We develop a mathematical and numerical framework to solve state estimation problems for applications that present variations in the shape of the spatial domain. This situation arises typically in a biomedical context where inverse problems are posed on certain organs or portions of the body which inevitably involve morphological variations. If one wants to provide fast reconstruction methods, the algorithms must take into account the geometric variability. We develop and analyze a method which allows to take this variability into account without needing any a priori knowledge on a parametrization of the geometrical variations. For this, we rely on morphometric techniques involving Multidimensional Scaling, and couple them with reconstruction algorithms that make use of reduced model spaces pre-computed on a database of geometries. We prove the potential of the method on a synthetic test problem inspired from the reconstruction of blood flows and quantities of medical interest with Doppler ultrasound imaging.

中文翻译：

---

具有模型缩减和形状可变性的状态估计。应用于生物医学问题

我们开发了一个数学和数值框架来解决存在空间域形状变化的应用程序的状态估计问题。这种情况通常出现在生物医学环境中，在这种情况下，某些器官或身体部位会出现逆问题，这些问题不可避免地涉及形态变化。如果想要提供快速重建方法，算法必须考虑几何可变性。我们开发并分析了一种方法，该方法允许将这种可变性考虑在内，而无需任何有关几何变化参数化的先验知识。为此，我们依靠涉及多维缩放的形态测量技术，并将它们与重建算法结合使用，这些算法利用在几何数据库上预先计算的简化模型空间。