Joint Fourier Research Articles

On Joint Fourier–Laplace Transforms

Weak convergence of random variables is characterized by pointwise convergence of the Fourier transform of the respective distributions, and in some cases can also be characterized through the Laplace transform. For some distributions, the Laplace transform is easier to compute and provides an alternative approach to the method of characteristic functions that facilitates proving weak convergence. We show that, for a bivariate distribution, a joint Fourier–Laplace transform always characterizes the distribution when the second variate is positive almost surely.

Dynamic holographic joint fourier transform correlator based on a CdF2:Ga crystal

A joint Fourier transform correlator based on a bulk holographic medium—a CdF2:Ga crystal serving a dynamic matched spatial-frequency filter—is implemented. Specific features of the correlator design are discussed. Correlatorf’s possibilities for selective recognition of objects of different type are demonstrated. Experimental data, results of mathematical modeling, and their analysis are reported. Suggestions for modifying the correlator scheme in order to improve its characteristics (operating speed, stability with respect to orientational and scale changes in objects, signal-to-noise ratio, correlation peak sharpness) are formulated.

Impulsive effect on an elastic solid with generalized thermodiffusion

The theory of generalized thermoelastic diffusion with one relaxation time is employed to study the distribution of temperature, displacement components, stresses, concentration and chemical potential in a semi-infinite medium having an impulsive mechanical load at the origin. Using the joint Laplace and Fourier transforms, the governing equations are transformed into a vector–matrix differential equation which is then solved by the eigenvalue approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace and Fourier transforms. Results of this work are presented graphically and are compared with the results of generalized thermoelasticity and classical elasticity deduced as special cases.

Biopolymer-based material used in optical image correlation

We investigate the possible application of a modified deoxyribonucleic acid (DNA)-dye system for dynamic processing of optical information, e.g., optical correlation. The system consists of a biopolymeric matrix made of DNA substituted with the cationic surfactant molecule cetyltrimethyl-ammonium chloride (CTMA) and doped with a photochromic Disperse Red 1 dye. Fast dynamics (millisecond range of rise and fall times) of output correlation signal formation was measured in a joint Fourier transform optical correlator experimental setup. Full reversibility of the correlation signal and reproducibility were observed even after long-time exposures.

FLEXIBLY PROGRAMMABLE OPTO-ELECTRONIC ANALOGIC CNN COMPUTER (POAC) IMPLEMENTATION APPLYING AN EFFICIENT, UNCONVENTIONAL OPTICAL CORRELATOR ARCHITECTURE

To overcome shortcomings of the currently available optical computer architectures a programmable opto-electronic CNN array computer (POAC) is introduced. It combines the optical correlators' supreme resolution and complexity with the versatility of the single chip adaptive visual CNN-UM sensor and processor arrays. The system is grounded on an unconventional optical correlator architecture, which is a modified joint Fourier transform correlator. This architecture uses the angular coding of the template pixels and applies the template not in the recording phase, but in the reconstruction phase of correlation. Feasibility of this concept has been successfully demonstrated by a breadboard model. It computes correlation in two consecutive steps: First a hologram of the input image is recorded in a dynamic holographic material (we used chemically-modified Bacteriorhodopsin film samples). Next, angularly-coded read-out beams reconstruct the appropriately weighted and shifted replicas of the stored input image. Superposition of these copies produces the correlogram. A programmable adaptive sensor array, a special Visual CNN-UM chip detects the resulting correlogram. So, both the sensing and the required post-processing steps are accomplished in a parallel way. The introduced opto-electronic architecture provides exceptional synergy and good compromise of the applicable technologies and the system performance.

Two-dimensional average contrast analysis of the joint Fourier spectrum under rotation of an input object

We present a detailed study of the intensity variations of the joint Fourier spectrum of two input objects when one of them is rotated while the other remains fixed by determining the 2-D average contrast of the spectrum as a function of the angle of rotation. The contrast is found to be object dependent, decreases rapidly for small rotations, and has an almost flat response for larger rotations. We explain this behavior in both theoretical and numerical ways for two simple rectangular objects and only numerically using the fast Fourier transform, for binary and grayscale test objects whose analog rotations are acquired with an optomechanical system that uses a high-precision rotatory stage. As an application of the 2-D average contrast analysis a pair of similar cross section images of a biological organ are used, to obtain the lowest rotation angle between them, and as a consequence a quantitative aligning method for fine angular movements is presented that uses a 2 f optical-digital coherent processor to obtain the Fourier transform of the input objects. Experimental results are presented.

