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Abstract The averaged Hausdorff distance Δ p is an inframetric, recently introduced in evolutionary multiobjective optimization (EMO) as a tool to measure the optimality of finite size approximations to the Pareto front associated to a multiobjective optimization problem (MOP). Tools of this kind are called performance indicators, and their quality depends on the useful criteria they provide to evaluate the suitability of different candidate solutions to a given MOP. We present here a purely theoretical study of the compliance of the Δ p -indicator to the notion of Pareto optimality. Since Δ p is defined in terms of a modified version of other well-known indicators, namely the generational distance GDp , and the inverted generational distance IGDp , specific criteria for the Pareto compliance of each one of them is discussed in detail. In doing so, we review some previously available knowledge on the behavior of these indicators, correcting inaccuracies found in the literature, and establish new and more general results, including detailed proofs and examples of illustrative situations.
Resumen La distancia promedio de Hausdorff Δp es una inframétrica recientemente introducida en optimización multiobjetivo evolutiva (EMO) como una herramienta para medir la optimalidad de aproximaciones finitas al frente de Pareto asociado con un problema de optimización multiobjetivo (MOP). Presentamos aquí un estudio puramente teórico sobre la sujeción del indicador Δp a la noción de optimalidad de Pareto. Puesto que Δp está definida en términos de una versión modificada de otros indicadores bien conocidos como lo son la distancia generacional GD p y la distancia generacional invertida IGD p , discutimos en detalle criterios específicos para la sujeción de tipo Pareto de cada uno de ellos. Adicionalmente, presentamos una revisión del comportamiento previamente conocido de estos indicadores, corrigiendo imprecisiones que se encuentran en la literatura y establecemos resultados nuevos y más generales, incluyendo pruebas detalladas y ejemplos ilustrativos.
Resumo A distância média de Hausdorff Δp é uma inframétrica introduzida recentemente em otimização multiobjetivo evolucionária (EMO) como uma ferramenta para medir a otimalidade de aproximações finitas para o frente de Pareto associado com um problema de optimização multiobjetivo (MOP). Apresentamos aqui um estudo puramente teórico sobre a sujeição do indicador Δp à noção de otimalidade de Pareto. Desde Δp é definido em termos de uma versão modificada de outros indicadores bem conhecidos, tais como a distância geraçional GD p ∈ a distância geraçional invertida IGD p , discutimos em detalhes critérios específicos para a sujeição de Pareto de cada um deles. Além disso, apresentamos uma revisão do comportamento previamente conhecido desses indicadores, corrigindo imprecisões encontradas na literatura ∈ estabelecemos novos ∈ mais gerais resultados, incluindo testes detalhados ∈ exemplos ilustrativos.
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Objective@#To calculate out the Hausdorff distance based on the scripting in RayStation treatment planning system, which was then applied in measuring the deformation error of brain stem during image automatic registration between CT and MR.@*Methods@#Scripting was edited in RayStation system (version 4.7) by using IronPython. The set of point coordinates on the contour of any two region of interest (ROI) had been found firstly, then the Hausdorff distance between the two point sets was calculated out. A graphical user interface (GUI) was designed by using XAML to acquire the visualized output of Hausdorff distance. GUI appeared when the script was run, where two ROIs was selected, then the corresponding Hausdorff distance and the running time were displayed by pressing the "Calculate" button.@*Results@#The mean Hausdorff distance of brain stem in 20 patients with head and neck neoplasms was 1.20 cm while the mean elapsed time was 11.01s.@*Conclusions@#Hausdorff distance of any two ROIs can be calculated out by using the developed method. GUI is designed to realize the visual interaction with RayStation system. Therefore, the RayStation system satisfies the demands of Hausdorff distance calculation in both clinical and research work.
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Objective To demonstrate the relationship between modified Hausdorff distance (MH) and the spatial change of arterial vasculature in residuum. Methods Data of the vascular model were obtained from CT scans of clinical cases, and a centerline model of the residuum vasculature was established by setting the appropriate parameter of distance between control points (DCP) in MIMICS 10.0. Deformation was applied on the model through compiling the program in MATLAB. The MH values before and after the deformation were calculated and analyzed. According to different locations of the rotation axis, the deformation was divided into four cases (C1, C2, C3, C4) on the basis of growth order in node numbers and total length of the vasculature. Results MH increased with the increment of the vascular rotation angle, but due to different maximum rotation radius and total length of the vascular vessel, 4 deformed models showed different growth trends. With the rotation angle increasing, MH gradually increased from C1 to C4. The maximum change occurred from C2 to C3, and minimal change occurred from C3 to C4. Specifically, the maximum rotation radius and total length of the vessels from C1 to C2 increased by 22.2% and 91.3%, while those from C3 to C4 increased by 14.1% and 26.8%, respectively. Moreover, the maximum change of MH value was 60.4% from C1 to C2 and 4.5% from C3 to C4. Conclusions MH can be an effective parameter to indicate spatial change of the residuum vasculature. MH of the residuum vasculature is proportional to deformation angle of the vessel, and the maximum rotation angle has a significant influence on MH. Moreover, when DCP is smaller than 3 mm, the influence of parameter setting can be avoided effectively. In addition, MH can also be applied in the studies of vasculature change and comparison.
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This paper presents a new method for QRS complex self-assortment based on mathematical morphology, which measures the comparability of QRS complex with weighted Hausdorff distance. Beginning with the traditional Hausdorff distance, this paper introduces the weighted Hausdorff distance of planar object and the detailed algorithm of self-assortment of QRS complex. The evaluation results of this algorithm by MIT-BIHECG database show that this method has high self-assortment capability.