\documentclass[11pt]{article} \usepackage{times,amsmath,amsfonts} \usepackage[dvips]{graphicx,graphics} \setlength{\textheight}{10in} \setlength{\textwidth}{7.1in} \setlength{\columnsep}{0.3in} \setlength{\topmargin}{0in} \setlength{\headheight}{-0.3in} \setlength{\headsep}{0in} \setlength{\oddsidemargin}{-0.6in} % Centers text. % Add the period after section numbers. Adjust spacing. %\newcommand{\Section}[1]{\vspace{-5pt}\section{\hskip -1em.~~#1}\vspace{-3pt}} %\newcommand{\SubSection}[1]{\vspace{-2pt}\subsection{\hskip -1em.~~#1} % \vspace{-3pt}} \newcommand{\bqn}{\begin{eqnarray*}} \newcommand{\eqn}{\end{eqnarray*}} \newcommand{\bq}{\begin{eqnarray}} \newcommand{\eq}{\end{eqnarray}} \newcommand{\ben}{\begin{enumerate}} \newcommand{\een}{\end{enumerate}} \begin{document} \title{Stat 992: Lecture 41\\ Image Analysis via Random Fields and Kernel Smoothing: Overview} \author{Moo K. Chung \tt{mchung@stat.wisc.edu}} \maketitle \thispagestyle{empty} %\begin{figure} %\centering %\renewcommand{\baselinestretch}{1} %\includegraphics[scale=0.8]{homework05-1.eps} %\end{figure} Although it may be difficult to formulate all the problems in image analysis in terms of random fields and kernel smoothing, it can be done non the less. I put references that I am familiar with so these are not the complete list of bibliography. Most references are cross-listed. \begin{description} \item[3. Gaussian mixture.] This is widely used statistical model for image segmentation. Most methods combines EM algorithm with a Bayesian prior. \ben \item Dempster {\em et al.} J. Roy. Statist. Soc. 1997. \item Robert and Casella's Monte Carlo Statistical Methods for the introduction to EM. \item Flury's A First Course in Multivariate Statistics. \item Ashburner, J., Friston, K.J. 2001. Why Voxel-based morphometry should be used. {\em NeuroImage} {\bf 14}:1238-1243. \item Ashburner, J., Friston, K.J. 2000. Voxel-based morphometry-the method, {\em NeuroImage.} {\bf 11}:805-821. \item T. Hastie, R. Tibshirani, J. Friedman, The element of statistical learning. Springer. \item Convex Optimization by S. Boyd and L. Vandenberghe, 2004. Combridge University Press \een \item[4 and 5 Non mixture-based segmentation.] Neural-network based segmentations are often used. Deformable surface tend to segment geometrically complex boundaries nicely. \ben \item Vasken, K. \newblock {\em Performance Analysis of Automatic Techniques for Tissue Classificatio in Magnetic Resonance Images of the Human Brain.} \newblock Master's Thesis, Computer Science, Concordia University, Canada. 1996. \item Sled, J.G., Zijdenbos, A.P., and Evans, A.C. \newblock A Nonparametric Method for Automatic Correction of Intensity Nonuniformity in MRI Data. \newblock {\em IEEE Transactions on Medical Imaging}, 17:87--97, 1988. \item Ozkan, M., Dawant, B.M., and Maciunas, R.J. \newblock Neural-Network-Based Segmentation of Multi-Modal Medical Images: a Comparative and Prospective Study. \newblock {\em IEEE Trans. Med. Imaging}, 12:534--544, 1993. \item MacDonald, J.D. (1997). \newblock {\em A Method for Identifying Geometrically Simple Surfaces from Three Dimensional Images} \newblock Ph. D. thesis, School of Computer Science, McGill University, Canada. \item Sethian., J. A. \newblock {\em Level Set Methods and Fast Marching Methods}. \newblock Cambridge University Press, 1996. \item Lorensen, W.E. and Cline, H.E. \newblock Marching Cubes: A Hight Resolution 3D Surface Construction Algorithm. \newblock {\em Computer Graphics}, 21:21, 1987. \item Davatzikos, C. and Bryan, R.N. (1995). \newblock Using a Deformable Surface Model to Obtain a Shape Representation of the Cortex. \newblock In {\em Proceedings of the IEEE International Conference on Computer Vision}, pages 2122--2127. \item Khaneja, N., Miller, M.I., and Grenander, U. \newblock Dynamic Programming Generation of Curves on Brain Surfaces. \newblock {\em IEEE Transactions on Pattern Analysis and Machine Intelligence}, 20:1260--1265, 1998. \een \item[6. Similarity measure] This is an interesting concept. \ben \item C.B. Shannon A Mathematical Theory of Communication, The Bell System Technical Journal. 