In this paper we present a first non-iterative imaging method for nonlinear materials, based on Monotonicity Principle. Specifically, we deal with the inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly embedded in linear known background. The Monotonicity Principle (MP) is a general property for various class of PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it states a monotone relation between the point-wise value of the unknown material property and the boundary measurements. It is at the foundation of a class of non-iterative imaging methods, characterized by a very low execution time that makes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of the peculiar difficulties in practical application of MP to imaging of nonlinear materials, preserving the feasibility for real-time applications. For the sake of clarity, we focus on a specific application, i.e. the Magnetostatic Permeability Tomography where the goal is retrieving the unknown (nonlinear) permeability by boundary measurements in DC operations. This choice is motivated by applications in the inspection of boxes and containers for security. Reconstructions from simulated data prove the effectiveness of the presented method.

翻译：本文提出了一种基于单调性原理的首个非迭代非线性材料成像方法。具体而言，我们处理的是逆障碍问题，目标是恢复嵌入在线性已知背景中的非线性异常体。单调性原理是各类偏微分方程的通用性质，最近已推广至非线性椭圆型偏微分方程。其本质在于建立了未知材料属性点值与边界测量值之间的单调关系。该原理构成了一类非迭代成像方法的基础，这类方法具有极低的执行时间，使其成为实时应用的理想候选方案。在本研究中，我们开发了一种反演方法，克服了将单调性原理应用于非线性材料成像时的一些特殊困难，同时保持了实时应用的可行性。为清晰起见，我们聚焦于特定应用场景，即静磁磁导率层析成像，其目标是通过直流工作状态下的边界测量恢复未知（非线性）磁导率。这一选择源于箱体与容器安检应用的需求。通过模拟数据重建的结果验证了所提方法的有效性。