Victor Paul Pauca

Engineering the pupil phase to improve image quality

By suitably phase-encoding optical images in the pupil plane and then digitally restoring them, one can greatly improve their quality. The use of a cubic phase mask originated by Dowski and Cathey to enhance the depth of focus in the images of 3-d scenes is a classic example of this powerful approach. By using the Strehl ratio as a measure of image quality, we propose tailoring the pupil phase profile by minimizing the sensitivity of the quality of the phase-encoded image of a point source to both its lateral and longitudinal coordinates. Our approach ensures that the encoded image will be formed under a nearly shift-invariant imaging condition, which can then be digitally restored to a high overall quality nearly free from the aberrations and limited depth of focus of a traditional imaging system. We also introduce an alternative measure of sensitivity that is based on the concept of Fisher information. In order to demonstrate the validity of our general approach, we present results of computer simulations that include the limitations imposed by detector noise.

High-Resolution Imaging Using Integrated Optical Systems

Certain optical aberrations, such as defocus, can significantly degrade the signal collected by an imaging system, producing images with low resolution. In images with depth‐dependent detail, such degradations are difficult to remove due to their inherent spatially varying nature. In 1995, Dowski and Cathey introduced the concept of wavefront coding to extend the depth of field. They showed that wavefront coding and decoding enables quality control of such images using integrated optical‐digital imaging systems. With wavefront coding, a high‐resolution image can be efficiently obtained without the need to resort to expensive algorithms for spatially varying restoration. In this article, we discuss a novel and effective multiple‐design‐parameter approach for optimizing the processes of encoding and decoding the wavefront phase in integrated optical‐digital imaging systems. Our approach involves the use of information metrics, such as the Strehl ratio and Fisher information, for determining the optimal pupil‐phase distribution for which the resulting image is insensitive to certain aberrations, such as focus errors. The effectiveness of this approach is illustrated with a number of numerical simulation experiments, and applications to the development of iris recognition systems with high‐resolution capabilities are briefly discussed.

High-resolution iris image reconstruction from low-resolution imagery

We investigate the use of a novel multi-lens imaging system in the context of biometric identification, and more specifically, for iris recognition. Multi-lenslet cameras offer a number of significant advantages over standard single-lens camera systems, including thin form-factor and wide angle of view. By using appropriate lenslet spacing relative to the detector pixel pitch, the resulting ensemble of images implicitly contains subject information at higher spatial frequencies than those present in a single image. Additionally, a multi-lenslet approach enables the use of observational diversity, including phase, polarization, neutral density, and wavelength diversities. For example, post-processing multiple observations taken with differing neutral density filters yields an image having an extended dynamic range. Our research group has developed several multi-lens camera prototypes for the investigation of such diversities. In this paper, we present techniques for computing a high-resolution reconstructed image from an ensemble of low-resolution images containing sub-pixel level displacements. The quality of a reconstructed image is measured by computing the Hamming distance between the Daugman4 iris code of a conventional reference iris image, and the iris code of a corresponding reconstructed image. We present numerical results concerning the effect of noise and defocus blur in the reconstruction process using simulated data and report preliminary work on the reconstruction of actual iris data obtained with our camera prototypes.

Integrated optical-digital approaches for enhancing image restoration and focus invariance

A novel and successful optical-digital approach for removing certain aberrations in imaging systems involves placing an optical mask between an image-recording device and an object to encode the wavefront phase before the image is recorded, followed by digital image deconvolution to decode the phase. We have observed that when appropriately engineered, such an optical mask can also act as a form of preconditioner for certain deconvolution algorithms. It can boost information in the signal before it is recorded well above the noise level, leveraging digital restorations of very high quality. In this paper, we 1) examine the influence that a phase mask has on the incoming signal and how it subsequently affects the performance of restoration algorithms, and 2) explore the design of optical masks, a difficult nonlinear optimization problem with multiple design parameters, for removing certain aberrations and for maximizing restorability and information in recorded images.

PERIODIC: Integrated Computational Array Imaging Technology

An array imaging system, dubbed PERIODIC, is presented, capable of exploiting diversities, including subpixel displacement, phase, polarization, and wavelength, to produce superresolution images. The hardware system and software interface described, and sample results are shown.

