Optical pattern recognition using circular harmonic expansion

OAM-based optical wavelet using a single pixel detection system for probing dynamic environments with application to real-time measurements of strong atmospheric turbulence

This paper presents a novel method for optical probing by generating optical fields with characteristics of wavelets. The optical wavelets form a basis of rotated asymmetric beams with scaled orbital angular momentum (OAM) and beam sizes. The probing method was used experimentally to measure the continuous wavelet transform of a turbulent propagation path, giving insight into the angular properties about a fixed radius. The wavelet transform of a three-dimensional turbulence distribution was measured; the measurements are much faster than the turbulence changes, allowing characterization of an instantaneous realization of turbulence over time. Results show highly localized regions of OAM in space through the turbulence and characteristics of the turbulence can be extracted from the wavelet transforms.

Rotation Invariant Parallel Signal Processing Using a Diffractive Phase Element for Image Compression

We propose a new rotation invariant correlator using dimensionality reduction. A diffractive phase element is used to focus image data into a line which serves as input for a conventional correlator. The diffractive element sums information over each radius of the scene image and projects the result onto one point of a line located at a certain distance behind the image. The method is flexible, to a large extent, and might include parallel pattern recognition and classification as well as further geometrical invariance. Although the new technique is inspired from circular harmonic decomposition, it does not suffer from energy loss. A theoretical analysis, as well as examples, are given.

The Fourier-Argand Representation: An Optimal Basis of Steerable Patterns

Computing the convolution between a 2D signal and a corresponding filter with variable orientations is a basic problem that arises in various tasks ranging from low level image processing (e.g. ridge/edge detection) to high level computer vision (e.g. pattern recognition). Through decades of research, there still lacks an efficient method for solving this problem. In this paper, we investigate this problem from the perspective of approximation by considering the following problem: what is the optimal basis for approximating all rotated versions of a given bivariate function? Surprisingly, solely minimising the L2-approximation-error leads to a rotation-covariant linear expansion, which we name Fourier-Argand representation. This representation presents two major advantages: 1) rotation-covariance of the basis, which implies a "strong steerability" - rotating by an angle α corresponds to multiplying each basis function by a complex scalar e-ikα; 2) optimality of the Fourier-Argand basis, which ensures a few number of basis functions suffice to accurately approximate complicated patterns and highly direction-selective filters. We show the relation between the Fourier-Argand representation and the Radon transform, leading to an efficient implementation of the decomposition for digital filters. We also show how to retrieve accurate orientation of local structures/patterns using a fast frequency estimation algorithm.

Half a century of optics in computing--a personal perspective [Invited]

Optical signal processing and computing was triggered by the invention of the laser. Starting practically in 1960, it really took off with the introduction of the spatial-matched filter in 1964. Almost half a century later, research and engineering activity in the field continues unabated but in directions that could not have been anticipated in those early days. This paper presents an overview of the developments in the field, discussing the advantages, disadvantages, and limitations of optics in computing paradigms to indicate where and how optics can be exploited in this area. Initially, optical methods were introduced for processing analog signals. Early attempts to extend optical methods toward digital processing failed because the differences between photons and electrons were not properly appreciated. In the last part of the paper we show that some novel concepts and advanced technology may revitalize also optical processes within the digital computing world. This latter development is demonstrated by digital logic functions implemented on simple electro-optic networks. (My personal perspective on the role of optics in computing is deeply rooted in many years of collaboration with my late friend, H. John Caulfield, and I dedicate this paper to his memory.).

Phase-space rotations and orbital Stokes parameters

We introduce the orbital Stokes parameters as a linear combination of a beam's second-order moments. Similar to the ones describing the field polarization and associated with beam energy and its spin angular momentum, the orbital Stokes parameters are related to the total beam width and its orbital angular momentum. We derive the transformation laws for these parameters during beam propagation through first-order optical systems associated with phase-space rotations. The values of the orbital Stokes parameters for Gaussian modes and arbitrary fields expressed as their linear superposition are obtained.

