There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Modern digital cameras employ silicon focal plane array (FPA) image sensors featuring millions of pixels. However, it is possible to make a camera that only needs one pixel. In these cameras a spatial light modulator, placed before or after the object to be imaged, applies a time-varying pattern and synchronized intensity measurements are made with a single-pixel detector. The principle of compressed sensing then allows an image to be generated. As the approach suits a wide a variety of detector technologies, images can be collected at wavelengths outside the reach of FPA technology or at high frame rates or in three dimensions. Promising applications include the visualization of hazardous gas leaks and 3D situation awareness for autonomous vehicles. Rather than requiring millions of pixels, it is possible to make a camera that only needs one pixel. This Review details the working principle, advantages, technical considerations and future potential of single-pixel imaging.

Principles and prospects for single-pixel imaging

Terahertz (THz) radiation is poised to have an essential role in many imaging applications, from industrial inspections to medical diagnosis. However, commercialization is prevented by impractical and expensive THz instrumentation. Single-pixel cameras have emerged as alternatives to multi-pixel cameras due to reduced costs and superior durability. Here, by optimizing the modulation geometry and post-processing algorithms, we demonstrate the acquisition of a THz-video (32 × 32 pixels at 6 frames-per-second), shown in real-time, using a single-pixel fiber-coupled photoconductive THz detector. A laser diode with a digital micromirror device shining visible light onto silicon acts as the spatial THz modulator. We mathematically account for the temporal response of the system, reduce noise with a lock-in free carrier-wave modulation and realize quick, noise-robust image undersampling. Since our modifications do not impose intricate manufacturing, require long post-processing, nor sacrifice the time-resolving capabilities of THz-spectrometers, their greatest asset, this work has the potential to serve as a foundation for all future single-pixel THz imaging systems. Terahertz imaging is promising in many applications, but still relies on complex equipment. Here, the authors develop a simplified solution that enables terahertz real-time imaging using a single-pixel detector and rapid reconstruction methods.

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Real-time terahertz imaging with a single-pixel detector

Whereas modern digital cameras use a pixelated detector array to capture images, single-pixel imaging reconstructs images by sampling a scene with a series of masks and associating the knowledge of these masks with the corresponding intensity measured with a single-pixel detector. Though not performing as well as digital cameras in conventional visible imaging, single-pixel imaging has been demonstrated to be advantageous in unconventional applications, such as multi-wavelength imaging, terahertz imaging, X-ray imaging, and three-dimensional imaging. The developments and working principles of single-pixel imaging are reviewed, a mathematical interpretation is given, and the key elements are analyzed. The research works of three-dimensional single-pixel imaging and their potential applications are further reviewed and discussed.

https://www.mdpi.com/1424-8220/19/3/732/pdf

Single-Pixel Imaging and Its Application in Three-Dimensional Reconstruction: A Brief Review

We demonstrate enhanced light funneling through subwavelength features at optical frequencies using an epsilon near zero (ENZ) material layer. Transmission through a subwavelength slit filled with the epsilon near zero material is characterized.

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Funneling Light Through a Subwavelength Aperture Using Epsilon Near Zero Materials

We present a closed-form image reconstruction method for single-pixel imaging based on the generalized inverse of the measurement matrix. Its numerical cost scales proportionally with the number of measured samples. Regularization of the inverse problem is obtained by minimizing the norms of the convolution between the reconstructed image and a set of spatial filters. The final reconstruction formula can be expressed in terms of matrix pseudoinverse. At high compression, this approach is an interesting alternative to the methods of compressive sensing based on l1-norm optimization, which are too slow for real-time applications. For instance, we demonstrate experimental single-pixel detection with real-time reconstruction obtained in parallel with measurement at a frame rate of 11 Hz for highly compressive measurements with a resolution of 256 × 256. To this end, we preselect the sampling functions to match the average spectrum obtained with an image database. The sampling functions are selected from the Walsh-Hadamard basis, from the discrete cosine basis, or from a subset of Morlet wavelets convolved with white noise. We show that by incorporating the quadratic criterion into the closed-form reconstruction formula, we can use binary rather than continuous sampling and reach similar reconstruction quality as is obtained by minimizing the total variation. This makes it possible to use cosine- or Morlet-based sampling with digital micromirror devices without advanced binarization methods.

