**1. Introduction**

All bodies at temperature above absolute zero (0 K) emit electromagnetic radiation. If the temperature of a body has the same magnitude as the ambient temperature, then the emission is mainly relegated to the infrared (IR) range of the spectrum and can be sensed and displayed by a thermal camera as a false-color image, called a thermogram. In addition, by knowing a series of parameters such as the inspected object emissivity and the apparent reflected temperature (i.e., the ambient temperature), an approximated temperature map of the object can be computed in output. Infrared thermography is a non-invasive (non-contact and non-destructive) imaging method, which makes it a widely applicable technique. For example, it has a vast range of applications in research and industry, building and infrastructure, electrical installation inspection, microsystems engineering, but also in biology, medicine, life sciences and cultural heritage [1,2].

Like thermal imaging, 3D reconstruction techniques are nowadays widespread in many different fields and are commonly used to acquire the object geometry and to provide easy 3D documentation. They are a powerful tool to improve the identification, monitoring, conservation and restoration of objects and structures [3].

The utility of the integration is largely due to the fact that the ability to combine both temperature and geometric data together can lead to several advantages: it enhances and speeds up the interpretation of the results; it offers the possibility to select the region of interest by taking into account the geometry; it allows the easy segmentation of the 2D data from the background. One of the most important advantages, however, is that it allows one to overcome a significant limitation of 2D thermograms, namely the systematic error in the measured temperature due to the dependence of the emissivity on the viewing angle [4–6].

Several works in the literature have shown the strong potential of 3D thermal mapping (commonly known in the literature as 3D thermography). The thermal data are obtained with a thermal camera, whereas the way in which 3D data are acquired varies: a concise overview can be found in the work of G. Chernov et al. [4]. For example, in the medical field some applications of 3D thermography are the work of X. Ju et al. [7] and the one of K. Skala et al. [8]. In [7], the process of 3D capture relied upon stereo photogrammetry, whereas in [8] the system consisted of a high-resolution offline 3D laser scanner and a real-time low-resolution 3D scanner, both paired together with a thermal imaging camera, for human body 3D thermal model comparison and analysis. In [9], S. Vidas and P. Moghadam presented "HeatWave", a handheld, low-cost 3D thermography system, which allows non-experts to generate detailed 3D surface temperature models for energy auditing. The core technology of this device is obtained by combining a thermal camera and an RGB-D sensor (depth sensing device coupled with an RGB camera). In several other latest applications, such as [10–12], the spatial data were obtained by using a depth camera such as the Microsoft Kinect, which has become one of the top choices for 3D thermography, because of its large versatility and the capability to be exploited to perform real-time integrations. In [13], the integration was carried out on the data acquired by two smartphones arranged in a stereo configuration and a thermal camera. In [14], a fully automatic system that generated 3D thermal models of indoor environments was presented; it consisted of a mobile platform equipped with a 3D laser scanner, an RGB camera and a thermal camera.

In the cultural heritage field, spatial and multispectral data have usually been fused together for documentation reasons, historical studies, restoration plans and visualization purposes; several examples can be found in [15–19].

One advantage of 3D thermal models is that, for each 3D point, one can compute the so-called viewing angle (i.e., the angle between the surface normal vector in that point and the vector joining the point and the optical center). This information can be used to correct the error in the temperature caused by the dependence of the emissivity on the viewing angle. Indeed, for a given material, the emissivity is usually not constant, but depends on several factors, such as the surface condition, the wavelength, the temperature, the presence of concavities and the viewing angle (a viewing angle-dependent emissivity is often called "directional emissivity"). A detailed explanation of how these factors affect the emissivity can be found in [1] (pp. 35–45). Whereas the role of many of these factors can be in general considered negligible, the dependence on the viewing angle is normally relevant, and can bias the results, as outlined, for example, in [20] and [5]. Therefore, by knowing both the directional emissivity and viewing angle, it is possible to correct the temperature accordingly. Examples can be found in [4,6] and moreover in [21], where the internal reflections due to concave surfaces of a complex test setup were also taken into account.

It is worth noticing that in these publications the different sensors were rigidly linked together (mainly for calibration purposes and real-time data integration), and the trend was to strengthen this physical union (until obtaining, in the final form, a unique device such as "HeatWave" [9]). However, in some cases, it can be more convenient to keep the two devices decoupled and independent. This is especially true in outdoor surveys, where there is often the need to perform the thermographic analysis at a specific day-time or night-time and/or weather conditions, which requires high versatility (e.g., for the assessment of the damages and energy efficiency of the building envelope [18]). Laser scanners, on the other hand, can be bulky and heavy; their handling and the regulation of their position and orientation (usually are mounted on a tripod) may be time consuming and requires caution. Fixing a thermal camera to this type of scanner would make them even more di fficult to regulate, and the easiness of handling of the thermal camera would be compromised. Furthermore, the two devices may have very di fferent optics, which make the optimal distance of acquisition distinct. Conversely, with a decoupled acquisition, the integration can be applied in a flexible way, namely only when it is useful, based on previously recorded data, and does not a ffect the stand-alone capability of each device. In the literature, works based on this approach are uncommon. One exception is the work of A. G. Krefer et al. [22], which consisted of a method for generating 3D thermal models with decoupled acquisition, which relies on structure from motion and particle swarm optimization. Our paper focuses as well on a decoupled type of integration, but it di ffers from [22] in several aspects, such as the calibration method, the data fusion technique and the managemen<sup>t</sup> of the superimposition of multiple thermograms.

This paper is organized as follows: the system architecture is outlined in Sections 2–4 explain the geometrical calibration and data fusion procedure adopted, respectively, and Section 5 presents the results. In the first experimental case of Section 5, the e ffectiveness of our approach is tested on a 3D-printed object properly designed. The second example of application belongs to the cultural heritage field. In the last few years, we have carried out a many-sided research project aimed at preserving and restoring the ancient sanctuary of Santa Maria delle Grazie, built toward the middle of the 15th century, in the place of Fornò near the city of Forlì (Italy). In particular, we have applied thermal imaging and laser scanning both to the building at large and to the ornamental elements. One example, presented in this paper, is the application to a marble statue called Madonna with the Child, an admirable work of the Florentine sculptor Agostino di Duccio (1418–1481), made up of four superimposed blocks. Up to the year 2000, this sculpture was in a niche on the entrance arch to the prothyrum of the sanctuary. Afterwards, since it showed clear signs of deterioration, especially due to rainwater and air pollution, it was carefully restored and then moved permanently to a grea<sup>t</sup> hall in the Bishop's palace in Forlì, where our surveys were performed.
