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Utilizing nonlinear optical pattern formation for a simple image-processing task
Abstract
A novel scheme for all-optical image processing is suggested, which is based on spontaneous pattern-formation processes. The
spatiotemporal instabilities, connected with these self-organization effects, appear in many nonlinear optical systems and
mostly obstruct the intended application. We propose to utilize the particular features of pattern formation for parallel
image processing, namely the sensitivity to external influences, the selection of well-defined final states, and the rotational
and translational invariance. We report here on a single-feedback experiment with a liquid crystal light valve as the optical
nonlinearity. In this experiment, the recognition of hexagonal structures is realized. We characterize the response dynamics
and study associative properties of this scheme. The extension to square patterns is discussed, and first steps towards a
practical implementation are undertaken in providing a simple post-processing scheme and testing the concept on realistic
input images.
... An investigation of the temporal evolution of spontaneously formed optical patterns from an initial seed pattern is also interesting in the context of optical parallel image processing, but only a few studies have been reported. Recently, the formation of such patterns from seed optical patterns, which are continuously supplied by an external source, has been investigated in some nonlinear optical systems, and a spatial frequency filtering effect, spatial locking, and spatial synchronization were observed [19,20]. ...
Spontaneous pattern formation in an optoelectronic system with an optical diffractive feedback loop exhibits a contrast enhancement effect, a spatial filtering effect, and filling-up of vacant space while maintaining surrounding structures. These effects allow image processing with defect tolerance. Aberrations and slight misalignments that inevitably exist in optical systems distort the spatial structures of the formed patterns. Distortion also increases due to a small aspect ratio difference between a display device and an image sensor. We experimentally demonstrate that the spatial distortion of the optoelectronic feedback system is reduced by electronic distortion correction and the initial structure of a seed optical pattern is preserved for a long time. We also demonstrate image processing of a fingerprint pattern based on seeded spontaneous optical pattern formation with electronic distortion correction.
... Recently the formation of such patterns from seed optical patterns that are continuously supplied by an external source was investigated in some nonlinear optical systems, and a spatial frequency filtering effect, spatial locking, and spatial synchronization were observed. 13,14 In this paper we analyze the temporal evolution of optical patterns spontaneously formed from a seed pattern initially supplied to an optoelectronic feedback system (OEFS). The OEFS is given the seed pattern just once at the first frame; that is, it is not continuously forced by the pattern. ...
Spontaneous optical pattern formation from an initial seed optical pattern in an optoelectronic system with optical diffractive feedback is investigated experimentally. We demonstrate that the temporal evolution of the spontaneously formed patterns exhibits a contrast enhancement effect, a spatial filtering effect, and filling of vacant space while the surrounding structures are maintained. These effects allow us to perform image processing of natural fringe patterns, i.e., in our experiments, fingerprint patterns. We also demonstrate image processing with defect invariance for fingerprint patterns.
We investigate an extended, nonlinear optical experiment, exhibiting the spontaneous formation of hexagonal patterns out of a stationary bifurcation. The system is exposed to a two-dimensional spatially periodic forcing, namely, static hexagonal patterns, under variation of their spatial periodicity. Parameters are the strength of the forcing and the distance to pattern forming threshold. The system response is quantitatively characterized with different methods. We observe several locking regimes, where the system is entrained by the forcing. Most of the locking regimes can be related to resonances between the different critical wave numbers and the forcing wave number or its spatial harmonics. One particular locking appears to result from two of these simple resonances in a kind of generalized order m:n synchronization. The width of the locking regimes increases with forcing strength, apparently representing a spatial analog of Arnold tongues.
A spatial perturbation method for the stabilization of an unstable state of a pattern forming system is presented. The perturbation, as a weak driving signal, mimics the target pattern. Stabilization, selection, and tracking of unstable rolls, squares, hexagons, and honeycombs are demonstrated by numerical simulations through the example of an optical system. The application of an external spatial perturbation also results in critical slowing down and resonant phenomenon.