Automatic detection of magnetic flux emergings in the solar atmosphere from full-disk magnetogram sequences

In this paper, we present a novel method to detect Emerging Flux Regions (EFRs) in the solar atmosphere from consecutive full-disk Michelson Doppler Imager (MDI) magnetogram sequences. To our knowledge, this is the first developed technique for automatically detecting EFRs. The method includes several steps. First, the projection distortion on the MDI magnetograms is corrected. Second, the bipolar regions are extracted by applying multiscale circular harmonic filters. Third, the extracted bipolar regions are traced in consecutive MDI frames by Kalman filter as candidate EFRs. Fourth, the properties, such as positive and negative magnetic fluxes and distance between two polarities, are measured in each frame. Finally, a feature vector is constructed for each bipolar region using the measured properties, and the Support Vector Machine (SVM) classifier is applied to distinguish EFRs from other regions. Experimental results show that the detection rate of EFRs is 96.4% and of non-EFRs is 98.0%, and the false alarm rate is 25.7%, based on all the available MDI magnetograms in 2001 and 2002.

Intensity-invariant nonlinear filtering for detection in camouflage

We introduce a method based on an orthonormal vector space basis representation to detect camouflaged targets in natural environments. The method is intensity invariant so that camouflaged targets are detected independently of the illumination conditions. The detection technique does not require one to know the exact camouflage pattern, but only the class of patterns (e.g., foliage, netting, woods). We use nonlinear filtering and the calculation of several correlations. The nonlinearity of the filtering process also allows high discrimination against false targets. Several experiments confirm the target detectability where strong camouflage might delude even human viewers.

Generalization of the Jared and Ennis method of complex transmittance objects for the generation of synthetic discriminant function filters

We present a simple method of constructing synthetic discriminant function filters optimized to take into account the modulation of liquid-crystal devices. This relaxation algorithm, a generalization of the Jared and Ennis method, is an iterative method that includes arbitrary modulations for both scene and filter, extending the problem to the complex plane. Simulated and experimental results obtained in a VanderLugt correlator are presented for a two-class recognition problem. The optimal number of images needed to describe an object in a filter generated in this way is discussed, and the influence of the spatial light modulation resolution on the correlation is studied.

Detection of three-dimensional objects under arbitrary rotations based on range images

In this paper a unique map or signature of three dimensional objects is defined. The map is obtained locally, for every possible rotation of the object, by the Fourier transform of the phase-encoded range-image at each specific rotation. From these local maps, a global map of orientations is built that contains the information about the surface normals of the object. The map is defined on a unit radius sphere and permits, by correlation techniques, the detection and orientation evaluation of three dimensional objects with three axis translation invariance from a single range image.

Invariant pattern recognition by use of a spatial code division multiplexing approach

Invariant pattern recognition can be achieved by use of harmonic decomposition, for example circular harmonics are used for rotation invariant recognition. A common problem with such methods is that often only a single term of the harmonic decomposition is used, and it does not contain a sufficient amount of the reference energy. Thus discrimination capability is limited, especially in the presence of noise or other disturbances. By using several terms of the harmonic decomposition together this problem can be solved; this can be achieved by the use of code division filter multiplexing. Several harmonic terms are encoded onto a single filter, and the signal is simultaneously correlated with all of them, hence producing enhanced discrimination capabilities. Here two methods are suggested for such encoding. The first involves multiplexing the filters in the Fourier plane, while the second involves multiplexing in the image plane.

Complex encoding of rotation-invariant filters onto a single phase-only spatial light modulator

The accuracy and flexibility of the technique proposed by Davis et al. [Appl. Opt. 35, 2488 (1996)] for the encoding of the amplitude and the phase of a filter onto a single liquid-crystal spatial light modulator operating in a phase-only regime has been exploited to implement several filter designs in a convergent optical correlator. We have selected some filters, that given their mathematical structure showing some degree of rotational invariance, or having a parameter to regulate their behavior, require amore precise encoding. We present correlation results of outstanding quality for various rotationally invariant filter designs that have never been previously implemented with a real-time optical correlator.

Rotation invariant color pattern recognition by use of a three-dimensional Fourier transform

Recently, the use of three-dimensional correlation for multichannel pattern recognition has been introduced. In this work we propose the use of circular harmonic components with this new technique to obtain invariance under target rotations. The differences between this method and the previous use of circular harmonic filters for multichannel images are discussed. Also the problem of determining the proper center is studied and, to our knowledge, a new and more understandable criterion to locate it is introduced. Some simulation results to verify the successful operation of the method are included.

