Phase-only Filter Research Articles

Generalization of the linear matched filter concept to nonlinear matched filters

A nonlinear matched filter based image correlator is investigated. The linear matched filter is expressed as a bandpass function containing the amplitude and phase of the Fourier transform of the reference signal. The bandpass filter function is then applied to a kth law nonlinear device to produce the nonlinear matched filter function. Analytical expressions for the nonlinear matched filter are provided. The effects of the nonlinear transfer characteristics on the correlation signals at the output plane are investigated. The correlation signals are determined in terms of the nonlinear characteristics used to transform the filter. We show that the nonlinear filter results in a sum of infinite harmonic terms. Each harmonic term is envelope modulated due to the nonlinear characteristics of the device, and phase modulated by m times the phase modulation of the linear filter function. The correct phase information of the filter is recovered for the first-order harmonic of the series. The envelope of each harmonic term is proportional to the kth power of the Fourier transform magnitude of the reference signal. We show that various types of filter such as the continuous phase-only filters can be produced simply by varying the severity of the nonlinearity. Nonlinear filters provide higher correlation peak intensity and a better defined correlation spot.

Rotation and scale sensitivities of one-dimensional differentiation correlation

Both symmetrical and nonsymmetrical input patterns are used to study the rotation and scale sensitivities of 1D differention correlation. Computer simulation results show that 1D differentiation correlation has a good performance of rotation and scale sensitivities, almost the same as phase-only matched filter.

Improved correlation discrimination using an amplitude-modulated phase-only filter

A novel amplitude-modulated phase-only filter (AMPOF) is proposed for achieving improved correlation discrimination. The proposed AMPOF has an amplitude spectrum which is the inverse of a biased amplitude spectrum of the object function and a phase spectrum which is a complex conjugate of the phase spectrum of the object function. When compared with the phase-only filters, the AMPOF is found to have significantly superior correlation discrimination capability.

Bounds on the performance of continuous and quantized phase-only matched filters

The classical matched-filter definition of the signal-to-noise ratio (SNR) is used as the performance measure to present bounds for the performances of phase-only filters (POF’s), binary phase-only filters (BPOF’s), and N-ary phase-only filters (NPOF’s). Specifically, signal-dependent bounds on the SNR of the POF relative to the ideal matched spatial filter for additive white noise are developed. In addition, the tightest possible bounds on the SNR of the BPOF relative to the POF for any type of additive noise are developed. These bounds are then extended to develop the tightest possible bounds on the SNR of the NPOF relative to the POF. The extension to the generalized BPOF and the generalized NPOF is also discussed.

Average amplitude matched filter

We propose a simple method of determining whether an unknown object is a member of a class of objects. We modify the standard 4f optical correlator by inserting an amplitude mask in front of the phaseonly filter (POF) in the Fourier transform plane. Our computer simulations show that this changes the output from a POF correlation response to a classical matched filter (CMF) response. The CMF is less sensitive to an exact pattern match than the POF, which can be an advantage in detecting a class of objects. We took the Fourier transform of an ensemble average of five similar input images to make the amplitude mask, thus creating an average amplitude matched filter (AAMF). The AAMF emulated the response ofthe CMF in all autocorrelation and cross-correlation simulations. We also simulated binary phase-only versions of the CMF and AAMF. Again, in all cases the binary AAMF (BAMF) emulated the correlation response of the binary CMF (BCMF). A binary POF contains both a correlation and a convolution component. In the correlation plane, the correlation response of the binary POF is much larger than the convolution response. Both the BCMF and the BAMF correlation response contain a marked convolution as well as correlation response. The amplitude function forces more energy into the convolution response. The result is that the convolution component of the output is nearly as large as the correlation component.

