%!TEX root = ../dissertation.tex
\chapter{Introduction}
\label{introduction}

Humans are naturally visual creatures. To a perhaps surprising degree, what we study is determined by what we can look at. It is not surprising, then, that there have been many  Nobel prizes awarded for tools to help us see things better (Betzig, Hell and Moerner for super-resolved flouresence microscopy; Boyle and Smith for the charge-coupled device (CCD); Lauterbur and Mansfield for magnetic resonance imaging (MRI); Ruska, Binning, and Rohrer for the Electron Microscope and the scanning tunneling microscope; Klug for crystallographic electron microscopy; Gabor for Holography; Cormack and Hounsfield for computer assisted tomography)\cite{nobelprize_physics_2014} and several more for related technologies. This thesis was made possible by the work leading to two of those imaging prizes.

Holography, invented by Dennis Gabor in 1948 \cite{gabor_new_1948}, is a way to capture a three-dimensional image of an object from the phase of light. Because light oscillates at frequencies around $10^{14}\text{~Hz}$ it is not possible to measure the phase directly; therefore, a hologram captures information about the phase from interference. The interference pattern records the electric field at a specific plane. To recover the field, Gabor proposed what is now known as optical reconstruction, which involves shining light back through the hologram. Light that is diffracted by the hologram forms a three-dimensional real image of the object captured by the hologram.

There are several challenges in holography that have limited it to mostly niche applications. Because holography records information in very fine interference fringes, one has to record holograms at very high resolution. Until recently that meant using specialized photographic media. Besides requiring specialized equipment, these media generally require large light irradiance for exposure, necessitating long exposure times. This requirement imposes strict constraints on the stability of the apparatus, since  any movement of order the wavelength of the light over the exposure can blur out the fringes in the hologram. Also, it is difficult to obtain quantitative information from optical reconstructions, further limiting their use in science.

CCD and CMOS digital sensors solve the first of these problems. Modern digital sensors have the resolution and pixel density to be useful for capturing holograms, enabling a new era of digital holography~\cite{schnars_digital_2002}. CCD and CMOS sensors can be much more light sensitive than the old photographic media, which, combined with easy availability of high powered lasers, allows us to record holograms more quickly. The enhanced speed shortens the time over which motion can blur the fringes during recording and relaxes the constraints on the stability of the apparatus. It has taken holography from a relatively slow way to record an image to one of the fastest ways to record a three-dimensional scene. My colleagues have captured holograms at rates in excess of 1000 frames per second~\cite{kaz_physical_2012}.

Unlike many other three-dimensional imaging techniques, such as confocal microscopy, holographic microscopes need not be complex. They require high precision and careful alignment, but only a few optical elements. Therefore, as I show in Chapter~\ref{chap:simple-dhm} it is possible to build simple holographic instruments. Although these offer no particular advantage in quantitative analysis, they show that new ways of making useful measurements with holograms can make the technique more widely applicable.

Recording holograms digitally also makes it possible to do digital processing which addresses the problem of making quantitative measurements with holograms. For example, there is a digital analog of optical reconstruction called numerical reconstruction~\cite{schnars_digital_2002} (Chapter~\ref{chap:simple-dhm}). This technique enables the use of holography as a kind of large-depth-of-field microscopy for imaging swimming microorganisms\cite{jericho_submersible_2006, lewis_swimming_2006, sheng_digital_2007}, tracking microspheres\cite{kreuzer_digital_2001, repetto_lensless_2004, xu_tracking_2003, sheng_digital_2006}, tracking flow~\cite{satake_measurements_2006}, imaging cells~\cite{choi_three-dimensional_2009, seo_lensfree_2009} other imaging other biological samples~\cite{xu_digital_2001, moon_automated_2009, dubois_digital_2006}. However, for objects of size comparable to the wavelength of light, this method leads to imaging artifacts~\cite{pu_intrinsic_2003}. These artifacts and other difficulties in working with reconstructions have not prevented experts from doing useful work with holograms, but they do limit the wider application of holography.

The need for higher precision measurements led Ovryn and Izen~\cite{ovryn_imaging_2000, ovryn_three-dimensional_2000} and later Lee, Grier and coworkers~\cite{lee_characterizing_2007} to an approach that involves predicting holograms from Lorenz--Mie scattering theory and solving an inverse scattering problem to characterize the particle recorded in the hologram. The approach put forth by Lee and coworkers is to use the Levenberg--Marquardt algorithm~\cite{marquardt_algorithm_1963} to minimize a $\chi^2$ error metric (sum over squared pixel differences) and fit for the parameters of the Lorenz--Mie theory (particle position, radius, and refractive index) that best match the recorded hologram. These parameters provide a high-precision measurement of that particle.

