Hausdorff Distance evaluation of orthodontic accessories' streaking artifacts in 3D model superimposition

The aim of this study was to determine whether image artifacts caused by orthodontic metal accessories interfere with the accuracy of 3D CBCT model superimposition. A human dry skull was subjected three times to a CBCT scan: at first without orthodontic brackets (T1), then with stainless steel brackets bonded without (T2) and with orthodontic arch wires (T3) inserted into the brackets' slots. The registration of image surfaces and the superimposition of 3D models were performed. Within-subject surface distances between T1-T2, T1-T3 and T2-T3 were computed and calculated for comparison among the three data sets. The minimum and maximum Hausdorff Distance units (HDu) computed between the corresponding data points of the T1 and T2 CBCT 3D surface images were 0.000000 and 0.049280 HDu, respectively, and the mean distance was 0.002497 HDu. The minimum and maximum Hausdorff Distances between T1 and T3 were 0.000000 and 0.047440 HDu, respectively, with a mean distance of 0.002585 HDu. In the comparison between T2 and T3, the minimum, maximum and mean Hausdorff Distances were 0.000000, 0.025616 and 0.000347 HDu, respectively. In the current study, the image artifacts caused by metal orthodontic accessories did not compromise the accuracy of the 3D model superimposition. Color-coded maps of overlaid structures complemented the computed Hausdorff Distances and demonstrated a precise fusion between the data sets.

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.

Continuity of Lyapunov functions and of energy level for generalized gradient semigroup

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions)-defined on the whole phase space- which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results.