Generalized Image Reconstruction in Optical Coherence Tomography Using Redundant and Non-Uniformly-Spaced Samples.

In this paper, we use Frame Theory to develop a generalized OCT image reconstruction method using redundant and non-uniformly spaced frequency domain samples that includes using non-redundant and uniformly spaced samples as special cases. We also correct an important theoretical error in the previously reported results related to OCT image reconstruction using the Non-uniform Discrete Fourier Transform (NDFT). Moreover, we describe an efficient method to compute our corrected reconstruction transform, i.e., a scaled NDFT, using the Fast Fourier Transform (FFT). Finally, we demonstrate different advantages of our generalized OCT image reconstruction method by achieving (1) theoretically corrected OCT image reconstruction directly from non-uniformly spaced frequency domain samples; (2) a novel OCT image reconstruction method with a higher signal-to-noise ratio (SNR) using redundant frequency domain samples. Our new image reconstruction method is an improvement of OCT technology, so it could benefit all OCT applications.

Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals.

Sparse signal representations have gained much interest recently in both signal processing and statistical communities. Compared to orthogonal matching pursuit (OMP) and basis pursuit, which solve the L0 and L1 constrained sparse least-squares problems, respectively, least angle regression (LARS) is a computationally efficient method to solve both problems for all critical values of the regularization parameter λ. However, all of these methods are not suitable for solving large multidimensional sparse least-squares problems, as they would require extensive computational power and memory. An earlier generalization of OMP, known as Kronecker-OMP, was developed to solve the L0 problem for large multidimensional sparse least-squares problems. However, its memory usage and computation time increase quickly with the number of problem dimensions and iterations. In this letter, we develop a generalization of LARS, tensor least angle regression (T-LARS) that could efficiently solve either large L0 or large L1 constrained multidimensional, sparse, least-squares problems (underdetermined or overdetermined) for all critical values of the regularization parameter λ and with lower computational complexity and memory usage than Kronecker-OMP. To demonstrate the validity and performance of our T-LARS algorithm, we used it to successfully obtain different sparse representations of two relatively large 3D brain images, using fixed and learned separable overcomplete dictionaries, by solving both L0 and L1 constrained sparse least-squares problems. Our numerical experiments demonstrate that our T-LARS algorithm is significantly faster (46 to 70 times) than Kronecker-OMP in obtaining K-sparse solutions for multilinear leastsquares problems. However, the K-sparse solutions obtained using Kronecker-OMP always have a slightly lower residual error (1.55% to 2.25%) than ones obtained by T-LARS. Therefore, T-LARS could be an important tool for numerous multidimensional biomedical signal processing applications.

Massively parallel simulator of optical coherence tomography of inhomogeneous turbid media.

BACKGROUND AND OBJECTIVE: An accurate and practical simulator for Optical Coherence Tomography (OCT) could be an important tool to study the underlying physical phenomena in OCT such as multiple light scattering. Recently, many researchers have investigated simulation of OCT of turbid media, e.g., tissue, using Monte Carlo methods. The main drawback of these earlier simulators is the long computational time required to produce accurate results. We developed a massively parallel simulator of OCT of inhomogeneous turbid media that obtains both Class I diffusive reflectivity, due to ballistic and quasi-ballistic scattered photons, and Class II diffusive reflectivity due to multiply scattered photons. METHODS: This Monte Carlo-based simulator is implemented on graphic processing units (GPUs), using the Compute Unified Device Architecture (CUDA) platform and programming model, to exploit the parallel nature of propagation of photons in tissue. It models an arbitrary shaped sample medium as a tetrahedron-based mesh and uses an advanced importance sampling scheme. RESULTS: This new simulator speeds up simulations of OCT of inhomogeneous turbid media by about two orders of magnitude. To demonstrate this result, we have compared the computation times of our new parallel simulator and its serial counterpart using two samples of inhomogeneous turbid media. We have shown that our parallel implementation reduced simulation time of OCT of the first sample medium from 407 min to 92 min by using a single GPU card, to 12 min by using 8 GPU cards and to 7 min by using 16 GPU cards. For the second sample medium, the OCT simulation time was reduced from 209 h to 35.6 h by using a single GPU card, and to 4.65 h by using 8 GPU cards, and to only 2 h by using 16 GPU cards. Therefore our new parallel simulator is considerably more practical to use than its central processing unit (CPU)-based counterpart. CONCLUSIONS: Our new parallel OCT simulator could be a practical tool to study the different physical phenomena underlying OCT, or to design OCT systems with improved performance.

Monte Carlo simulation of optical coherence tomography for turbid media with arbitrary spatial distributions.

We developed a Monte Carlo-based simulator of optical coherence tomography (OCT) imaging for turbid media with arbitrary spatial distributions. This simulator allows computation of both Class I diffusive reflectance due to ballistic and quasiballistic scattered photons and Class II diffusive reflectance due to multiple scattered photons. It was implemented using a tetrahedron-based mesh and importance sampling to significantly reduce computational time. Our simulation results were verified by comparing them with results from two previously validated OCT simulators for multilayered media. We present simulation results for OCT imaging of a sphere inside a background slab, which would not have been possible with earlier simulators. We also discuss three important aspects of our simulator: (1) resolution, (2) accuracy, and (3) computation time. Our simulator could be used to study important OCT phenomena and to design OCT systems with improved performance.

Fast calculation of multipath diffusive reflectance in optical coherence tomography.

