What is this course really about?

This course is not about traditional methods for imaging, it is about how to develop new and better methods for imaging. During this course, many classical methods for imaging or migration will be derived as special cases of a very general theory based on the wave equation, which we call the "full-wave" method, but the focus of this course is not about those special cases, but about this general theory itself.

What do we mean by "better" imaging methods?

There are two aspects about our meaning of "better". On one aspect, "better" means "more general", a more general method that can be applied to many different types of data gathered under very different circumstances. For example, in traditional imaging methods, we have the distinction between "reflection" and "transmission". For transmission, we usually call our imaging methods "tomography", for reflection, we usually call our imaging methods "migration". The reason that we are having these distinctions is that some imaging methods work better for some types of data but do not work for other types of data. This course is about the unification of specialized methods under a general framework. A more general method has its advantages. First of all, as the amount of data available for analysis is exploding exponentially, it is getting more and more difficult to condition the data to meet the special needs of different imaging methods, so we'd better come up with a more general method that can account for different types of data. Second of all, and more importantly, we would like to obtain an image or model that is consistent with different types of data and the best way to achieve this goal is to including those different types of data in a unified and self-consistent imaging method.

A second aspect about being "better" is in terms of resolution of the obtained image. Strictly speaking, the imaging problem is usually ill-posed, in the sense that the observation gathered on a discrete grid is not sufficient to constrain a continuous model uniquely. At some point, we always need to incorporate a-priori information into the imaging process in order to pick a model from the large number of models that are consistent with the data. The a-priori information essentially extrapolates or interpolates the portion of the image that is constrained by the data to the portion that is not constrained by the data. By using a more general imaging method that can account for many different types of data, we can essentially incorporate more constraints to our model and minimize the use of a-priori information for model interpolation or extrapolation. Another important factor that limits the resolution in traditional imaging methods is the mathematical approximations, which, in essence, are adopted to reduce the cost for obtaining the solution. With the exponential growth in computing technology, in particular, distributed-memory parallel computing, our capability for obtaining solutions is dramatically improved compared with two decades ago. As a result, many or all of the mathematical approximations can now be dropped and we can now employ an "exact" imaging method that does not sacrifice resolution for solvability.

Why is this course useful?

From the practical perspective, the need for developing better imaging methods is motivated by the oil exploration industry. On one hand, the oil companies are making record profits from the high price of oil. On the other hand, they have to drill in increasingly complex geological environment under which traditional methods often fail to provide satisfactory images. In the past decade, oil companies have been investing more and more money into developing new imaging methods and the "full-wave" method is in the spotlight.

Another very important application of "full-wave" imaging is in medicine. So far, X-ray tomography has been the working-horse in medical imaging. The underlying physics of X-ray tomography is essentially the so-called "straight-ray" tomography. The problem with X-ray tomography, in addition to the radiation introduced by the imaging process, is that it is primarily sensitive to density contrasts, but we need images of other types of properties in addition to density in order to reach reliable diagnoses. This problem is more evident in cancer imaging, in which case, the density of the tumor is not too different from ordinary tissue. In the past decade, people have been building new instruments for cancer imaging using acoustic waves, which can provide images of acoustic wave speed, attenuation property in addition to density. Accurate imaging of acoustic scattering in human body will require a "full-wave" approach.

What is the approach for teaching this course?

A general approach that I will adopt for teaching this class is a "top-down" approach. I'll introduce the most general concepts at the beginning of the course and derive more specific imaging algorithms as the course progresses. The development history of imaging methods is rather the opposite and follows a "bottom-up" thread, in which specific migration algorithms were designed based on the intuitions of geophysicists and more general concepts about imaging were extracted from those individual migration algorithms later on. Among those intuitions, the geometric ray theory plays an important role. The reason that I am adopting a "top-down" approach is three-folded. First of all, the most general concepts do not necessary require more advanced mathematics. In places where we do require results from more advanced mathematics, for example, functional analysis, we can often draw analogies with results from more fundamental mathematics such as basic calculus and resort to our intuition to extrapolate those fundamental results to more advanced results. Second of all, a "top-down" approach allows us to understand the classical imaging methods in a way that reveals the inherent assumptions that were not explicitly stated by the early creators of those classical methods. And third of all, the latest development in the imaging community is often based on those most general concepts.

Lecture Notes

Pre-class Survey on Math

Bayes' Theorem
Generalized Inverse
Lesson1 (PowerPoint 2007) Lesson1 (PDF) Explanation (PDF) (incomplete)

Lesson2 Lesson2 (PDF) explanation

Lesson2.1 (PDF) (review of Lesson 2 and case study: X-ray tomography)

Lesson 2 Survey

Lesson3 (PDF) (Green's function)

Lesson3 HW1 Solution

Lesson3 HW2 Solution

Lesson3.1 (PDF) (Green's function in multi-dimension)

Lesson3.2 (Born series, Born approxmiation)

Lesson3.3 (Adjoint of Born modeling operator, Kirchhoff migration, Diffraction tomography)

Lesson 4 (Rytov approximation, Rytov kernel, seismogram perturbation kernel)

Lesson 5 (Summary)

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