Multi-Scale Surface Descriptors
Gregory Cipriano, Student Member, IEEE, George N. Phillips Jr., and Michael Gleicher, Member, IEEE
Abstract—Local shape descriptors compactly characterize regions of a surface, and have been applied to tasks in visualization,
shape matching, and analysis. Classically, curvature has be used as a shape descriptor; however, this differential property character-
izes only an infinitesimal neighborhood. In this paper, we provide shape descriptors for surface meshes designed to be multi-scale,
that is, capable of characterizing regions of varying size. These descriptors capture statistically the shape of a neighborhood around a
central point by fitting a quadratic surface. They therefore mimic differential curvature, are efficient to compute, and encode anisotropy.
We show how simple variants of mesh operations can be used to compute the descriptors without resorting to expensive parameteri-
zations, and additionally provide a statistical approximation for reduced computational cost. We show how these descriptors apply to
a number of uses in visualization, analysis, and matching of surfaces, particularly to tasks in protein surface analysis.
Index Terms—Curvature, descriptors, npr, stylized rendering, shape matching.
1 INTRODUCTION
Local shape descriptors distill the shape of a region of a surface into
a short vector of numbers, each corresponding to a property of the
region. These descriptors have broad application when working with
shapes: for example, they are used in visualizing and analyzing sci-
entific data, shape matching, and in stylized rendering. While various
local shape descriptors and methods for computing them exist, their
inability to summarize the shape of larger regions limits their utility.
Our goal is a local surface shape descriptor that is applicable at
different scales to summarize the shapes of differently sized neigh-
borhoods. This allows it to be applied to smaller regions to capture
small-scale detail, or to larger neighborhoods to summarize their over-
all shape. Regions of the surface may have one shape at a small scale,
but a different shape at a larger scale (e.g. a small bump within a large
bowl). This paper introduces an approach called multi-scale surface
descriptors that meets this goal. We present a local shape descriptor
that can be applied at multiple scales, along with techniques for com-
puting them efficiently on a triangle mesh.
The shape of any finite region may contain arbitrary amounts of
detail, therefore a shape descriptor can only provide a summary. For an
infinitesimal region, the amount of detail is limited, so the shape can be
completely described by its curvature. Curvature provides a compact
local descriptor: three or four numbers are sufficient to characterize
the shape for an infinitesimally small region. For finite-sized regions,
however, the mathematics of curvature do not apply.
We provide a descriptor that captures the most significant features
of the shape of a local surface region. The descriptor considers a lo-
cal neighborhood around a central point with a roughly circular area
specified by radial distance. It measures the degree and type of non-
planarity of the region, for example encoding whether something is a
steep bump or a shallow bowl. It also captures the degree and direc-
tion of anisotropy, identifying troughs and ridges. A key insight of our
approach is that while these quantities are not sufficient to capture all
details of the shape of a finite region, they do capture the most sig-
nificant aspects of shape. We introduce robust and practical methods
for computing these larger-scale surface descriptors, and show their
usefulness in a number of applications.
One of our key motivating applications is the matching of molecular
surface regions to identify potentially similar chemical functionality.
An important aspect of this functionality is surface shape complemen-
• Authors are with the Department of Computer Sciences, University of
Wisconsin, Madison, E-mail: gregc@cs.wisc.edu,
phillips@biochem.wisc.edu, gleicher@cs.wisc.edu.
Manuscript received 31 March 2009; accepted 27 July 2009; posted online
11 October 2009; mailed on 5 October 2009.
For information on obtaining reprints of this article, please send
email to: tvcg@computer.org .
tarity: a binding partner for a protein will often have locally comple-
mentary shape to its region of binding. Much as a key fits only its
matching lock, complementarity implies that binding is highly stere-
ospecific. Therefore, by characterizing the shape of a known binding
pocket, and then using this information to identify similar regions in
other proteins, we may find new targets for a given partner.
This application, which we discuss in more detail in §5.2, high-
lights many of the requirements for practical, effective local shape de-
scriptors and the methods to compute them: they must operate over
large enough neighborhoods to be chemically significant; they must
be efficient, as we need to compute the descriptors for all points on
each molecule in a database; they must be robust against poorly tes-
selated surfaces; and they must correspond to domain scientists’ intu-
ition about shape and neighborhood. We provide the first shape de-
scriptors that we feel are able to meet these needs.
1.1 Contribution
The contribution of this paper is an approach to local shape descriptors
that provides a method for characterizing neighborhoods at multiple
scales on the surface of a mesh. Our approach is the first to address all
of the following goals for such descriptors:
• It scales to describe larger neighborhoods. Curvature captures
only infinitesimal regions and prior approaches to curvature com-
putation focus on minimizing the size of regions to better ap-
proximate the differential case. In §3 we present methods that
summarize a non-trivial region statistically.
• It corresponds to intuitions about curvature. Like curvature, our
descriptor captures the degree and type of non-planarity of a re-
gion, and the degree and direction of anisotropy.
• It allows for control of scale in a simple way. Neighborhoods are
specified by their center and a radius, so regions are controlled by
physically relevant quantities of the surface. In contrast, methods
such as mesh pre-filtering require a less direct specification of
neighborhood size in terms of frequency (which can be difficult
to explain to domain scientists), and filtering causes points to
move, precluding localized assessment.
• It compensates for issues in tessellation. The methods of §3.1
account for discretization, allowing our descriptor to be robust to
poorly tesselated surfaces.
• It affords efficient computation. Throughout §3 we present meth-
ods for computing the descriptors efficiently. In particular, our
approach avoids expensive parameterizations and our approxi-
mations can provide good performance without resorting to ex-
pensive mesh operations such as exact geodesic computations.
• It works in applications. In §5 we describe how our descriptor is
effective in applications. By providing a larger-scale summariza-
tion of surface regions, it allows for simpler algorithms to make
effective use of descriptors across a number of applications.
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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 15, NO. 6, NOVEMBER/DECEMBER 2009