This technique has proved useful for tracking~\cite{kaz_physical_2012, moyses_robustness_2013}, characterizing~\cite{wang_holographic_2016, hannel_holographic_2015, moyses_robustness_2013, cheong_holographic_2011} and monitoring~\cite{wang_stimulus-responsive_2015, wang_celebrating_2015} particles and for microrefractometry~\cite{shpaisman_holographic_2012} but it suffers from some limitations. Lorenz--Mie theory is only valid for spherical particles, and in fact for only a single sphere. It is possible to use the theory for multiple spheres by linearly superimposing the scattered electric fields from each. However, this approach neglects any multiple scattering or near-field effects of scatterers in close proximity and thus is only valid for well-separated particles. With my assistance, my colleagues Jerome Fung and Rebecca Perry overcame this limitation by implementing an exact multi-sphere superposition scattering theory that enabled them to make new measurements of dense clusters of particles~\cite{perry_real-space_2012, fung_imaging_2012}.

As the reader is hopefully well aware, only a small fraction of things in the world are spheres, or even well approximated by spheres. Jokes about spherical cows aside, most physicists are not satisfied with considering only spherical objects. Fortunately, it is possible, though somewhat more difficult, to model light scattering from other shapes. One general technique for modelling light scattering from arbitrarily shaped objects is the discrete dipole approximation (DDA)~\cite{purcell_scattering_1973}. In the DDA objects are represented as a discrete three-dimensional array of point dipoles. One then solves for a self-consistent polarization of these dipoles an incident electric field. This solution can be remarkably accurate; the only error introduced is from the discretization. I have adapted the DDA for use in modeling and fitting holograms, and with my colleague Anna Wang, have used this technique to study the diffusion of non-spherical particles, as described in Chapter~\ref{chap:dda}.

Unfortunately, these scattering models can take a significant amount of time to compute. Because of the thousands of evaluations required to find optimal parameters, fitting holograms can take a prohibitively long time; particularly when working with the DDA or with larger clusters of spheres. To help address this problem, I developed a technique for fitting holograms using only a randomly selected fraction of its pixels. Although one must capture holograms at high resolutions to record fine fringes, the resulting holograms have a large amount of redundant data. Randomly selecting pixels can sample all of the useful information while avoiding many wasteful evaluations. I describe this technique in detail in Chapter~\ref{chap:random-subset} and demonstrate its effectiveness and accuracy.

So far I have glossed over a detail that significantly limits these scattering-model-based approaches to analyzing holograms. The space covered by the many-dimensional parameterizations of scattering objects is vast and can contain local minima of the $\chi^2$ error metric. Thus, algorithms such as Levenberg--Marquardt will only converge if given an initial guess sufficiently close to the correct answer. This turns out to be a significant impediment; it is a difficult task to determine such an initial guess from a hologram. The need to manually initialize fits is a major limiting factor in analysis of holographic data, particularly for more complex systems. A skilled expert can find, with heroic effort, satisfactory initial guesses to fit a cluster of six particles~\cite{perry_real-space_2012}, but that expertise took years to develop; therefore, manual guessing is hardly a scalable approach to wide application.

There is another problem with the parameterizations used in these fits. Lee and coworkers~\cite{lee_characterizing_2007} were forced to introduce a non-physical parameter $\alpha$ to account for some optical effects in the recording of the hologram. Furthermore, one must always fit for all parameters of the scattered object or objects, even if one is interested in, say, only the position or size of a particle. Covariance between the parameters can mean that the parameters not of interest can affect the uncertainty of those that are. Parameters that are required in the model but are not of interest are often called ``nuisance parameters'' because they interfere with analysis. These parameters are such a nuisance because, in classical frequentist statistics, there isn't really anything you can do about them.

Fortunately there is a way to deal with both of these problems, and I describe it in Chapter~\ref{chap:bayesian}. Bayesian statistical techniques provide a more powerful way to include prior information, such as the initial guesses used in fitting holograms, and enable one to integrate over, or marginalize, nuisance parameters. Bayesian priors are a natural way to describe what one knows about a parameter. Combined with a subset based tempering approach I introduce in Section~\ref{sec:tempering}, the method allows us to infer parameter values starting with only information that is easy to obtain. Then, by marginalizing over $\alpha$ and any other parameters not of interest, one can accurately assess the uncertainty of parameters of interest and the covariances between them.

These techniques greatly expand the space of phenomena that can be investigated using powerful scattering-based approaches. Combined with my open source implementation in the software package HoloPy\cite{dimiduk_holopy_2016}, they also greatly reduce the expertise required to use these techniques, making them available to a much wider variety of scientists interested in dynamics of microscopic in soft matter physics and biophysics.
