Statistical Inference using the Morse-Smale Complex

The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare multivariate functions. In this paper, we present some statistical results on estimating Morse-Smale complexes. This allows us to derive new results for two existing methods: mode clustering and Morse-Smale regression. We also develop two new methods based on the Morse-Smale complex: a visualization technique for multivariate functions and a two-sample, multivariate hypothesis test.Comment: 45 pages, 13 figures. Accepted to Electronic Journal of Statistic

Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing

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 $X_i, i=1,...,n,$ that are independent and uniformly distributed in $G.$ The problem is to estimate the integral curve of the differential equation $$\frac{dx(t)}{dt}=v(x(t)),\qquad t\geq 0,x(0)=x_0\in G,$$ 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.Comment: Published in at http://dx.doi.org/10.1214/009053607000000073 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

CT diagnosis of early stroke : the initial approach to the new CAD tool based on multiscale estimation of ischemia

Background: Computer aided diagnosis (CAD) becomes one of the most important diagnostic tools for urgent states in cerebral stroke and other life-threatening conditions where time plays a crucial role. Routine CT is still diagnostically insufficient in hyperacute stage of stroke that is in the therapeutic window for thrombolytic therapy. Authors present computer assistant of early ischemic stroke diagnosis that supports the radiologic interpretations. A new semantic-visualization system of ischemic symptoms applied to noncontrast, routine CT examination was based on multiscale image processing and diagnostic content estimation. Material/Methods: Evaluation of 95 sets of examinations in patients admitted to a hospital with symptoms suggesting stroke was undertaken by four radiologists from two medical centers unaware of the final clinical findings. All of the consecutive cases were considered as having no CT direct signs of hyperacute ischemia. At the first test stage only the CTs performed at the admission were evaluated independently by radiologists. Next, the same early scans were evaluated again with additional use of multiscale computer-assistant of stroke (MulCAS). Computerized suggestion with increased sensitivity to the subtle image manifestations of cerebral ischemia was constructed as additional view representing estimated diagnostic content with enhanced stroke symptoms synchronized to routine CT data preview. Follow-up CT examinations and clinical features confirmed or excluded the diagnosis of stroke constituting 'gold standard' to verify stroke detection performance. Results: Higher AUC (area under curve) values were found for MulCAS -aided radiological diagnosis for all readers and the differences were statistically significant for random readers-random cases parametric and non-parametric DBM MRMC analysis. Sensitivity and specificity of acute stroke detection for the readers was increased by 30% and 4%, respectively. Conclusions: Routine CT completed with proposed method of computer assisted diagnosis provided noticeable better diagnosis efficiency of acute stroke according to the rates and opinions of all test readers. Further research includes fully automatic detection of hypodense regions to complete assisted indications and formulate the suggestions of stroke cases more objectively. Planned prospective studies will let evaluate more accurately the impact of this CAD tool on diagnosis and further treatment in patients suffered from stroke. It is necessary to determine whether this method is possible to be applied widely