Spatiotemporal instabilities of a single-feedback-mirror device are investigated with the help of the linear stability analysis of the stationary regime for a saturable medium that has a complex nonlinear susceptibility; the outcome is compared with experimental results. The first striking result of the analysis is the occurrence of an oscillation with a period nearly twice the round-trip time, regardless of its value with respect to the atomic linewidth for given physical situations, when only longitudinal grating effects are considered. This result cannot be interpreted as a pure phase-conjugate resonator operation, which is expected to occur only when transverse grating effects are included. When such effects are included, the second striking feature is the prediction of the spontaneous breaking of the revolution symmetry; symmetry is broken by rotating light spots, with an instability threshold that depends on the number of spots; this prediction agrees with experiment.
We report the observation of two-dimensional periodic and quasiperiodic structures in the transverse profile of an optical beam circulating in a loop which contains a nonlinear medium. The symmetries of the patterns are imposed by an image rotation within the loop. We propose a simple model which shows the existence of two unstable bands of transverse wave vectors. Close to threshold, the predictions of the model are qualitatively and quantitatively confirmed by the experiment.
In a single-feedback setup with a Liquid Crystal Light Valve as optical nonlinearity, the formation of stationary hexagonal patterns can be observed. Under increase of pump power, the order of these regular patterns breaks down and a complex dynamics sets in. This transition is investigated for experimentally observed patterns in the self-focusing case with help of different tools, namely time-average patterns, symmetry autocorrelation functions and temporal correlation functions. All methods indicate the existence of a second pump threshold, where order and dynamical properties of the patterns change.
We show that a control technique, based on feedback filtered at a
Fourier plane, can stabilize the spatio-temporally disordered output of
a nonlinear optical system. We demonstrate this in an experiment with a
LCLV feedback system and in a theoretical model. We stabilize the system
and select square and roll patterns. The technique is non-invasive in
that the control signal becomes small when control is achieved. A
combination of real- and Fourier-space filtering can stabilize patterns
in any chosen region of the transverse space.
We discuss an optical associative memory architecture, which combines a nonlinear part — the laser — and a linear part formed by lenses, holograms and a pinhole mask. The laser works as the discriminator element in the system, and operates in a regime of spatial multistability, i.e. of coexistence of different stable stationary states. Numerical calculations in the simplest situations indicate that the laser is able to decide which of its stationary states is most similar to a field pattern that is injected into the laser itself. The memory is constituted by a certain number of images which are stored in one of the holograms and are in one-to-one correspondence with the stationary states of the laser. The task of the linear part is (i) to convert an arbitary image, which is offered to the system, into an appropriate field pattern which is injected into the laser and (ii) to convert the stationary beam, which emerges from the laser, into the corresponding image in memory. We show that the system as a whole works as an associative memory. We illustrate some experimental results, obtained using a HeNe laser, which represent an initial step in the direction of a realization of this system.
Optical dependence of spatial frequency of spontaneously formed patterns in a nonlinear optical ring resonator on focusing deviation of the feedback optical system is experimentally investigated. It has identical square root dependence for positive and negative deviations, when the nonlinear device is set near an inversion property. To explain the behavior of the resonator, a model based on the Talbot effect for a fringe pattern is applied. The model is supported by the experimental result that the input pattern of the nonlinear device is the inversion of the output one in the resonator.
Optical patterns generated in a nonlinear optical feedback system composed of an optically addressable spatial light modular and two-dimensional optical feedback contain many stable components of a spatial frequency. The spatial frequency components are supported by the Talbot effect model. We demonstrate that each component in the optical patterns is individually selected by a direct modulation method for a spatial frequency. From our findings, we discuss that the main spatial frequency of the generated optical pattern is decided by competition between the stable solutions, based on the modulation transfer functions of the spatial light modulator and the optical feedback with a spatial frequency filter.
We consider spatiotemporal dynamics in an optical system containing a thin film of nonlinear medium having binary-type refractive nonlinear response and a feedback mirror. Steady-state output transverse patterns consist of a set of localized states, and strongly depend on the initial spatial phase modulation (seeded phase image). These steady-state solution properties can be used for nonlinear image processing and edge detection. System dynamics are studied both theoretically and through numerical simulation.
