STA 4274H -- STATISTICAL INVERSE PROBLEMS

`Statistical inverse problems' refers to situations where we can get data in one `domain', but our interest lies in another domain where measurements cannot be taken. This subject has a high-tech flavour and is the focus of much current research. Two typical kinds of statistical inverse problems, both of which will be discussed in this course, are as follows:

(1) In a typical problem of STEREOLOGY physiologists must estimate the distribution of the radii of spherical `neuron vesicles' distributed throughout tissue. Since it is not possible to measure these directly, a thin slice of tissue is taken and examined under an electron microscope. This two-dimensional slice intersects with some of the spheres, and we can measure the radii of the `circular sections' seen in the slice. However the distribution of radii of circles in the 2-D slice differs from the distribution of radii of the spheres in 3-D space. These two distributions are related through integral equations, but it turns out that estimating the two-dimensional distribution and simply `transforming' is not statistically effective.

So how does one attack such problems?

(2) A typical problem of MEDICAL TOMOGRAPHIC IMAGING is POSITRON EMISSION TOMOGRAPHY (or PET). Here a radioactive substance is injected and travels to the brain, where about one million `positron emissions' occur. What is needed is the density function of the points where these emissions occur, since this gives a picture of the structure of tissue in the brain. However due to the physics of the situation it is not possible to observe the points of emission. We can however observe about one million randomly oriented lines, each of which which passes through one of the emission points. Reconstructing PET scan images from data of this type is an important statistical problem.

GRADING: Based on individual projects. Students may choose either a theoretical or an applied (e.g. computational) topic (or a combination of these) depending on their interests. An example of such a project would be to read a paper in the field and give a written (or oral) exposition on its contents.

Permission of the instructor is required.

Some References:

Bertero, M. and Boccacci, P. (1998). Introduction to Inverse Problems in Imaging. Institute of Physics Publishing, Philadelphia, Pa.
Bracewell, R.N. (1968). The Fourier Transform and its Applications. McGraw-Hill.
Deans, S.R. (1983). The Radon Transform and Some of its Applications. Wiley.
Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist., 19, 1257-1272.
Feuerverger and Vardi. (2000). Positron emission tomography and random coefficients regression. Annals of the Institute of Statistical Mathematics (Tokyo). Vol. 52, No. 1, 123-138.
Herman, G.T. (1980). Image Reconstruction From Projections: The Fundamentals of Computerized Tomography. Academic.
Hochstadt, H. (1973). Integral Equations. Wiley.
Kak, A.C. and Slaney, M. (1988). Principles of Computerized Tomographic Imaging. IEEE Press, New York.
Lewitt, R.M. (1983). Reconstruction algorithms: Transform methods. Proc. IEEE, 71, 390-408.
O'Sullivan, F. (1986). A statistical perspective on ill-posed inverse problems (with discussion). Statistical Science, 1, 1986, 502-527.
Shepp, L.A. and Kruskal, J.B. (1978). Computerized tomography: The new medical X-ray technology. Amer. Math. Monthly, 420-438.
Vardi, Y., Shepp. L.A. and Kaufman, L. (1985). A statistical model for positron emission tomography (with discussion). J. Amer. Statist. Assoc., 80, 8-37.
Walther, G. (1997). On a conjecture concerning a theorem of Cramer and Wold. J. Mult. Anal., 63, 313-319.
Wicksell, S.D. (1925, 1926). The corpuscle problem, Pt. I: A mathematical study of a biometric problem. Biometrica, 17, 84-99; Pt. II 1926, 18, 151-172.