Towards accurate recovery of shapes from shading
The human visual system can rapidly and accurately derive the three-dimensional shape of surfaces from the brightness variations alone in a grey scale image. Making progress in understanding visual perception in humans was the scientific challenge that framed the work of INSIGHT2+ project as it has potential impact on the performance evaluation of artificial vision systems. The three-dimensional shape of surfaces was rigorously computed by researchers at the University of Nice solely from the brightness of their black and white images. This was achieved by using variational methods and non-linear partial differential equations of brightness. The existence and uniqueness of reliable solutions for this 'Shape from shading' problem was systematically proven by modelling it with a pinhole camera and a single light source at its optical centre. The innovation of the approach proposed lays on the notion of singular viscosity solutions for the Hamilton-Jacobi equations, providing a mathematical framework for developing a numerical algorithm for their computation. Moreover, an extension of the notion of singular viscosity solutions provided for a new characterisation of discontinuous solutions and allowed the unification of various theoretical results of the 'Shape from shading' problem. The potential applicability of the 'Shape from shading' method in document restoration has been demonstrated and validated by removing perspective and photometric distortions generated by the classical photocopy of a bulky book. The stability of the approximation schemes and the convergence of the numerical solutions computed have been even proven on images containing discontinuities, non-smooth surfaces and black shadows. Further real-life applications include face reconstruction, modifications and improvement of images, as well as medical imaging.