Performance modeling of adaptive-optics imaging systems using fast Hankel transforms

Real-time adaptive-optics is a means for enhancing the resolution of ground based, optical telescopes beyond the limits previously imposed by the turbulent atmosphere. One approach for linear performance modeling of closed-loop adaptive-optics system involves calculating very large covariance matrices whose components can be represented by sums of Hankel transform based integrals. In this paper we investigate approximate matrix factorizations of discretizations of such integrals. Two different approximate factorizations based upon representations of the underlying Bessel function are given, the first using a series representation due to Ellerbroek and the second an integral representations. The factorizations enable fast methods for both computing and applying the covariance matrices. For example, in the case of an equally spaced grid, it is shown that applying the approximated covariance matrix to a vector can be accomplished using the derived integral-based factorization involving a 2D fast cosine transform and a 2D separable fast multiple method. The total work is then O(N log N) where N is the dimensions of the covariance matrix in contrast to the usual O(N2) matrix-vector multiplication complexity. Error bounds exist for the matrix factorizations. We provide some simple computations to illustrate the ideas developed in the paper.

A mathematical framework for the linear reconstructor problem in adaptive optics

Abstract The wave front field aberrations induced by atmospheric turbulence can severely degrade the performance of an optical imaging system. Adaptive optics refers to the process of removing unwanted wave front distortions in real time, i.e., before the image is formed, with the use of a phase corrector. The basic idea in adaptive optics is to control the position of the surface of a deformable mirror in such a way as to approximately cancel the atmospheric turbulence effects on the phase of the incoming light wave front. A phase computation system, referred to as a reconstructor, transforms the output of a wave front sensor into a set of drive signals that control the shape of a deformable mirror. The control of a deformable mirror is often based on a linear wave front reconstruction algorithm that is equivalent to a matrix–vector multiply. The matrix associated with the reconstruction algorithm is called the reconstructor matrix. Since the entire process, from the acquisition of wave front measurements to the positioning of the surface of the deformable mirror, must be performed at speeds commensurate with the atmospheric changes, the adaptive optics control imposes several challenging computational problems. The goal of this paper is twofold: (i) to describe a simplified yet feasible mathematical framework that accounts for the interactions among main components involved in an adaptive optics imaging system, and (ii) to present several ways to estimate the reconstructor matrix based on this framework. The performances of these various reconstruction techniques are illustrated using some simple computer simulations.

Some computational problems arising in adaptive optics imaging systems

Abstract Recently there has been growing interest and progress in using numerical linear algebra techniques in adaptive optics imaging control computations. Real-time adaptive optics is a means for enhancing the resolution of ground based, optical telescopes beyond the limits previously imposed by the turbulent atmosphere. An adaptive optics system automatically corrects for light distortions caused by the medium of transmission. The system measures the characteristics of the phase of the arriving wavefront and corrects for the degradations by means of one or more deformable mirrors controlled by special purpose computers. No attempt is made in this paper to give a comprehensive survey of recent numerical linear applications in optical imaging. Rather, two fairly representative applications are discussed in some detail. The following research topics in the area of adaptive optics control systems, each involving the formulation and numerical solution of difficult problems in numerical linear algebra, are described: (1) Jacobi-like eigenvalue computations for multiple bandwidth deformable mirror control methods, and (2) covariance matrix computations for performance modeling of adaptive optics systems using fast Hankel transforms.

Structured matrix representations of two-parameter Hankel transforms in adaptive optics

Abstract We derive efficient approaches for two-parameter Hankel transforms. Such transforms arise, for example, in covariance matrix computations for performance modeling and evaluation of adaptive optics (AO) systems. Fast transforms are highly desirable since the parameter space for performance evaluation and optimization is large. They may be also applicable in real-time control algorithms for future AO systems. Both approaches exploit the analytical properties of the Hankel transform and result in structured matrix representations of approximate transforms. The approximations can be made to satisfy any pre-specified accuracy requirement. The matrix structures can then be exploited in subsequent computations to significantly reduce computation cost.

Pupil phase encoding for multi-aperture imaging

Digital super-resolution refers to computational techniques that exploit the generalized sampling theorem to extend image resolution beyond the pixel spacing of the detector, but not beyond the optical limit (Nyquist spatial frequency) of the lens. The approach to digital super-resolution taken by the PERIODIC multi-lenslet camera project is to solve a forward model which describes the effects of sub-pixel shifts, optical blur, and detector sampling as a product of matrix factors. The associated system matrix is often ill-conditioned, and convergence of iterative methods to solve for the high-resolution image may be slow. We investigate the use of pupil phase encoding in a multi-lenslet camera system as a means to physically precondition and regularize the computational super-resolution problem. This is an integrated optical-digital approach that has been previously demonstrated with cubic type and pseudo-random phase elements. Traditional multi-frame phase diversity for imaging through atmospheric turbulence uses a known smooth phase perturbation to help recover a time series of point spread functions corresponding to random phase errors. In the context of a multi-lenslet camera system, a known pseudo-random or cubic phase error may be used to help recover an array of unknown point spread functions corresponding to manufacturing and focus variations among the lenslets.