White matter (WM) fiber tractography (i.e., the reconstruction of the 3D architecture of WM fiber pathways) is known to be an important application of diffusion tensor magnetic resonance imaging (DT-MRI). For the quantitative evaluation of several fiber-tracking properties, such as accuracy, noise sensitivity, and robustness, synthetic ground-truth DT-MRI data are required. Moreover, an accurate simulated phantom is also required for optimization of the user-defined tractography parameters, and objective comparisons between fiber-tracking algorithms. Therefore, in this study a mathematical framework for simulating DT-MRI data, based on the physical properties of WM fiber bundles, is presented. We obtained a model of a WM fiber bundle by parameterizing the various features that characterize this bundle. We then evaluated three different synthetic DT-MRI models using experimental data in order to test the proposed methodology, and to determine the optimum model and parameter settings for constructing a realistic simulated DT-MRI phantom. Diffusion tensor magnetic resonance imaging (DT-MRI) is currently the only method available to obtain quantitative information about the three-dimensional (3D) anisotropic diffusion of water molecules in biological tissue (1,2). This DT anisotropy reflects the presence of spatially oriented microstructures (e.g., neural fibers in the central nervous system), where the mobility of the diffusing particles is mainly determined by the fiber pathway (3). On the basis of this intrinsic property, which assumes that the orientation of the DT field matches the orientation of the corresponding underlying fiber system, DT-MRI has been applied in several studies to infer microstructural characteristics and obtain valuable diagnostic information regarding various neuropathological conditions. Excellent reviews on white matter (WM) and neuropsychiatric diseases can be found in Refs. 4 and 5.It is known that ambiguous results are obtained when DT-MRI is used to study regions in which WM fibers cross or multiple fibers merge (e.g., Ref. 6). In such regions, the second-rank DT model is incapable of describing multiple fiber orientations within an individual voxel. To overcome this problem, a number of new techniques have been proposed, such as high-angular-resolution diffusion-weighted imaging (HARDI) (7), Q-ball imaging (QBI) (8), diffusion spectrum imaging (DSI) (9), persistent angular structure (PAS) reconstruction (10), and generalized DT imaging (GDTI) (11). Although these recently developed techniques can provide more accurate and unambiguous results, for simplicity the focus in this paper is confined to classic DT-MRI.An important application of DT-MRI is the reconstruction of the 3D WM fiber network, which is referred to as DT tracking (DTT) or fiber tractography. This technique is based on the assumption that it can accurately retrieve the spatial information of the underlying fiber network, using the available diffusion information of the corresponding tensor field. DTT provides exciting...

In graph mining, a frequency measure for graphs is anti-monotonic if the frequency of a pattern never exceeds the frequency of a subpattern. The efficiency and correctness of most graph pattern miners relies critically on this property. We study the case where frequent subgraphs have to be found in one graph. Vanetik et al. (Data Min Knowl Disc 13(2):243-260, 2006) already gave sufficient and necessary conditions for anti-monotonicity of graph measures depending only on the edge-overlaps between the instances of the pattern in a labeled graph. We extend these results to homomorphisms, isomorphisms and homeomorphisms on both labeled and unlabeled, directed and undirected graphs, for vertex-and edge-overlap. We show a set of reductions between the different morphisms that preserve overlap. As a secondary contribution, we prove that the popular maximum independent set measure assigns the minimal possible normalized frequency and we introduce a new measure based on the minimum clique partition that assigns the maximum possible normalized frequency. In that way, we obtain that all normalized anti-monotonic overlap graph measures are bounded from above and below. We also introduce a new measure sandwiched between the former two based on the polynomial time computable Lovász θ -function.

In graph mining, a frequency measure is anti-monotonic if the frequency of a pattern never exceeds the frequency of a subpattern. The efficiency and correctness of most graph pattern miners relies critically on this property. We study the case where the dataset is a single graph. Vanetik, Gudes and Shimony already gave sufficient and necessary conditions for anti-monotonicity of measures depending only on the edge-overlaps between the intances of the pattern in a labeled graph. We extend these results to homomorphisms, isomorphisms and homeomorphisms on both labeled and unlabeled, directed and undirected graphs, for vertex and edge overlap. We show a set of reductions between the different morphisms that preserve overlap. We also prove that the popular maximum independent set measure assigns the minimal possible meaningful frequency, introduce a new measure based on the minimum clique partition that assigns the maximum possible meaningful frequency and introduce a new measure sandwiched between the former two based on the poly-time computable Lovász θ-function. 1. We study systematically all 24 combinations of iso-, homo-, or homeomorphism, on labeled or unlabeled, directed or undirected graphs, with edge-or vertexoverlap and extend the anti-monotonicity results. 2. In our proofs, we use reductions which are also of interest in their own right, as they allow to transfer results for different types of morphisms and overlap from one setting to another. For an overview of the different reductions, see Figure 1. 3. An interesting consequence of the reductions is that any unlabeled, undirected graph is a potential vertexand edge-overlap graph in all considered settings.