Finite Conjugate Research Articles

Probabilistic Inference from Arbitrary Uncertainty using Mixtures of Factorized Generalized Gaussians

This paper presents a general and efficient framework for probabilistic inference and learning from arbitrary uncertain information. It exploits the calculation properties of finite mixture models, conjugate families and factorization. Both the joint probability density of the variables and the likelihood function of the (objective or subjective) observation are approximated by a special mixture model, in such a way that any desired conditional distribution can be directly obtained without numerical integration. We have developed an extended version of the expectation maximization (EM) algorithm to estimate the parameters of mixture models from uncertain training examples (indirect observations). As a consequence, any piece of exact or uncertain information about both input and output values is consistently handled in the inference and learning stages. This ability, extremely useful in certain situations, is not found in most alternative methods. The proposed framework is formally justified from standard probabilistic principles and illustrative examples are provided in the fields of nonparametric pattern classification, nonlinear regression and pattern completion. Finally, experiments on a real application and comparative results over standard databases provide empirical evidence of the utility of the method in a wide range of applications.

A combined finite element-nonlinear conjugate gradient spatial method for the reconstruction of unknown scatterer profiles

The inversion of microwave imaging data is treated by means of a novel iterative spatial domain reconstruction technique. The latter combines the Finite Element Method (FEM) and a Nonlinear Conjugate Gradient (NCG) optimization scheme. The method is applied to the reconstruction of the complex permittivity profile of cylindrical inhomogeneous objects, that are illuminated by transverse magnetic incident waves, and is based on scattered far field measurements. Starting from an initial guess of the scatterer profile, the FEM is used to solve the forward problem, resulting in an estimate of the scattered field. The inverse process of updating the constitutive parameters of the scatterer is performed by the Polak-Ribiere algorithm by the use of which the difference between the measured and the estimated scattered data is minimized. The minimization procedure is carried out by a sensitivity analysis scheme based on the FEM and the introduction of the adjoint state vector. The proposed method is applied to various cases of scatterer profiles.

Conjugate Image Plane Correlator with Holographic Disk Memory

A conjugate image plane correlator with holographic disk memory is proposed. Optical correlation between conjugate images reconstructed from a holographic disk and an input image on liquid-crystal television is executed with the rotation of the disk. Regardless of Fourier hologram recording with the pseudorandom diffuser, it is found possible to take out the diffuser from the original hologram recording scheme using an image reconstruction process and to get correlation signals between input and reconstructed conjugate images in the output plane of a two-lens imaging system. Generation of conjugate replicas with high contrast causes exact matching with an input image which results in high recognition performance for autocorrelation signals. The transfer function of an optical system can be controlled by adjustment of either hologram size or hologram area illuminated with a laser beam. Hence, the output intensity distribution can be adjusted by selecting a proper pupil function and the size of an output pupil defined by the input pupil size and the optical system magnification factor. The real-time character recognition by optical parallel high-speed processing for two dimensional images with position normalization is demonstrated.

Application of the see-saw method to all refracting optical systems

The optical see-saw diagram is a method that describes image correction to third-order approximation over a finite field of view in rotationally symmetric systems that employ aspheric surfaces. The aim of this paper is to describe the correction of aberrations caused by plane surfaces in all refracting optical systems in terms of the see-saw diagram. A lens correction algorithm based on the see-saw method is described to correct analytically the Seidel aberrations, primary spherical aberration, coma, astigmatism, and distortion, in such systems. We then apply this lens correction algorithm to the design of equivalent configurations by aspherizing different surfaces of the system, and the high-order aberrations of the equivalent configurations are evaluated by means of transverse-ray-aberration plots. Results indicate that this method gives information on what the contribution must be to the third-order aberrations that each component should provide to the system to give a better balance of high-order aberrations. Examples of the lens correction algorithm applied to lenses with six refracting surfaces and working for both finite and infinite object conjugates are given.

INTEGRATED MICRO MO HEAD USING LAMINATED GLASS PLATES

We have already proposed two types of integrated micro MO heads for Mini Disc and reported on their characteristics [1],[2],[3]. These heads are made by forming optical elements such as holograms and beam splitters in multiple parallel glass plates. The type 1 head is made by laminating a 44 μm thick half-wave plate as an azimuth rotator[1],[2]. On the other hand, the type 2 head includes, instead of a half-wave plate, a polarization beam splitter which has a sloping incident plane making an angle of 45 deg. to the incident plane of polarization [3]. In this paper, we report mainly on the detailed structure of the integrated MO head developed for the Mini Disc, the characteristics of each optical thin film for the finite conjugate optical system used in this device, and the simulation results for a number of optical characteristics.

