TECHNICAL NOTES Experiments on nonlinear joint transform correlator using an optically addressed spatial light modulator in the Fourier plane Bahram Javidi, Qing Tang, Don A. Gregory, and T. D. Hudson Don Gregory and T. D. Hudson are with U.S. Army Missile Command, Research Directorate, Redstone Arsenal, Ala- bama 35898; the other authors are with University of Con- necticut, Department of Electrical Engineering, Storrs, Connecticut 06269-3157. Received 4 August 1990. Correlation experiments for the images used in the pro- posed setup indicate that nonlinear compression of the joint power spectrum may be necessary to produce good correlation performance and a peak-to-sidelobe ratio of larger than unity. Theoretical and experimental studies show that the non- linear joint transform correlator (JTC) produces good corre- lation performance. 1–3 The nonlinear JTC uses nonlinearity at the Fourier plane to transform nonlinearly the joint power spectrum. 1,2 We provide an experimental investigation of the effects of nonlinear transformation of the joint power spectrum on the performance of the JTC. Experiments are used to determine the correlation peak intensity, signal-to- noise ratio (SNR), peak-to-sidelobe ratio (PSR), and correla- tion width for various degrees of the nonlinearity used at the Fourier plane. The experiments are performed in both the absence and the presence of the input scene noise. The spatial light modulator (SLM) used at the Fourier plane is a Hughes liquid crystal light valve 4 (LCLV). The LCLV is operated in different degrees of nonlinear mode by adjusting its bias supply frequency. The experiments indicate that, as the severity of the nonlinear transformation of the joint power spectrum increases, the peak intensity increases, the SNR increases, the PSR increases, and the correlation width . decreases. The experiments also show that for the images used here, nonlinear compression of the joint power spec- trum is necessary to pull the signal out of scene noise and produce a PSR of larger than unity. The implementation of the nonlinear JTC using an optically addressed SLM is shown in Fig. 1 (a). Plane P 1 is the input plane that contains the reference signal r(x – x 0, y) and the input signal s(x + xo,y). The joint power spectrum of the reference signal and input signal is where (a,β) are the spatial frequency coordinates, and S(a,β) exp[iφs(α,β)] and R(a,β) exp[iφ R (α,β)] correspond to the Fourier transforms of the input signal s(x,y) and the refer- ence signal r(x,y), respectively. In the conventional case, the inverse Fourier transform of Eq. (1) produces the linear correlation signals at the output plane. The first two terms in Eq. (1) produce the autocorrelation terms. The terms of 1772 APPLIED OPTICS / Vol. 30, No. 14 / 10 May 1991 interest are the third and the fourth terms, which produce the cross-correlations of the reference signal with the input signal. In Fig. 1(a), the Fourier transform interference is displayed at the input of the LCLV to obtain the intensity of the Fourier transform interference. The LCLV compresses the joint power spectrum according to the nonlinear charac- teristic of the device. The compressed joint power spectrum can be considered as the output of a nonlinear system. 1,5,6 The nonlinear char- acteristic of the device is denoted by g(E), where E is the Fourier transform's interference intensity. An expression for the nonlinearity transformed interference intensity can be obtained using the transform method of communication theory 1 : where Here G(ω) is the Fourier transform of the nonlinearity. It can be seen that for v = 1 the nonlinear system has preserved the phase of the cross-correlation term [φs(a,β) – φ R (a,β)], and only the amplitude is affected. This results in the good correlation properties of the first-order correlation signal at the output plane. Varying the severity of the nonlinearity will produce correlation signals with different characteris- tics. 1 For highly nonlinear transformations, the high spatial frequencies are emphasized and the correlation becomes more sensitive in discrimination. In our experiments, the optically addressed SLM operating in a nonlinear mode may be used to transform nonlinearly the joint power spectrum. The experimental correlator