Fast and Robust Surface Normal Integration by a Discrete Eikonal Equation

Abstract

The integration of surface normals is a classic and fundamental task in computer vision. In this paper we deal with a highly efficient fast marching (FM) method to perform the integration. In doing this we build upon a previous work of Ho and his coauthors. Their FM scheme is based on an analytic model that incorporates the eikonal equation. Our method is also built upon this equation, but it makes use of a complete discrete formulation for constructing the FM integrator (DEFM). We not only provide a theoretical justification of the proposed method, but also illustrate at hand of a simple example that our approach is much better suited to the task. Several more sophisticated tests confirm the robustness and higher accuracy of the DEFM model. Moreover, we present an extension of DEFM that allows to integrate surface normals over non-trivial domains, e.g. featuring holes. Numerical results confirm desirable qualities of this method.

Normal Integration: A Survey

“…In [27], Ho, Lim, Yang and Kriegman derive from (20) the eikonal equation ∇z 2 = p 2 + q 2 , and aspire to use the fast marching method for its resolution. Unfortunately, this method requires that the unknown z has a unique global minimum over Ω.…”

“…From now on, we do not care more about the projection model. We just have to solve the generic equation (20).…”

Fast and accurate surface normal integration on non-rectangular domains

“…Complexity BC Non-rect. FFT [13] n log n periodic no DCT [43] n log n free no FM [14] n log n free yes Sylvester [18] free no CG-Poisson [20] n 3 7 free yes…”

“…where λ > 0 and f : R → R are user-defined. Let us note that the FM integrator requires parameter λ to be tuned, yet is not a crucial choice as any large number λ 0 will work [14]. 2 Using PDE (3) we do not compute the depth function v directly, but instead we solve in a first step for w. Let us note that this intermediate step is necessary for the successful application of the FM method, in order to avoid local minima and ensure that any initial point can be considered [21].…”

“…At this point it should be noted that the stencils in (12), (13) and the subsequent equation (14) are only valid for inner points of the computational domain. Indeed, when pixel (i, j) is located near the border of Ω, some of the four neighbour values (14) refer to the depth outside Ω.…”