Exact controllability of a spherical cap: Numerical implementation of HUM

This paper deals with the numerical implementation of the exact boundary controllability of the Reissner model for shallow spherical shells (Ref. 1). The problem is attacked by the Hilbert uniqueness method (HUM, Refs. 2–4), and we propose a semidiscrete method for the numerical approximation of the minimization problem associated to the exact controllability problem. The numerical results compare well with the results obtained by a finite difference and conjugate gradient method in Ref. 5.

Merging: a new method for tomography through random media

Multiple laser backscatter references (or guide stars) and optimal processing of the data are used to extend the useful diameter that is obtained with the use of a single laser backscatter reference for imaging at infinite conjugates. The results illustrate that the mean-square error in the estimated wave-front distortion E2 is proportional to D5/3, where D is the telescope diameter. For all the cases that are considered the results are expressible in the form E2 = (D/d0)5/3, where the quantity d0 is a measure of the useful diameter of the telescope. When nine references are used d0 is triple the value that would be achieved with a single reference when a backscatter altitude of 20 km is used. The results illustrate the impact of varying the altitude, changing the number of refrences, having references at two altitudes, and introducing reference-position uncertainty.

The density of states and anisotropic optical absorption of a disordered polymer film

Motivated by a three-dimensional LDA band structure calculation for trans-polyacetylene (PA) we construct an effective two-dimensional model for a film of conjugated polymers. In real materials a non-perfect structural order is found. It has its origin in the finite conjugation length of single chain segments, impurities and crystalline defects. We model these effects through a fluctuating contribution to the interchain hopping amplitude. Using a supersymmetric path integral formalism we obtain the averaged one-particle Green function giving the electronic density of states (DOS) and furthermore the current correlation function determining the optical absorption coefficient and DC conductivity. As a consequence of the interchain coupling we find a broad shoulder at the band edge in the DOS and the optical absorption. The main effect of disorder is seen in a broadening of the interband transition peak and an increasing intraband absorption. The anisotropy in the absorption coefficient parallel and perpendicular to the chain direction allows us to fix the interchain coupling strength; the line shape contains information about the effective disorder strength.

Visualization of the photorefractive grating in BSO

A technique based on self-imaging at finite conjugates has been experimentally demonstrated for visualization of laser-induced photorefractive gratings in BSO. A simple model theory has been used to provide the correct rules of geometrical optics of self-imaging. A novel technique has been used for photoinducing the periodic structure. An application has been made to determine the harmonic components of the nonsinusoidal refractive index modulation in the BSO for a grating period of 10 μm and for external DC voltages of 6.0 and 3.0 kV. The technique is directly applicable to the visualization and study of laser-induced gratings (LIG) in thin nonlinear optical media.

Delta Methods in Enveloping Algebras of Lie Superalgebras

Let $L$ be a Lie superalgebra over a field $K$ of characteristic $\ne 2$ . We define \[ \Delta (L) = \{ l \in L|{\dim _K}[L,l] < \infty \}. \] Then $\Delta (L)$ is a Lie ideal of $L$ and is restricted if $L$ is restricted. $\Delta (L)$ is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products. In this paper we examine $U(L)$, where depending upon $\operatorname {char}K$, $U(L)$ denotes either the enveloping algebra or the restricted enveloping algebra of $L$. We show that $\Delta (L)$ controls certain properties of $U(L)$. Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.

The Finite-conjugate Self-imaging and the Generalized Lau Effect Due to Complex Gratings

Abstract Laser induced gratings in nonlinear optical media, which are complex or phase gratings, are the key structures in many applications involving laser beam-nonlinear material interactions. The concept of coded imaging of self-imaging structures, developed elsewhere, has been utilized to make a comprehensive study of the self-imaging properties, at finite conjugates, of model complex or phase gratings. The same approach has been used to extend our previous work on the generalized Lau effect. An interesting application of the self-imaging process has been made to visualize, at a certain primary magnification, laser induced thin photorefractive grating in a BSO crystal. Possible applications of this technique include visualization of dynamic or developed laser induced gratings in thin nonlinear optical materials.

Optical design with Wood lenses 1: infinite conjugate systems

Spherical aberration in a flat-surfaced radial gradient-index lens (a Wood lens) with a parabolic-index profile can be corrected by altering the profile to include higher-order terms. However, this results in a large amount of third-order coma. An alternative method of aberration correction similar to that used in the catadioptric Schmidt system is presented. A Wood lens with a parabolic-index profile is used to provide all or most of the optical power. Coma is corrected by stop shifting, and spherical aberration is corrected by placing a powerless Wood lens corrector plate at the stop.

