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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 11 — Jun. 2, 2014
  • pp: 13515–13530
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Adaptive millimeter-wave synthetic aperture imaging for compressive sampling of sparse scenes

Alex Mrozack, Martin Heimbeck, Daniel L. Marks, Jonathan Richard, Henry O. Everitt, and David J. Brady  »View Author Affiliations


Optics Express, Vol. 22, Issue 11, pp. 13515-13530 (2014)
http://dx.doi.org/10.1364/OE.22.013515


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Abstract

We apply adaptive sensing techniques to the problem of locating sparse metallic scatterers using high-resolution, frequency modulated continuous wave W-band RADAR. Using a single detector, a frequency stepped source, and a lateral translation stage, inverse synthetic aperture RADAR reconstruction techniques are used to search for one or two wire scatterers within a specified range, while an adaptive algorithm determined successive sampling locations. The two-dimensional location of each scatterer is thereby identified with sub-wavelength accuracy in as few as 1/4 the number of lateral steps required for a simple raster scan. The implications of applying this approach to more complex scattering geometries are explored in light of the various assumptions made.

© 2014 Optical Society of America

1. Introduction

One of the greatest challenges facing the millimeter wave (MMW) and especially the terahertz (THz) imaging communities is the restriction posed by the requirement to use expensive point detectors. The impressive scans of obscured objects frequently reported in the MMW and THz literature are usually obtained through slow raster scanning of a source, object, or detector, often taking hours or days to complete [1

1. M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]

3

3. L. Li, W. Zhang, and F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010). [CrossRef]

]. Although source power and detector sensitivity are improving, the rate-limiting factor remains the desired signal-to-noise ratio (SNR) of the scattered signal coupled with the limited mechanical scanning speed and/or the associated mechanical settling time before an acquisition can begin. Although mechanical scanning is often the only practical strategy for obtaining an image of a complex scene with diverse spatial content, there are many problems where the imager is only being used to find isolated or sparsely-distributed scatterers in a visually opaque host. For example, one might wish to find nails behind wallpaper or metallic plumbing behind sheetrock. For such problems, it is impractical to raster scan large areas.

Coherent sources are common in the MMW and THz imaging bands, so synthetic aperture imaging may be used to overcome the limitation of sensing with a single, large, and expensive transceiver. The synthesized aperture can either use a diverging beam, as is typically done in synthetic aperture radar [3

3. L. Li, W. Zhang, and F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010). [CrossRef]

], or a quasi-optical system with a converging beam, as is done in optical coherence tomography (OCT) to increase penetration depth in scattering media [1

1. M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]

, 2

2. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006). [CrossRef]

], but the formalism is equivalent for both. The scanning time for synthetic aperture systems using classical processing techniques [e.g. [1

1. M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]

3

3. L. Li, W. Zhang, and F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010). [CrossRef]

]] could be reduced if more powerful sources and more sensitive detectors were used, but that would greatly increase system cost. However, if the number of spatial samples could be reduced, it becomes more practical to use less expensive sources and detectors, especially if rapid mechanical scanning and efficient data processing are combined to reduce the time required to estimate a scene.

2. Adaptive sensing

2.1. Method

Our goal is to locate a sparse array of scatterers as efficiently as possible using a synthetic aperture system with a single transceiver capable of sweeping frequency over a wide bandwidth attached to a linear stage. The transceiver produces a beam which is focused via a quasi-optical system into a sample, creating a Gaussian beam. The most efficient sampling occurs when the object is translated through a defocused portion of the beam as in inverse-SAR. For a synthetic aperture system of this nature, a measurement includes both input (or system) parameters un such as system NA, wavenumber, etc., and object parameters w such as scatterer position and scattering strength. The nth measurement is then
g(un)=M(un;w)+ηn
(1)
where M is the measurement forward model which maps the input and object parameters to the dataset, and η={ηn}n=1N is a signal independent and identically distributed (i.i.d.) white Gaussian noise source such that 𝔼[ηn] = 0, 𝔼 [|ηn|2] = 1/β, and 𝔼[ηnηmn]=0, where 𝔼 denotes expected value. The aforementioned Gaussian beam optics, and first Born approximation scattering theory, allow for M(un; w) to be specified by M0(xn,kn,NA;{xi,zi,qi}i=1P) defined by
M0(xn,kn,NA;{xi,zi,qi}i=1P)=i=1P[W0Wexp[(xixn)2W2jkzijk(xixn)22R+jtan1zizR]]2qi
(2)
where x is the lateral dimension scanned mechanically, z is the depth direction scanned by frequency sweep, W0 is the beam waist in the z = 0 plane, W0(kn,NA)=2knθ, θ is the divergence angle of the beam, θ (NA) = sin−1 NA, zR is the Rayleigh range, zR(kn,NA)=kn2W02, W is the beam waist for arbitrary z, W(zi)=W01+(zizR)2, and R is the radius of curvature of the beam, R(zi)=zi[1+(zRzi)2] [2

2. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006). [CrossRef]

]. Therefore, the forward model depends on three system parameters - lateral location xn of the transceiver, wavenumber kn of the interrogating beam, numerical aperture NA of the interrogating optics - and three object parameters wi consisting of the complex scattering strength qi and the two dimensional location (x′i, z′i) for each scatterer in the field [2

2. T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006). [CrossRef]

]. The square in the sum of Eq. (2) is from the fact that both the scene illumination and effective receiver gain pattern are assumed to have the same Gaussian beam mode. The measurement system used in this study is shown in Fig. 1, and is discussed in detail in Sec. 3.1. For the purposes of this section all that is necessary is knowledge of the analytical acquisition model M. In the case of extremely high SNR and well-corrected lenses where sidelobes are strong relative to the background, the model could be changed without loss of generality.

Fig. 1 The system diagram. A transceiver generates a Gaussian beam transmit and receive gain pattern focused by two lenses in a 4f configuration. A translation stage then moves the target through the beam to generate multiple lateral measurements.

The collection of all N measurements defines the dataset DN={g(un)}n=1N for which the probability density function is defined as
p(DN|w)exp[β2n=1N|M(un;w)g(un)|2].
(3)

2.2. Model approximations

The specified adaptation procedure will only be successful if an optimization routine can find a reliable solution to Eq. 5. Figure 2(a) shows a cut through the objective function specified in Eq. (5) after 8 simulated adaptive measurements, with M specified by M0 in Eq. (2) using a W-band system with a bandwidth of 75 – 110GHz and NA of .28. The cut is the height of the objective function for varying estimates of x′1 and z′1 while holding the estimate of qr1 and qi1 constant. The true location of the scatterer is x′1 = 3 mm and z′1 = 4 cm. Ideally this function would be a smooth bowl, allowing for a simple optimizer to use the gradient to find the minimum at the true location. Unfortunately, this is not the case for the objective function with M represented by M0 as the measurement model oscillates strongly because the measurement is made far from baseband (i.e. k ≫ 0).

Fig. 2 Cross sections of the three objective functions in Eq. (5) with the model replaced with the model in Eq. (2) (plots (a) & (b)), the depth demodulated model in Eq. (15) (plots (c) & (d)), and the fully-demodulated model in Eq. (16) (plots (e) & (f)). The true location of the scatterer is indicated by the star. The left column is for a single point scatterer in which the scattering density is held constant and the two position variables of the objective function are varied. The right column searches for the first of two point scatterers holding constant the scattering density of the first scatterer and all parameters of the second scatterer. The true location of the sought scatterer is x1 = 3 mm and zi = 4 cm for all scenarios. Note the reduced oscillations in (b) and (c) relative to (a), and in (e) and (f) relative to (d), as well as the inaccurate lateral location of the minimum in (f).

This can be corrected if we make the common OCT assumption that the scatterers are nondispersive, allowing us to demodulate the model and perform data matching against a smoother function. Figure 2(b) shows cuts through the objective function for the same scenario as for Fig. 2(a) with M being specified not by M0, but by
M^(xn,kn,NA;{xi,zi,qi}i=1P)=i=1P[W0Wexp[(xixn)2W2j(kkmin)zijk(xixn)22R+jtan1zizR]]2qi
(15)
which demodulates the axial plane-wave component of the Gaussian beam by subtracting kminz from the phase, where kmin is the wavenumber corresponding to the lowest frequency sampled. Clearly this has greatly reduced axial oscillations in the objective function.