Real-time single-pixel video imaging with Fourier domain regularization.

We demonstrate numerically the diffraction-free propagation of sub-wavelength sized optical beams through simple elements built of metal-dielectric multilayers. The proposed metamaterial consists of silver and a high refractive index dielectric, and is designed using the effective medium theory as strongly anisotropic and impedance matched to air. Further it is characterised with the transfer matrix method, and investigated with FDTD. The diffraction-free behaviour is verified by the analysis of FWHM of PSF in the function of the number of periods. Small reflections, small attenuation, and reduced Fabry–Perot resonances make it a flexible diffraction-free material for arbitrarily shaped optical planar elements with sizes of the order of one wavelength.

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Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers

We optimize the effective skin-depth and resolution of Ag-TiO2, Ag-SrTiO3, and Ag-GaP multilayers for imaging with sub-wavelength resolution. In terms of transmission and resolution, the optimized multilayers outperform simple designs based on combined use of effective medium theory, impedance matching and Fabry–Perot resonances. For instance, an optimized Ag-GaP multilayer consisting of only 17 layers, operating at the wavelength of 490 nm and having a total thickness equal to one wavelength, combines 78% intensity transmission with a resolution of 60 nm. It is also shown that use of the effective medium theory leads to sub-optimal multilayer designs with respect to the trade-off between the skin depth and resolution already when the period of the structure is on the order of 40 nm or larger.

Optimized low-loss multilayers for imaging with sub-wavelength resolution in the visible wavelength range

Single-pixel imaging is an indirect imaging technique which utilizes simplified optical hardware and advanced computational methods It offers novel solutions for hyper-spectral imaging, polarimetric imaging, three-dimensional imaging, holographic imaging, optical encryption and imaging through scattering media The main limitations for its use come from relatively high measurement and reconstruction times In this paper we propose to reduce the required signal acquisition time by using a novel sampling scheme based on a random selection of Morlet wavelets convolved with white noise While such functions exhibit random properties, they are locally determined by Morlet wavelet parameters The proposed method is equivalent to random sampling of the properly selected part of the feature space, which maps the measured images accurately both in the spatial and spatial frequency domains We compare both numerically and experimentally the image quality obtained with our sampling protocol against widely-used sampling with Walsh-Hadamard or noiselet functions The results show considerable improvement over the former methods, enabling single-pixel imaging at low compression rates on the order of a few percent

/pdf/single-pixel-imaging-with-morlet-wavelet-correlated-random-ovpavresc0.pdf

Single-pixel imaging with Morlet wavelet correlated random patterns

We characterize the sensitivity of imaging properties of a layered silver-TiO2 flat lens to fabrication inaccuracies. The lens is designed for approximately diffraction-free imaging with subwavelength resolution at distances in the order of a wavelength. Its operation may be attributed to self-collimation with a secondary role of Fabry-Perot resonant transmission, even though the first order effective medium description of the structure is inaccurate. Super-resolution is maintained for a broad range of overall thicknesses and the total thickness of the multilayer is limited by absorption. The tolerance analysis indicates that the resolution and transmission efficiency are highly sensitive to small changes of layer thicknesses.

Anna Pastuszczak

Papers

Real-time single-pixel video imaging with Fourier domain regularization.

Sub-wavelength diffraction-free imaging with low-loss metal-dielectric multilayers

Optimized low-loss multilayers for imaging with sub-wavelength resolution in the visible wavelength range

Single-pixel imaging with Morlet wavelet correlated random patterns

Sensitivity of imaging properties of metal-dielectric layered flat lens to fabrication inaccuracies