We show how to efficiently calculate the signal in optical coherence tomography (OCT) systems due to the ballistic photons, the quasi-ballistic photons, and the photons that undergo multiple diffusive scattering using Monte Carlo simulations with importance sampling. This method enables the calculation of these three components of the OCT signal with less than one hundredth of the computational time required by the conventional Monte Carlo method. Therefore, it can be used as a design tool to characterize the performance of OCT systems, and can also be used in the development of novel signal processing techniques that can extend the imaging range of OCT systems. We investigate the parameter dependence of our importance sampling method and we validate it by comparison to an existing method.

Improved importance sampling for Monte Carlo simulation of time-domain optical coherence tomography.

We developed an importance sampling based method that significantly speeds up the calculation of the diffusive reflectance due to ballistic and to quasi-ballistic components of photons scattered in turbid media: Class I diffusive reflectance. These components of scattered photons make up the signal in optical coherence tomography (OCT) imaging. We show that the use of this method reduces the computation time of this diffusive reflectance in time-domain OCT by up to three orders of magnitude when compared with standard Monte Carlo simulation. Our method does not produce a systematic bias in the statistical result that is typically observed in existing methods to speed up Monte Carlo simulations of light transport in tissue. This fast Monte Carlo calculation of the Class I diffusive reflectance can be used as a tool to further study the physical process governing OCT signals, e.g., obtain the statistics of the depth-scan, including the effects of multiple scattering of light, in OCT. This is an important prerequisite to future research to increase penetration depth and to improve image extraction in OCT.

Eigenfunction expansion of the electric fields in the focal region of a high numerical aperture focusing system.

The Debye-Wolf electromagnetic diffraction integral is now routinely used to describe focusing by high numerical (NA) lenses. We obtain an eigenfunction expansion of the electric vector field in the focal region in terms of Bessel and generalized prolate spheroidal functions. Our representation has many optimal and desirable properties which offer considerable simplification to the evaluation and analysis of the Debye- Wolf integral. It is potentially also useful in implementing two-dimensional apodization techniques to synthesize electromagnetic field distributions in the focal region of a high NA lenses. Our work is applicable to many areas, such as optical microscopy, optical data storage and lithography.

Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region.

The forward problem of focusing light using a high numerical aperture lens can be described using the Debye-Wolf integral, however a solution to the inverse problem does not currently exist. In this work an inversion formula based on an eigenfunction representation is derived and presented which allows a field distribution in a plane in the focal region to be specified and the appropriate pupil plane distribution to be calculated. Various additional considerations constrain the inversion to ensure physicality and practicality of the results and these are also discussed. A number of inversion examples are given.

Statistics of the depth-scan photocurrent in time-domain optical coherence tomography.

We derive the time-variant second-order statistics of the depth-scan photocurrent in time-domain optical coherence tomography (TD-OCT) systems using polarized thermal light sources and superluminescent diodes (SLDs). Since the asymptotic-joint-probability-distribution function (JPDF) of the photocurrent due to polarized thermal light is Gaussian and the signal-noise-ratio in TD-OCT is typically high (>80 dB), the JPDF of the depth-scan photocurrent could be approximated as a Gaussian random process that is completely determined by its second-order statistics. We analyze both direct and differential light detection schemes and include the effect of electronic thermal fluctuations. Our results are a necessary prerequisite for future development of statistical image processing techniques for TD-OCT.

Photon statistics in single molecule orientational imaging.

Optical techniques in single molecule imaging rely heavily on photon counting for data acquisition. Extraction of information from the recorded readings is often done by means of statistical signal processing, however this requires a full knowledge of the photoelectron statistics. In addition to counting statistics we include a specific form of random signal variations namely reorientational dynamics, or wobble to derive the general probability density function of the number of detected photons. The relative importance of the two factors is dependent upon the total number of photons in the system and results are given in all regimes.

Effect of detector noise in incoherent hybrid imaging systems.

Hybrid imaging systems involve the joint design of an optical image-gathering module and digital processing algorithms to obtain a required final image. They have the potential to achieve imaging performance hitherto unobtainable by conventional imaging techniques. A reduction in the signal-to-noise ratio of the final image is one of their main disadvantages when one is considering linear signal processing. We analyze the effect of additive white noise at the detector on the performance of hybrid imaging systems under quasi-monochromatic incoherent illumination. We also show numerical results and computer-simulated images for an extended depth-of-field hybrid system.

Phase plate to extend the depth of field of incoherent hybrid imaging systems.

A hybrid imaging system combines a modified optical imaging system and a digital postprocessing step. We describe a spatial-domain method for designing a pupil phase plate to extend the depth of field of an incoherent hybrid imaging system with a rectangular aperture. We use this method to obtain a pupil phase plate to extend the depth of field, which we refer to as a logarithmic phase plate. Introducing a logarithmic phase plate at the exit pupil of a simulated diffraction-limited system and digitally processing the detector's output extend the depth of field by an order of magnitude more than the Hopkins defocus criterion. We also examine the effect of using a charge-coupled device optical detector, instead of an ideal optical detector, on the extension of the depth of field. Finally, we compare the performance of the logarithmic phase plate with that of a cubic phase plate in extending the depth of field of a hybrid imaging system with a rectangular aperture.

Reduced depth of field in incoherent hybrid imaging systems.

A hybrid imaging system combines a modified optical imaging module and a digital postprocessing step. We define what to our knowledge is a new metric to quantify the blurring of a defocused image that is more suitable than the defocus parameter for describing defocused hybrid imaging systems. We use this metric to design a pupil phase grating to reduce the depth of field, thereby increasing the axial resolution, of an incoherent hybrid imaging system using quasi-monochromatic illumination. By introducing this grating at the exit pupil and digitally processing the output of the detector, we reduce the depth of field by more than a factor of 2. Finally, we examine the effect of using a CCD optical detector, instead of an ideal optical detector, on the reduction of the depth of field.