However, there is another strongly contributing phase term in the Gaussian beam equation: the transverse quadratic term in x. OCT usually works in the confocal region of the beam and does not have this term. To demodulate both the axial plane-wave and lateral quadratic terms of the Gaussian beam, kminz + kminx2 are subtracted from the phase. Figure 2(c) shows cuts through the objective function for the same scenario as for Fig. 2(a) and (b) with M being specified not by M0, but by
M˜(xn,kn,NA;xi,zi,qi)=i=1P[W0Wexp[(xixn)2W2j(kkmin)zij(kkmin)(xixn)22R+jtan1zizR]]2qi.
(16)
This has further reduced the oscillations in the objective function and produces the bowl-like shape desired for rapid convergence of an optimization routine.

Now that we have an objective function for the P = 1 case, we need to verify that our assumptions will hold for two scatterers, especially when they are close to each other and the interference between them is strong. Consider a second scatterer at x′2 = 7 mm and z′2 = 4 cm, approximately one beam waist away from the first scatterer still at x′1 = 3 mm and z′1 = 4 cm. Figure 2(d) shows a cut through the objective function with M0 as M in Eq. (5) after 12 simulated adaptive measurements. The cut plots the height of the objective function for varying x′1 and z′1 while holding qr1, qi1, x′2, z′2, qr2, and qi2 constant. As in the P = 1 case, the objective function rapidly oscillates. Although substituting for M reduces these oscillations as before (Fig. 2(e)), substituting for M (Fig. 2(f)) produces an erroneous result: the minimum of the function shifts from 3 mm to 5 mm. While this shift is less than a beam waist, lateral resolution is the most important criterion for imaging at these relatively large wavelengths. Since provides the smoothest objective and accurately locates the single scatterer, M is represented by for the P = 1 case. However, since the scatterers cannot be reliably located in simulation for the P = 2 case if M is represented by , M is instead represented by . In general, the number of scatterers is not known. The model can be generalized to many scatterers and should be used in most cases. The P = 1 case is an important special case to illustrate the maximum capability of adaptive sensing for SAR.

3. Experiment

3.1. Setup

The W-band transceiver shown in Fig. 1 consists of a 6x frequency multiplication chain that upconverts a 12.50 – 18.33 GHz frequency sweep from an Agilent N5222A vector network analyzer (VNA) to a 75–110 GHz output. A small portion of the output signal is down-converted with a reference mixer and coupled into the VNAs reference input for phase sensitive measurements. The detector portion consists of another mixer which down-converts the received W-band signal for the VNAs measurement input. This monostatic source-reference-detector setup is realized with packaged frequency extenders from Virginia Diodes VNAX TXRX WR10.

A focused source was generated by a two lens confocal imaging system. The first lens collimated the 20 degree diverging beam from the W-band source, and the second lens focused the beam in the vicinity of the object depth. Note that the scatterer is placed near but not in the focal plane of this confocal configuration, and the ideal location depends on considerations of the siganl-to-scatter ratio (i.e. the ratio of returned signal power to background signal power, analogous to signal-to-clutter ratio in radar). The scatterer is more detectable if placed closer to the focal plane, but it is in the detector’s field-of-view for fewer lateral locations and requires more measurements to locate. Conversely, placing the scatterer farther from the focal plane reduces the received signal but permits it to be located in fewer measurements. Clearly the NA of the interrogating beam plays a critical role in system considerations: higher NA offers greater resolution throughout the imaging volume and higher signal intensity at the focus, but the beam diverges more quickly and the signal-to-scatter ratio becomes low for a shorter depth-of-field. Using the data from the 10 lateral locations farthest to one side of the scatterrer to estimate the background energy, the signal-to-scatter ratio for this scenario was estimated to be 33dB for the main dataset used in this paper. It is beyond the scope of this manuscript to consider how the performance of this algorithm depends on the SNR of the system. For the purposes of comparing the advantage of the adaptive approach to a simple raster scan, it is sufficient that both have the same peak SNR as determined by this signal to background measurement.

