##
**Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing.**
*(English)*
Zbl 1195.62040

Summary: Let \(v\) be a vector field in a bounded open set \(G\subset \mathbb R^d\). Suppose that \(v\) is observed with a random noise at random points \(X_i\), \(i=1,\dots,n\), that are independent and uniformly distributed in \(G\). The problem is to estimate the integral curve of the differential equation

\[ dx(t)/dt= v(x(t)), \quad t\geq 0,\;x(0)= x_0\in G, \]

starting at a given point \(x(0)=x_0\in G\) and to develop statistical tests for the hypothesis that the integral curve reaches a specified set \(\Gamma\subset G\). We develop an estimation procedure based on a Nadaraya-Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface \(\Gamma\subset G\). Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches \(\Gamma\).

The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.

\[ dx(t)/dt= v(x(t)), \quad t\geq 0,\;x(0)= x_0\in G, \]

starting at a given point \(x(0)=x_0\in G\) and to develop statistical tests for the hypothesis that the integral curve reaches a specified set \(\Gamma\subset G\). We develop an estimation procedure based on a Nadaraya-Watson type kernel regression estimator, show the asymptotic normality of the estimated integral curve and derive differential and integral equations for the mean and covariance function of the limit Gaussian process. This provides a method of tracking not only the integral curve, but also the covariance matrix of its estimate. We also study the asymptotic distribution of the squared minimal distance from the integral curve to a smooth enough surface \(\Gamma\subset G\). Building upon this, we develop testing procedures for the hypothesis that the integral curve reaches \(\Gamma\).

The problems of this nature are of interest in diffusion tensor imaging, a brain imaging technique based on measuring the diffusion tensor at discrete locations in the cerebral white matter, where the diffusion of water molecules is typically anisotropic. The diffusion tensor data is used to estimate the dominant orientations of the diffusion and to track white matter fibers from the initial location following these orientations. Our approach brings more rigorous statistical tools to the analysis of this problem providing, in particular, hypothesis testing procedures that might be useful in the study of axonal connectivity of the white matter.

### MSC:

62G08 | Nonparametric regression and quantile regression |

62G20 | Asymptotic properties of nonparametric inference |

92C55 | Biomedical imaging and signal processing |

62E20 | Asymptotic distribution theory in statistics |

62M40 | Random fields; image analysis |

### Keywords:

kernel regression estimator; vector fields; integral curves; functional central limit theorem
PDFBibTeX
XMLCite

\textit{V. Koltchinskii} et al., Ann. Stat. 35, No. 4, 1576--1607 (2007; Zbl 1195.62040)

### References:

[1] | Basser, P. J., Pajevic, S., Pierpaoli, C. and Aldroubi, A. (2002). Fiber tract following in the human brain using DT-MRI data. IEICE Trans. Information and Systems E85-D 15-21. |

[2] | Basser, P. J., Pajevic, S., Pierpaoli, C., Duda, J. and Aldroubi, A. (2000). In vivo fiber tractography using DT-MRI data. Magnetic Resonance in Med. 44 625–632. |

[3] | Chung, M. K., Lazar, M., Alexander, A. L., Lu, Y. and Davidson, R. (2003). Probabilistic connectivity measure in diffusion tensor imaging via anisotropic kernel smoothing. Technical report, Univ. Wisconsin-Madison. |

[4] | Efromovich, S. (1999). Nonparametric Curve Estimation. Methods, Theory and Application . Springer, New York. · Zbl 0935.62039 · doi:10.1007/b97679 |

[5] | Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and its Applications . Chapman and Hall, London. · Zbl 0873.62037 |

[6] | Gössl, C., Fahrmeir, L., Pütz, B., Auer, L. M. and Auer, D. P. (2002). Fiber tracking from DTI using linear state space models: Detectability of the pyramidal tract. NeuroImage 16 378–388. |

[7] | Guye, M., Parker, G. J. M., Symms, M., Boulby, P., Wheeler-Kingshott, C. A. M., Salek-Haddadi, A., Barker, G. J. and Duncan, J. S. (2003). Combined functional MRI and tractography to demonstrate the connectivity of the human primary motor cortex in vivo. NeuroImage 19 1349–1360. |

[8] | Hille, E. (1969). Lectures on Ordinary Differential Equations . Addison-Wesley, Reading, MA. · Zbl 0179.40301 |

[9] | Ibragimov, I. and Has’minskii, R. (1981). Statistical Estimation. Asymptotic Theory . Springer, New York. · Zbl 0467.62026 |

[10] | Jones, D. K. (2003). Determining and visualizing uncertainty in estimates of fiber orientation from diffusion tensor MRI. Magnetic Resonance in Med. 49 7–12. |

[11] | Koltchinskii, V., Sakhanenko, L. and Cai, S. (2005). Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: Nonparametric kernel estimation and hypotheses testing. Technical report. Available at www.stt.msu.edu/ luda/diffusion4fwpics.pdf. · Zbl 1195.62040 |

[12] | O’Donnell, L., Haker, S. and Westin, C.-F. (2002). New approaches to estimation of white matter connectivity in diffusion tensor MRI: Elliptic PDEs and geodesics in a tensor warped space. Image Computing and Computer-Assisted Intervention MICCAI 2002. Lecture Notes in Comput. Sci. 2488 459–466. · Zbl 1028.68862 |

[13] | Pajevic, S., Aldroubi, A. and Basser, P. J. (2002). A continuous tensor field approximation of discrete DT-MRI data for extracting microstructural and architectural features of tissue. J. Magnetic Resonance 154 85–100. |

[14] | Parker, G. J. M., Barker, G. J. and Buckley, D. L. (2002). A probabilistic index of connectivity (PICo) determined using a Monte Carlo approach to streamlines. ISMRM Workshop on Diffusion MRI (Biophysical Issues) Saint-Malo, France 245–255. |

[15] | Parker, G. J. M., Stephan, K. E., Barker, G. J., Rowe, J. B., MacManus, D. G., Wheeler-Kingshott, C. A. M. et al. (2002). Initial demonstration of in vivo tracing of axonal projections in the macaque brain and comparison with the human brain using diffusion tensor imaging and fast marching tractography. NeuroImage 15 797–809. |

[16] | Parker, G. J. M., Wheeler-Kingshott, C. A. M. and Barker, G. J. (2002). Estimating distributed anatomical connectivity using fast marching methods and diffusion tensor imaging. IEEE Trans. Medical Imaging 21 505–512. |

[17] | Polzehl, J. and Spokoiny, V. (2001). Functional and dynamic magnetic resonance imaging using vector adaptive weights smoothing. Appl. Statist. 50 485–501. JSTOR: · Zbl 1112.62323 · doi:10.1111/1467-9876.00249 |

[18] | Poupon, C., Clark, C. A., Frouin, V., Régis, J., Bloch, I., Le Bihan, D. and Mangin, J.-F. (2000). Regularization of diffusion-based direction maps for the tracking of brain white matter fascicles. NeuroImage 12 184–195. |

[19] | Shafie, K., Sigal, B., Siegmund, D. and Worsley, K. J. (2003). Rotation space random fields with an application to fMRI data. Ann. Statist. 31 1732–1771. · Zbl 1043.92019 · doi:10.1214/aos/1074290326 |

[20] | Weinstein, D., Kindlmann, G. and Lundberg, E. (1999). Tensorlines: Advection-diffusion based propagation through diffusion tensor fields. In Proc. Visualization’99 249–253. IEEE, Piscataway, NJ. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.