Fortunately, the information content of most real-world images is much lower than that theoretically possible to obtain at the same resolution. A simple calculation shows that a full measurement with the dimension equal to the number of pixels would take more than half a minute and require 77 GB of memory to store the binary representation of the sampling functions ψ i, which is impractical. For instance, in this work we are using a state-of the art binary spatial light modulator with a maximum resolution of 1024 × 768 and the maximum frame rate of 22 kHz. This may be seen as a way to capture an encoded and compressed representation of the image which is useful for transmission or storage, and at the same time to deal with the relatively low operation frequencies of current spatial light modulators. Usually, the size of measurement is much smaller than the number of pixels of the image at full resolution. Mathematically, this is a sequence of dot-products of the measured image X with some sampling functions ψ i which are used for modulation. As a result, the detector captures a sequence of average intensities of the modulated image. Images measured by a single-pixel detector are modulated either with structured illumination or using a structured aperture within the detector. The reconstruction problem of the full-dimensional image from such a compressive measurement is an ambiguous inverse problem consisting in solving an underdetermined system of linear equations. The branch of mathematics known as compressive sensing 1, 14, 15, 16(CS) brings the tools needed to restore the image from an indirect lower dimensional measurement. Indirect imaging is mostly limited by the increased time of image acquisition and by the high computational requirements for image reconstruction after the measurement. Currently single-pixel cameras can not compete with the low-cost widely available cameras for the visible wavelength range, however their development offers new possibilities for hyperspectral maging 3, 4, polarimetric imaging 5, 6, holographic imaging 4, 7, 8 THz imaging 9, 3D imaging 10, 11, 12 or imaging though scattering media 13, to mention just some applications. Single-pixel imaging 1, 2 is a technique which makes use of a single detector, such as a photodiode or photomultiplier, and utilizes spatial and temporal modulation of the optical signal to measure an indirect, compressed and encrypted representation of an image. Simplifying the optoelectronic hardware of cameras is one of the reasons for the development of indirect imaging techniques. Nonetheless, these components, which in some cases tend to be very sophisticated and costly, are not indispensable elements of imaging systems. High resolution detector arrays together with high quality optics constitute the most important parts of any classical camera. The results show considerable improvement over the former methods, enabling single-pixel imaging at low compression rates on the order of a few percent. 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 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. While such functions exhibit random properties, they are locally determined by Morlet wavelet parameters. 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. The main limitations for its use come from relatively high measurement and reconstruction times. It offers novel solutions for hyper-spectral imaging, polarimetric imaging, three-dimensional imaging, holographic imaging, optical encryption and imaging through scattering media. Single-pixel imaging is an indirect imaging technique which utilizes simplified optical hardware and advanced computational methods.
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