For our experiments, operating characteristics similar to those in [1

1. M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]

] were chosen. Point scatterers in 2D space were either 1 or 2 wires suspended behind the focus. These scatterers were placed several centimeters beyond the focus of a .28NA beam, providing a cm-scale beam waist and a cm-scale depth-of-field while still retaining enough detectable signal. The transmitter/optical system/receiver were held fixed, and the wire scatterers were laterally scanned by a Newport translation stage with 1 mm resolution. At each location, the data collected consisted of the scattered signal received in response to a complete 75 – 110 GHz frequency sweep.

After setting up the system, three calibration tasks were performed. First, the VNA output was calibrated so that reflection from the horn impedance mismatch at the output of the frequency extender did not create large reflections and reduce the dynamic range of the detector. All subsequent scans, calibration or otherwise were taken with the calibrated VNA. Second, the data from a single, full, calibration scan was rotated such that the focus of the beam was in the zero phase plane of the data. Third, the amplitude of the calibration data was altered to match the spectrum assumed by the model, and the gain correction. The amount of rotation and gain correction were saved for later application to adaptively acquired data. The correct rotation for the data was determined by performing the ISAM algorithm on the full scan for various rotations until the tightest focus was achieved [1

1. M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]

]. The amplitude correction was performed by empirically ascertaining a function that matched the amplitude of synthetically generated data for the same assumed system parameters. In this case, the amplitude correction applied was the following gain correction:
g(xn,kn,NA;x1,z1,q1)=g(xn,kn,NA;x1,z1,q1)G(kn)
(17)
G(kn)=exp[kmax2kn2].
(18)

3.2. Simulated scans from real data: 1 and 2 points

Fig. 3 The optimal hopping geometry showing the previously sampled region (red), the oversampled region (blue), the newly interrogated region of interest (green), and the region of possible interference for P = 2(orange). Operating farther from the focus yields more efficient sampling, but lower SNR.

Fig. 4 A typical measurement path for a simulated (a) and fully adaptive acquisition (c), and the magnitude of the first and second derivative of the precision matrix versus measurement for that path (b) and (d), respectively.

To ascertain the advantage gained from adaptive sampling, consider the number of simulated steps required for convergence for the optimal hopping and wide window cases with an object at a depth of 5 cm. In the optimal hopping window case, the scanner could be started from as far away as 9 mm laterally from the scatterer. The scanner was started at each 1 mm increment in this range, and 10 trials were performed at each starting location, constraining the estimate to the green region of Fig. 3. In the wide window case, the scanner was started at each 2 mm increment between −2 cm and +2 cm, and 10 trials were performed at each starting location. Figure 5(a) and (b) show the lateral estimates for all trials at all starting locations after 8 lateral measurements for the optimal hopping and wide window cases, respectively. The variation for different runs at the same starting position were caused by the optimizer which provided slightly different solutions in each iteration of each run, from which statistical inferences may be made. From the scatter plots it is clear that within 8 lateral measurements the estimate was confined to less than a ±1 mm lateral resolution for 84% of the runs for the wide window case and 87% of the runs for the optimal hopping window. Considering all trials, adaptive sampling produces an approximate 30% reduction in sampling for the optimal hopping window, given that the leftmost corner of the green parallelogram in Fig. 3 is 9 mm to the left of the beam center and the rightmost corner is 12 mm to the right of the beam center for an approximately 21 mm field-of-view. Because the wide window is larger, adaptive sampling produces a much greater 60% reduction in sampling. We note that the gain correction of Eq. (18) had little effect on the single scatterer results. With no interfering scatterer present, accurate phase information was sufficient to locate a single scatterer.

Fig. 5 The lateral estimates of a single point located at 0 mm from an optimal hopping window(a) and wide window(b) after eight lateral measurements, and of the weaker (greater distance from focus) of two point scatterers from an optimal hopping window (c) after six lateral measurements and a wide window (d) after seven lateral measurements.