1948. \item P. Baldi and S. Brunak, Bioinformatics, The machine learning appraoch, 2nd edition, MIT press. \item Ehsan S. Soofi Journal of the American Statistical Association, Vol. 89, No. 428. (Dec., 1994), pp. 1243-1254. Flexible Discriminant Analysis by Optimal Scoring (in Theory and Methods) \item Trevor Hastie; Robert Tibshirani; Andreas Buja Journal of the American Statistical Association, Vol. 89, No. 428. (Dec., 1994), pp. 1255-1270. \item T. Hastie, R. Tibshirani, J. Friedman, The element of statistical learning. Springer. \een \item[8. Image Registration.] Most interesting image registration algorithms find applications in medical imaging.\ben \item Incorporating Prior Knowledge into Image Registration Ashburner J, Neelin P, Collins DL, Evans AC, Friston KJ (1997) NeuroImage 6:344-352. \item Nonlinear Spatial Normalization using Basis Functions Ashburner J, Friston KJ (1999) Human Brain Mapping 7(4):254-266 \item Spatial Registration and Normalization of Images Friston KJ, Ashburner J, Poline JB, Frith CD, Heather JD, Frackowiak RSJ (1995) Human Brain Mapping 2:165-189. \item Ashburner, J., Friston, K.J. 1997. Multimodal image coregistration and partitioning - a unified framework, {\em NeuroImage} {\bf 6}:209-217. \item Bookstein, F.L. 2001. Voxel-based morphometry should not be used with imperfectly registered images. {\em NeuroImage} {\bf 14}:1454-1462. \item Shen, D., Davatzikos, C. 2003. Very High-Resolution Morphometry Using Mass-Preserving Deformations and HAMMER Elastic Registration {\em NeuroImage} {\bf 18}:28-41. \item Thompson, P.M., Giedd, J.N., Woods, R.P., MacDonald, D., Evans, A.C., and Toga, A.W. \newblock Growth Patterns in the Developing Human Brain Detected Using Continuum-Mechanical Tensor Mapping. \newblock {\em Nature}, 404:190--193, 2000. \item Davatzikos, C. and Bryan, R.N. \newblock Using a Deformable Surface Model to Obtain a Shape Representation of the Cortex. \newblock In {\em Proceedings of the IEEE International Conference on Computer Vision}, pages 2122--2127, 1995 \item Subsol, G. \newblock Crest Lines for Curve-Based Warping. \newblock {\em Brain Warping}, 241--262, 1999. \item S.C. Joshi's PhD thesis (1998) Large Deformation Diffeomorphisms and Gaussian Random Fields \een \item[9. Random Fields.] It is natural to model image as continuous functional data due to extremely high sampling density. Average brain image consists of $200 \times 200 \times 100 = 2$ million voxels or more. \begin{enumerate} \item Adler, Geometry of Random Fields. 1980 \item R. J. Adler, J. E. Taylor. Random fields and their Geometry In preparation -- first 4 chapters available as .pdf. To be published by Birkhauser. \item R.J. Adler's On Excursion Sets, Tube Formulae, and Maxima of Random Fields (1999) \item Worsley, K.J. (1996). An unbiased estimator for the roughness of a multivariate Gaussian random field. Technical Report, Department of Mathematics and Statistics, McGill University . \item Siegmund, D.O and Worsley, K.J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Annals of Statistics, 23:608-639. \item Dougherty, E.R. (1999). \newblock {\em Random Processes for Image and Signal Processing}. \newblock IEEE Press. \item Rosenfeld, A., and Kak., A. C. (1982) \newblock {\em Digital Picture Processing}, Vol. 2. \newblock Academic Press, Orlando. \item Gikhman and Skorokhod, Introduction to the Theory of Random Processes. 1969 \item Bernt Oksendal's Stochastic Differential Equations (5th edition, 2000) \item Courant and Hilbert. 1953 \item F. Riesz and B. Sz.-Nagy's Functional Analysis (1955) \item Rudin, Real and Complex Analysis. 1986. \item Rudin, Functional Analysis. 1991 \item Conway, A Course in Functional Analysis. 