General analysis of two-mirror relay systems

The general two-mirror system used at finite conjugates is examined here. Relations for first-order geometric properties and third-order aberrations are given in terms of five design parameters: object distance, image distance, exit pupil size, and the two mirror magnifications. The conditions for aplanatic solutions are derived for conic mirrors. The curvatures of the astigmatic image surfaces are given, and the condition for anastigmatic solutions is derived. The relations are applied to infinite conjugate systems and spherical mirror systems as special cases.

Predicting the Behavior of Finite Precision Lanczos and Conjugate Gradient Computations

It is demonstrated that finite precision Lanczos and conjugate gradient computations for solving a symmetric positive definite linear system $Ax = b$ or computing the eigenvalues of A behave very similarly to the exact algorithms applied to any of a certain class of larger matrices. This class consists of matrices $\hat{A} $ which have many eigenvalues spread throughout tiny intervals about the eigenvalues of A. The width of these intervals is a modest multiple of the machine precision times the norm of A. This analogy appears to hold, provided only that the algorithms are not run for huge numbers of steps. Numerical examples are given to show that many of the phenomena observed in finite precision computations with A can also be observed in the exact algorithms applied to such a matrix $\hat{A} $.

空間フィルタを用いたウエハ異物検出技術

An automated particle detection system for patterned wafers has been developed. A specific area of the wafer is illuminated with a laser at 45° diagonal in reference to an orientation-flat shaped on the wafer. The diffracted light is detected by an objective lens. The signal obtained by the memory-mat on a wafer is reduced to 1/8-1/15 by setting the printed spatial filter on the conjugate image plane of a Fourier transformed plane in an objective lens. By using the subtracted image signal with the spatial filter, 0.5μm standard particles on a multi-layer pattern can be detected.

Delta methods in enveloping algebras of Lie superalgebras

Let L L be a Lie superalgebra over a field K K of characteristic ≠ 2 \ne 2 . We define \[ Δ ( L ) = { l ∈ L | dim K [ L , l ] &gt; ∞ } . \Delta (L) = \{ l \in L|{\dim _K}[L,l] &gt; \infty \}. \] Then Δ ( L ) \Delta (L) is a Lie ideal of L L and is restricted if L L is restricted. Δ ( L ) \Delta (L) is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products. In this paper we examine U ( L ) U(L) , where depending upon char ⁡ K \operatorname {char}K , U ( L ) U(L) denotes either the enveloping algebra or the restricted enveloping algebra of L L . We show that Δ ( L ) \Delta (L) controls certain properties of U ( L ) U(L) . Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.

An electron optical tool kit for transmission and surface electron microscopy

This speculative discussion is offered in memory of Prof. Ben Siegel's fascination with the science of microscope instrumentation. It views the microscope configuration as a set of tools for creating observational opportunities. The central element is a condensor/objective lens in which the specimen is immersed, and along whose axis lie conjugate image and diffraction planes. An illumination system sets up beam conditions, in mode l1, forming a parallel beam, and Mode l2 forming a probe at the center of the lens. A projection system determines the plane within the objective that is magnified onto a recording detector surface. Projection mode P1, views the plane at the lens center, while projection mode P2 views the back focal or diffraction plane behind the lens. These two lens systems, along with apertures, scan coils, and detectors establish various modes of operation which make up the microscopist “tool kit”.

Four-wave mixing critical slowing down

Four-wave mixing with cross-polarised pump beams is known to produce infinite phase conjugate reflectivities at a coupling strength of ΓL=4 in the undepleted pumps approximation. Theoretical results presented here show that within this approximation an effective grating time constant also tends to infinity at this coupling strength i.e. the system exhibits critical slowing down. It can also be seen that the system is analogous to a positive feedback amplifier with a constant gain-bandwidth product. Experimental results are presented using the cubic photorefractive material BTO, these results support the theoretical predictions.

Design of diffractive singlets for monochromatic imaging

The Seidel aberrations of a rotationally-symmetric diffractive lens with an arbitrary phase profile are presented. It is shown that by a proper choice of phase function and aperture stop position, third-order coma and astigmatism can be eliminated for any chosen conjugate ratio. Since a diffractive lens has an inherent zero value for the Petzval sum, the image plane is flat in both tangential and sagittal meridians. The substrate curvature of the lens may be chosen to introduce a prescribed amount of distortion to allow for use as a Fourier transform lens or a laser scan lens. Examples are given of lens performance in finite conjugate imaging and laser scanning, where the f - theta condition is satisfied.

Current status of aplanatic gradient-index lens systems

A gradient index (GRIN) lens, when processed by a spherical curvature at its endface, yields the equivalent aspheric effect, if the curvature is adequately optimized with refractive index distribution coefficients. Aplanatic lens systems of infinite or finite conjugates can be made by utilizing the new type of GRIN lens.