A method for enforcing integrability in shape from shading algorithms

Abstract

Several recently developed techniques for reconstructing surface shape from shading information estimate surface slopes without ensuring that they are integrable. This paper presents a n approach for enforcing integrability, a particular implementation of the approach, a n example of its application to extending a n existing shapefrom-shading algorithm, and experimental results showing the improvement that results from enforcing integrability. A possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions, and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes. This projection maps closed convex sets into closed convex sets and, hence, is attractive as a constraint in iterative algorithms. The same technique is also useful for noniterative algorithms since it provides a least-squares fit of integrable slopes to nonintegrable slopes in one pass of the algorithm. The special case of Fourier basis functions is also formulated. This provides an intuitive frequency domain interpretation of shape from shading, a computationally efficient implementation using fast Fourier transforms, and a convenient method for introducing low-resolution information into the shape-from-shading solution. Reconstruction of surface height by integrating surface slope estimates is obtained as a byproduct of the integrability constraint. The integrability projection constraint was applied to extending a n iterative shape-from-shading algorithm of Brooks and Horn. Experimental results show that the extended algorithm converges faster and with less error than the original version. Good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces. Simulation examples show that the algorithm is robust with respect to large (but known) changes in illumination geometry, obtaining high-quality reconstructions even in the presence of significant shadowing. Other possible applications of this method to computer vision problems such as shape from texture and surface reconstruction from synthetic aperture radar imagery are discussed.

Surface-from-Gradients with Incomplete Data for Single View Modeling

“…In this section, we present cases to illustrate that our closed-form solution has advanced previous results, using a classical approach [5] and a new approach [9] for comparison on surface-from-gradients, where no heights are available.…”

“…The energy functional is non-linear and complex, so incomplete height field is needed for initialization. Other regularization approaches also minimize a Needle map Fourier [5] Shapelets [9] Closed-form Figure 1. Normals lie on the image plane.…”

“…One classical approach that makes use of basis functions was proposed by Frankot and Chellappa [5], where Fourier basis functions were used. More recently, shape from shapelets was proposed [9].…”

“…Normals lie on the image plane. For the methods proposed in [5,9], these normals have to be altered before running the algorithms. Our method produces a faithful 'staircase' surface without manually changing input normals.…”

Shifted base functions: An efficient and versatile new tool in optics

“…These methods can coarsely be divided into surface measurements, like e.g. mechanical scanning or interferometry and slope or gradient measurements, such as Hartmann-Shack [2] or Shape-fromShading [3]. Many of these methods provide data on a regular grid, but there are also methods, where the data are spaced non-uniformly.…”