1997 \end{enumerate} \item[10. Linear additive model and other statistical models] Most references are on fMRI time series analysis but should be applicable on other imaging modalities. \ben \item Worsley, K.J. (1994). Quadratic tests for local changes in random fields with applications to medical images. Technical Report, Department of Mathematics and Statistics, McGill University, 94-08. http://www.math.mcgill.ca/keith/quadratic/quadratic.pdf \item Statistical Parametric Mapping Friston KJ (2002) To appear in Human Brain Function II. \item Estimating Smoothness in Statistical Parametric Maps: Confidence Intervals on p-Values Poline J-B, Friston KJ, Worsley KJ and Frackowiak RSJ (1995) Journal of Computer Assisted Tomography 19(5):788-796. \item Statistical Parametric Maps in Functional Imaging: A General Linear Approach Friston KJ, Holmes AP, Worsley KJ, Poline JP, Frith CD, Frackowiak RSJ (1995) Human Brain Mapping 2:189-210 \item Analysis of fMRI Time Series Revisited Friston KJ, Holmes AP, Poline JB, Grasby PJ, Williams SCR, Frackowiak RSJ, Turner R (1995) Neuroimage 2:45-53. \item Analysis of fMRI Time-Series Revisited - Again Worsley KJ, Friston KJ (1995) Neuroimage 2:173-181. \item Characterizing Dynamic Brain Responses with fMRI: A Multivariate Approach Friston KJ, Frith CD, Frackowiak RSJ, Turner R (1995) Neuroimage 2:166-172. \item Characterizing Evoked Hemodynamics with fMRI Friston KJ, Frith CD, Turner R, Frackowiak RSJ (1995) Neuroimage 2:157-165. \item A Multivariate Analysis of PET Activation Studies Friston KJ, Poline J-B, Holmes AP, Frith CD, Frackowiak RSJ (1996) Human Brain Mapping 4:140-151. \item An overview and some new developments in the statistical analysis of PET and fMRI data Worsley KJ(1997) Human Brain Mapping 5:254-258. \een \item[11. Curve modeling] It could mean many things. Since a curve can be characterized by curvature, it includes the estimation of curvature and stochastic modeling on curvature function. \ben \item Hoffmann, T.J., Chung, M.K., Dalton, K.M., Alexander, A.L., Wahba, G., Davidson, R.J. (2004) Subpixel curvature estimation of the corpus callosum via splines and its application to autism, 10th Annual Meeting of the Organization for Human Brain Mapping. \item Wang, K. and Gasser, Y. Alignment of Curves by Dynamic Time Warping. Annals of Statistics, 25:1251-1276, 1997 \item Kass, Witkin and Terzopoulos (International Journal of Computer Vision 1:321-31, 1987 \item G. Sapiro's Geometric Partial Differential Equations and Image Analysis. \item Taubin's seminal paper on the shrinkage effect \item Bartesaghi, A. and Sapiro, G. \newblock A System for the Generation of Curves on 3D Brain Images. \newblock {\em Human Brain Mapping}, 14:1--15, 2001. \item Sethian., J. A. \newblock {\em Level Set Methods and Fast Marching Methods}. \newblock Cambridge University Press, 1996. \item Osher, Level set. \item Betounes, 1998 \een \item[12. Surface modeling] Surface modeling is much harder than curve modeling due to the non Euclidean nature of most surfaces \ben \item Jones, S.E., Buchbinder, B.R., and Aharon, I. (2000). \newblock Three-dimensional Mapping of Cortical Thickness Using Laplace's Equation. \newblock {\em Human Brain Mapping}, 11:12--32. \item Angenent, S., Hacker, S., Tannenbaum, A., and Kikinis, R. (1999). \newblock On the Laplace-Beltrami Operator and Brain Surface Flattening. \newblock {\em IEEE Transactions on Medical Imaging}, 18:700--711. \item Davatzikos, C. and Bryan, R.N. (1995). \newblock Using a Deformable Surface Model to Obtain a Shape Representation of the Cortex. \newblock In {\em Proceedings of the IEEE International Conference on Computer Vision}, pages 2122--2127. \item MacDonald, J.D., Kabani, N., Avis, D., and Evans, A.C. \newblock Automated 3-D Extraction of Inner and Outer Surfaces of Cerebral Cortex from MRI. \newblock {\em NeuroImage}, 12:340--356, 2000. \item Dale, A.M., and Fischl, B. \newblock Cortical