Recognition of unsegmented targets invariant under transformations of intensity

Images taken in noncooperative environments do not always have targets under the same illumination conditions. There is a need for methods to detect targets independently of the illumination. We propose a technique that yields correlation peaks that are invariant under a linear intensity transformation of object intensity. The new locally adaptive contrast-invariant filter accomplishes this by combining three correlations in a nonlinear way. This method is not only intensity invariant but also has good discrimination and resistance to noise. We present simulation results for various intensity transformations with and without random and correlated noise. When the noise is high enough to threaten errors, the method trades off intensity invariance in order to achieve the optimum signal to noise ratio, and the peak to sidelobe ratio in the presence of clutter is always greater than one. In the presence of random disjoint noise, the signal to noise ratio is independent of the target contrast and of the level of the noise.

Distortion tolerant image recognition receiver by use of a multiple-hypothesis method

A multiple-hypothesis method is used to detect a target or a reference signal in the presence of additive noise with unknown statistics. The receiver is designed to detect the target and to be tolerant of the variations in rotation and illumination of the target. A multiple-hypothesis test with unknown-noise parameters is used to locate the target position. The proposed method does not use any specific distortion-invariant-filtering technique, but it relies on a multiple-hypothesis approach. Maximum-likelihood estimates of the illumination constant and the unknown noise parameters are obtained. Computer simulations are presented to evaluate the performance of the receiver for various distorted noisy true-class targets with varying illumination and false-class objects.

Optimum nonlinear composite filter for distortion-tolerant pattern recognition

We describe a nonlinear distortion-tolerant filter for pattern recognition that is optimum in terms of tolerance to input noise and discrimination capability. This filter was derived by minimization of the output energy that is due to the overlapping additive noise and the input scene, and the output of the filter meets the design constraints obtained from the training data set. The performance of this filter was tested with an input scene containing one of the training data sets, a nontraining true target, and a false object in the presence of overlapping additive noise and nonoverlapping background noise. We carried out Monte Carlo runs to measure the statistical performance of the filter and obtained receiver operating characteristics curves to show the detection capabilities of the filter.

Projection-invariant pattern recognition with logarithmic harmonic function and wavelet transform

A logarithmic harmonic filter can detect objects at different projection angles. The Mexican-hat wavelet function can extract edges of equal width for objects, regardless of their sizes. Hence incorporating wavelet filtering in the logarithmic harmonic filter can improve its performance. The theory is presented together with computer simulation. Finally, an experiment using a joint transform correlator is presented to verify the capability of the proposed filter.

Rotation-invariant optical recognition of three-dimensional objects

An automatic method for rotation-invariant three-dimensional (3-D) object recognition is proposed. The method is based on the use of 3-D information contained in the deformed fringe pattern obtained when a grating is projected onto an object's surface. The proposed method was optically implemented by means of a two-cycle joint transform correlator. The rotation invariance is achieved by means of encoding with the fringe pattern a single component of the circular-harmonic expansion derived from the target. Thus the method is invariant for rotations around the line of sight. The whole experimental setup can be constructed with simple equipment. Experimental results show the utility of the proposed method.

Shift- and scale-invariant recognition of contour objects with logarithmic radial harmonic filters

The phase-only logarithmic radial harmonic (LRH) filter has been shown to be suitable for scale-invariant block object recognition. However, an important set of objects is the collection of contour functions that results from a digital edge extraction of the original block objects. These contour functions have a constant width that is independent of the scale of the original object. Therefore, since the energy of the contour objects decreases more slowly with the scale factor than does the energy of the block objects, the phase-only LRH filter has difficulties in the recognition tasks when these contour objects are used. We propose a modified LRH filter that permits the realization of a shift- and scale-invariant optical recognition of contour objects. The modified LRH filter is a complex filter that compensates the energy variation resulting from the scaling of contour objects. Optical results validate the theory and show the utility of the newly proposed method.

Nonlinear rotation-invariant pattern recognition by use of the optical morphological correlation

We introduce a modification of the nonlinear morphological correlation for optical rotation-invariant pattern recognition. The high selectivity of the morphological correlation is conserved compared with standard linear correlation. The operation performs the common morphological correlation by extraction of the information by means of a circular-harmonic component of a reference. In spite of some loss of information good discrimination is obtained, especially for detecting images with a high degree of resemblance. Computer simulations are presented, as well as optical experiments implemented with a joint transform correlator.

