As the quality of digitized range data available from active and passive sensors has improved with time, there has been a tremendous increase in computer vision research using this data. In this very extensive paper, the authors address the range-image recognition problem with the aim of developing a general purpose approach that handles arbitrary surface shapes and arbitrary viewing directions. The paper deals with the determination of invariant surface characteristics for 3D object recognition in range images. It is proposed that individual, isolated surface regions be matched with the surface function families of the individual objects in such a way that viewpoint-dependent effects are accounted for.

The key to the approach is the selection of view-independent surface characteristics that are robust enough to describe both polyhedra and curved objects. Two measurements, the “mean curvature” and the “Gaussian curvature” (which are referred to collectively as “surface curvature”), are selected as the surface characteristics that possess desirable invariance properties.

The mathematical theory of the differential geometry of surfaces is first reviewed, deriving equations for two “fundamental forms.” The first form is an intrinsic property that depends only on the surface itself, while the second form is dependent on the embedding of the surface in 3D space and is an extrinsic property of the surface. Two different surface curvature functions are obtained by mapping the two fundamental form matric functions into one scalar function. The surface curvature functions, the mean curvature H, and the Gaussian curvature K, are chosen as the natural algebraic invariants of the shape operator. The signs of the H and K surface curvatures are used to classify range image regions into one of eight basic viewpoint-independent surface types. For example, positive curvature of both H and K indicates a peak, zero curvature of both H and K indicates a flat, and H with positive curvature and K with zero curvature indicates a valley.

The authors then review the literature on surface characterization, and devote some time to a comparison of their technique with the somewhat related topographic primal sketch proposed by Haralick et al. [1], who label each pixel of an intensity image surface with one of ten possible topographic labels--e.g., peak, pit, flat, etc., based on gradients, Hessians, and first and second directional derivatives. The authors feel that the advantage of their method is that it treats edge-type and surface-type labeling as two distinctly different issues, while the topographic primal sketch mixes these surface labels together.

The next portion of their paper describes the range image computations of the first and second partial derivatives of the depth map. A continuous differentiable function that best fits the data is first determined, and then the derivatives of the continuous function are computed analytically. H, K, and other surface characteristics are then computed using these partial derivatives.

Experimental results for real and synthetic range images are presented to show the properties, usefulness, and importance of differential-geometric surface characteristics. The various surface parameters--such as zeros of the mean and Gaussian curvature, and magnitude of mean curvature--are displayed in image form and discussed.

The paper contains a wealth of detailed tutorial material on the differential geometry of surfaces and methods for computing the characteristics of these surfaces. While the experimental results seem to indicate the efficacy of their surface characterization approach, it remains to be seen whether these characteristics can be used for matching purposes.