Austria, July, 1996.

``Training Wheels for Encoder Networks'' in .ul 3 Proceedings of the International ICSC Symposium on Intelligent Industrial Automation (IIA'96) and Soft Computing (SOCO'96), March 26-28, Reading, U.K., ICSC Academic Press, ISBN 3-906454-01-0.

``Using the J Language for NN and GA Experiments'' in .ul 3 Proceedings of the International ICSC Symposium on Intelligent Industrial Automation (IIA'96) and Soft Computing (SOCO'96), March 26-28, Reading, U.K., ICSC Academic Press, ISBN 3-906454-01-0.

``Neural network fitness functions for a musical GA,'' with Al Biles and Laura Loggi, in .ul 3 Proceedings of the International ICSC Symposium on Intelligent Industrial Automation (IIA'96) and Soft Computing (SOCO'96), March 26-28, Reading, U.K., ICSC Academic Press, ISBN 3-906454-01-0.

``Using quasirandom numbers in neural networks,'' with Roger S. Gaborski, Ming Ge, Sanjay Raghavendra, and Mei-ling Lung, Proceedings of the International ICSC Symposium on Fuzzy Logic, May 26-27, 1995, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, Editor: N.C. Steele, Publication by ICSC Academic Press, International Computer Science Conventions, Canada/Switzerland, ISBN 3-906454-00-2, pp. A50 - A56.

``A Simple Proof of a Remarkable Continued Fraction Identity.'' with Tom C. Brown and Peter J.-S. Shiue, Proceedings of the American Mathematical Society, Vol. 13, No. 7, July 1995.

United States Patent Number 5,479,523: ``Constructing classification weights matrices for pattern recognition systems using reduced element feature sets,'' December 26, 1995.

``Advances in Linear Pixel Shuffling,'' Proceedings of the 1994 International Conference on Fibonacci Numbers and their applications.

``An algebraic mask for halftone dithering,'' Proceedings of the 47th Annular Conference of the Society for Imaging Science and Technology, May, 1994, pp. 487-489 Reprinted in Recent Progress in Digital Halftoning, Reiner Eschbach, ed., The Soc. for Imag. Sci. and Tech., 1995.

``Linear pixel shuffling applications,'' Proceedings of the 47th Annular Conference of the Society for Imaging Science and Technology, May, 1994, pp. 506-508 Reprinted in Recent Progress in Digital Halftoning, Reiner Eschbach, ed., The Soc. for Imag. Sci. and Tech., 1995.

``The polynomial classification system,'' with Roger S. Gaborski, Arun Rao Proceedings of the 47th Annular Conference of the Society for Imaging Science and Technology, May, 1994, pp. 508-510.

``A neural network approach to the histogram segmentation of digital radiographic images,'' with Roger Gaborski and Lori Barski, Intelligent Engineering Systems Through Artificial Neural Networks, Volume 3, Cihan H. Dagli, et al., eds. (Proceeding of the 1993 Conference Artificial Neural Networks in Engineering (``ANNIE 93'').), pp. 375-380.

``Linear Pixel Shuffling for Image Processing, an Introduction,'' The Journal of Electronic Imaging, April, 1993, pp. 147-154.

``Neural network training using a genetic algorithm and iterated pseudo-inverse,'' with Koji Ueda and John A. Biles, Intelligent Engineering Systems Through Artificial Neural Networks, Volume 3, Cihan H. Dagli, et al., eds. (Proceeding of the 1993 Conference Artificial Neural Networks in Engineering (``ANNIE 93'').), pp. 187-199

``Designing better keyboards via simulated annealing,'' with Lissa Light, A.I. Expert, September, 1993, pp. 20-27.

``A hardware polynomial feature net for hand-printed digit recognition,'' with Arun Rao, Roger, S. Gaborski, and K. S. Jaiswal, Proceedings of the Third IEE International Conference on Artificial Neural Networks, 1993, pp. 36-40.

``Genetic algorithm selection of features for hand-printed character identification,'' with Roger S. Gaborski, David G. Tilley, and Christopher T. Asbury, Artificial Neural Networks and Genetic Algorithms, Proceedings of the International Conference in Innsbruck, Austria, 1993, R. F. Albrecht, C. R. Reeves, and N. C. Steele (eds.), Springer-Verlag. pp. 101-106.

``The polynomial method augmented by supervised training for hand printed character recognition,'' with Roger S. Gaborski, Artificial Neural Networks and Genetic Algorithms, Proceedings of the International Conference in Innsbruck, Austria, 1993, R. F. Albrecht, C. R. Reeves, and N. C. Steele (eds.), Springer-Verlag. pp. 417-422.

Fast Rendering, Computer Language. February, 1993, pp. 40-48.

``Multidimensional golden means,'' Presented at the Fifth International Conference on Fibonacci Numbers and their Applications, Summer, 1992. Published in: Applications of Fibonacci Numbers, Vol. 5, G. Bergum, N. A. Philippou, A. F. Horodam, ed., Kluwer, 1993, pp. 1-10.

``Neural network Applications to the Color Scanner and Printer Calibrations,'' with Henry Kang, Journal of Electronic Imaging, April 1992, pp. 125-135

``Fibonaccene,'' Fibonacci Numbers with Applications, Proceedings of the 1984 Conference of the Fibonacci Association, D. Reidel Publishing Company, 1985.

``Redundancy Techniques for Software Quality,'' Proceedings of the 1978 Annual Reliability and Maintainability Symposium

``On the Formula pi = 2 SUM arcot f[2k+1],'' The Fibonacci Quarterly, Vol. 16, No. 2, April 1978, pp. 118.

``The Interpretation of i.r. and Raman Spectra Using Pattern Recognition,'' with J. M. Comerford, W. H. Snyder, and H. S. Kimmel, Spectrochimica Acta, Vol. 33A, pp. 651-667, Pergamon Press, 1977.

``Another Proof of the Theorem on Pattern Reproduction in Tessellation Structures,'' Journal of Computer and System Sciences, Vol. 12, No. 3, June 1976, pp. 394-398

``Computational Aspects of Deciding If All Roots of a Polynomial Lie within the Unit Circle,'' with L. E. Heindel and M. R. Garey, Computing, Vol. 16, 1976, pp. 293-304.

``A Generalization of Baudet's Conjecture (Van der Waerden's Theorem),'' American Mathematical Monthly, Vol. 83, No. 5, May 1976, pp. 359-361

``Cobordism Classes of Squares of Orientable Manifolds,'' Annals of Mathematics, Vol. 83, No. 1, January 1966, pp. 47-53.

``Cobordism Classes of Squares of Orientable Manifolds,'' Bulletin of the American Mathematical Society, November 1964, Vol. 70, No. 6, pp. 818-819

1.7417517417617E+47