Fast and accurate surface normal integration on non-rectangular domains

Abstract

The integration of surface normals for the purpose of computing the shape of a surface in 3D space is a classic problem in computer vision. However, even nowadays it is still a challenging task to devise a method that is flexible enough to work on non-trivial computational domains with high accuracy, robustness, and computational efficiency. By uniting a classic approach for surface normal integration with modern computational techniques, we construct a solver that fulfils these requirements. Building upon the Poisson integration model, we use an iterative Krylov subspace solver as a core step in tackling the task. While such a method can be very efficient, it may only show its full potential when combined with suitable numerical preconditioning and problem-specific initialisation. We perform a thorough numerical study in order to identify an appropriate preconditioner for this purpose. To provide suitable initialisation, we compute this initial state using a recently developed fast marching integrator. Detailed numerical experiments illustrate the benefits of this novel combination. In addition, we show on real-world photometric stereo datasets that the developed numerical framework is flexible enough to tackle modern computer vision applications.

Optimisation of Classic Photometric Stereo by Non-convex Variational Minimisation

“…where ∇z is the gradient of z. Then, the best smooth surface explaining the computed normals can be estimated in several ways [1], for instance by solving the variational problem:…”

“…The latter approach makes it possible to compute the 3D shape directly whereas in most methods following the classic PS setting a field of surface normals is computed which needs to be integrated in another step; see e.g. [1] for a recent discussion of integration techniques.…”

“…In practice, λ can be set to any small value, so that a solution of (9) lies as close as possible to a minimiser of (8). In all our experiments we set λ := 10 −6 and z 0 as the classic PS solution followed by least-squares integration [1].…”

Variational Methods for Normal Integration

“…This condition is a linear PDE in z which can be discretized using finite differences. Yet, providing a consistent discretization on the boundary of Ω is not straightforward [26], especially when dealing with nonrectangular domains Ω where many cases have to be considered [6]. Hence, we follow a different track, based on the discretization of the functional itself.…”

“…Hence, rather than considering that we are given |Ω| observations p, our discretization handles these data as |Ω + u |+|Ω − u | observations, some of them being interpreted in terms of forward differences, some in terms of backward differences, some in terms of both forward and backward differences, the points without any neighbor in the u-direction being excluded. 6 In 3D-reconstruction applications such as photometric stereo [55], the assumption on the noise should rather be formulated on the images. This will be discussed in more details in Subsection 4.4.…”