A recently proposed method for the stabilization of unstable patterns uses an instantaneous feedback derived from the Fourier transform of the output [R. Martin et al., Phys. Rev. Lett. 66, 4007 (1996)]. We successfully extend this method to regimes of spatiotemporal disorder. By focusing on two different nonlinear optical systems, a laser and a Kerr slice with feedback, we demonstrate the usefulness, effectiveness, and generality of the technique. Our method allows for high power outputs without loss of spatial and temporal coherence. It also provides the possibility of pattern selection and, in some instances, tracking within a disordered regime.
The properties of patterns in an anisotropic system are studied experimentally and numerically in an optical pattern forming system with single-mirror feedback. A controlled anisotropy is introduced by a slit filter in Fourier space. If the slit width is reduced, a transition between hexagons and stripes via nonequilateral hexagons and rhomboids is observed. The scenario is reproduced in numerical simulations. The bifurcation to stripes is found to be supercritical whereas the one to hexagons is subcritical in accordance with expectation. The results illustrate the potential of Fourier filtering as a tool for investigating mechanisms of pattern formation.
A self-regulating method for the control of spontaneous instabilities of the transversal beam profile in nonlinear optical systems is experimentally realized. The control method is the high-dimensional analogue of classical negative feedback regulators. An all-optical implementation with its capabilities of parallel processing is essential for the experimental feasibility. Unstable system-inherent transversal patterns are stabilized, including also the stationary homogeneous state. Even spatio-temporal disorder is removed in favour of the highly symmetric, stationary patterns and nearly without losses of output power. The manipulation of the output state is demonstrated to be noninvasive. This allows investigations of the otherwise inaccessible unstable patterns, for which an example is given.
It is shown that a fluid heated from below can reconstruct an incomplete initial pattern in the sense of an associative memory. This property of the fluid is based on the formal similarity between the order parameter equations of the fluid close to its instability point and those used in pattern recognition by means of synergetic computers. Under certain external conditions the system may additionally show multistable behaviour and discriminate between two offered initial patterns with different symmetries, such as a square and a rectangle.
We investigate experimentally the generation of hexagonal patterns of filaments in a laser beam traversing a thin liquid crystal cell placed in front of a single ‘‘virtual mirror.’’ This mirror is the image of a real mirror and is placed either in front of or behind the cell. The results clearly support the physical idea that the Talbot effect governs the phenomenon. Edge effects due to the limited width of the pump beam are accounted for both theoretically and experimentally.
Spontaneous formation of optical patterns is studied in a system composed of a photorefractive phase-conjugating medium and a virtually internal feedback mirror. In addition to ordinary regular hexagons, squares and squeezed hexagons are reported for the first time in optical pattern formation experiments. The angle of the generated sideband beams plotted as a function of the mirror position lies on two different curves, which can theoretically be explained by the contributions from two different regions of the medium separated by the virtual feedback mirror.
Synergetics is an interdisciplinary field of research. It deals with open systems that are composed of many individual parts which interacts with each other and can form spatial, temporal, or functional structures by self-organization. The research goal of synergetics is three-fold: (1) are there general principles of self-organization? (2) are there analogies in the behavior of self-organizing systems? (3) can new devices be constructed because of the results (1) and (2)? From a mathematical point of view, synergetics deals with nonlinear partial stochastic differential equations and studies their solutions close to those points where the solutions change their behavior qualitatively. As will be shown in this article, synergetics in its present form is based on the concepts of stability and instability, control parameters, order parameters and the slaving principle. The slaving principle allows one to compress the information that is necessary to describe complex systems into a few order parameters. This is possible if systems are close to their instability points, but it appears that the order parameter concept is also applicable to situations away from such instabilities. At the level of the order parameter equations, profound analogies between otherwise quite different systems become visible. This allows one to realize the same process (for instance, dealing with information) on quite different material substrates. The order parameter concept and the slaving principle are explained and their extension to discrete noisy maps and to delay equations are mentioned. These results can be applied to pattern formation in fluids, lasers, semiconductors, plasmas and other fields. A section is devoted to the analysis of spatio-temporal patterns in terms of order parameters and the slaving principle. It is shown how the concepts of synergetics can be utilized to devise a new type of computer for pattern recognition. In connection with preprocessing, it can recognize patterns that are shifted, scaled or rotated in space and that are deformed. It can recognize scenes and also facial expressions as well as movement patterns. The learning procedure is briefly outlined. Because of the analogy principle of synergetics, this computer allows for hardware realizations by means of semiconductors and lasers. Decision making by humans or machines is interpreted by means of an analogy with pattern recognition. Further sections are devoted to recognition of dynamic processes and learning by machines. In this author's opinion, synergetics will find important applications in medicine, for instance in the analysis of MEG and EEG patterns, and in the development of devices with brain-like functions. Future tasks for synergetics will be the application of the order parameter concept and the slaving principle to the integration of specialized computers or computer algorithms, for instance for the recognition of faces, movement patterns and so on, into a computer network for scene analysis and decision making. This will hold also for complicated production processes, and so on. Generally speaking, the potentialities of synergetics are based on its self-organization principles.
