Advanced Monte Carlo Simulation and Machine Learning for Frequency Domain Optical Coherence Tomography

Optical Coherence Tomography(OCT) is a popular, rapidly growing imaging technique with an increasing number of bio-medical applications due to its noninvasive nature. However, there are three major challenges in understanding and improving an OCT system: (1) Obtaining an OCT image is not easy. It either takes a real medical experiment or requires days of computer simulation. Without much data, it is difficult to study the physical processes underlying OCT imaging of different objects simply because there aren't many imaged objects. (2) Interpretation of an OCT image is also hard. This challenge is more profound than it appears. For instance, it would require a trained expert to tell from an OCT image of human skin whether there is a lesion or not. This is expensive in its own right, but even the expert cannot be sure about the exact size of the lesion or the width of the various skin layers. The take-away message is that analyzing an OCT image even from a high level would usually require a trained expert, and pixel-level interpretation is simply unrealistic. The reason is simple: we have OCT images but not their underlying ground-truth structure, so there is nothing to learn from. (3) The imaging depth of OCT is very limited (millimeter or sub-millimeter on human tissues). While OCT utilizes infrared light for illumination to stay noninvasive, the downside of this is that photons at such long wavelengths can only penetrate a limited depth into the tissue before getting back-scattered. To image a particular region of a tissue, photons first need to reach that region. As a result, OCT signals from deeper regions of the tissue are both weak (since few photons reached there) and distorted (due to multiple scatterings of the contributing photons). This fact alone makes OCT images very hard to interpret.

This thesis addresses the above challenges by successfully developing an advanced Monte Carlo simulation platform which is 10000 times faster than the state-of-the-art simulator in the literature, bringing down the simulation time from 360 hours to a single minute. This powerful simulation tool not only enables us to efficiently generate as many OCT images of objects with arbitrary structure and shape as we want on a common desktop computer, but it also provides us the underlying ground-truth of the simulated images at the same time because we dictate them at the beginning of the simulation. This is one of the key contributions of this thesis. What allows us to build such a powerful simulation tool includes a thorough understanding of the signal formation process, clever implementation of the importance sampling/photon splitting procedure, efficient use of a voxel-based mesh system in determining photon-mesh interception, and a parallel computation of different A-scans that consist a full OCT image, among other programming and mathematical tricks, which will be explained in detail later in the thesis.

Next we aim at the inverse problem: given an OCT image, predict/reconstruct its ground-truth structure on a pixel level. By solving this problem we would be able to interpret an OCT image completely and precisely without the help from a trained expert. It turns out that we can do much better. For simple structures we are able to reconstruct the ground-truth of an OCT image more than 98% correctly, and for more complicated structures (e.g., a multi-layered brain structure) we are looking at 93%. We achieved this through extensive uses of Machine Learning. The success of the Monte Carlo simulation already puts us in a great position by providing us with a great deal of data (effectively unlimited), in the form of (image, truth) pairs. Through a transformation of the high-dimensional response variable, we convert the learning task into a multi-output multi-class classification problem and a multi-output regression problem. We then build a hierarchy architecture of machine learning models (committee of experts) and train different parts of the architecture with specifically designed data sets. In prediction, an unseen OCT image first goes through a classification model to determine its structure (e.g., the number and the types of layers present in the image); then the image is handed to a regression model that is trained specifically for that particular structure to predict the length of the different layers and by doing so reconstruct the ground-truth of the image. We also demonstrate that ideas from Deep Learning can be useful to further improve the performance.

It is worth pointing out that solving the inverse problem automatically improves the imaging depth, since previously the lower half of an OCT image (i.e., greater depth) can be hardly seen but now becomes fully resolved. Interestingly, although OCT signals consisting the lower half of the image are weak, messy, and uninterpretable to human eyes, they still carry enough information which when fed into a well-trained machine learning model spits out precisely the true structure of the object being imaged. This is just another case where Artificial Intelligence (AI) outperforms human. To the best knowledge of the author, this thesis is not only a success but also the first attempt to reconstruct an OCT image at a pixel level. To even give a try on this kind of task, it would require fully annotated OCT images and a lot of them (hundreds or even thousands). This is clearly impossible without a powerful simulation tool like the one developed in this thesis.

