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Metamaterial Apertures for Computational Imaging

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Science  18 Jan 2013:
Vol. 339, Issue 6117, pp. 310-313
DOI: 10.1126/science.1230054

Abstract

By leveraging metamaterials and compressive imaging, a low-profile aperture capable of microwave imaging without lenses, moving parts, or phase shifters is demonstrated. This designer aperture allows image compression to be performed on the physical hardware layer rather than in the postprocessing stage, thus averting the detector, storage, and transmission costs associated with full diffraction-limited sampling of a scene. A guided-wave metamaterial aperture is used to perform compressive image reconstruction at 10 frames per second of two-dimensional (range and angle) sparse still and video scenes at K-band (18 to 26 gigahertz) frequencies, using frequency diversity to avoid mechanical scanning. Image acquisition is accomplished with a 40:1 compression ratio.

Imaging systems can be characterized by object dimension [for instance, two-dimensional (2D) for photographs] and information dimension (for example, the number of pixels in an image). Conventional imaging systems are built around the assumption that the object dimension must be conserved in the information dimension, regardless of the inherent information content of the scene. Compressive measurement leverages the realization that measurements need not conserve form of dimension in this sense (13). Indeed, the concept of dimension indicates that measurements are well ordered in some space, implying that adjacent measurements sample similar object data. Information-transfer efficiency, however, is maximized if object data measured by successive measurements are as distinct as possible.

At the diffraction limit, the finite size of the aperture used to form an image imposes a maximum pixel dimension N equal to the space-bandwidth product (SBP), which represents the number of measurement modes needed to exactly reproduce an arbitrary scene (4, 5). In a conventional imaging system, the measurement modes might be thought of as diffraction-limited spots, each of which samples a small portion of the scene (Fig. 1A). Because these modes have little or no spatial overlap in the detector plane, they can be acquired nearly independently and simultaneously with N detectors, such as a charge-coupled device array. However, for natural scenes, many of the modes provide little to no useful data; therefore, they can be substantially compressed without excessive loss of image fidelity (6).

Fig. 1

Comparison of (A) conventional and (B and C) compressive imaging schemes. A conventional imager uses a lens to form measurement modes that effectively map all parts of an object to a detector/image plane. Each mode contributes highly specific and localized information, and all modes can be captured simultaneously with a pixel array or other detector. Within single-pixel schemes, many types of modes can be used to form the image, with measurements being captured sequentially. In the example shown in (B), a random mask and two lenses are used to project incoherent modes that sample the entire scene. The microwave metamaterial imager reported here (C) makes use of a planar waveguide that feeds a holographic array of ELCs, removing the need for lenses. The waveguide acts as a coherent single-pixel device, with the array of ELCs serving to produce the illuminating complex spatial modes.

The concept of a measurement mode can be generalized, such that the imaging process can be expressed mathematically by the relation g = Hf, where g is a collection of measurements, H is the measurement matrix (a row-wise array of all measurement modes), and f is the sampled scene. To form a completely determined data set of measurements (thus enabling a unique linear solution for f), the rank of H must equal the scene’s data dimension (7). Compressive sampling allows reconstruction of underdetermined scenes, finding f by solving the minimization problem argminfgH f22+λR(f), where arg minf denotes the value of f that minimizes the expression, λ is a scalar weighting factor, and R(f) expresses some prior knowledge about the likely composition of the scene. Typically in compressive sampling, R is the l1-norm of the scene, represented in an appropriate basis, which reflects the inherent sparsity that exists in natural scenes. This nonlinear minimization problem is rigorously solvable, even with highly underdetermined measurement data sets (810).

Imaging systems for radio frequency and millimeter-wave electromagnetics have generally been of two types: scanned single-pixel systems and multielement phased-arrays (or synthetic phased arrays). The measurement modes used by classical single-pixel systems are typically inefficient at collecting imaging data. For instance, a rasterizing scanned beam collects information about only one point in space at a time. Multielement phased array systems have much more flexibility in the measurement modes they can access, but these systems sacrifice the size, weight, power, and price advantages of single-pixel systems.

New approaches to imaging at these frequencies have made use of lenses and spatially modulated masks combined with single-pixel detectors to make compressed measurements. One of the pioneering implementations of this form of compressive imaging, depicted in Fig. 1B, was carried out at optical (11) and terahertz frequencies with the use of random static (12) and dynamic (13) masks. Other groups have presented additional ways to introduce mode diversity (14). We show that metamaterial apertures have distinct advantages for compressive imaging because they can be engineered to support custom-designed complex measurement modes that vary with frequency. Leveraging the same electromagnetic flexibility that metamaterials have shown in many other contexts (1520), we can construct an imaging aperture suitable for single-pixel operation that can project nearly arbitrary measurement modes into the far-field, constrained only by the size of the aperture and resonant elements.

We use a 1D metamaterial aperture to perform compressed imaging of various 2D (one angle plus range) canonically sparse scenes. Our imaging device consists of a leaky waveguide, formed by patterning the top conductor of a standard microstrip line with complementary electric-inductor-capacitors (cELCs) (21, 22) metamaterial elements (Fig. 2C). This configuration is equivalent to the schematic of the compressive imager in Fig. 1C, except that the aperture becomes one-dimensional. Each cELC acts as a resonant element that couples energy from the waveguide mode to free space. The resonance frequency and spectral shape of each cELC controls the amplitude and phase of the transmitted wave, such that the far-field modes can be designed by modifying the geometry of the cELCs along the microstrip. By controlling the design and distribution of the individual elements, which affect both the scattered-field characteristics as well as the guided-wave characteristics, nearly any desired aperture mode can be created.

