# Non-Invasive Testing of Physical Systems Using Topological Sensitivity
## 1 Introduction  
Non-destructive testing of a physical system aims at identifying anomalies or degradations in some properties of the system from its response to (natural or forced) excitation. In this sense, this is an _inverse problem_ that somewhat opposites the associated _direct problem_, which consists of analyzing the response of the system when its properties are known. Direct and inverse problems can be clearly illustrated using a drum as a physical system and questioning whether one can \"hear the shape of the drum\" [1, 2]. In this example, the direct problem consists of calculating the sound produced by a drum of a given shape, while the inverse problem intends guessing the drum shape from the sound it produces. For a drum, this formulation of the inverse problem could be called a _full inverse problem_. The main difficulty in solving full inverse problems is that they are usually extremely ill-conditioned, even if the associated direct problem is very well-conditioned. Thus, full inverse problems are frequently almost impossible to solve with precision. _Perturbed inverse problems_, instead, are usually less ill-conditioned. In the case of the drum, for the perturbed inverse problem, the unperturbed shape of the drum and the associated sound are both known beforehand, and only small or localized perturbations of the drum shape from its 'healthy' shape, known as _degradations_, are considered. In this case, the perturbed inverse problem consists of identifying these small degradations from the measured (small) perturbations in the produced sound. _Diagnosis/prognosis_ of a biological (e.g., medical) or an engineering (e.g., an engine) system consists of identifying the present degradations in the relevant properties of the system (_diagnosis_), which permits predicting (forecasting or _prognosis_) the likely behavior of the system over time using its present degraded (or perturbed) state from healthy conditions, measured by its response to (natural or forced) excitation. In this sense, diagnosis (which is a first step to perform prognosis) also requires solving a perturbed inverse problem. Thus, inverse problems are of interest in a variety of fields, including, among others, structural health monitoring (SHM) for the non-destructive testing of structures [3], acoustic waves [4; 5], electromagnetic waves [6; 7], medical diagnosis [8], natural resources exploration [9], location of anti-personnel landmines [10], and oceanography exploration [11]. In this paper, we shall concentrate in SHM. In this field, the aim is identifying possible (usually, localized) defects/flaws in a given structure using its response to excitation, measured in some _sensors_ distributed along the structure surface. Two type of excitations will be considered in this paper: (i) guided _Lamb waves_ produced by some localized _vibrating emitters_, monitoring the elastodynamic response in some localized _sensors_, and (ii) _thermal excitation_ produced by one or more infrared (IR) lamps, using as input data the distributed temperature at a part of the structure surface, captured by an _IR_ thermographic camera. This camera produces _radiation intensity_ images, which are readily translated into color maps for the _temperature_. Concerning SHM via ultrasonic guided waves, some classical methods [3; 12; 13; 14] are based on the _'flight time'_ of a concentrated wave packet. The flight time accounts for the time shift between the emission of the packet and its reception after being reflected/scattered by defects. Since the propagation velocity of the wave packet is known, distances to the defects are estimated from the time shifts. Thus, the more concentrated the wave packet, the better performance of the method. However, reflection/scattering at defects may occur after one or more reflections at the outer boundary of the structure, which means that locating defects is not easy, specially when the boundary is curved.