Synergetic computers form a class of self-organized algorithms. Due to their close similarity to nonlinear self-organized systems in physics and chemistry they are potential candidates for a new sort of image processing hardware. We will study the performance of an unsupervised synergetic learning algorithm with classification problems on both artifical and real texture data and will show that unsupervised synergetic learning can be successfully used for unsupervised pattern classification.
Actual excitable media, and discrete models of excitable media, may be used to process images. Discrete time, discrete space models of excitable media are shown to be examples of synchronous concurrent algorithms and so may be considered as formal computational systems.
Presented are the first experimental results on transverse pattern formation with a liquid crystal light valve under optical feedback. The setup is reduced to the essential elements, retaining maximum symmetry. Formation of regular and perturbed hexagonal structures and optical turbulence is found. Analogy to an existing model, predicting such pattern formation, can be shown.
The information processing capabilities of biomolecular excitable media based on nonlinear dynamic mechanisms are discussed. Given even the simplest medium geometry, dynamics and information processing features inherent in biomolecular excitable media proves to be diverse and sophisticated. For the case of pseudo two-dimensional versions these media can be described in terms of neural networks having lateral connections. The main responses of shunting on-center off-surround feedback neural networks and pseudo two-dimensional excitable systems to the external excitations are surprisingly similar. The excitable media are capable of short-time memory, of contour enhancement and quenching or amplifying small features depending on medium state. The analogies discussed reaffirm specific neural net characteristics of excitable media and give the opportunity to estimate more accurate excitable medium characteristics.
Nonlinear optical systems using a Kerr slice and two-dimensional feedback are analyzed. A Neumann-series approach is used to study pattern formation. We show that interactions between spatial modes (rolls) occur in the form of ‘‘winner-takes-all’’ dynamics and cause formation of hexagon patterns.
Optical transmission measurement on two similar two-dimensional (2D) triangular lattices of air rods with different lattice constants of the order of 1.0 mum has revealed that the central frequencies of photonic band gaps show a reasonable shift relative to each other. The respective gap frequencies observed in the 1.02-mum lattice are found to be consistent with the theoretical ones. These two facts provide firm evidence for the existence of a common gap for H polarization in the entire 2D Brillouin zone.
A nonlinear optical system, which spontaneously forms hexagonal patterns, is exposed to a weak, spatially modulated forcing. As forcing, stationary hexagonal patterns are used under variation of their transverse wave number. In the experiment, we observe a locking when the forcing wave number is close to one of the critical wave numbers of the unforced system. Outside the locking regimes, forcing provokes spatiotemporal disorder. The system response is characterized quantitatively with respect to its dynamics and to its spatial order.
The experimental characterization of pattern forming bifurcations is difficult, since the unstable solutions are generally not accessible in experiments. In nonlinear optics, a novel control scheme allows one to select and stabilize generic patterns and thus to track these solutions in parameter space. This Fourier space scheme is applied to a single-feedback system and the amplitudes of roll, square and hexagon patterns are determined experimentally. Even though the bifurcation is imperfect, the coefficients of a prototype amplitude equation are recovered. The coefficients show satisfactory agreement with theory and with numerical simulations, which are performed for comparison. The simulations also clarify the origin of the imperfect bifurcation in the experiment: boundaries and speckles appear to have an unexpected strong influence.