Equiareal Shape-from-Template

This paper studies the 3D reconstruction of a deformable surface from a single image and a reference surface, known as the template. This problem is known as Shape-from-Template and has been recently shown to be well-posed for isometric deformations, for which the surface bends without altering geodesics. This paper studies the case of equiareal deformations. They are elastic deformations where the local area is preserved and thus include isometry as a special case. Elastic deformations have been studied before in Shape-from-Template, yet no theoretical results were given on the existence or uniqueness of solutions. The equiareal model is much more widely applicable than isometry. This paper brings Monge's theory, widely used for studying the solutions of nonlinear first-order PDEs, to the field of 3D reconstruction. It uses this theory to establish a theoretical framework for equiareal Shape-from-Template and answers the important question of whether it is possible to reconstruct a surface exactly with a much weaker prior than isometry. We prove that equiareal Shape-from-Template has a maximum of two local solutions sufficiently near an initial curve that lies on the surface. In addition, we propose an analytical reconstruction algorithm that can recover the multiple solutions. Our algorithm uses standard numerical tools for ODEs. We use the perspective camera model and give reconstruction results with both synthetic and real examples.