setup is shown in Fig. 1(a). An argon-ion laser beam (λ = 514 nm) is expanded and collimated. It is passed through a photographic transparen- cy containing the reference and input images. The images used in the correlation tests are shown in Fig. 1(b). The reference image is the tank without noise. The sizes of the two images were ~ 2 X 3 mm and 3 × 5 mm, respectively. The separation of the input and reference images is ~3 mm. A Fourier transform lens FTL 1 with focal length f 1 = 1000 mm is employed. The use of a long focal length lens is due to the limited resolution of the LCLV used in the experiments. A He–Ne laser beam (λ = 633 nm) is expanded, collimated, and used as the readout beam of the LCLV. The intensity of the readout beam is ~25 µW/cm 2 over the aperture size of 25 × 25 mm of the LCLV. The reflected beam is amplitude modulated by passing it through a polarizing beam splitter and an analyzer/polarizer. A second Fourier transform lens FTL 2 with focal length f 2 = 400 mm is placed behind the beam splitter. The correlation outputs are detected with a CCD camera interfaced with a computer and viewed with a video monitor. The input-output characteristics of the
Transcript
TECHNICAL NOTES
Experiments on nonlinear joint transform correlator using an optically addressed spatial light modulator in the Fourier plane Bahram Javidi, Qing Tang, Don A. Gregory, and T. D. Hudson
Don Gregory and T. D. Hudson are with U.S. Army Missile Command, Research Directorate, Redstone Arsenal, Alabama 35898; the other authors are with University of Connecticut, Department of Electrical Engineering, Storrs, Connecticut 06269-3157. Received 4 August 1990.
Correlation experiments for the images used in the proposed setup indicate that nonlinear compression of the joint power spectrum may be necessary to produce good correlation performance and a peak-to-sidelobe ratio of larger than unity.
Theoretical and experimental studies show that the nonlinear joint transform correlator (JTC) produces good correlation performance.1–3 The nonlinear JTC uses nonlinearity at the Fourier plane to transform nonlinearly the joint power spectrum.1,2 We provide an experimental investigation of the effects of nonlinear transformation of the joint power spectrum on the performance of the JTC. Experiments are used to determine the correlation peak intensity, signal-to-noise ratio (SNR), peak-to-sidelobe ratio (PSR), and correlation width for various degrees of the nonlinearity used at the Fourier plane. The experiments are performed in both the absence and the presence of the input scene noise. The spatial light modulator (SLM) used at the Fourier plane is a Hughes liquid crystal light valve4 (LCLV). The LCLV is operated in different degrees of nonlinear mode by adjusting its bias supply frequency. The experiments indicate that, as the severity of the nonlinear transformation of the joint power spectrum increases, the peak intensity increases, the SNR increases, the PSR increases, and the correlation width . decreases. The experiments also show that for the images used here, nonlinear compression of the joint power spectrum is necessary to pull the signal out of scene noise and produce a PSR of larger than unity. The implementation of the nonlinear JTC using an optically addressed SLM is shown in Fig. 1 (a). Plane P1 is the input plane that contains the reference signal r(x – x0,y) and the input signal s(x + xo,y). The joint power spectrum of the reference signal and input signal is
where (a,β) are the spatial frequency coordinates, and S(a,β) exp[iφs(α,β)] and R(a,β) exp[iφR(α,β)] correspond to the Fourier transforms of the input signal s(x,y) and the reference signal r(x,y), respectively. In the conventional case, the inverse Fourier transform of Eq. (1) produces the linear correlation signals at the output plane. The first two terms in Eq. (1) produce the autocorrelation terms. The terms of
interest are the third and the fourth terms, which produce the cross-correlations of the reference signal with the input signal. In Fig. 1(a), the Fourier transform interference is displayed at the input of the LCLV to obtain the intensity of the Fourier transform interference. The LCLV compresses the joint power spectrum according to the nonlinear characteristic of the device.
