## 1 Introduction  
Thermospraphic techniques have proven to be very efficient nondestructive testing methods for the detection of defects in a wide variety of materials [1]. They use infrared (IR) video-cameras to detect anomalies caused by flaws in the surface temperature distribution of a material after some kind of excitation. Optical excitation is the most common method used to induce a heat flux within a sample. In opaque materials, the optical energy absorbed at the surface produces a temperature elevation and a subsequent heat flow. The presence of a defect distorts heat diffusion and produces an anomaly in the surface temperature distribution that is measured with an IR camera. Therefore, the presence of the flaw represents just a perturbation of an existing temperature field. On the contrary, in other thermographic modalities with ultrasonic [2] or inductive excitation [3] (vibrothermography and inductive thermography, respectively), heat is mainly generated at the defects. In the case of cracks or delaminations excited with ultrasounds, the main mechanism for heat production is the friction between the asperities of the moving faces of the defect under ultrasonic vibration [4], which turns the flaw into a heat source in a cold environment. The defect is detected as a temperature elevation at the surface, above the defect. In consequence, the technique is defect-selective, as the (excess) temperature field in the specimen is only due to the presence of the defect. Similarly, the detection of electrically conducting inclusions within isolating materials by inductive heating of the foreign material makes the inclusion behave as a heat source in a cold environment. However, the most common application of inductive thermography is the detection of cracks in electrically conducting parts [3]. In these applications, Eddy currents induced in the material generate heat by the Joule effect and the presence of cracks may distort either heat propagation or Eddy currents distribution, which often entails an additional source of heat. The ultimate goal of nondestructive techniques is the characterization of defects, i.e., determination of size, location, and orientation. In the case of cracks excited by ultrasounds and foreign metallic inclusions embedded in electrical insulators excited inductively, the physical magnitude that can be characterized is the heat flux generated at the defect: dimensions of active area, location, orientation, and absolute values of the flux distribution. In the last decade, many efforts have been devoted to the characterization of planar heat sources perpendicular to the surface of the material, representing vertical fatigue cracks excited with ultrasounds [5; 6; 7; 8; 9; 10; 11; 12]. For a broad overview, see [13]. In these works, the spatial extension and depth of vertical cracks was retrieved by inverting lock-in surface temperature data obtained under modulated excitation [5; 6; 7], as well as time domain data obtained under burst excitation [8; 9; 10]. In the latter modality, quantitative values of the local heat flux were determined locally along the crack surface [11; 12]. Some authors have focused on the evaluation of the crack length using minimization algorithms [14] or image processing methods [15]. Recently, the virtual wave concept has been applied to the identification of volumetric heat sources generated inductively by metallic balls embedded in epoxy [16]. The works mentioned above are mostly addressed at characterizing vertical heat sources typically generated by fatigue cracks in ultrasound-excited thermography. However, cracks may feature any (unknown) inclination with respect to the sample surface. Very few works have addressed the problem of tilted heat sources in thermographic experiments. The virtual wave concept approach has been applied to determine the orientation of planar heat sources but not the size or emitted flux [17]. The influence of the crack inclination has also been analyzed in inductive thermography experiments in metallic samples [18].