Average amplitude matched filter

We propose a simple method of determining whether an unknown object is a member of a class of objects. We modify the standard 4f optical correlator by inserting an amplitude mask in front of the phaseonly filter (POF) in the Fourier transform plane. Our computer simulations show that this changes the output from a POF correlation response to a classical matched filter (CMF) response. The CMF is less sensitive to an exact pattern match than the POF, which can be an advantage in detecting a class of objects. We took the Fourier transform of an ensemble average of five similar input images to make the amplitude mask, thus creating an average amplitude matched filter (AAMF). The AAMF emulated the response ofthe CMF in all autocorrelation and cross-correlation simulations. We also simulated binary phase-only versions of the CMF and AAMF. Again, in all cases the binary AAMF (BAMF) emulated the correlation response of the binary CMF (BCMF). A binary POF contains both a correlation and a convolution component. In the correlation plane, the correlation response of the binary POF is much larger than the convolution response. Both the BCMF and the BAMF correlation response contain a marked convolution as well as correlation response. The amplitude function forces more energy into the convolution response. The result is that the convolution component of the output is nearly as large as the correlation component.

Off-axis phase-only filter for pattern recognition

A new kind of amplitude-encoded phase-only filter (AE POF) is presented. The interference of the Fourier phase value with a tilted plane reference wave encodes the pure Fourier phase information of the object wave and results in an off-axis phase-only filter (OA POF). After the binarization step the filter function is recorded directly into the recording medium in final size using a laser scanner. Comparing them with the conventional matched filter, the conventional phase-only filter, and the AE POF, as well as their binary versions, we discuss the performances of OA POFs (OA BPOFs), present computer simulations, and demonstrate experimental results.

Generation of terahertz-rate trains of femtosecond pulses by phase-only filtering

We describe a simple linear filtering technique for transforming individual femtosecond light pulses into terahertzrepetition-rate bursts of femtosecond pulses. By using phase-only filtering, high efficiency is achieved. Pulse repetition rates approaching 6 THz are obtained.

Statistical filtering of time-sequenced peak correlation responses for distortion-invariant recognition of multiple input objects

A distortion invariant correlation plane filter (DICPF) based on the analysis of variance technique used in experimental statistics is used to analyze peak correlation responses from a time sequenced coherent optical correlator and identify multiple input objects rapidly. A mathematical development of the filter for invariance to an arbitrary number of distortions, such as rotation, scale, aspect, or any other distortion that can be quantized, is presented. This statistical correlation plane filtering technique can be used to process output correlation data from an array of simple or composite filters that are presented either sequentially or simultaneously (space multiplexed at the Fourier plane of a coherent optical correlator). Results are presented from computer simulations and experiments using time sequenced simple binary phase-only reference filters (BPOFs) to achieve rotation and scale invariant recognition of multiple objects. Results from experiments using sequences of composite BPOFs that are invariant over a range of target rotations are also presented to demonstrate the general applicability of this technique. These results indicate that the DICPF can correctly identify multiple objects within the correlator's target group more than 90% of the time, even in the presence of clutter and noise (up to 20% noise), while maintaining a false alarm rate of 6% or less using single BPOFs.

Two-focal-length optical correlator

A two-focal-length coherent optical correlator design is presented that uses simple, single element lenses. Experimental autocorrelation results using a binary phase-only filter verify the performance of this compact design.

Correlation with a spatial light modulator having phase and amplitude cross coupling

In correlation filtering a spatial light modulator is traditionally modeled as affecting only the phase or only the amplitude of light. Usually, however, a single operating parameter affects both phase and amplitude. An integral constraint is developed that is a necessary condition for optimizing a correlation filter having single parameter coupling between phase and amplitude. The phase-only filter is shown to be a special case.