Optimal tradeoff circular harmonic function correlation filter methods providing controlled in-plane rotation response

Correlation methods are becoming increasingly attractive tools for image recognition and location. This renewed interest in correlation methods is spurred by the availability of high-speed image processors and the emergence of correlation filter designs that can optimize relevant figures of merit. In this paper, a new correlation filter design method is presented that allows one to optimally tradeoff among potentially conflicting correlation output performance criteria while achieving desired correlation peak value behavior in response to in-plane rotation of input images. Such controlled in-plane rotation response is useful in image analysis and pattern recognition applications where the sensor follows a pre-arranged trajectory while imaging an object. Since this new correlation filter design is based on circular harmonic function (CHF) theory, we refer to the resulting filters as optimal tradeoff circular harmonic function (OTCHF) filters. Underlying theory, OTCHF filter design method, and illustrative numerical results are presented.

Filter implementation technique for multicriteria characterization of coding domains in the joint transform correlator

An improved method for implementing correlation filters in the joint transform correlator architecture is proposed. We derived the method from computer-generated holography techniques. It allows us to use any correlation filters, especially ones that provide an optimal trade-off between noise robustness, peak sharpness, and optical efficiency, with any spatial light modulator (SLM). This method also allows for an objective comparison of the performance of the coding domains of various SLM's.

Rotation-invariant and controllable space-variant optical correlation

We propose a method for designing a correlator for achieving rotation-invariant and controllable space-variant optical correlation. The design concept is based on a combination of fractional correlation and circular-harmonic decomposition of the reference object. The suggested method is described and analyzed in detail. Numerical simulations show that this new correlator might provide potential applications in practice.

Parallel joint transform correlator applied to devanagari script recognition

A novel, to our knowledge, scheme for a parallel joint transform correlator system is demonstrated by the introduction of a five-channel zone-plate image multiplicator combined with photographic referencing and is applied to the recognition of Devanagari script. For considering the script's specific features, a partial image was obtained by means of windowing with 25 x 25 pixels out of a 64 x 64 pixel image of full-size characters and was employed for correlation. A comprehensive measurement of all 38 fundamental characters, namely, 1444 correlation pairs, showed satisfactory discrimination performance.

Sidelobeless multiple-object discriminant filters recorded as discrete-type computer-generated holograms

The computer generation of sidelobeless multiple-object discriminant correlation filters has been stressed. We propose to synthesize the filter functions by use of the simulated-annealing algorithm. By this method the filters can be obtained as discrete-type computer-generated holograms. The filters can suppress the sidelobes and provide sharp correlation peaks. A computer simulation and an optical experiment were performed, and the expected correlation responses were obtained.

Coordinate-transformed filter for shift-invariant and scale-invariant pattern recognition

A variable radial coordinate transformation of the phase-only filter (POF) that is dependent on the energy's angular distribution of the target spectrum is used to perform shift- and scale-invariant pattern recognition. The POF of a basic size target and the cumulative energy of its angular distribution are calculated. The filter function is then transformed by means of stretching along the radial coordinate so that the same energy contribution to the correlation peak is provided for any size target. The maximum ratio for recognizing scaled objects is 1:1.5. Computer simulations and optical experiments showing the performances of the filter are presented.

Partial rotation-invariant pattern matching and face recognition with a joint transform correlator

We describe the circular-harmonic (CH) image CH(mr), which is based on CH components for rotationally invariant pattern recognition. CH components of the order m, derived from an image in polar coordinates, are used to form a two-dimensional space together with the radial variable r. Filtering the CH(mr) image leads to a reference image with some rotational invariance. For a narrow-pass filter we have a single CH component with full rotation invariance; for an all-pass filter we have the original image with no rotational invariance; for a low-pass filter we form a reference image containing multiple circular harmonics with partial rotation invariance. Results of numerical simulations and optical experiments with a joint transform correlator are given that illustrate partial-rotation-invariant recognition for human face images.

Invariant pattern recognition by use of wavelength multiplexing

Rotation-invariant pattern recognition can be achieved with circular-harmonic decomposition. A common problem with such a filter is that, because it is only a single term out of the circular decomposition, it does not contain much of the reference object's energy. Thus, the obtained correlation selectivity is low. This problem is solved by use of wavelength multiplexing. First, different harmonic terms are encoded by different wavelengths, and then they all are added incoherently in the output correlation plane. This process leads to rotation-invariant pattern recognition with a higher discrimination ability.

