## 1 Introduction  
Classical photogrammetric techniques have been used for reverse engineering in buildings for decades [1], and currently, due to UAVs, their applications are even more common [2, 3]. The mathematical control of the photographic images allows the orthorectification and measurement of elements on 2D images or the volumetric measurement of 3D elements by stereoscopy. These techniques, widely known and described in numerous publications [4, 5, 6, 7], have been complemented for years with the appearance of laser scanners [8, 9, 10, 11]. Although a photograph usually provides the interpretation of any visible element (there are many cases where other types of images result in being more convenient [12]), the laser contributes its geometry with a very high degree of precision, automatically and a short space of time [13, 14, 15, 16, 17]. Due to the laser scanner, the application of both techniques together allows for non-invasive reverse engineering processes where no contact with the element is necessary [18, 19, 20, 21, 22], which is especially relevant in heritage studies [23]. Cultural heritage is one of the fields with more different applications of combined techniques (as many different sciences are applied together in restoration works). Recently, augmented reality applications have become of great interest too [24, 25]. One of the buildings that encompasses the most architectural conservation and restoration activities, due to their high patrimomial value, are churches [26, 27, 28] and caverns with Rock Arts [29]. No contact with the object is of great importance when the surface to be restored contains works of art (usually pictorial). In these cases, the work of art depends directly on the geometry of the structural surface on which it is located. For the restoration of paintings, the surface is usually modelled and developed mathematically in 2D (using cartographic projections) to perform the restoration on a plane and then transfer it to the surface of the structure [30; 31]. However, there are cases in which it is not possible, mainly because there are surfaces that cannot be developed mathematically [32; 33]. Sometimes it happens because the construction has not been well executed (mainly due to the constructive means of the era in question), because the surface is not mathematically developable (mainly irregular spherical surfaces), or for both things at the same time [34]. In these cases, in which it is not possible to apply the mathematical transformations that allow a two-dimensional development without metric and/or angular deformations, the only possible option is to act directly on the surface. This study shows the restoration by video projection of the paintings that were destroyed by a fire on the vault of the church of Santos Juanes in Valencia. For this, there are only two black and white analogue frames taken before the fire, whose geometric origin is unknown (no data are available on the camera parameters or the shooting position). Due to the three-dimensional model obtained with the laser scanner, we will have a precise geometry of the surface. Given that the ceiling of our case study is an irregular hemispherical vault, it is impossible to mathematically develop its 2D surface to reconstruct the painting and then transfer it to the structure. The solution will be to perform the reconstruction directly on the element by projecting images on its surface (with video cannons) in the original position and without metric deformation. This implies being able to define the geometry with the greatest accuracy so that we can project partial images on the real surface by mathematically controlling the deformations of the projection and the scale of each of them. For this, it will be necessary to first calculate a complete 3D virtual image of paintings on the mathematical modelling of this irregular surface (from the original frames) and then, calculate 3D partial virtual images (which will be projected by each video cannon).