# Ascertaining the Ideality of Photometric Stereo Datasets under Unknown Lighting
## 1 Introduction  
Photometric stereo (which we will denote by PS) is a classic computer vision approach for reconstructing the shape of a three-dimensional object [1, 2]. It is considered a _shape from shading_ (SfS) technique, as it takes as input images that embed the shape and color information of the observed object. However, while the original SfS problem only considers a single two-dimensional image of the object [3, 4] and it is known to be ill-posed, PS stands on the use of a set of images acquired from a fixed point of view under varying lighting conditions. If the lighting positions are known, [PERSON] [5] proved that, under suitable assumptions, the PS problem is well-posed when at least two pictures are available. [PERSON]'s approach consists of modeling the problem by a system of first-order Hamilton-Jacobi partial differential equations (PDEs), for which various numerical methods have been proposed (see, e.g., [6]). The solution process that we will consider is slightly different and operates in two steps. First, the normal vectors at each discretization point of a rectangular domain, namely, the digital picture, are determined by solving an algebraic matrix equation. Then, after numerically differentiating the normal vector field, the solution of a Poisson PDE leads to the approximation of the object surface. This approach has been used in [7] to reconstruct rock art carvings found in _Domus de Janas_ (fairy houses), a specific typology of Neolithic tombs found in Sardinia, Italy [8]. While this computational scheme requires at least three images with different lighting conditions, the availability of a larger datasetleads to a least-squares solution method, which allows for a more efficient treatment of measurement errors. Though in the research community, the position of the light sources is generally assumed to be known [9; 10; 11], real-world applications of PS often deal with datasets acquired under unknown light conditions [12; 13]. In [14], [PERSON] has shown that the lighting directions can be identified directly from the data when at least six images with different illumination are available (see [14] and Section 2 for a proof), yielding the possibility to apply PS to field measurement scenarios, as in the case of archaeological excavations [15]. For example, the bas-relief displayed in Figure 1 is found in the _Domus de Janas_ of Corongiu, in Pimentel (South Sardinia, Italy), dated 4 th millennium BC. Figure 2 shows the small lobby where the engraving is located. While the camera can be placed sufficiently far from the rock surface, it is impossible to illuminate it with a flashlight at a distance suitable for proper lighting. Moreover, measuring with precision the position of the light source relative to the engraving is very difficult. This situation is common in many archaeological sites. PS application implies that certain assumptions are met in the acquisition process. In general, a perspective camera projection model may be used, with parameters estimated by camera calibration methods (see [16] for an introduction to the use of perspective and [10; 17] for applications). In this work, we consider an orth  
Figure 1: Engraving found in the _Domus de Janas_ of Corongiu, in Pimentel (Sardinia, Italy). Figure 2: Lobby of the Neolithic tomb located in the _Domus de Janas_ of Corongiu (Sardinia, Italy). The engraving is over the entrance to the internal chamber. which simplifies the presentation and is suitable for situations where the camera can be positioned at a relatively far distance from the observed object. The mathematical formulation of PS assumes that the object surface is Lambertian, meaning that [PERSON]'s cosine law can be employed to describe the object reflectance. This condition implies that the surface is matte and free from specular reflections, which is seldom verified in practice (see [18; 19; 20], where the Oren-Nayar model has been used to preprocess a dataset originated by a non-Lambertian object).