## Adaptive millimeter-wave synthetic aperture imaging for compressive sampling of sparse scenes |

Optics Express, Vol. 22, Issue 11, pp. 13515-13530 (2014)

http://dx.doi.org/10.1364/OE.22.013515

Acrobat PDF (4454 KB)

### 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

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. 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]

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]

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. 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]

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. 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]

**37**, 1316–1318 (2012). [CrossRef] [PubMed]

11. E. Lebed, P. J. Mackenzie, M. V. Sarunic, and F. M. Beg, “Rapid volumetric oct image acquisition using compressive sampling,” Opt. Express **18**, 21003–21012 (2010). [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]

*a priori*optimized code such as in [4

4. P. Potuluri, M. Gehm, M. Sullivan, and D. Brady, “Measurement-efficient optical wavemeters,” Opt. Express **12**, 6219–6229 (2004). [CrossRef] [PubMed]

*a posteriori*estimate. An optimized sampling strategy for scanning large areas is later presented, and experimental results are then discussed. The method is shown to be capable of resolving the scatterers in a number of measurements less than the number of resolvable points in the embedding space. The accurate results and the compressive nature are shown to be preserved even when the number of scatterers is unknown. Finally, the extension of this work to more complex targets is considered in light of the results and assumptions made.

## 2. Adaptive sensing

### 2.1. Method

*u**such as system NA, wavenumber, etc., and object parameters*

_{n}**such as scatterer position and scattering strength. The**

*w**n*

^{th}measurement is then where

*M*is the measurement forward model which maps the input and object parameters to the dataset, and

*η*] = 0, 𝔼 [|

_{n}*η*|

_{n}^{2}] = 1/

*β*, and

*M*(

*u**;*

_{n}**) to be specified by**

*w**x*is the lateral dimension scanned mechanically,

*z*is the depth direction scanned by frequency sweep,

*W*

_{0}is the beam waist in the

*z*= 0 plane,

*θ*is the divergence angle of the beam,

*θ*(NA) = sin

^{−1}NA,

*z*is the Rayleigh range,

_{R}*W*is the beam waist for arbitrary

*z*,

*R*is the radius of curvature of the beam,

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]

*x*of the transceiver, wavenumber

_{n}*k*of the interrogating beam, numerical aperture NA of the interrogating optics - and three object parameters

_{n}

*w**consisting of the complex scattering strength*

_{i}*q*and the two dimensional location (

_{i}*x′*,

_{i}*z′*) for each scatterer in the field [2

_{i}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]

*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.

*M*, the goal is to measure adaptively until the scene (

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]

*N*measurements defines the dataset

**for**

*ŵ***, a maximum**

*w**a posteriori*(MAP) estimate is used. Given the posterior distribution the estimate must be where

*p*

_{0}(

**) is a prior distribution on the object. In general**

*w**p*

_{0}(

**) may be any appropriate distribution for the object. For practical reasons to be explained shortly, the prior distribution is often chosen to be uniform or Gaussian with mean zero. The experiments presented in this paper are concerned with detection of extremely sparse scatterers in two dimensions, lateral and depth, with the number of scatterers**

*w**P*equal to 1 or 2. When

*P*= 1, the objective function is a measure of how well a single point with lateral position

*x′*

_{1}, depth position

*z′*

_{1}, and complex scattering strength

*q*

_{1}=

*q*

_{r1}+

*i**q*

_{i1}match the data given the model

*M*

_{0}, posing a 4-dimensional estimation problem. For

*P*= 2 an 8 dimensional estimation problem must be solved, and for arbitrary

*P*4

*P*parameters must be estimated. Now given a current estimate, the adaptive method needs to decide where to measure next given the current estimate and

*D**. To achieve this goal, two primary assumptions will be made. The first assumption is that the posterior distribution defined by Eq. (4) is approximately Gaussian about mean*

_{N}**with inverse covariance matrix**

*ŵ*

*A**Other assumptions could be made as the circumstances warrant, but the Gaussian assumption is a likely distribution for most scenarios, and allows us to illustrate the technique with a closed form solution. This is necessary because maximizing the determinant of this precision matrix*

_{N}

*A**has information theoretic significance [19*

_{N}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]

*C*is an arbitrary constant. From Eq. (4), Eq. (8) becomes

*p*(

*D**|*

_{N}**), it could be substituted directly into Eq. (9) yielding a final form for the precision matrix. However, for this application it is necessary for the matrix to depend only on the current estimate of the object and the locations of the measurements previously taken. This can be achieved by the second necessary assumption that the model is linear about the estimate. Taylor expanding**

