Eduardo R. Caianiello

C-calculus: An elementary approach to some problems in pattern recognition

Abstract This work utilizes the concept of “Composite set” ( C -set) and of the related C -calculus to study some standard problems of pattern analysis and general processing of signals. After some basic definitions and notations about composite sets are briefly stipulated, it is shown how a family of C -sets can be associated with a digitized picture. Each element in the family conveys partial information about the picture itself, yet it is possible to combine the various contributions from each C -set in such a way as to completely retrieve the image. Conditions that guarantee such “convergence” are theoretically investigated: the cases of nonconvergence are also proved to be of some interest. C -calculus is concretely applied to the extraction of significant regions in a digitized picture, of contours, etc. An application to texture discrimination and analysis is also outlined.

Maximal acceleration and Sakharov’s limiting temperature

SummaryIt is shown that Sakharov’s maximal temperature, derived by him from astrophysical considerations, is a straightforward consequence of the maximal acceleration introduced by us in previous works.

Neural Networks, Fuzziness and Image Processing

A fuzzy neural network system suitable for image analysis is proposed. Each neuron is connected to a windowed area of neurons in the previous layer. The operations involved follow a method for representing and manipulating fuzzy sets, called Composite Calculus. The local features extracted by the consecutive layers are combined in the output layer in order to separate the output neurons in groups in a self-organizing manner by minimizing the fuzziness of the output layer. In this paper we focalize our attention on the application of the proposed model to the edge detection based segmentation, reporting results on real images and investigating the robustness of the system with noisy data.

Neural associative memories with minimum connectivity

Abstract In this paper a binary associative network model with minimal number of connections is examined and its microscopic dynamics exactly studied. The knowledge of its time behavior allows us to determine a learning rule which realizes a one-step recalling associative memory. Its storage capacity is also analyzed with randomly distributed patterns and is proved to be O(log n) in the worst case, n being the number of neurons and connections, but to increase considerably when the patterns to be memorized are correlated. Spurious states are also investigated.

An application of C-calculus to texture analysis: C-transforms

Abstract A method for the analysis and discrimination of textures, based on C-calculus, is proposed. The concepts of C-space and C-transform of a digitized signal are introduced as simple tools, well suited to the visualization of the filtering properties of C-calculus: C-filters are thus also defined and the “natural” role they seem to play in problems concerning textures is investigated in some practical instances. In particular, C-transforms of some sample textures are provided and texture classification in C-space is performed. Discrimination of objects against textural background is obtained by C-filtering, in an inherently parallel fashion. The philosophy involved in this approach is finally briefly discussed in a comparison with some extant methods.

Outline of a linear neural network

Abstract By utilizing a new definition of product, we develop a neural net model . The memorization and generalization capabilities are investigated in an Information Theory fashion. To show the memorization capabilities, we use it as a decoder, and prove the net reduces the error probability to zero in the range of the error correcting capacity of the used code. To show the generalization capabilities, we use it to infer a code from patterns received by a noisy channel. When the data are affected by independent random errors, this strategy is shown to require a small number of patterns to obtain a good identification with high probability of the code from the noisy data. We also address its use as an associative memory.

On Some Analytic Aspects of C-Calculus

In the description and analysis of various systems there are some situations in which properties or readings cannot be considered as sharp. For instance, this happens in hierarchical classifying, in describing physical phenomena for which the measurements have to be considered valid in ranges rather than in single points and, generally, when some data of the problems are given with tolerances.

Systems and Uncertainty: A Geometrical Approach

Classical mechanics is founded on the assumption that the quantities it studies have an existence of their own, which we may detect and measure, if we care, to any accuracy permitted by experimental know-how. Quantum mechanics sets theoretical limits to the accuracy we may achieve in the measurement of some quantities; their “existence” is not its object, the formalism deals with “phenomena”, not “noumena”. What about the general systems of Cybernetics (or “General Systems Theory”, or any of the more or less synonymic denominations that abound)? A major advance in scientific thought and philosophy has been the realization that all theories are (more or less successful) “models”: man-made, that is, so that the metaphysical notion of “truth”, attached in turn to Newton’s or Einstein’s descriptions of gravitation, to mention the most classic example, has lost its absolute connotation. The search for better, or just alternative models for gravitation and everything else fills all scientific journals.

Entropy, Information and Quantum Geometry

This Chinese apophthegm paraphrases here the equivalent one “Entropy is not Shannon Entropy”. There are in fact at least two things wrong with Shannon entropy. The first is its name: had Claude Shannon not accepted John von Neumann’s advice (“you should call it entropy and for two reasons: first, the function is already in use in thermodynamics under that name; second and more importantly, most people don’t know what entropy really is, and if you use the word “entropy in an argument, you will win every time (1)”) and just called it “uncertainty”, endless confusion would have been spared. “Entropy” is psychologically tied with “thermodynamics” in a physicist’s mind, so that the purely logical, far wider connotation of Shannon’s concept escapes attention. Shannon entropy is simply and avowedly the “measure of the uncertainty inherent in a preassigned probability scheme”; as such it has nothing whatever to do with thermodynamica1 entropy, except in the case in which that probability distribution is known, or proven to be, “canonical”.

Branching Equations of I and II Type and Their Connections with Finite Equations of Motion and the Callan-Symanzik Equation

The aim of this paper is to recall, and present in the context of today’s quantum field theory, some results which we found in a sequel of past works. Most of them were obtained many years ago1); our main objective was at the time to discriminate, by using appropriate techniques, combinatoric from analytic problems. The first were brought into a naturally compact form by introducing some algorithms, which need not be mentioned here; the second were treated, at the beginning of our research, by defining “finite parts” of all quantities of interest, together with the formal properties which had to be satisfied by them in order that combinatorics could apply. Specific definitions were also given for the computation of finite part integrals, but no special emphasis was then placed on this aspect of the problem.