surface-based analysis I. segmentation and surface reconstruction. \newblock {\em NeuroImage}, 9:179--194, 1999. \item Joshi, S.C., Wang, J. and Miller, M.I., Van Essen, D.C., Grenander, U. \newblock On the Differential Geometry of the Cortical Surface. \newblock {\em Vision Geometry IV}, 2573:304-311, 1995. \item Gurtin, M.E. and McFadden, G.B. (1994). \newblock {\em On the Evolution of Phase Boundaries}. \newblock Springer-Verlag, New York. \item Griffin, L.D. \newblock The intrinsic geometry of the cerebral cortex. \newblock {\em Journal of Theoretical Biology}, 166:261-273, 1994. \item Thompson, P.M., Mega, M.S., Woods, R.P., Zoumalan, C.I., Lindshield, C.J., Blanton, R.E., Moussai, J., Holmes, C.J., Cummings, J.L., and Toga, A.W. \newblock Cortical Change in Alzheimer's Disease Detected with a Disease-Specific Population-Based Brain Atlas. \newblock {\em Cerebral Cortex}, 11:1--16, 2001. \item Thompson, P.M., Giedd, J.N., Woods, R.P., MacDonald, D., Evans, A.C., and Toga, A.W. \newblock Growth Patterns in the Developing Human Brain Detected Using Continuum-Mechanical Tensor Mapping. \newblock {\em Nature}, 404:190--193, 2000. \item Sethian., J. A. \newblock {\em Level Set Methods and Fast Marching Methods}. \newblock Cambridge University Press, 1996. \item Taubin, G. \newblock Curve and surface smoothing without shrinkage \newblock {\em The Proceedings of the Fifth International Conference on Computer Vision}, 852-857, 1995. \item M.De Berg's Computational Geometry: Algorithms and Applications, 2nd rev. ed. Berlin: Springer-Verlag, pp. 147-163, 2000 \item H. Edelsbrunner's Geometry and Topology for Mesh Generation, \item Chung, M.K., Worsley, K.J., Robbins, S., Paus, P., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2003) Deformation-Based Surface Morphometry with an Application to Gray Matter Deformation, NeuroImage. 18:198-213. \item Chung, M.K., Worsley, K.J., Robbins, S., Evans, A.C. (2003) Tensor-based Brain Surface Modeling and Analysis, The proceeding of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Vol I 467-473. \item Chung, M.K,. Worsley, K.J., Paus, T., Robbins, S., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2002) Tensor-Based Surface Morphometry, Technical Report 1049. Department of Statistics, University of Wisconsin-Madison. \item Kreyszig, E. \newblock {\em Differential Geometry}. \newblock University of Toronto Press, Toronto, 1959. \item Boothby, W.M. (1986). \newblock {\em An Introduction to Differential Manifolds and Riemannian Geometry}. \newblock Academic Press, London, 2nd. edition. \item Do Carmo, M.P. (1992). \newblock {\em Riemannian Geometry}. \newblock Prentice-Hall, Inc. \een \item[13. Kernel smoothing.] Due to simplicity and speed, it is most widely used method for reducing noise. It is a form of integral transform. \begin{enumerate} \item J.S. Simonoff, Smoothing Methods in Statistics. \item N.S. Altman, Kernel smoothing of data with correlated errors, JASA 1990 58:411. \item Ramsay and Silverman, Functional Data Analysis. 1997 J.S. \item Simonoff's Smoothing Methods in Statistics (1996) \item Ruppert {\em et al.} (2003) Semiparametric Regression. \item Nadaraya, 1964 \item Watson, 1964. \item Dirac, P.A.M., Quantum Mechanics. 1958. \item Gelfand and Shilov, Generalized functions. 1964. \item Stakgold, Green's Functions and Boundary Value Problem. 1997. \end{enumerate} \item[14. Diffusion smoothing.] There are a lot of diffusion smoothing paper in computer vision but none of them use it as a way to improve Gaussianness. \ben \item Chung, M.K., Taylor, J. (2004) Diffusion Smoothing on Brain Surface via Finite Element Method, IEEE International Symposium on Biomedical Imaging (ISBI). \item Andrade, A., Kherif, F., Mangin, J., Worsley, K.J., Paradis, A., Simon, O., Dehaene, S., Le~Bihan, D., and Poline, J-B. \newblock Detection of Fmri Activation Using Cortical Surface Mapping. \newblock {\em Human Brain Mapping}, 12:79--93, 2001. \item Chung, M.K., Worsley, K.J., Robbins, S., Evans, A.C. (2003) Tensor-based Brain Surface Modeling and Analysis, The proceeding of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Vol I 467-473. \item Chung, M.K., Worsley, K.J., Robbins, S., Paus, P., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2003) Deformation-Based Surface Morphometry with an Application to Gray Matter Deformation, NeuroImage. 