*w**M*about

**where**

*ŵ***(**

*v*

*u**;*

_{n}**) is the slope of the model about the estimate**

*ŵ***at location**

*ŵ*

*u**. Of course general models include non-linearities, but the response is most often dominated by linear terms locally. Eq. (3) is then applied to Eq. (9) using the new form for the model, and the precision matrix becomes which depends only on the current estimate*

_{n}**and previous measurement locations**

*ŵ***does not change drastically from measurement to measurement, allow for the precision matrix from the next measurement to be approximated as before the next measurement is taken, assuming ∇**

*ŵ**∇*

_{w}*log*

_{w}*p*

_{0}(

**) is a constant or zero as happens when**

*w**p*

_{0}(

**) is uniform or Gaussian as specified earlier. An optimal choice of**

*w*

*u*_{N+1}would maximize the determinant of the precision matrix

*A*_{N+1}[19

**42**, 2535–2543 (2004). [CrossRef]

### 2.2. Model approximations

*M*specified by

*M*

_{0}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

*q*

_{r1}and

*q*

_{i1}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

*M*

_{0}as the measurement model oscillates strongly because the measurement is made far from baseband (

*i.e. k*≫ 0).

*M*being specified not by

*M*

_{0}, but by

*k*

_{min}

*z*from the phase, where

*k*

_{min}is the wavenumber corresponding to the lowest frequency sampled. Clearly this has greatly reduced axial oscillations in the objective function.

*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,

*k*

_{min}

*z*+

*k*

_{min}

*x*

^{2}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

*M*

_{0}, but by This has further reduced the oscillations in the objective function and produces the bowl-like shape desired for rapid convergence of an optimization routine.

*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

*M*

_{0}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

*q*

_{r1},

*q*

_{i1},

*x′*

_{2},

*z′*

_{2},

*q*

_{r2}, and

*q*

_{i2}constant. As in the

*P*= 1 case, the objective function rapidly oscillates. Although substituting

*M̂*for

*M*reduces these oscillations as before (Fig. 2(e)), substituting

*M̃*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

*M̃*provides the smoothest objective and accurately locates the single scatterer,

*M*is represented by

*M̃*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

*M̃*,

*M*is instead represented by

*M̂*. In general, the number of scatterers is not known. The model

*M̂*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

**37**, 1316–1318 (2012). [CrossRef] [PubMed]

**37**, 1316–1318 (2012). [CrossRef] [PubMed]

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

*x′*

_{1}= 0mm and

*z′*

_{1}= 4.8cm, and two scenarios were simulated. The first scenario adaptively scanned to estimate the location of a scatterer somewhere within a wide window (−2

*cm*<

*x′*

_{1}< 2

*cm*and 3

*cm*<

*z′*

_{1}< 7

*cm*). The second scenario used an optimal hopping approach where the scanner would take large jumps until a single point scatterer entered the field-of-view. This optimal hopping strategy could only locate a point scatterer within the parallelogram-shaped region illustrated in Fig. 3. For both scenarios, adaptation selected the next best lateral location to measure, and the frequency sweep at that new location will be added to the data previously collected to estimate the location and scattering strength of the target. (Only the lateral dimension was sampled adaptively because frequency sweeping is much faster than the time required to move the scanner.)

*A**and the derivatives of the*

_{N}

*A**with respect to lateral measurement shown in Fig. 4(b). For the first few lateral measurements, the change in the determinant of*

_{N}

*A**was large, but with each successive measurement, the change in det(*

_{N}

*A**) became smaller. After only 7 measurements the second derivative of det(*

_{N}

*A**) vs. iteration was approximately zero, and little more information could be obtained about the scene. At this point, a practical implementation of the algorithm would have stopped acquiring data. To see what would happen if it did not stop, Fig. 4 indicates that eventually there are no more locations to measure close to the scatterer, so the scanner begins to take larger steps oscillating around the scatterer (lateral measurements 12–15).*

_{N}*x′*

_{1}= 0.6,

*z′*

_{1}= 6.9 cm and scatterer 2 at

*x′*

_{2}= 2.1,

*z′*

_{2}= 3.8 cm. Because of its greater depth, the optimal hopping window would see scatterer 1 first, and according to Fig. 3 the acceptable starting locations of the scanner were from 12 mm to 3 mm to the left of scatterer 1 (

*i.e.*−6 mm to +3 mm in the coordinates of the dataset). Again, the scanner was started at 1 mm increments within this range and 10 trials were performed at each starting location. For comparison, the wide window case searched a range 5.5 cm wide spanning the range −1.5 to 4 cm in the coordinates of the dataset. Figure 5 (c) and (d) show the lateral estimates for scatterer 1, which is deeper and scatters more weakly, for all trials at all starting locations after 6 or 7 lateral measurements for the optimal hopping or wide window cases, respectively. The optimal hopping window confines 93% of the estimates for the weaker scatterer (scatterer 1) and 83% of the estimates for the stronger scatterer (scatterer 2) to within 1 mm laterally after 6 measurements. This represents a 74% reduction in lateral sampling for the optimal hopping window. The wide window confined 84% of the estimates for scatterer 1 and all of the estimates for scatterer 2 to within 1 mm laterally after 7 measurements. This slightly larger window required one more measurement and therefore maintained the same lateral sampling reduction of 74%.

