A unified approach to the well-posedness of some non-Lambertian models in Shape-from-Shading theory

TL;DR

This paper introduces an attenuation factor into perspective Shape-from-Shading models, ensuring the associated differential problems are well-posed by leveraging viscosity solution theory, and discusses boundary conditions for the related Hamilton-Jacobi equations.

Contribution

It presents a unified approach using viscosity solutions to establish well-posedness of non-Lambertian Shape-from-Shading models with attenuation factors.

Findings

Attenuation factor makes the differential problems well-posed.

Unique viscosity solutions exist for the modified brightness equations.

Detailed analysis of boundary conditions for Hamilton-Jacobi equations.

Abstract

In this paper we show that the introduction of an attenuation factor in the %image irradiance brightness equations relative to various perspective Shape from Shading models allows to make the corresponding differential problems well-posed. We propose a unified approach based on the theory of viscosity solution and we show that the brightness equations with the attenuation term admit a unique viscosity solution. We also discuss in detail the possible boundary conditions that we can use for the Hamilton-Jacobi equations associated to these models.