A refractive lens is one of the simplest, cost-effective and easily available imaging elements. With a spatially incoherent illumination, a refractive lens can faithfully map every object point to an image point in the sensor plane, when the object and image distances satisfy the imaging conditions. However, static imaging is limited to the depth of focus, beyond which the point-to-point mapping can be only obtained by changing either the location of the lens or the imaging sensor. In this study, the depth of focus of a refractive lens in static mode has been expanded using a recently developed computational reconstruction method, Lucy-Richardson-Rosen algorithm (LRRA). The technique consists of three steps. In this first step, the point spread functions (PSFs) were recorded along different depths and stored in the computer as PSF library. In the next step, the object intensity distribution was recorded. The LRRA was then applied to deconvolve the object information from the recorded intensity distributions in the final step. The results of LRRA were compared against two well-known reconstruction methods namely Lucy-Richardson algorithm and non-linear reconstruction.

The retrieval of tree height percentiles with satellite lidar waveforms over the mountainous ar-eas is greatly challenged due to slanted and rugged topographic relief of underlying ground. A novel algorithm is proposed to extract the height percentiles based on the target response waveform (TRW) that is resolved by using a Richardson–Lucy deconvolution algorithm with an adaptive iterative time. The height metrics of the energy percentiles of 25%, 50%, 75% and 95% for the cumulative TRW distribution are determined by their vertical distances relative to the ground elevation in this study. To validate the proposed algorithm, we select the received waveforms of the Global Ecosystem Dynamics Investigation (GEDI) lidar over the Pahvant Mountains of central Utah, USA. The results reveal that the resolved TRWs resemble closely to the actual target response waveforms from the coincident airborne lidar data, with the mean values of the coefficient of correlation, total bias and root mean square error (RMSE) taking val-ue of 0.92, 0.0813 and 0.0016. In addition, the accuracies of the derived height percentiles from the proposed algorithm have been greatly improved compared with the conventional Gaussian de-composition method and the slope-adaptive waveform metrics method. The RMSE values de-crease by the mean values of 1.96 m and 2.72 m, respectively. It demonstrates that the proposed algorithm has good potential in the extraction of forest structure parameter over rugged moun-tainous areas.

Direct imaging systems that create an image of an object directly on the sensor in a single step are prone to many constraints as a perfect image is required to be recorded within this step. In designing high resolution direct imaging systems with a diffractive lens, the outermost zone width either reaches the lithography limit or the diffraction limit itself imposing challenges in fabrication. However, if the imaging mode is switched to an indirect one consisting of multiple steps to complete imaging, then different possibilities open up. One such methods is the widely used indirect imaging method with Golay configuration telescopes. In this study, a Golay-like configuration has been adapted to realize a large area diffractive lens with three sub-aperture diffractive lenses. The sub-aperture diffractive lenses are not required to collect light and focus them to a single point as in a direct imaging system but to focus independently on different points within the sensor area. This approach of Large Area Diffractive lens with Integrated Sub-Apertures (LADISA) relaxes the fabrication constraints and allows the sub-aperture diffractive elements to have a larger outermost zone width and smaller area. The diffractive sub-apertures were manufactured using photolithography. The fabricated diffractive element has been implemented in indirect imaging mode using non-linear reconstruction and Lucy-Richardson-Rosen algorithm with synthesized point spread functions. The computational optical experiments revealed an improved optical and computational imaging resolutions compared to previous studies.

Coded aperture 3D imaging techniques have been rapidly evolving in the recent years. The two main directions of evolution are in aperture engineering to generate the optimal optical field and in development of computational reconstruction to reconstruct the object’s image from the intensity distribution with a minimal noise. The goal is to find the ideal aperture-reconstruction method pair and if not, to optimize one to match the other for designing an imaging system with required 3D imaging characteristics. Lucy-Richardson-Rosen algorithm (LR²A), a recently developed computational reconstruction method was found to perform better than its predecessors such as matched filter, Weiner filter, phase-only filter, Lucy-Richardson algorithm and non-linear reconstruction (NLR) for certain apertures when the point spread function (PSF) is a real and symmetric function. For other cases of PSF, NLR performed better than the rest of the methods. In this tutorial, LR²A has been presented as a generalized approach for any optical field along with MATLAB codes for reconstruction of any image when the PSF is known. The common problems and pitfalls in using LR²A has been discussed. Simulation and experimental studies for common optical fields such as spherical, Bessel, vortex beams and exotic optical fields such as Airy, scattered and self-rotating beams have been presented. From this study, it can be seen that it is possible to transfer the 3D imaging characteristics from non-imaging type exotic fields to indirect imaging systems faithfully using LR²A. The application of LR²A to medical images such as colonoscopy images and cone beam computed tomography images with synthetic PSF has been demonstrated. We believe that the tutorial will provide a deeper understanding of computational reconstruction using LR²A.

Pattern recognition techniques form the heart of most, if not all, incoherent linear shift-invariant systems. When an object is recorded using a camera, the object information gets sampled by the point spread function (PSF) of the system, replacing every object point with the PSF in the sensor. The PSF is a sharp Kronecker Delta-like function when the numerical aperture (NA) is large with no aberrations. When the NA is small, and the system has aberrations, the PSF appears blurred. In the above case, if the PSF is known, then the object information can be obtained by scanning the PSF over the recorded object intensity pattern and looking for pattern matching conditions through a mathematical process called correlation. In this study, a recently developed deconvolution method, the Lucy-Richardson-Rosen algorithm (LR2A), has been implemented to computationally refocus images recorded in the presence of spatio-spectral aberrations. The performance of LR2A was compared against the Lucy-Richardson algorithm and non-linear reconstruction. LR2A exhibits a superior deconvolution capability even in extreme cases of spatio-spectral aberrations and blur. Experimental results of deblurring a picture captured using high-resolution smartphone cameras are presented. LR2A was implemented to significantly improve the performances of the widely used deep convolutional neural networks for image classification.