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.

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.

Gaussian

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)