Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Description for Spring 2002

This is the first time that this class is offered as follow-up to 2.710 (Introductory Graduate Optics). In its older incarnation, 2.717 itself was introductory. Not any more; as of the academic year ‘01-‘02, 2.717 requires familiarity with at least the basics of Geometrical Optics, and Fourier Optics (aka “Physical Optics”).

The focus (no pun intended!) this year is on two topics: Statistical Optics and Inverse Problems (i.e. Theory of Imaging). The two are very closely related, but seldom taught in conjunction. We will attempt to cover them with equal weight, but emphasizing the connections rather than the peculiarities of each. In particular, we will see how some topics in coherence (Statistical Optics) enter in the design of imaging systems and algorithms; inversely (again, no pun!) we will explore how image quality, indeed the very ability to form images, depend on randomness in optical fields. The ultimate objective is to understand imaging systems; at the same time, the very basic concepts of wave optics are turned inside-out leading, hopefully, to fluent, deep understanding of the subject of Optics itself. By its nature, the topic also involves probability and the theory of stochastic processes; these will be reviewed briefly at the beginning of the class, and will be recurring thereafter.

In a nutshell, the topics covered this semester are:

Review of Fourier Optics, Probability and Stochastic Processes (~4 weeks)
Light Statistics and Theory of Light Coherence (~2 weeks)
The van Cittert-Zernicke Theorem and Applications of Statistical Optics (~3 weeks)
Basic Concepts of Inverse Problems and Examples (~2 weeks)
Information-theoretic View of Inverse Problems (~2 weeks)

Requirements

4 Homeworks (one per week during the first four weeks)
2 Projects, organized as follows:
Project 1: Calculation of the Properties of Integrated Intensity (~2 weeks, 1-page report)
Project 2: Study of Special Topics Relating to Coherence in Imaging Systems (~4 weeks, lecture-style presentation)

Alternative project topics (e.g. of special interest for your research projects) are also possible by prior arrangement. Projects are collaborative (teams of ~3-4). There are no quizes or final exam.

Grading

ACTIVITIES	PERCENTAGES
Homeworks	33.3%
Project 1	33.3%
Project 2	33.3%

Textbooks

Goodman, Joseph W. Statistical Optics. Hoboken, NJ: Wiley-Interscience, 2000. ISBN: 9780471399162.

Bertero, Mario, and Patrizia Boccacci. Introduction to Inverse Problems in Imaging. London, NY: Taylor & Francis, 1998. ISBN: 9780750304351.

Inverse Problems…

… is a fancy way of saying “Study of Imaging.” In the context of wave propagation, if you are given a source and some boundary conditions, you can work out the wave structure and evolution in pretty much every location in space-time. This is the “Forward Problem.” Generally, forward problems are solvable (albeit often with difficulty). The inverse problem is posed as follows: if I am given some observations of a wave, say along one or several surfaces, can I infer the source? As you can imagine, this is not always possible. If the information given by the measurement is incomplete, the inverse problem is, in general, “ill-posed.” This means that there might be several possible sources that would give rise to the same observation and we have no way to determine which one among them is the real one. (The terms ill-posed and incomplete are used in a loose sense here). Inverse problems is the field of study that tries to quantify when a problem is ill-posed and to what degree, and extract maximum information (again, in the loose, every-day sense of the word) under practical circumstances. For instance, an astronomer observing the sky with a telescope only might think that a blob of light originated from a single star; but if she applies an algorithm called CLEAN (or its variants), she might actually discern (the official term is “resolve”) two or more stars in a constellation. This is the typical mode in which imaging system design benefits from knowledge of inverse problem theory. Note that this kind of imaging system is hybrid (telescope + computer). In addition to astronomy, these techniques benefit several application domains such as biomedical imaging and industrial inspection. The most spectacular success in the field of inverse problems was the invention of an inversion algorithm for Computed Tomography by Cormack (1963) and its experimental demonstration by Hounsfield (1973). The two shared the Nobel prize in Physiology or Medicine in 1979.

