Enhancing the Visualization of the Peripheral Retina with Wide Field-of-View Optical Coherence Tomography

The goal of my Ph.D. thesis is to enhance the visualization of the peripheral retina using wide-field optical coherence tomography (OCT) in a clinical setting.

OCT has gain widespread adoption in clinical ophthalmology due to its ability to visualize the diseases of the macula and central retina in three-dimensions, however, clinical OCT has a limited field-of-view of 300. There has been increasing interest to obtain high-resolution images outside of this narrow field-of-view, because three-dimensional imaging of the peripheral retina may prove to be important in the early detection of neurodegenerative diseases, such as Alzheimer's and dementia, and the monitoring of known ocular diseases, such as diabetic retinopathy, retinal vein occlusions, and choroid masses.

Before attempting to build a wide-field OCT system, we need to better understand the peripheral optics of the human eye. Shack-Hartmann wavefront sensors are commonly used tools for measuring the optical imperfections of the eye, but their acquisition speed is limited by their underlying camera hardware. The first aim of my thesis research is to create a fast method of ocular wavefront sensing such that we can measure the wavefront aberrations at numerous points across a wide visual field. In order to address aim one, we will develop a sparse Zernike reconstruction technique (SPARZER) that will enable Shack-Hartmann wavefront sensors to use as little as 1/10th of the data that would normally be required for an accurate wavefront reading. If less data needs to be acquired, then we can increase the speed at which wavefronts can be recorded.

For my second aim, we will create a sophisticated optical model that reproduces the measured aberrations of the human eye. If we know how the average eye's optics distort light, then we can engineer ophthalmic imaging systems that preemptively cancel inherent ocular aberrations. This invention will help the retinal imaging community to design systems that are capable of acquiring high resolution images across a wide visual field. The proposed model eye is also of interest to the field of vision science as it aids in the study of how anatomy affects visual performance in the peripheral retina.

Using the optical model from aim two, we will design and reduce to practice a clinical OCT system that is capable of imaging a large (800) field-of-view with enhanced visualization of the peripheral retina. A key aspect of this third and final aim is to make the imaging system compatible with standard clinical practices. To this end, we will incorporate sensorless adaptive optics in order to correct the inter- and intra- patient variability in ophthalmic aberrations. Sensorless adaptive optics will improve both the brightness (signal) and clarity (resolution) of features in the peripheral retina without affecting the size of the imaging system.

The proposed work should not only be a noteworthy contribution to the ophthalmic and engineering communities, but it should strengthen our existing collaborations with the Duke Eye Center by advancing their capability to diagnose pathologies of the peripheral retinal.

Math 412 - Topology with Applications

Highlights of Data Expedition: • Students explored daily observations of local climate data spanning the past 35 years. • Topological Data Analysis, or TDA for short, provides cutting-edge tools for studying the geometry of data in arbitrarily high dimensions. • Using TDA tools, students discovered intrinsic dynamical features of the data and learned how to quantify periodic phenomenon in a time-series. • Since nature invariably produces noisy data which rarely has exact periodicity, students also considered the theoretical basis of almost-periodicity and even invented and tested new mathematical definitions of almost-periodic functions. Summary The dataset we used for this data expedition comes from the Global Historical Climatology Network. “GHCN (Global Historical Climatology Network)-Daily is an integrated database of daily climate summaries from land surface stations across the globe.” Source: https://www.ncdc.noaa.gov/oa/climate/ghcn-daily/ We focused on the daily maximum and minimum temperatures from January 1, 1980 to April 1, 2015 collected from RDU International Airport. Through a guided series of exercises designed to be performed in Matlab, students explore these time-series, initially by direct visualization and basic statistical techniques. Then students are guided through a special sliding-window construction which transforms a time-series into a high-dimensional geometric curve. These high-dimensional curves can be visualized by projecting down to lower dimensions as in the figure below (Figure 1), however, our focus here was to use persistent homology to directly study the high-dimensional embedding. The shape of these curves has meaningful information but how one describes the “shape” of data depends on which scale the data is being considered. However, choosing the appropriate scale is rarely an obvious choice. Persistent homology overcomes this obstacle by allowing us to quantitatively study geometric features of the data across multiple-scales. Through this data expedition, students are introduced to numerically computing persistent homology using the rips collapse algorithm and interpreting the results. In the specific context of sliding-window constructions, 1-dimensional persistent homology can reveal the nature of periodic structure in the original data. I created a special technique to study how these high-dimensional sliding-window curves form loops in order to quantify the periodicity. Students are guided through this construction and learn how to visualize and interpret this information. Climate data is extremely complex (as anyone who has suffered from a bad weather prediction can attest) and numerous variables play a role in determining our daily weather and temperatures. This complexity coupled with imperfections of measuring devices results in very noisy data. This causes the annual seasonal periodicity to be far from exact. To this end, I have students explore existing theoretical notions of almost-periodicity and test it on the data. They find that some existing definitions are also inadequate in this context. Hence I challenged them to invent new mathematics by proposing and testing their own definition. These students rose to the challenge and suggested a number of creative definitions. While autocorrelation and spectral methods based on Fourier analysis are often used to explore periodicity, the construction here provides an alternative paradigm to quantify periodic structure in almost-periodic signals using tools from topological data analysis.