Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) are two well-known recovery algorithms in compressed sensing. To recover a $d$ -dimensional $m$ -sparse signal with high probability, OMP needs $O(m\ln{d})$ number of measurements, whereas BP needs only $O\left(m\ln{{d\over m}}\right)$ number of measurements. In contrary, OMP is a practically more appealing algorithm due to its superior execution speed. In this piece of work, we have proposed a scheme that brings the required number of measurements for OMP closer to BP. We have termed this scheme as ${\rm OMP}_{\alpha}$ , which runs OMP for $(m+\lfloor\alpha{m}\rfloor)$ -iterations instead of $m$ -iterations, by choosing a value of $\alpha\in[0,1]$ . It is shown that ${\rm OMP}_{\alpha}$ guarantees a high probability signal recovery with $O\left(m\ln{{d\over\lfloor\alpha{m}\rfloor+1}}\right)$ number of measurements. Another limitation of OMP unlike BP is that it requires the knowledge of $m$ . In order to overcome this limitation, we have extended the idea of ${\rm OMP}_{\alpha}$ to illustrate another recovery scheme called ${\rm OMP}_{\infty}$ , which runs OMP until the signal residue vanishes. It is shown that ${\rm OMP}_{\infty}$ can achieve a close to $\ell_{0}$ -norm recovery without any knowledge of $m$ like BP.

Multispectral imaging plays an important role in many applications, from astronomical imaging and earth observation to biomedical imaging. However, current technologies are complex with multiple alignment-sensitive components and spatial and spectral parameters predetermined by manufacturers. Here, we demonstrate a single-shot multispectral imaging technique that gives flexibility to end users with a very simple optical setup, thanks to spatial correlation and spectral decorrelation of speckle patterns. These seemingly random speckle patterns are point spread functions (PSFs) generated by light from point sources propagating through a strongly scattering medium. The spatial correlation of PSFs allows image recovery with deconvolution techniques, while the spectral decorrelation allows them to play the role of tunable spectral filters in the deconvolution process. Our demonstrations utilizing optical physics of strongly scattering media and computational imaging present a cost-effective approach for multispectral imaging with many advantages.

Super-resolution imaging has been revolutionizing technical analysis in various fields from biological to physical sciences. However, many objects are hidden by strongly scattering media such as biological tissues that scramble light paths, create speckle patterns and hinder object’s visualization, let alone super-resolution imaging. Here, we demonstrate non-invasive super-resolution imaging through scattering media based on a stochastic optical scattering localization imaging (SOSLI) technique. After capturing multiple speckle patterns of photo-switchable point sources, our computational approach utilizes the speckle correlation property of scattering media to retrieve an image with a 100-nm resolution, an eight-fold enhancement compared to the diffraction limit. More importantly, we demonstrate our SOSLI to do non-invasive super-resolution imaging through not only static scattering media, but also dynamic scattering media with strong decorrelation such as biological tissues. Our approach paves the way to non-invasively visualize various samples behind scattering media at nanometer levels of detail.

Recent dictionary training algorithms for sparse representation like K-SVD, MOD, and their variation are reminiscent of K-means clustering, and this letter investigates such algorithms from that viewpoint. It shows: though K-SVD is sequential like K-means, it fails to simplify to K-means by destroying the structure in the sparse coefficients. In contrast, MOD can be viewed as a parallel generalization of K-means, which simplifies to K-means without perturbing the sparse coefficients. Keeping memory usage in mind, we propose an alternative to MOD; a sequential generalization of K-means (SGK). While experiments suggest a comparable training performances across the algorithms, complexity analysis shows MOD and SGK to be faster under a dimensionality condition.

Benefiting from the memory effect (ME) for speckle intensity correlations, only one single-shot speckle pattern can be used for the high-quality recovery of objects. However, ME gives a restriction to the field of view (FOV) for imaging with scattering media. Objects beyond the ME region cannot be recovered and produce unwanted speckle patterns, which cause reduction in the speckle contrast and recovery quality. Nevertheless, all the spatial information from a large object is embedded in a single speckle image. Here, we extract the spatial information from these unavoidable speckle patterns and enlarge the FOV of the imaging system with scattering media. Regional point spreading functions, which are fixed and only need to be recorded once for all-time use, are employed to recover corresponding spatial regions of an object by deconvolution. Then, an automatic weighted averaging in an iterative process is performed to obtain the object with significantly enlarged FOV. Our results present an important advancement of imaging techniques with strongly scattering media.

Non-invasive optical focusing inside scattering media is still a big challenge because inhomogeneous media scatter incoming photons for focusing and outgoing photons for observation. Various approaches, utilizing non-linear fluorescence or ultrasound, have been reported to address this difficulty. However, implementation of these methods is complicated and highly expensive, as ultrafast laser systems or photo-acoustic equipment must be employed. Here, we demonstrate a wavefront shaping technique to achieve non-invasive focusing inside scattering media using only a linear fluorescent signal. The contrast and mean of incoherent speckles, produced by the linear fluorescence, are utilized as feedback signals to optimize the input wavefront. While increasing speckle contrast makes the focus tighter and increasing the speckle mean enhances the intensity, fine-tuning the contribution of these two factors in our two-step optimization is essential. An optimal wavefront is found to achieve simultaneously both a micrometer focal spot size (down to 20 μm diameter) and high intensity (more than a 100-fold enhancement) inside the scattering media. Our method promises a route in life science toward focusing, imaging, or manipulating deep into biological tissues with linear fluorescent agents.