Circular harmonic filters for the recognition of marine microorganisms

We present an application of circular-harmonic filters (CHF's) for the recognition of planktonic microorganisms. CHF's discriminated both genera Acartia and Calanus. The symmetry of genus Acartia permitted discrimination to the species and sex levels, whereas the asymmetry of the genus Calanus permitted discrimination only to the generic level. The differences among organisms of different sex of the genus Calanus could not be detected by these particular CHF's. More research needs to be carried out with more complex CHF's to enhance their performance and to permit the implementation of an automated optodigital system to identify and count marine microorganisms.

Projection-invariant pattern recognition with a phase-only logarithmic-harmonic-derived filter

A phase-only filter based on logarithmic harmonics for projection-invariant pattern recognition is presented. This logarithmic-harmonic-derived filter is directly calculated in the Fourier plane. With respect to normal logarithmic-harmonic filters it provides a smaller variation of the correlation intensity with the projection factor of the target. Computer and optical experiments are presented.

Double-multiplexed computer-generated holograms

One of the detour-phase computer-generated-hologram encoding methods was improved for obtaining two objects simultaneously without increasing the plotter resolution. The encoded patterns were reconstructed along two orthogonal diffraction directions. Such a hologram could be useful for two-channel coherent or incoherent optical correlation. Laboratory experiments demonstrate the suggested technique.

Convolution-kernel-based optimal trade-off filters for optical pattern recognition

An architecture for the implementation of optical pattern recognition is proposed that makes use of convolution-kernel-based optimal trade-off filters to allow for an increased speed of operation and filter storage capability. The derivation of these new convolution-kernel-based optimal trade-off filters is presented, and their noise robustness and discrimination capabilities are discussed.

Wavelet-transform-based composite filters for invariant pattern recognition

A wavelet-transformation-based optical processor for performing invariant pattern recognition is suggested. It contains a composite filter that consists of several wavelet daughter functions derived from the reference object. The intensity of the correlation peak is determined to be invariant to various deformations of the reference object. Computer simulations show explicitly the promising capability of the new technique. Laboratory experimental results are given.

Optimum rotation-invariant filter for disjoint-noise scenes

We introduce an optimum matched filter for rotation-invariant pattern recognition in scenes with disjoint noise. The optimum filter maximizes the ratio between the correlation-peak intensity and the correlation-output energy; in order to ensure rotation invariance, we assume the filter to take the form of a circular-harmonic function. The performance of this new filter is tested by computer simulations with a variety of natural and artificial backgrounds. The results obtained from these tests show an excellent performance much better than the classical circular-harmonic function (the filter that maximizes the signal-to-noise ratio while maintaining rotation invariance).

Multiple circular-harmonic-function correlation filter providing specified response to in-plane rotation

The circular-harmonic-function correlation filter originally proposed by Hsu and Arsenault [Appl. Opt. 21, 4016 (1982)] for in-plane rotation invariance uses only one harmonic, which results in poor discrimination capability of the filter. Various methods to use multiple harmonics were explored previously by different researchers. We present a new method to combine multiple circular harmonics into a single filter that can provide the desired correlation response to in-plane rotation while minimizing the correlation-plane energy. Since multiple harmonics are included, the filter can discriminate well, and since correlation-plane energy is minimized, correlation peaks tend to be sharp. Since the designer can specify the desired in-plane rotation response, a variety of filter behaviors (including complete invariance to input rotations) can be obtained. Underlying theory is discussed, and simulation results are presented.

Image preprocessing for rotation-invariant pattern recognition in the presence of signal-dependent noise

I propose a new method that ensures efficient rotation-invariant pattern recognition in the presence of signal-dependent noise by combining the application of rotation-invariant correlation filters with preprocessing of the noisy input images. The preprocessing uses local suboptimal estimators derived from estimation theory and implies an a priori knowledge of a model describing the noise source. The image noise sources considered are speckle and film-grain noise. Pour different metrics are used to analyze the correlation performance of the circular-harmonic filter, the phase-only circular-harmonic filter, and the binary phase-only circular-harmonic filter, with and without a preprocessing. Computer simulations show that signal-dependent noise can seriously degrade the performance of the phase-only circular-harmonic filter and the binary phase-only circular-harmonic filter. The most severe indication of correlation-performance degradation is the occurrence of false alarms in 15% to 20% of noise realizations of the correlation. Preprocessing increases the correlation-peak signal-to-noise ratio significantly and reduces the false-alarm probability by one to two orders of magnitude.

