Abstract
Shape-from-Shading represents the problem of computing the three-dimensional shape of a surface given a single gray-value image of it as input. In a recent paper, we showed that the introduction of an attenuation factor in the brightness equations related to various perspective Shape-from-Shading models allows us to ensure the well-posedness of the corresponding differential problems. Here, we propose a unified convergence result of a numerical scheme for several non-Lambertian reflectance models. This result is interesting since it can be easily extended to other non-Lambertian models in a unified and, therefore, powerful framework.
Highlights
The Shape-from-Shading (SfS) problem consists of computing the three-dimensional shape of an object starting from a single gray-level image of it
We show the convergence of a numerical scheme for the perspective SfS problem associated with different non-Lambertian models, based on the method proposed in [32] and the theoretical results contained in [29]
We have shown a convergence result for the recent unified formulation of the perspective SfS models proposed in [29], in which the authors considered an attenuation term in order to achieve the well-posedness of the problem in the context of viscosity solutions
Summary
In [18,19], a different setup for the perspective model was proposed, with a pinhole camera and light source located at the optical center All these works are limited to the assumption of a Lambertian reflectance model, which is known to be not always suitable for describing real-world surfaces. In order to achieve the uniqueness of solution starting from a single input image, in [18,28], the authors introduced an attenuation factor in the Lambertian brightness equation under perspective projection This factor takes into account the distance between the surface and the light source and, thanks to this attenuation term, the associated HJ equation admits a unique viscosity solution. We show the convergence of a numerical scheme for the perspective SfS problem associated with different non-Lambertian models, based on the method proposed in [32] and the theoretical results contained in [29].
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