The fact that the ±1 mm resolution could be achieved is of great significance. The beam waist for the mean frequency sampled was 2.6 mm. This nearly factor of three enhancement is a benefit of a recent insight in compressed sensing to move “off of the grid” [20

20. G. Tang, B. Bhaskar, P. Shah, and B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

]. The nonlinear model specified in Eq. (16) estimated the location of the scatterer in continuous space, for which the resolution is determined by the quality and number of measurements taken, not a sampling grid determined by the linear unbiased estimation bounds of the space-bandwidth product [21

21. D. J. Brady, Optical Imaging and Spectroscopy (Wiley-interscience, New Jersey, USA, 2008).

].

The gain correction in Eq. (18) was needed for good estimation accuracy of the two point scatterers. If the amplitude of one scatterer could not be accurately removed, some of the phase from it would be left in the data while estimating the location of the other scatterer, thereby producing inaccurate results. While the locations of the scatterers are estimated concurrently, the data amplitude is dominated by the stronger scatterer, so estimating its location is more important for minimizing the objective function. In the wide window case, the locations of the two scatterers were unconstrained within the volume. Without the gain correction, the weaker scatterer was often estimated to be at the location of the stronger scatterer (2.1 cm laterally) due to this latent phase problem. This problem was greatly reduced by gain correcting the data to match the model M0 more accurately. The optimal hopping window case did not suffer as greatly from latent phase since the weaker scatterer was detected first, and its lateral location therefore strongly constrained by the geometry of Fig. 3. Nevertheless, its gain corrected estimates exhibited a lower variance than the uncorrected estimates.

In a more important example of the over-specified case with P = 1, we allowed the algorithm to assume not one but = 5 scatterers are present in the estimation volume, then attempted to estimate their locations from a single scan. Here the scanner was started at 4 mm increments within the same 4 cm window, with 1 trial per location. Ideally, the algorithm would find all five possible scatterers to be located at the actual position of the one true scatterer, and over J = 11 trials, 87% of the five scatterers were in fact found within 1mm of the one true scatterer after only eight lateral measurements. For points estimated to be in the correct lateral location, the average estimated scattering energy |i|2 was 28dB larger than for the erroneous scatterers. The energy weighted observation frequency, defined as
ξt=j=1Ji=1P˜rect((xi,jbt)/bw)|qi,j|2,
(19)
calculates how often and with how much energy a given lateral estimate is likely to appear within the tth resolution bin of width bw located at bt. The resulting histogram in Fig. 6 thus represents an average estimation over all trials when = 5 is assumed. The ability of adaptive sensing to estimate locations accurately when the number of scatterers is over-specified leads us to an important insight: the operator of a practical system should specify to be as large as is computationally feasible to ensure no scatterers are missed (i.e. P < ).

Fig. 6 A weighted histogram showing an energy distribution of the estimated points when P = 5 is assumed and P = 1 is the true model representing the quantity ξ defined in Eq. (19). The bin width is 2mm (±1mm) An overwhelming amount of the energy is placed in the true object location.

3.3. Fully adaptive scans

A final verification of the method was performed by adaptively driving the scanner itself during acquisition. The goals were the same as in the simulated adaptation case: to ascertain how many lateral scan measurements are required to locate a single scatterer and to what accuracy its position may be found. To generate these statistics, the scatterer was started at various locations within the fixed field. In a scenario similar to the wide window case presented in Sec. 3.2, thirty adaptive measurements were made for each starting position in a window laterally constrained to be within 2 cm of the starting location and to a depth between 0 – 9 cm from the focus. A typical adaptive path is shown in Fig. 4(c). Clearly the algorithm followed a similar decision pattern to the simulated adaptation runs and quickly converged on the lateral location of the scatterer. The convergence of det(AN) also followed the same pattern as shown in Fig. 4(d), starting out large and becoming smaller as the estimate converges to zero in approximately seven measurements.