18:198-213. \item Chung, M.K,. Worsley, K.J., Paus, T., Robbins, S., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2002) Tensor-Based Surface Morphometry, Technical Report 1049. Department of Statistics, University of Wisconsin-Madison. \item Chung, M.K., Worsley, K.J., Taylor, J., Ramsay, J.O. (2001) Diffusion Smoothing in Brain Imaging, The proceedings of the 53rd Session of the International Statistical Institute \item Tang, B., Sapiro, G. and Caselles, V. \newblock Direction Diffusion \newblock {\em The Proceedings of the Seventh IEEE International Conference on Computer Vision}, 2:1245-1252, 1999. \item Sochen, N., Kimmel, R. and Malladi, R. \newblock A General Framework for Low Level Vision. \newblock {\em IEEE Transactions on Image Processing}, 7:310-318, 199 \item Perona, P., Malik, J. \newblock Scale-space and Edge Detection Using Anisotropic Diffusion. \newblock {\em IEEE Trans. Pattern Analysis and Machine Intelligence}, 12:629-639, 1990. \item Betounes, D. 1998. {\em Partial Differential Equations for computational Science.} Springer-Verlag New York, Inc. \item Egorov, Y.V. and Shubin, M.A. (1991). \newblock {\em Partial Differential Equations I: Foundations of the Classical Theory}. \newblock Springer-Verlag. \een \item[15. Heat kernel smoothing.] This is a differential geometry problem. \ben \item S. Rosenberg's The Laplacian on a Riemannian Manifold (1997 Cambridge University Press) \item I Chavel's Eigenvalues in Riemannian geometry (Academic Press, 1984) \item E.B. Davies' Heat kernels and spectral theory (Cambridge University Press, 1989). \item P. Olver and C. Shakiban's Fundamentals of Applied Mathematics, Prentice-Hall, in preparation. \item Malladi, R and Ravve, I. \newblock Fast Difference Schemes for Edge Enhancing Beltrami Flow. \newblock Computer Vision-ECCV 2002 Part I 343-357. 7th European Conference on Computer Vision. \newblock Springer, 2002. \item Ekholm, T., 2004 \item Schoen, R. and Yau S.-T. Lectures on Differential Geometry (Vol I. Conference Proceedings and Lecture notes in Geometry and Topology. International Press. 1994) \item Diffusion kernels on statistical manifolds by Lafferty, J. and Lebanon, G. Technical Report CMU-CS-04-101, School of Computer Science, CMU, 2004 \item Chen {\em et al.}, 2001, CVPR. \item Chu {\em et al.} (1998. JASA) \item Huber in Robust statistics 1981 \item Kreyszig, E. \newblock {\em Differential Geometry}. \newblock University of Toronto Press, Toronto, 1959. \een \item[16 Directional Smoothing and other smoothing techniques] Smoothing vector and tensor fields \ben \item Chung et al., Probabilistic Connectivity Measure in Diffusion Tensor Imaging via Anisotropic Kernel Smoothing. TR 1081. 2003. \item Grace Wahba, Spline Models for Observational Data. \item L. Amodei and M.N. Benbourhim, A vector spline approximation, Journal of Approximation Theory 67: 51-79. 1991. \item Redescending M-estimators in regression analysis, cluster analysis and image analysis. Christine Muller. \een \item[17 False discovery rate (FDR) and 19 Permutation test] This is a somewhat new method for performing multiple comparisons. It is not as popular as the random field theory in neuroimaging. \ben \item Non-Parametric Analysis of Statistic Images From Functional Mapping Experiments,Holmes AP, Blair RC, Watson JDG, Ford I (1996) Journal of Cerebral Blood Flow and Metabolism 16:7-22. \item Genovese, C.R., Lazar, N.A., Nichols, T.E. 2002. Thresholding of statistical maps in functional neuroimaging