The compressed joint power spectrum can be considered as the output of a nonlinear system. 1,5,6 The nonlinear characteristic of the device is denoted by g(E), where E is the Fourier transform's interference intensity. An expression for the nonlinearity transformed interference intensity can be obtained using the transform method of communication theory1:
where
Here G(ω) is the Fourier transform of the nonlinearity. It can be seen that for v = 1 the nonlinear system has preserved the phase of the cross-correlation term [φs(a,β) – φR(a,β)], and only the amplitude is affected. This results in the good correlation properties of the first-order correlation signal at the output plane. Varying the severity of the nonlinearity will produce correlation signals with different characteristics.1 For highly nonlinear transformations, the high spatial frequencies are emphasized and the correlation becomes more sensitive in discrimination. In our experiments, the optically addressed SLM operating in a nonlinear mode may be used to transform nonlinearly the joint power spectrum.
The experimental correlator setup is shown in Fig. 1(a). An argon-ion laser beam (λ = 514 nm) is expanded and collimated. It is passed through a photographic transparency containing the reference and input images. The images used in the correlation tests are shown in Fig. 1(b). The reference image is the tank without noise. The sizes of the two images were ~ 2 X 3 mm and 3 × 5 mm, respectively. The separation of the input and reference images is ~3 mm. A Fourier transform lens FTL1 with focal length f1 = 1000 mm is employed. The use of a long focal length lens is due to the limited resolution of the LCLV used in the experiments. A He–Ne laser beam (λ = 633 nm) is expanded, collimated, and used as the readout beam of the LCLV. The intensity of the readout beam is ~25 µW/cm2 over the aperture size of 25 × 25 mm of the LCLV. The reflected beam is amplitude modulated by passing it through a polarizing beam splitter and an analyzer/polarizer. A second Fourier transform lens FTL2 with focal length f2 = 400 mm is placed behind the beam splitter. The correlation outputs are detected with a CCD camera interfaced with a computer and viewed with a video monitor. The input-output characteristics of the
Fig. 1. (a) Nonlinear joint transform correlator using a LCLV at the Fourier plane. (b) Images used in the correlation tests. Tank and the tank in the input scene noise.
LCLV were tested for several different power supply frequencies applied to the LCLV. Figure 2(a) shows the readout light amplitude values of the LCLV vs the writing light intensity for the bias supply voltage of 10 V rms and three different supply frequencies applied to the LCLV. When the LCLV is used in the Fourier plane of the JTC, these curves represent an approximation of the nonlinear transformation in the Fourier plane. The curves were measured over an area of ~2 mm2 at the center of the LCLV. The nonlinear characteristics of the device vary spatially across the surface of the device. Also, the orientation of the device polarization axis as well as the settings of the polarizer/analyzer plates affects the nonlinear characteristics of the LCLV.
The experimental analysis of the nonlinear JTC is presented using two methods. In one method, we use the saturation property of the input–output characteristics of the LCLV shown in Fig. 2(a) to compress the joint power spectrum. In another method, we test the performance of the JTC when the LCLV operates in three different nonlinear modes corresponding to the three input-output characteristics of Fig. 2(a). For the first method, we choose the top curve of Fig. 2(a) as our working curve, which corresponds to
the bias supply voltage of 10 V rms and supply frequency of 200 Hz applied to the LCLV. This curve remains linear over a range of input light intensities and then starts to saturate. For the case of a low intensity write beam, the joint power spectrum is placed in the linear region of the input-output characteristics of the LCLV. An approximation of a linear JTC is achieved. However, for a large intensity write beam, some pixel values of the joint power spectrum fall in the saturation region of the curve and are compressed. The joint power spectrum of objects normally have a large dynamic range, and the low spatial frequency pixel values are much larger than the high spatial frequency pixel values. When the write beam light intensity increases, the low spatial frequency components of the joint spectrum will exceed the linear dynamic range and fall into the saturation region. In this case, the low spatial frequency components are compressed, and in contrast the high spatial frequency components are enhanced. Thus compression of the joint power spectrum may improve the correlation performance. The correlation tests were performed for the images of the tank, and the tank and the images of the tank and the tank in the input scene noise [see Fig. 1(b)]. The intensity of the joint
* Ip is the correlation peak intensity. SNR is the correlation signal to noise ratio. PSR is the peak to sidelobe ratio. FWHM is the full width of the correlation signal at its half value. FWHM is determined in both x and y directions.
power spectrum was adjusted by a neutral density filter inserted in the write beam to implement different degrees of nonlinear compression in the Fourier plane. Figure 2(b) illustrates the increase in the correlation peak intensity for different values of the write beam which correspond to different levels of the joint power spectrum compression. In Fig. 2(b), the top curve corresponds to the correlation of the tank and the tank, and the bottom curve corresponds to the correlation of the tank and the tank in the input scene noise.