*M*

_{0}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.

*P̃*= 2 scatterers present.

*P̃*is the assumed number of scatterers, not the actual number of scatterers

*P*= 1. Ten trials were performed for each starting location, beginning with the scanner 2 cm to the left of the scatterer. Each time, after 8 lateral measurements the scanner was able to find the true location of the single scatterer. Although the algorithm tried to find a second scatterer, in 80% of the trials it estimated this nonexistent scatterer to be in the same location as the first, albeit weaker by an average of 6 dB.

*P*= 1, we allowed the algorithm to assume not one but

*P̃*= 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 |

*q̂*|

_{i}^{2}was 28dB larger than for the erroneous scatterers. The energy weighted observation frequency, defined as calculates how often and with how much energy a given lateral estimate is likely to appear within the

*t*

^{th}resolution bin of width bw located at

*b*. The resulting histogram in Fig. 6 thus represents an average estimation over all trials when

_{t}*P̃*= 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

*P̃*to be as large as is computationally feasible to ensure no scatterers are missed (i.e.

*P*<

*P̃*).

*P*= 2 experiment but let

*P̃*= 1. Ten adaptive scanning trials were performed for each starting location, each time starting the scanner 2 cm to the left of the weaker scatterer. After 8 measurements the weaker scatterer was always located to within 1.3 mm laterally. The slight loss in resolution can be attributed to the interference from the other scatterer which is not correctly taken into account. Overall, the adaptive algorithm can be considered robust, even when the number of points is mis-specified.

### 3.3. Fully adaptive scans

*A**) 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.*

_{N}*x*were plotted against the associated relative initial starting locations

_{f}*x*, and a regression line was fit to ascertain how close the algorithm came to the ground truth

_{i}*x*= −

_{f}*x*. (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

_{i}*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 1

*σ*-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 2

*σ*-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.

## 4. Implications and assumptions

*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.

*k*→

*k*−

*k*

_{min}). 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.

## References

1. | M. S. Heimbeck, D. L. Marks, D. Brady, and H. O. Everitt, “Terahertz interferometric synthetic aperture tomography for confocal imaging systems,” Opt. Lett. |

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 |

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 |

4. | P. Potuluri, M. Gehm, M. Sullivan, and D. Brady, “Measurement-efficient optical wavemeters,” Opt. Express |

5. | E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique |

6. | D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

7. | E. Candes and M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. |

8. | W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. |

9. | D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express |

10. | C. F. Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J. Brady, “Millimeter-wave compressive holography,” Appl. Opt. |

11. | E. Lebed, P. J. Mackenzie, M. V. Sarunic, and F. M. Beg, “Rapid volumetric oct image acquisition using compressive sampling,” Opt. Express |

12. | E. Candes and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory |

13. | M. Duarte and Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. |

14. | W. U. Bajwa, R. Calderbank, and S. Jafarpour, “Why gabor frames? two fundamental measures of coherence and their role in model selection,” J. Commun. Netw. |

15. | J. Duarte-Carvajalino and G. Sapiro, “Learning to sense sparse signals: Simultaneous sensing matrix and sparsifying dictionary optimization,” IEEE Trans. Image Process. |

16. | Z. Zhang and T. Buma, “Adaptive terahertz imaging using a virtual transceiver and coherence weighting,” Opt. Express |

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. |

18. | D. J. MacKay, “Information-based objective functions for active data selection,” Neural Computation |

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 |

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, |

22. | M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. |

23. | S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Sig. Process. |

**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

- 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]
- 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]
- 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]
- P. Potuluri, M. Gehm, M. Sullivan, D. Brady, “Measurement-efficient optical wavemeters,” Opt. Express 12, 6219–6229 (2004). [CrossRef] [PubMed]
- E. Candes, “The restricted isometry property and its implications for compressed sensing,” Comptes Rendus Mathematique 346, 589–592 (2008). [CrossRef]
- D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
- E. Candes, M. Wakin, “An introduction to compressive sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008). [CrossRef]
- 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]
- D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009). [CrossRef] [PubMed]
- 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]
- 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]
- E. Candes, T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006). [CrossRef]
- M. Duarte, Y. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans. Sig. Proc. 59, 4053–4085 (2011). [CrossRef]
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