using the false discovery rate. {\em NeuroImage} {\bf 15}:870-878. \item Nichols, T.E. and Holmes, A.P. 2002. Nonparametric permutation tests for functional neuroimaging experiments: a primer with examples. {\em Human Brain Mapping} {\bf 15}:1-25. \item Miller, R. G. Jr. Simultaneous Statistical Inference. New York: Springer-Verlag, 1991. \item J. D. Storey, J. E. Taylor, D. O. Siegmund. Strong control, conservative point estimation, and simultaneous conservative consistency of false discovery rates: A unified approach. Journal of the Royal Statistical Society, Series B. In press. \een \item[18. Random Fields Theory (RFT) (Corrected $P$-value)] It usually refers to random fields based multiple comparison correction methodology in neuroimaging. $P$-value that takes care of multiple comparison is called the {\em corrected P-value}. \ben \item Worsley, K.J. (2003). Detecting activation in fMRI data. Statistical Methods in Medical Research, 12:401-418. \item Worsley, K.J. (2003). An improved theoretical P-value for SPMs based on discrete local maxima. NeuroImage, submitted. \item Cao, J. and Worsley, K.J. (2001). Applications of random fields in human brain mapping. In M. Moore (Ed.) Spatial Statistics: Methodological Aspects and Applications, Springer Lecture Notes in Statistics, 159:169-182. \item Worsley, K.J., Andermann, M., Koulis, T., MacDonald, D., and Evans, A.C. (1999). Detecting changes in non-isotropic images. Human Brain Mapping, 8:98-101. \item Cao, J. and Worsley, K.J. (1999). The detection of local shape changes via the geometry of Hotelling's $T^2$ fields. Annals of Statistics, 27, 925-942. \item Worsley, K.J., Marrett, S., Neelin, P., Vandal, A.C., Friston, K.J., and Evans, A.C. (1996). A unified statistical approach for determining significant signals in images of cerebral activation. Human Brain Mapping, 4:58-73. \item Worsley, K.J., Marrett, S., Neelin, P. and Evans, A.C. (1995). A unified statistical approach for determining significant signals in location and scale space images of cerebral activation. Quantification of brain function using PET, Eds. R. Myers, V.J.Cunningham, D.L. Bailey and T. Jones , Academic Press, San Diego, 327-333. \item Siegmund, D.O and Worsley, K.J. (1995). Testing for a signal with unknown location and scale in a stationary Gaussian random field. Annals of Statistics, 23:608-639. \item Worsley, K.J. (1995). Estimating the number of peaks in a random field using the Hadwiger characteristic of excursion sets, with applications to medical images. Annals of Statistics, 23:640-669. \item Worsley, K.J. (1994). Local maxima and the expected Euler characteristic of excursion sets of $\chi2$, $F$ and $t$ fields. Advances in Applied Probability, 26:13-42. \item Worsley, K.J., Marrett, S., Neelin, P., and Evans, A.C. (1992). A three-dimensional statistical analysis for CBF activation studies in human brain. Journal of Cerebral Blood Flow and Metabolism, 12:900-918. \item J. E. Taylor, R. J. Adler. Euler characteristics for Gaussian fields on manifolds. Annals of Probability, 31(2), 533-563, 2003. \item Chung, M.K., Dalton, K.M., Alexander, A.L., Davidson, R.J. (2004) Less white matter concentration in autism: 2D Voxel-Based Morphometry. NeuroImage, in press. ({\em F-fields in 2D Euclidean space}) \item Chung, M.K., Worsley, K.J., Robbins, S., Paus, P., Taylor, J., Giedd, J.N., Rapoport, J.L., Evans, A.C. (2003) Deformation-Based Surface Morphometry with an Application to Gray Matter Deformation, NeuroImage. 18:198-213. ({\em T-fields on 2D manifolds}) \item Chung, M.K., Worsley, K.J. , Paus, T., Cherif, C., Giedd, J.N., Rapoport, J.L, Evans, A.C. (2001) A Unified Statistical Approach to Deformation-Based Morphometry, NeuroImage 14:595-606. ({\em T-fields in 3D Euclidean space}) \item Schmidt and E. Spodarev's Joint Estimators for the Specific Intrinsic Volumes of Stationary Random Sets (2003). \een \end{description} \end{document}