What do I gain by learning about inverse problems?

Understand how the performance of imaging systems can be quantified; learn how to properly use terms such as “resolution,” and “space-bandwidth product.”
Build intuition about the effects of noise on imaging, and how some of these can be survived.
Learn about popular imaging techniques such Computed Tomography (CT) and functional Magnetic Resonance Imaging (fMRI) which rely heavily on the theory of inverse problems. See the example below.
Share the instructor’s fascination of blending inverse problems with information theory, through which information transfer from a physical object through an imaging system can be precisely quantified.

Magnetic Resonance Imaging (MRI) and functional Magnetic Resonance Imaging (fMRI) are two non-invasive imaging techniques that rely on data measured around a part of a subject’s body (e.g. the head) to acquire information about what’s going on inside the body part (e.g. the brain). The data are processed using a mathematical technique called Filtered Backprojection, which is a computationally efficient and stable way of computing the Inverse Radon transform. The MRI machine shown above is made by GE Medical Systems. An example of the kind of data that can be acquired via this technique is the image of a patient’s brain shown below.

Image of patient’s brain acquired via ‘Filtered Backprojection’ technique. (Image courtesy of Prof. Christof Koch. Used with permission.)

Statistical Optics…

… is the study of randomness in Optical Waves. In most introductory classes, we treat Optical Waves as deterministic quantities, described by electric and magnetic fields that obey Maxwell’s Equations. This treatment is sufficient for forming the foundation of understanding Optical phenomena. However, a wealth of additional phenomena and applications can be discovered by taking the next step of re-formulating optical fields as statistical entities. Quantum physicists were the first to discover that the haven of determinism, while comfortable for lawful citizens averse to gambling, is still much deprived in terms of successfully describing Nature. Randomness is apparent all around us, and dominates the micro- and nano-scales where optical phenomena originate (light-matter interactions typically occur at the scale of wavelengths, ~0.5 micrometer for visible light). We will not venture into the quantum world in this class; instead, we will formulate Statistical Optics based on classical, phenomenological models sufficient to describe randomness for our purposes. We will use the standard formulation of stochastic processes to describe Random Optical Fields, and the Fourier Optics approach of block-diagram models of optical elements (except with random inputs).

What do I gain by learning Statistical Optics?

Deep understanding of optics, since the simple, deterministic concepts are significantly reinforced when seen “belly-up” from the statistical point of view.
Insight into other random processes in Nature, e.g. random mechanical/structural vibrations, noise in electrical circuits, etc. Not surprisingly, the formulations and properties among these diverse topics are essentially identical.
Proficiency in cutting edge applications of Optics, primarily in imaging (but also in communications and computing). The traditional models of image formation (e.g., geometrically faithful and focus-conscious design of multi-element lens systems) are rapidly collapsing under the progress of optical science and fast, affordable digital processing. Statistical optics has led this revolution because it exploits the properties of Optical Fields in ways richer than either Geometrical or (deterministic) Wave Optics. Two examples of novel, unconventional imaging instruments are given below.

Very Large Array (VLA) Radio Telescope. (Image courtesy of the National Radio Astronomy Observatory.)

The VLA in Socorro, NM, utilizes 27 antennae with an overall aperture between 1.3km and 13km to observe very distant stars and galaxies. The mathematical foundation of the VLA’s operation is the van Cittert-Zernicke theorem which allows us to associate the statistical autocorrelation between radio signals received at different antennae (pair-wise) with the structure of the remote radiowave sources.

Optical Coherence Tomography (OCT). (Image courtesy of LightLab Imaging LLC. Used with permission.)

OCT is a non-invasive medical imaging technique which exploits the coherence properties of light to obtain depth information about tissue. It is based on “white light interferometry,” a term which at first may sound like a contradiction (how can white light interfere?). When properly exploited, white light interference can provide excellent depth resolution (better than 1 micron) and decent lateral resolution (~10-30 microns, limited by the irregularities of the tissue).

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Optical Engineering