Roland Kempf

Regularity and chaos in recurrent fuzzy systems

Abstract In this paper, we shall present a mathematical definition of recurrent fuzzy systems and begin to systematically investigate the underlying theory involved. Unlike static fuzzy systems, recurrent fuzzy systems are equipped with time-delayed feedback of their output and allow representing knowledge-based dynamic processes that may be stated in the form of “if …, then …” rules. We study their relationship to automata and show that they have an automaton-like behavior when appropriately designed. In other cases, recurrent fuzzy system may exhibit chaotic behavior. We present sufficient conditions for the occurrence of chaos in recurrent fuzzy systems that can easily be checked solely on the basis of the qualitative, linguistically formulated models. We also discuss the extent to which state graphs may be used for describing the behaviors of recurrent fuzzy systems.

Equilibria of recurrent fuzzy systems

Unlike static fuzzy systems, recurrent fuzzy systems are equipped with feedback loops and thus exhibit dynamic behaviors. The dynamics of a recurrent fuzzy system is largely determined by its rule base. The dynamic behavior of a significant subclass of recurrent fuzzy systems may be immediately deduced from their rule base, without need for analyzing their mathematical description. Their equilibrium points may be readily identified and their stability behaviors investigated based on their rule base. The investigations involved lead to convergence theorems and other statements that preclude chaotic dynamics.

Contact-Based Stiffness Sensing of Human Eye

Goldmann applanation tonometry is commonly used for measuring intraocular pressure (IOP) to diagnose glaucoma. However, the measured IOP by conventional applanation tonometry is valid only under the assumption that all subjects have the same structural eye stiffness. This paper challenges in vivo measurement of eye stiffness with a noninvasive approach and investigates individual differences of eye stiffness. Eye stiffness is defined by the applied force and displacement of the cornea. The displacement is detected based on captured images by a high resolution camera. The experimental results show that the measured stiffness nicely matches the analytical result that is derived from a simple spherical deformation model with an internal pressure. However, some subjects have different eye stiffness even with the same IOP. IOP with abnormal stiffness may be over/underestimated by conventional applanation tonometry. The proposed eye stiffness measurement can help detect the misestimated eye and it contributes to the early detection of glaucoma.

On Ω-limit Sets of Discrete-time Dynamical Systems

The paper investigates z -limit sets for discrete-time dynamical systems of the form x n +1 = f n +1 ( x n ), n S 0, with each f n mapping an interval I of R into itself. For autonomous systems, i.e. f n = f for all n , and f continuous on I =[ a , b ], the case that all z -limit sets consist of one point only is characterized by several equivalent conditions, one being that f has no 2-periodic points. The non-autonomous case assumes that the functions f n converge uniformly to a continuous function f X that has no 2-periodic points. It is shown that the z -limit sets are closed intervals consisting of fixed points of f X only. Under certain conditions these closed intervals contain exactly one point each. This allows a treatment of certain discrete-time dynamical systems in R n .

Sequential pattern recognition employing recurrent fuzzy systems

Sequential pattern-recognition systems check whether data strings, e.g., time series, exhibit certain pattern primitives in a specified order. As in the case of most other pattern-recognition methods, either conventional methods or fuzzy systems may be used here. This paper presents a sequential pattern-recognition system employing recurrent fuzzy systems that is employed as a monitoring system on continuous-casting systems in the steel industry worldwide. Taking that application as a starting point, a general method for sequential pattern recognition in time series that uses recurrent fuzzy systems is described.

Understanding eye deformation in non-contact tonometry

Non-contact tonometers are widely used to measure the internal eye pressure, i.e. the IntraOcular Pressure (IOP), which is an important parameter for the diagnosis and treatment of glaucoma. During the measurement, the eye is deformed by a short air pulse. Commonly the pressure dependent deformation is estimated from the time when the eye becomes flat, which is derived from the monitored reflection of an incident infrared light. We used a high speed camera to capture the complete motion of the eye directly and obtain more data during the pressure measurement. Assuming a simple eye model with non-linear material properties of the cornea, we extend our previous analysis of the motion of the eye, and obtain a similar principle shape of the eye deformation as observed in the experiments

Dynamic sensing of human eye using a high speed camera

The eye pressure is an important index for judging whether an eye suffers from glaucoma or not. Conventional eye pressure measurements provide us with meaningful data only under the assumption that all the subjects have the same eye stiffness. Our former work, however, says that the assumption is not valid. This work is an extended version. By combining both a high speed camera with 5000[fps] and an air puff supply system, the eye surface can be precisely tracked with respect to time. We introduce a new parameter, which is the displacement at a predetermined applied force. We found that the parameter can strongly emphasize the characteristic of the individual eye stiffness and keeps a high correlation with the measured external eye pressure, so called IOP (intraocular pressure).

Measurement and Analysis of Human Eye Excited by an Air Pulse

For the diagnosis and treatment of glaucoma, a reliable and non-invasive measurement method of the internal eye pressure, i.e. the intraocular pressure (IOP), is desired. The IOP resists a deformation of the eye and thus non-contact tonometers, which deform the eye by a short air pulse, are suitable to estimate the IOP. By monitoring the reflection of an incident infrared light, the tonometer measures the time an applied air jet flattens part of the eye. The relationship between flattening time and IOP are provided by calibration. In our work, we used an additional high speed camera to capture the eye deformation directly and obtain more data during the pressure measurement. We show that the principle shape of the deformation can be understood by assuming simple non-linear material properties of the cornea in an eye model. Furthermore, we discuss which sources of error are still needed to be overcome to give a medical meaningful diagnosis for the examined eye by the data from the high speed camera

Dynamic Properties of Human Eyes

This paper presents novel details on the dynamic behavior of human eyes. A high speed camera is used to capture the movement of the eye surface, which is excited by an air jet. For one group of subjects the dynamic response of the eyes ends shortly after the air jet stops. For another group of subjects a distinct offset in the displacement remains, which takes a significantly longer time to vanish. The two distinct phases in the eye movement are the result of the dynamic response of the cornea and the total eye, respectively. A deeper understanding of the eye dynamics is important for obtaining a higher reliability of diagnostic tools for glaucoma

Eye Stiffness Measurement by Probe Contact Method

The internal eye pressure is an important index for judging whether an eye suffers from glaucoma or not. The conventional eye pressure measurement is valid only under the condition that all subjects have the same structural eye stiffness. This paper challenges the practice of measuring the stiffness of a human eye by pressing the cornea with a contact probe. The displacement of the eye is captured by a camera with high resolution. Experimental results suggest that the measured eye stiffness nicely matches with the theoretical estimation. Based on the experimental results, the difference between the eye stiffness measured by the contact method and the non-contact method is discussed