Distortion range of filter synthetic discriminant function binary phase-only filters

This paper considers the performance of filter synthetic-discriminant-function binary-phase-only filters (fSDF-BPOFs) with images in the distortion range that were not members of the training set. This evaluation was performed to understand better the distortion range that can be effectively covered by fSDF-BPOFs. The peak correlation and peak clutter responses were measured over various distortion ranges for in-plane and out-of-plane rotations. The peak correlation initially decreased rapidly as the distortion range increased but decreased gradually as the distortion range became larger for the images used in this study. The distortion ranges that can be effectively covered by fSDF-BPOFs appear to be modest: 0-30 degrees for inplane rotation and 0-10 degrees for out-of-plane rotation. The results indicate that a temporal multiplexing scheme will be required to cover a large distortion range when BPOFs are used.

Quad-phase-only filter implementation

Some of the fastest commercially available coherent spatial light modulators are binary. This fact has motivated research on the binary phase-only filter, which is unfortunately unable to represent a complex filter function. A technique is introduced here with which a single binary device can be used to represent a complex filter with independent binary representation of real and imaginary parts. The technique is analyzed theoretically, and simulation and laboratory results are presented.

Phase-only entropy-optimized filter generated by simulated annealing

Highly selective and efficient optical pattern recognition is implemented by using phase-only entropy-optimized filters generated by simulated annealing. An electro-optic architecture is proposed for filter generation.

Smoothing of the edge-enhanced impulse response from binary phase-only filters using random binary patterns

We report a technique for reducing the edge-enhanced impulse response from binary phase-only filters (BPOFs). In this technique, the object used to make the filter is multiplied by a random binary pattern. Then the BPOF is constructed. Experimental details are presented.

Fresnel lens-encoded binary phase-only filters for optical pattern recognition

We describe a technique for reducing false signals from optical correlators using binary phase-only filters (BPOF's). Since the impulse-response function of the BPOF contains both the target object and the inverted version of it, the output consists of the correlation of the input object with both of these objects. In our research a Fresnel phase plate lens is encoded onto the BPOF. As a result the correlation with the desired target is on a converging beam, and the correlation with the undesired inverted image of the target is on a diverging beam. Experimental results show that these filters reject inverted input objects.

Phase-only circular harmonic matched filtering

The idea of a phase-only matched filter [J. L. Horner and P. D. Gianino, Appl. Opt. 23, 812-816 (1984)] is incorporated in the scheme of circular harmonic component matched filtering [Y. N. Hsu and H. H. Arsenault, Appl. Opt. 21, 4016-4019 (1982)]. Computer simulation showed that the phase-only circular harmonic matched filter possesses the rotational invariant property for 2-D recognition. Furthermore, it does a better job in recognizing closely placed targets compared with traditional circular harmonic filters. The simulation also demonstrated that a pure phase circular harmonic filter is less liable to give rise to false alarms in recognition because the sidelobe produced by it is much less pronounced and much less extensive than that produced by a traditional circular harmonic filter.

Optimum encoding of binary phase-only filters with a simulated annealing algorithm

A simulated annealing algorithm is introduced to encode binary phase-only filters optimally for image recognition. Similar patterns that cannot be distinguished with conventional filter encoding methods are clearly distinguished with the optimized filter. The computational requirements for optimizing the filter are not excessive.

Efficient algorithm for designing a ternary valued filter yielding maximum signal to noise ratio

An efficient algorithm for designing a ternary valued filter yielding the highest signal to noise ratio (SNR) is utlined. Numerical evaluations using the image of a tank indicate that using such a filter can provide an improvement in SNR of ~5 dB over the conventional binary phase-only filter (BPOF). This is superior to the 1-dB improvement obtained for that image by varying the threshold line angle (TLA) in filter binarization. Simulation results are presented. They agree with the numerically computed SNRs.

Amplitude encoded phase-only filters

The phase-only filter (POF) and binary phase-only filter (BPOF) have been shown to have high optical efficiency, good target discrimination, and virtually no sidelobes. We present a simple method of amplitude encoding the signal phase information. These new filters have most of the same advantages of a POF or BPOF so that amplitude modulating spatial light modulators or photographic film can be used.