Signal Recovery and computational sensing and imaging: introduction to the feature issue.

The fall 2009 conference of the Optical Society of America was held in San Jose, California, 11-15 October. The collocation of topical meetings on Computational Optical Sensing and Imaging (COSI), and Signal Recovery and Synthesis (SRS) with the Frontiers in Optics (FiO) Annual Meeting brought together a diverse group of scientists and engineers sharing a common interest in the processing of information carried by optical fields. The papers featured in this issue highlight several important trends.

The Aggregate behavior of branch points--measuring the number and velocity of atmospheric turbulence layers.

Optical waves propagating through atmospheric turbulence develop spatial and temporal variations in their phase. For sufficiently strong turbulence, these phase differences can lead to interference in the propagating wave and the formation of branch points; positions of zero amplitude. Under the assumption of a layered turbulence model, we show that these branch points can be used to estimate the number and velocities of atmospheric layers. We describe how to carry out this estimation process and demonstrate its robustness in the presence of sensor noise.

Fast and optimal multiframe blind deconvolution algorithm for high-resolution ground-based imaging of space objects.

We report a multiframe blind deconvolution algorithm that we have developed for imaging through the atmosphere. The algorithm has been parallelized to a significant degree for execution on high-performance computers, with an emphasis on distributed-memory systems so that it can be hosted on commodity clusters. As a result, image restorations can be obtained in seconds to minutes. We have compared and quantified the quality of its image restorations relative to the associated Cramér-Rao lower bounds (when they can be calculated). We describe the algorithm and its parallelization in detail, demonstrate the scalability of its parallelization across distributed-memory computer nodes, discuss the results of comparing sample variances of its output to the associated Cramér-Rao lower bounds, and present image restorations obtained by using data collected with ground-based telescopes.

Biased Cramér-Rao lower bound calculations for inequality-constrained estimators.

Unbiased Cramér-Rao lower bound (CRB) theory can be used to calculate lower bounds to the variances of unbiased estimates of a set of parameters given only the probability density function of a random vector conditioned on the true parameter values. However, when the estimated parameter values are required to satisfy inequality constraints such as positivity, the resulting estimator is typically biased. To calculate CRBs for biased estimates of the parameter values, an expression for the bias gradient matrix must also be known. Unfortunately, this expression often does not exist. Because expressions for biased CRBs are preferable to sample variance calculations, alternative methods for deriving biased CRB expressions associated with inequality constraints are needed. We present an alternative approach that is based upon creating the probability density function associated with a given biased estimate of these parameters using the available knowledge of the estimator properties. We apply this approach to the calculation of biased CRBs for estimators that use a positivity constraint with and without a support constraint for a specific measurement model and discuss the benefits and limitations of this approach.

Primary and secondary superresolution by data inversion.

Superresolution by data inversion is the extrapolation of measured Fourier data to regions outside the measurement bandwidth using post processing techniques. Here we characterize superresolution by data inversion for objects with finite support using the twin concepts of primary and secondary superresolution, where primary superresolution is the essentially unbiased portion of the superresolved spectra and secondary superresolution is the remainder. We show that this partition of superresolution into primary and secondary components can be used to explain why some researchers believe that meaningful superresolution is achievable with realistic signal-to-noise ratios, and other researchers do not.

Signal-to-noise-ratio expressions in optical diffusion tomography.

Optical diffusion tomography is a technology that is employed to obtain images of the heterogeneous nature of turbid media by using optical radiation. Noise ultimately limits the achievable spatial resolution in these reconstructed images; therefore it is of interest to develop signal-to-noise-ratio expressions that relate spatial resolution in the images to the underlying system and material properties. In this study, Fourier-domain signal-to-noise-ratio expressions are derived for two types of optical diffusion tomography systems: those that use amplitude-modulated illumination sources and those that use continuous-wave illumination sources. The signal-to-noise-ratio expressions are compared for these two types of systems and are validated by laboratory data.

Blind deconvolution in optical diffusion tomography.

We use blind deconvolution methods in optical diffusion tomography to reconstruct images of objects imbedded in or located behind turbid media from continuous-wave measurements of the scattered light transmitted through the media. In particular, we use a blind deconvolution imaging algorithm to determine both a deblurred image of the object and the depth of the object inside the turbid medium. Preliminary results indicate that blind deconvolution produces better reconstructions than can be obtained using backpropagation techniques. Moreover, it does so without requiring prior knowledge of the characteristics of the turbid medium or of what the blur-free target should look like: important advances over backpropagation.

Image transfer through cirrus clouds. II. Wave-front segmentation and imaging.

A hybrid technique to simulate the imaging of space-based objects through cirrus clouds is presented. The method makes use of standard Huygens-Fresnel propagation beyond the cloud boundary and a novel vector trace approach within the cloud. At the top of the cloud, the wave front is divided into an array of input gradient vectors, which are in turn transmitted through the cloud model by use of the Coherent Illumination Ray Trace and Imaging Software for Cirrus. At the bottom of the cloud, the output vector distribution is used to reconstruct a wave front that continues propagating to the ground receiver. Images of the object as seen through cirrus clouds with different optical depths are compared with a diffraction-limited image. Turbulence effects from the atmospheric propagation are not included.