Fig. 2

Simulated far-field profiles for the metamaterial aperture at two frequencies: 18.5 GHz (A) and 21.8 GHz (B). Resonant metamaterial apertures (C) can create mode distributions that vary with frequency. (D) Measured magnitude of the measurement matrix, H, as a function of angle and frequency.

For canonically sparse scenes, an efficient set of measurement modes are those that distribute energy randomly across both the amplitude and phase space of the scene. The dispersion present in resonant metamaterial elements makes frequency a natural choice for indexing the modes, creating a mapping between the measurement modes and frequency. Thus, by sweeping the frequency of the illuminating signal across the available bandwidth, we access the aperture modes sequentially, without having to move or reconfigure the aperture. For image reconstruction schemes that use an arbitrary set of measurement modes, it is essential that the modes be as orthogonal as possible to each other, which places demands on the sharpness and separation of the resonances, with sharper resonances yielding less correlated modes.

We have fabricated a random-mode metamaterial aperture, 40 cm in length, designed to operate in the K band from 18.5 to 25 GHz. Two samples of the measurement modes for this design are plotted in Fig. 2, A and B, and the complete measurement matrix is plotted in Fig. 2C. In our demonstration, scenes are illuminated by far-field radiation from a single source: a low-directivity horn antenna. Backscattered radiation from objects in the scene floods the metamaterial aperture, which selectively admits only one specific mode at each measured frequency. The resulting (complex) signal is measured by a vector network analyzer.

We formed several simple sparse scenes inside an anechoic chamber with dimensions of 4 m by 4 m by 3 m. Each scene contained two or three scattering objects (retroreflectors), 10 cm in diameter, located at arbitrary positions in the chamber. Figure 3B shows one such scene. All scenes were reconstructed using the TwIST code (23).

Fig. 3

(A) Reconstructions of four different static scenes (differentiated by color), consisting of two (black, green, and blue) or three (red) 10-cm scattering objects. The solid "+" symbols show the actual location of objects, and the pixels show the reconstructed image. Pixel size reflects the maximum instrument resolution. The image has been cropped from the full field of view of ±70°. (B) Photograph of a single scene corresponding to the black markers in (A).

For an aperture of this size and bandwidth, the diffraction-limited angular resolution is 1.7°, and the bandwidth-limited range resolution is 4.6 cm. Across a field of view of ±70° in angle and 1.5 to 4 m in range, the equivalent SBP = 4475. Our measurement, however, contains only 101 values, representing a compression ratio of more than 40:1. We note that the acquisition of a complete data set for the scenes in Fig. 3 requires only 100 ms, which makes the imaging of moving scenes a tantalizing possibility.

To demonstrate the imaging of moving scenes, we performed repeated 100-ms sweeps while moving an object through the scene. The acquisition speed in this experiment was limited by the signal-to-noise-ratio—primarily due to the network analyzer, which is designed for operational flexibility rather than high dynamic range or sweep speed. We imaged a single scattering object moving through the scene on a linear path at ~0.2 m/s. Figure 4 depicts the reconstructed scene. The object position in angle and range, mapped as a function of time, is observed in the retrieved scene. These data are presented in video format along with a camera recording of the object motion in movie S1.

Fig. 4

(Left) Image of a single moving object at 10-Hz frequency. Each voxel is sized to match the spatio-temporal resolution of the metamaterial aperture, and the amplitude of the reconstructed scattering density is mapped to the transparency of each voxel. Voxels are also color-coded in time, from blue to red. (Right) Data projections in two dimensions.

A major advantage of metamaterial radiators in this application is the ability to incorporate tuning (2427). Dynamically varying the resonance of the metamaterial elements would enable reconfigurable measurement modes. Dynamic tuning also frees the frequency bandwidth, effectively enabling hyperspectral imaging (28).

The imaging system we present here combines a computational imaging approach with custom aperture hardware that allows compression to be performed on the physical layer that is used to do the illumination and/or recording. The use of metamaterials is a convenient tool for the creation of such apertures, as metamaterial techniques offer a well-understood design path. Leveraging the resonant nature of metamaterial elements also creates frequency diversity of the measurement modes, giving an all-electrical method of quickly sweeping through a mode set. This metamaterial approach scales linearly with frequency through terahertz frequencies with correspondingly higher resolutions. Due to their small form factor and lack of moving parts, similar systems may extend microwave and millimeter-wave imaging capabilities.

Supplementary Materials

www.sciencemag.org/cgi/content/full/339/6117/310/DC1

Supplementary Text

Figs. S1 to S3

Movie S1

References and Notes

  1. Acknowledgments: This work was supported by a grant from the Air Force Office of Scientific Research (no. FA9550-09-1-0539). Data and code are available at http://people.duke.edu/~jdh51/Data_Metamaterial_Apertures_for_Computational_Imaging.zip.

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