In the experimental results, the PSR is defined as the ratio of the correlation peak intensity [I(xityj)]max to the noise intensity average values around the correlation peak
where Iixuyj) is the correlation peak intensity, n(xi,yj) is the noise intensity outside the 50% response portion of the correlation peak intensity, N\ and N2 are the total number of pixels of the area where the correlation peak is measured, and N1 and N2 are the number of pixels under the 50% response portion of the correlation spot. Here we use N1 = N2 = 64 pixels. The SNR is defined as the ratio of the correlation peak intensity [I(xi,yj)],max to the standard deviation of the noise intensity
Here n(xi,yj) is the average value of the n(xi,yj) for the N1N2 pixels. Here we take N\ = N2= 64 pixels.
Figure 2(c) shows the increase in the PSR in the correlation plane with the increase of the write beam intensity. Circles correspond to the correlation of the tank and the tank, and stars correspond to the correlation of the tank and the tank in the input scene noise. It can be seen from this figure that as the compression of the joint power spectrum increases, the PSR increases. When the input scene noise is present, a more severe compression of the joint power spectrum is required. Below a certain level of nonlinear com
pression, the signal is not detected and a PSR of less than unity is produced (not shown in Fig. 2).
In a different method, we test the effects of the LCLV nonlinearity on the JTC performance when the LCLV operates in three different nonlinear modes. In this case, the degree of nonlinearity is controlled by the bias supply frequency applied to the LCLV. It is seen from Fig. 2(a) that lowering the power supply frequency increases the nonlinearity of the input-output characteristics of the LCLV. The correlation results in the presence of the input scene noise are shown in Table I. Here the average write intensity to the LCLV is 1200 µW/cm2. Ip is the correlation peak intensity normalized to that of the correlation peak intensity obtained by operating the LCLV with a bias frequency of 200 Hz. FWHM is the full correlation width at half-maximum. It can be seen from the table that the best results are obtained for the bias frequency of 60 Hz. For the correlation of the tank and the tank in the input scene noise, the bias frequency of 200 Hz produces the worst performance including false correlation peaks and a PSR of less than unity. When the LCLV is operated with 120 Hz, it results in a more nonlinear transformation of the joint power spectrum (compared with the 200-Hz case) and the correlation performance is improved. In this case, Ip is 1.5, the SNR is 25, and the PSR is 3.5. When the LCLV is operated with 60 Hz, a more nonlinear transformation of the joint power spectrum occurs compared with the two previous cases and the correlation performance is further improved. In this case, Ip is 5.1, the SNR is 35, and the PSR is 6.4. The peak width for the 60-Hz case decreases 50% relative to that of the 120-Hz case. The autocorrelation results without the input scene noise are also shown in Table I. The photographs and the 3-D mesh plots of the correlation planes of the tank and the tank in the input scene noise obtained by operating the LCLV with 200, 120, and 60 Hz are shown in Figs. 3(a)-(c), respectively. A frame grabber was used in conjunction with a CCD camera and a computer to obtain 3-D mesh plots for the correlation peaks. The plots are the cross-correlation signals and are normalized to a maximum value of unity. The dc terms are not shown in these 3-D mesh plots.
Fig. 3. Experimental correlation planes of the tank and the tank in the input scene noise. The LCLV is operated with a frequency of (a) 200, (b) 120, and (c) 60 Hz. The 3-D plots of the cross-correlation peak intensity are shown for each case. The peak values are normalized to a maximum of unity. The dc terms are not shown in the 3-D plots.
Experiments presented in this paper indicate that for some noisy input scenes used in the proposed setup, it may be necessary to transform nonlinearly the joint power spectrum to suppress the output noise term and produce a PSR of larger than unity.
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