High frequency wave propagation in the earth: theory and observation

The wave-theoretical analysis of acoustic and elastic waves refracted by a spherical boundary across which both velocity and density increase abruptly and thence either increase or decrease continuously with depth is formulated in terms of the general problem of waves generated at a steady point source and scattered by a radially heterogeneous spherical body. A displacement potential representation is used for the elastic problem that results in high frequency decoupling of P-SV motion in a spherically symmetric, radially heterogeneous medium. Through the application of an earth-flattening transformation on the radial solution and the Watson transform on the sum over eigenfunctions, the solution to the spherical problem for high frequencies is expressed as a Weyl integral for the corresponding half-space problem in which the effect of boundary curvature maps into an effective positive velocity gradient. The results of both analytical and numerical evaluation of this integral can be summarized as follows for body waves in the crust and upper mantle:

1) In the special case of a critical velocity gradient (a gradient equal and opposite to the effective curvature gradient), the critically refracted wave reduces to the classical head wave for flat, homogeneous layers.

2) For gradients more negative than critical, the amplitude of the critically refracted wave decays more rapidly with distance than the classical head wave.

3) For positive, null, and gradients less negative than critical, the amplitude of the critically refracted wave decays less rapidly with distance than the classical head wave, and at sufficiently large distances, the refracted wave can be adequately described in terms of ray-theoretical diving waves. At intermediate distances from the critical point, the spectral amplitude of the refracted wave is scalloped due to multiple diving wave interference.

These theoretical results applied to published amplitude data for P-waves refracted by the major crustal and upper mantle horizons (the Pg, P*, and Pn travel-time branches) suggest that the 'granitic' upper crust, the 'basaltic' lower crust, and the mantle lid all have negative or near-critical velocity gradients in the tectonically active western United States. On the other hand, the corresponding horizons in the stable eastern United States appear to have null or slightly positive velocity gradients. The distribution of negative and positive velocity gradients correlates closely with high heat flow in tectonic regions and normal heat flow in stable regions. The velocity gradients inferred from the amplitude data are generally consistent with those inferred from ultrasonic measurements of the effects of temperature and pressure on crustal and mantle rocks and probable geothermal gradients. A notable exception is the strong positive velocity gradient in the mantle lid beneath the eastern United States (2 x 10^-3 sec^-1), which appears to require a compositional gradient to counter the effect of even a small geothermal gradient.

New seismic-refraction data were recorded along a 800 km profile extending due south from the Canadian border across the Columbia Plateau into eastern Oregon. The source for the seismic waves was a series of 20 high-energy chemical explosions detonated by the Canadian government in Greenbush Lake, British Columbia. The first arrivals recorded along this profile are on the Pn travel-time branch. In northern Washington and central Oregon their travel time is described by T = Δ/8.0 + 7.7 sec, but in the Columbia Plateau the Pn arrivals are as much as 0.9 sec early with respect to this line. An interpretation of these Pn arrivals together with later crustal arrivals suggest that the crust under the Columbia Plateau is thinner by about 10 km and has a higher average P-wave velocity than the 35-km-thick, 62-km/sec crust under the granitic-metamorphic terrain of northern Washington. A tentative interpretation of later arrivals recorded beyond 500 km from the shots suggests that a thin 8.4-km/sec horizon may be present in the upper mantle beneath the Columbia Plateau and that this horizon may form the lid to a pronounced low-velocity zone extending to a depth of about 140 km.