Liquid crystals for photonic applications

In this paper we describe application of liquid crystals in optical imaging and processing. Electrically and optically addressed liquid crystal spatial light modulators are key elements in real-time holographic devices. Their implementation for beam steering and hologram formation is briefly discussed. The Joint Fourier transform optical correlator for pattern recognition is presented as well as the use of liquid crystals for the adaptive optics purposes is discussed.

Defect Detection in Fabrics with a Joint Transform Correlation Technique: Theoretical Basis and Simulation

The theoretical basis of a technique for real time defect detection in fabrics is presented using a joint transform correlator. This correlation technique is an extension of Fourier transform analysis and is extremely useful for real time pattern recognition. The regular periodic nature of a woven fabric makes it possible to use the Fourier transform technique to detect defects. However, classifying various defect types is difficult from the Fourier analytical and experimental results. A solution to this problem is to use a joint Fourier transform of a reference pattern and the test pattern, and the joint power spectrum is further processed. Cross- and auto-correlation peaks, generated after the execution of the second Fourier transform on the filtered joint power spectrum, indicate the existence of a particular defect type. Because of the parallel processing ability of the optical system, implementing the joint transform correlation technique in an optical domain is advanta geous. A fractional power fringe adjusted filter is used for efficient detection of defects. The mathematical formulation of the technique is supported by the simulated results for identifying some defects such as the existence of thick yarns, knots, and missing yarns.

Synthetic optical processor for generalized logic realization problems

A synthetic joint Fourier transform (SJFT) correlator is proposed for the realization of generalized logic problems. As a case study, a multi-channel SJFT correlator is employed to realize a multi-output logic unit. For a given correlator set-up, this particular scheme ensures the maximal utilization of space bandwidth product. This work establishes a pragmatic approach for implementing real-time programmable content addressable memory for information processing.

Measurement of focal length for a spherical lens with a joint Fourier transform spectrum interferometer

A method of testing the monochromatic focal length of a spherical lens by using a joint Fourier transform spectrum interference technique is proposed and experimentally demonstrated. The focal length is determined by the spacing of the interference fringes resulting from two identical input images and is measured at the back focal plane of the tested lens. The system is easy to adjust and realizes automatic measurement with high accuracy.

Joint Fourier transform correlation with phase thresholding in the Fourier domain

Wright State UniversityComputer Science and EngineeringDepartmentDayton, Ohio 45435E-mail: aawwal@cs.wright.eduAbstract. A new implementation of the joint Fourier transform correla-tion (JFTC) utilizing the phase information in the Fourier plane is pro-posed. When a phase-based thresholding scheme is used on the jointFourier spectrum, the computation is significantly accelerated. We obtainalmost double normalized correlation peaks with comparable discrimina-tion peak ratio as that of binarized JFTC.

Optical wavelet-matched filtering with bacteriorhodopsin films

We present here the experimental and numerical results to demonstrate the superior performance of optical wavelet-matched filtering over conventional matched filtering. For this purpose, the biomolecule material bacteriorhodopsin (bR) as the recording media and the improved dual-axis joint transform correlator configuration as the preferred optical setup have been used. Compared with the dual-axis joint Fourier transform correlator, the dual-axis joint wavelet transform correlator provides better correlation performance.

Joint transform correlator subtracting a modified Fourier spectrum

The optically generated joint Fourier transform (JFT) of a test image and a reference image is processed using a new method: the JFT is recorded twice. In the second recording the reference image is phase shifted by π with respect to the first recording. The two JFT's are subtracted and binarized with a threshold of zero. Strong correlation peaks are obtained, and correlations within the test image are suppressed. Some results of optical implementation are presented, using a ferroelectric liquid crystal display with 128 × 128 pixels for data input. The phase shift of the reference was implemented by the contrast-inverted reference input on the binary light-modulating device. Processing of the JFT is done by a CCD camera, a frame grabber, and a personal computer.