Synthetic circular-harmonic phase-only filter for shift, rotation, and scaling-invariant correlation

A synthetic circular-harmonic phase-only filter is described. With this filter and a Fourier-transform correlator it is possible to obtain shift, rotation, and scaling-invariant correlations.

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.

Polynomial expansion for shift- and one- or two-dimensional scale-invariant pattern recognition

A polynomial expansion is suggested for achieving optical invariant pattern recognition. The expansion results in a real function and thus is theoretically able to be implemented under both coherent and spatially incoherent illumination. One obtains the expansion after applying the Gram-Schmidt algorithm on the Laurent's series in order to achieve orthonormality. The initial Laurent term with which we apply the Gram-Schmidt procedure is chosen according to the desired expansion order. The use of the polynomial expansion is demonstrated for shift- and one-dimensional scale-invariant pattern recognition as well as for shift-and two-dimensional scale-invariant recognition.

Adaptive pattern recognition with rotation, scale, and shift invariance

Mutually orthogonal pattern distortions are handled by an adaptive optical recognition system. A double-channel system is presented that implements pattern recognition with rotation, scale, and shift invariance. The recognition process is based on a two-stage operation: An object-independent determination of one distortion parameter (the scale, in the example presented here) is performed, and then the recognition is completed by a shift- and rotation-invariant optical correlator that is adapted to the measured parameter. Thus, complete invariance to three distortion parameters is achieved by the combination of two channels. The overall process is performed efficiently and can be executed in real time.

Amplitude-modulated circular-harmonic filter for pattern recognition

An amplitude-modulated circular-harmonic filter is proposed for rotation-invariant pattern recognition. We investigate the filter characteristics by varying two design parameters, A(ρ) and B(ρ), and select optimum values to design an amplitude-modulated circular-harmonic filter. When compared with the phase-only circular-harmonic filter, the amplitude-modulated circular-harmonic filter is found to yield a sharper correlation peak, a better noise tolerance, and an improved correlation discrimination.

Composite harmonic filters for scale-, projection-, and shift-invariant pattern recognition

The Mellin radial harmonic filter and the logarithmic harmonic filter are useful for performing optical scale- and projection-invariant pattern recognition, respectively. To our knowledge, on the basis of the harmonic-function method, no one has been able to obtain more than one invariant property (in addition to the shift invariance) when using the matched-filter approach. A new method of combining the scale-, the projection-, and the shift-invariance properties is proposed, based on two decomposition stages of the input pattern. Computer simulations are presented as well as preliminary experimental results.

Circular-Mellin features for texture segmentation

Texture is an important cue in region-based segmentation of images. We provide an insight into the development of a new set of distortion-invariant texture operators. These "circular-Mellin" operators are invariant to both scale and orientation of the target and represent the spectral decomposition of the image scene in the polar-log coordinate system. Coupled with the unique shift invariance property of the correlator architecture, we show that these circular-Mellin operators can be used for rotation-and scale-invariant feature extraction. We note that while these feature extractors have a functional form that is similar to the Gabor operators, they have distortion-invariant characteristics unlike the Gabor functions that make them more suitable for texture segmentation. A detailed analytical description of these operators and segmentation results to highlight their salient properties are presented.

Optical correlation: influence of the coding of the input image

We analyze the influence of different optical coding methods of the input image in optical correlators. The noise robustness and the optical efficiency of the correlator are investigated. We show in particular that the signal-to-noise ratio is greatly dependent on the coding method. It decreases drastically for large phase modulation.

Rotation-invariant joint transform correlator

We propose a new method for a rotation-invariant joint transform correlator that uses a rotating reference image that can potentially be applied to machine vision.

Generation of continuous complex-valued functions for a joint transform correlator

The principle of representing continuous complex-valued functions by their decomposition into three positive-valued ones is proposed for the generation of complex reference functions for a joint transform correlator. Three basic approaches involving coherent and incoherent superposition of the component functions are analyzed. The potentials and limitations of the techniques are discussed.