A refractive lens is one of the simplest, most cost-effective and easily available imaging elements. Given 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 only be obtained by changing either the location of the lens, object 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 imaging process consists of three steps. In the first step, 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 during the final step. The results of LRRA were compared with two well-known reconstruction methods, namely the Lucy-Richardson algorithm and non-linear reconstruction.

Coded aperture 3D imaging techniques have been rapidly evolving in recent years. The two main directions of evolution are in aperture engineering to generate the optimal optical field and in the development of a computational reconstruction method to reconstruct the object’s image from the intensity distribution with minimal noise. The goal is to find the ideal aperture–reconstruction method pair, and if not that, to optimize one to match the other for designing an imaging system with the required 3D imaging characteristics. The Lucy–Richardson–Rosen algorithm (LR²A), a recently developed computational reconstruction method, was found to perform better than its predecessors, such as matched filter, inverse 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 when the PSF is known along with MATLAB codes for reconstruction. The common problems and pitfalls in using LR²A have 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.

Indirect-imaging methods involve at least two steps, namely optical recording and computational reconstruction. The optical-recording process uses an optical modulator that transforms the light from the object into a typical intensity distribution. This distribution is numerically processed to reconstruct the object’s image corresponding to different spatial and spectral dimensions. There have been numerous optical-modulation functions and reconstruction methods developed in the past few years for different applications. In most cases, a compatible pair of the optical-modulation function and reconstruction method gives optimal performance. A new reconstruction method, termed nonlinear reconstruction (NLR), was developed in 2017 to reconstruct the object image in the case of optical-scattering modulators. Over the years, it has been revealed that the NLR can reconstruct an object’s image modulated by an axicons, bifocal lenses and even exotic spiral diffractive elements, which generate deterministic optical fields. Apparently, NLR seems to be a universal reconstruction method for indirect imaging. In this review, the performance of NLR isinvestigated for many deterministic and stochastic optical fields. Simulation and experimental results for different cases are presented and discussed.

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 is 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 case of aberrations, if the PSF is known, then the blurred object image can be deblurred by scanning the PSF over the recorded object intensity pattern and looking for pattern matching conditions through a mathematical process called correlation. Deep learning-based image classification for computer vision applications gained attention in recent years. The classification probability is highly dependent on the quality of images as even a minor blur can significantly alter the image classification results. In this study, a recently developed deblurring method, the Lucy-Richardson-Rosen algorithm (LR²A), was implemented to computationally refocus images recorded in the presence of spatio-spectral aberrations. The performance of LR²A was compared against the parent techniques: Lucy-Richardson algorithm and non-linear reconstruction. LR²A exhibited a superior deblurring capability even in extreme cases of spatio-spectral aberrations. Experimental results of deblurring a picture recorded 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.

Coded aperture imaging (CAI) is a powerful imaging technology that has rapidly developed during the past decade. CAI technology and its integration with incoherent holography have led to the development of several cutting-edge imaging tools, devices, and techniques with widespread interdisciplinary applications, such as in astronomy, biomedical sciences, and computational imaging. In this review, we provide a comprehensive overview of the recently developed CAI techniques in the framework of incoherent digital holography. The review starts with an overview of the milestones in modern CAI technology, such as interferenceless coded aperture correlation holography, followed by a detailed survey of recently developed CAI techniques and system designs in subsequent sections. Each section provides a general description, principles, potential applications, and associated challenges. We believe that this review will act as a reference point for further advancements in CAI technologies.

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.