After all the runs, the set of final lateral estimates xf were plotted against the associated relative initial starting locations xi, and a regression line was fit to ascertain how close the algorithm came to the ground truth xf = −xi. (The coordinate system of the stage is flipped relative to the coordinate system of the estimation code, hence the negative slope.) The regression line shown in Fig. 7(a) reveals how close the algorithm came to this ideal, especially given that the wire was offset a fraction of a millimeter from the stage center x = 0. Using fits like this for each iteration, the rate of convergence and accuracy of the algorithm could be estimated by measuring the standard deviation of the data from the regression line. This standard deviation, shown in Fig. 7(b), shows that the algorithm converged after 7 lateral measurements with a 1σ-error of ∼ ± 0.5mm. Again, it is quite significant that a resolution of < 1 mm was achieved using this “off of the grid” nonlinear model, just as it was in the simulated experiments assuming the 2σ-error as the resolution. The number of measurements required for a given resolution were thereby dramatically reduced: if the linear unbiased estimator were able to achieve this resolution, it would require approximately three times as many lateral measurements.

Fig. 7 The regression line specifying the true location of the target (a) and the standard deviation from that line vs. iteration (b) for the fully adaptive experiment.

4. Implications and assumptions

The results presented here were for specific cases, showing resolution enhancement and reduction in the number of lateral measurements for synthetic aperture sensing. Although it is difficult to generalize these findings, rules of thumb from compressed sensing and the observations made here can at least provide some intuition. Compressed sensing theory is largely based on RIP which states that all measurements should give equal information about the scene (i.e. measure approximately the same amount of energy). For this to be true, the measurements must be widely distributed across the scene. In the optimal hopping window case this is indeed true. The farther away the volume of interest is from the focus, the wider the green parallelogram of Fig. 3 will be. This will allow for greater speedups in acquisition, but it comes at the cost of reduced SNRs. In turn, the ability to super-resolve the points will be degraded since small variations in the phase will be less distinguishable. Widening the window of estimation also allows for more compressed estimation of the object locations. However, in this case there is a finite probability of observing no energy, so RIP is violated. Our work shows that with good SNR this is acceptable to some extent; however, a system designer should be aware that expanding the window too far could lead to bad estimates. Exactly how far is too far largely depends on SNR and the number of points needed to be estimated simultaneously. Ascertaining these dependencies on the salient variables in this more generic problem will be addressed in a subsequent project.

The remaining assumptions are that the beam’s spatial distribution M(x,k) is Gaussian and may be demodulated by referencing the wavevector to the minimum frequency used (kkkmin). The case of a non-Gaussian spatial profile is easily addressed by changing M(x,k) to reflect the beam’s actual spatial profile, assuming this is known accurately. In this case, the algorithm should still identify the correct locations of the scatterer(s), even if the strongest signals occur when the scatterer is not in the center of the beam. By contrast, the demodulation correction was not made ab initio but in response to the quality of the convergence in the fits. It is likely that different scenarios will require different versions of this demodulation, but the approach used here will work for most non-dispersive scatterers since their behaviors are similar at base band as at the operational frequencies. The correction of the beam gain by Eq. (18) was in response to the experimental data recognizing that phase may not be altered and appropriately compensating the amplitude to match the antenna performance. Similar corrections may be required for other antenna structures.

Perhaps the most challenging assumptions involve the scatterers themselves; namely, that their scattering cross sections do not depend sensitively on angle, frequency, or polarization, and that the embedding medium scatters much more weakly by comparison. By using vertically oriented, sub-wavelength diameter wires as scatterers in two dimensions (x and z) and performing one-dimensional lateral scans, the dependence of scattering on angle and frequency in the measurement plane is removed, as is the polarization dependence because the source polarization was aligned with the wires. Generalizing our findings to the case of a three dimensional array of sparse, oriented, non-spherical scatterers, we see that all three effects may vary significantly as the aspect of the scatterer changes over the course of the two dimensional scan. These effects can be minimized by ensuring that the wavelength is significantly larger than the size of the scatterers and that circular polarization is used instead of linear. Such scatterers even more strongly demand a well conditioned Gaussian beam to simplify the analysis. Nevertheless, this work has shown that compressive sampling techniques are practical and may become increasingly indispensable for MMW and THz imaging applications, especially when used to locate or render scenes dominated by a few sparsely-arranged scatterers.

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J. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009). [CrossRef] [PubMed]

16.