Control of polarization signal distortion by frequency domain phase conjugation in optical fiber systems

Optical frequency domain phase conjugation (FDPC) is based on phase conjugation of spectrum of an input signal. It is equivalent to the phase conjugation and the time reversal of the temporal envelope of an input signal. The use of FDPC to control polarization signal distortion in birefringent optical fiber systems is proposed. Evolution of polarization signals in the system using midway FDPC is analyzed theoretically and simulated numerically. It is shown that the distortion of polarization signals can be controlled effectively by FDPC. The impairments due to dispersion and nonlinear effects can be suppressed simultaneously.

One-shot parallel complex Fourier-domain optical coherence tomography using a spatial carrier frequency

We propose a novel method of one-shot parallel complex Fourier-domain optical coherence tomography using a spatial carrier frequency for full range imaging. The spatial carrier frequency is introduced into the 2-D spectral interferogram in the lateral direction by using a tilted reference wavefront. This spatial-carrier- contained 2-D spectral interferogram is recorded with one shot of a 2-D CCD camera, and is Fourier-transformed in the lateral direction to obtain a 2-D complex spectral interferogram by a spatial-carrier technique. A full-range tomogram is reconstructed from the 2-D complex spectral interferogram. The principle of this method is confirmed by cross-sectional imaging of a glass slip object. (c) 2008 Society of Photo-Optical Instrumentation Engineers.

The ionic basis of frequency selectivity in hair cells of the bullfrog's sacculus

Hair cells from the bull frog's sacculus, a vestibular organ responding to substrate-borne vibration, possess electrically resonant membrane properties which maximize the sensitivity of each cell to a particular frequency of mechanical input. The electrical resonance of these cells and its underlying ionic basis were studied by applying gigohm-seal recording techniques to solitary hair cells enzymatically dissociated from the sacculus. The contribution of electrical resonance to frequency selectivity was assessed from microelectrode recordings from hair cells in an excised preparation of the sacculus.

Electrical resonance in the hair cell is demonstrated by damped membrane-potential oscillations in response to extrinsic current pulses applied through the recording pipette. This response is analyzed as that of a damped harmonic oscillator. Oscillation frequency rises with membrane depolarization, from 80-160 Hz at resting potential to asymptotic values of 200-250 Hz. The sharpness of electrical tuning, denoted by the electrical quality factor, Q_e, is a bell-shaped function of membrane voltage, reaching a maximum value around eight at a membrane potential slightly positive to the resting potential.

In whole cells, three time-variant ionic currents are activated at voltages more positive than -60 to -50 mV; these are identified as a voltage-dependent, non-inactivating Ca current (I_ca), a voltage-dependent, transient K current (I_a), and a Ca-dependent K current (I_c). The C channel is identified in excised, inside-out membrane patches on the basis of its large conductance (130-200 pS), its selective permeability to Kover Na or Cl, and its activation by internal Ca ions and membrane depolarization. Analysis of open- and closed-lifetime distributions suggests that the C channel can assume at least two open and three closed kinetic states.

Exposing hair cells to external solutions that inhibit the Ca or C conductances degrades the electrical resonance properties measured under current-clamp conditions, while blocking the A conductance has no significant effect, providing evidence that only the Ca and C conductances participate in the resonance mechanism. To test the sufficiency of these two conductances to account for electrical resonance, a mathematical model is developed that describes I_ca, I_c, and intracellular Ca concentration during voltage-clamp steps. I_ca activation is approximated by a third-order Hodgkin-Huxley kinetic scheme. Ca entering the cell is assumed to be confined to a small submembrane compartment which contains an excess of Ca buffer; Ca leaves this space with first-order kinetics. The Ca- and voltage-dependent activation of C channels is described by a five-state kinetic scheme suggested by the results of single-channel observations. Parameter values in the model are adjusted to fit the waveforms of I_ca and I_c evoked by a series of voltage-clamp steps in a single cell. Having been thus constrained, the model correctly predicts the character of voltage oscillations produced by current-clamp steps, including the dependencies of oscillation frequency and Q_e on membrane voltage. The model shows quantitatively how the Ca and C conductances interact, via changes in intracellular Ca concentration, to produce electrical resonance in a vertebrate hair cell.