Filter-feature-based rotation-invariant joint Fourier transform correlator

Rotation-invariant target detection using a trained filter-feature-based joint Fourier transform (JFT) correlator is investigated. First, a composite reference image is obtained from a training set of targets. An optimum filter formulation is then applied on this composite image to come up with a new feature that we refer to as a filter feature. This feature is then used in a JFT correlator, which results in a simple and robust rotation-invariant target recognition system.

All-optical nonlinear joint Fourier transform correlator

We present the first all-optical nonlinear joint transform correlator based on a square-law receiver in the Fourier plane. Our device uses a photorefractive limiting quadratic processor. The compressional nonlinearity associated with the transfer function of the limiting quadratic processor enables the correlator to detect signals embedded in Gaussian and non-Gaussian noise. In the limiting region this device correlates the phase-only information of the input. This is the first time to our knowledge that photorefractives or real-time holography has been used in the correlation of the phase-only information. We demonstrate the operation of this device experimentally, and we evaluate its performance throughcomputer simulation for various forms of noise.

Applying time modulation to the joint transform correlator

A major limitation on the optical joint transform correlator (JTC) is that the output plane is dominated by unwanted autocorrelation products. A new technique is proposed that uses time modulation and demodulation to separate the correlation components. Time modulation is applied to the JIG inputs, resulting in a time-modulated joint power spectrum. Demodulation of the transform plane separates the self-correlation components from the other terms. When the demodulated signal is the input to the second transform stage in the JTC the result is system peak-to-noise ratio/peak-to-secondary ratio (PNR/PSR) improvement, removal of input plane location constraints, and elimination of detection problems resulting from multiple targets. Two implementations are discussed. Using a general model for sinusoidal modulation, it is shown that amplitude, phase, and polarization modulation of the inputs all result in amplitude modulation of the correlation signals in the transform plane. The general solution is difficult to implement, because it requires temporal demodulation of the joint Fourier signal on a pixel-by-pixel basis. A more practical system results from the case of square-wave modulation, where it is shown that demodulation can be easily implemented through image subtraction using only two to four frames of data. This Fourier plane processing technique has been implemented using a binary JTC (BJTC). Performance improvement of 6 dB over the conventional BJTC is demonstrated through computer simulation and laboratory results.

Joint transform correlator performance enhancement by a logarithmic- polar transformation

Rotation and scale requirements imposed by the joint Fourier transform correlator (JFTC) are relaxed by means of a logarithmic-polar transformation of the input object. The resulting target recognition/tracking system utilizes the positional invariance of the Fourier image, whereas the logarithmic-polar mapping is performed on the Fourier plane instead of the intensity of the Fourier plane. The positional information is saved, and consequently, multiple input objects are processed simultaneously. Enhancements in the areas of input binarization and power spectrum enhancements to the liquid crystal television based JFTC are performed to improve its recognition and discrimination abilities.

Effects of sampling and binarization in the output of the joint Fourier transform correlator

We discuss a number of anomalies that can result in the output of a joint Fourier transform correlator. Some of these are caused by the writing of the joint power spectrum onto a spatial light modulator having a grid structure. Others result from the binary recording of the joint power spectrum. These effects can cause spurious output cross-correlation peaks and can affect the strengths of the desired cross correlation peaks.

An Efficient Implementation of Joint Fourier Transform Correlation Using a Modified LCTV

AbstractThe joint Fourier transform correlation (JFTC) algorithm is implemented using a modified Citizen liquid crystal television (LCTV) which acts as a spatial light modulator in the amplitude mode. Using a combination of optics and electronics, results are obtained in real time (1/30 of a second). The video output is introduced into a video processing system to eliminate the noise inherent when using a coherent, monochromatic source. An optical disk is used to eliminate the dc term in the correlation plane and obtain only the correlation peaks.

Single spatial light modulator joint transform correlator

We describe a joint Fourier transform image correlator that employs thresholding at both the input plane and the Fourier plane. This suggests using a single binary spatial light modulator (SLM) to read in sequentially the binarized input signal and the binarized Fourier transform interference intensity. The performance of the single SLM joint Fourier transform correlator (JTC) is compared with that of the classical JTC in the areas of correlation peak intensity, peak-to-sidelobe ratio, signal-to-noise ratio (SNR), and correlation width. We show that the single SLM JTC outperforms the classical JTC in all such areas. Using a single binary SLM results in significant reduction in cost, size and complexity of the system.