Z. Zhang and T. Buma, “Adaptive terahertz imaging using a virtual transceiver and coherence weighting,” Opt. Express 17, 17812–17817 (2009). [CrossRef] [PubMed]

17.

K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, and H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013). [CrossRef]

18.

D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation 4, 590–604 (1992). [CrossRef]

19.

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004). [CrossRef]

20.

G. Tang, B. Bhaskar, P. Shah, and B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.

21.

D. J. Brady, Optical Imaging and Spectroscopy (Wiley-interscience, New Jersey, USA, 2008).

22.

M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

23.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008). [CrossRef]

OCIS Codes
(120.4290) Instrumentation, measurement, and metrology : Nondestructive testing
(110.1085) Imaging systems : Adaptive imaging
(120.1088) Instrumentation, measurement, and metrology : Adaptive interferometry
(110.1758) Imaging systems : Computational imaging

ToC Category:
Imaging Systems

History
Original Manuscript: January 23, 2014
Revised Manuscript: April 21, 2014
Manuscript Accepted: May 8, 2014
Published: May 28, 2014

Citation
Alex Mrozack, Martin Heimbeck, Daniel L. Marks, Jonathan Richard, Henry O. Everitt, and David J. Brady, "Adaptive millimeter-wave synthetic aperture imaging for compressive sampling of sparse scenes," Opt. Express 22, 13515-13530 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13515


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References

  1. M. S. Heimbeck, D. L. Marks, D. Brady, H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. 37, 1316–1318 (2012). [CrossRef] [PubMed]
  2. T. S. Ralston, D. L. Marks, P. S. Carney, S. A. Boppart, “Inverse scattering for optical coherence tomography,” J. Opt. Soc. Am. A 23, 1027–1037 (2006). [CrossRef]
  3. L. Li, W. Zhang, F. Li, “Derivation and discussion of the sar migration algorithm within inverse scattering problem: Theoretical analysis,” Geoscience and Remote Sensing, IEEE Transactions on 48, 415–422 (2010). [CrossRef]
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  5. E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346, 589–592 (2008). [CrossRef]
  6. D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
  7. E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008). [CrossRef]
  8. W. L. Chan, M. L. Moravec, R. G. Baraniuk, D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008). [CrossRef] [PubMed]
  9. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009). [CrossRef] [PubMed]
  10. C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. 49, E67–E82 (2010). [CrossRef] [PubMed]
  11. E. Lebed, P. J. Mackenzie, M. V. Sarunic, F. M. Beg, “Rapid volumetric oct image acquisition using compressive sampling,” Opt. Express 18, 21003–21012 (2010). [CrossRef] [PubMed]
  12. E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006). [CrossRef]
  13. M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011). [CrossRef]
  14. W. U. Bajwa, R. Calderbank, S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. 12, 289–307 (2010). [CrossRef]
  15. J. Duarte-Carvajalino, G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. 18, 1395–1408 (2009). [CrossRef] [PubMed]
  16. Z. Zhang, T. Buma, “Adaptive terahertz imaging using a virtual transceiver and coherence weighting,” Opt. Express 17, 17812–17817 (2009). [CrossRef] [PubMed]
  17. K. Kim, D.-G. Lee, W.-G. Ham, J. Ku, S.-H. Lee, C.-B. Ahn, J.-H. Son, H. Park, “Adaptive compressed sensing for the fast terahertz reflection tomography,” IEEE trans. Terahertz Sci. Technol. 3, 395–401 (2013). [CrossRef]
  18. D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation 4, 590–604 (1992). [CrossRef]
  19. Y. Zhang, X. Liao, L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface uxo,” IEEE Trans. Geosci. Remote Sensing 42, 2535–2543 (2004). [CrossRef]
  20. G. Tang, B. Bhaskar, P. Shah, B. Recht, “Compressive sensing off the grid,” in “Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on,” (2012), pp. 778–785.
  21. D. J. Brady, Optical Imaging and Spectroscopy (Wiley-interscience, New Jersey, USA, 2008).
  22. M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).
  23. S. Ji, Y. Xue, L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. 56, 2346–2356 (2008). [CrossRef]

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