Fua -1 
Parametric Models are Versatile: 
The Case of Model Based Optimization 
P. Fua 
Artificial Intelligence Center 
SRI International 
333 Ravenswood Avenue 
Menlo Park, California 94025 
Abstract 
Model-Based Optimization (MBO) is a paradigm in which an objective function is used to express 
both geometric and photometric constraints on features of interest. A parametric model of a feature 
(such as a road, a building, or coastline) is extracted from one or more images by adjusting the model’s 
state variables until a minimum value of the objective function is obtained. The optimization procedure 
yields a description that simultaneously satisfies (or nearly satisfies) all constraints, and, as a result, is 
likely to be a good model of the feature. 
1 Introduction 
Model-Based Optimization (MBO) is a paradigm in which an objective function is used to express both 
geometric and photometric constraints on features of interest. A parametric model of a feature (such as a 
road, a building, or coastline) is extracted from one or more images by adjusting the model’s state variables 
until a minimum value of the objective function is obtained. The optimization procedure yields a description 
that simultaneously satisfies (or nearly satisfies) all constraints, and, as a result, is likely to be a good model 
of the feature. 
The deformable models we use here are extensions of traditional snakes [Terzopoulos et al., 1987, Kass et 
al ., 1988, Fua and Leclerc, 1990]. They are polygonal curves or facetized surfaces to which is associated an 
objective function that combines an “image term” that measures the fit to the image data and a regularization 
term that enforces geometric constraints. 
In this paper we demonstrate cartographic applications of this paradigm and show that a large variety of 
objects can be thus modeled. More specifically we use MBO to effectively delineate 2D and 3D features— 
such as roads, rivers and buildings—and to recover the shape of the surrounding terrain. The algorithms 
described below are implemented within the Radius Common Development Environment (R.CDE) [Mundy 
et al., 1992]. 
2 2-D and 3-D Delineation 
We model linear features as polygonal curves that may either be described as sequential list of vertices or, for 
more complex objects such as a road network or a 3-D extruded object, exhibit the topology of a network. In 
the latter case, to describe them completely, one must supply not only the list of their vertices but also a list 
of “edges” that defines the connectivity of those vertices. In addition, with some of these complex objects, 
one can also define “faces,” that is circular lists of vertices that must be constrained to remain planar.