A Fuzzy Model for Coverage Evaluation of Cameras and
Multi-Camera Networks
Aaron Mavrinac
mavrin1@uwindsor.ca
Jose L. Alarcon Herrera
alarconj@uwindsor.ca
Xiang Chen
xchen@uwindsor.ca
Department of Electrical and Computer Engineering
University of Windsor
401 Sunset Ave., Windsor, Ontario, Canada N9B 3P4
ABSTRACT
A comprehensive, intuitive, task-oriented three-dimensional
coverage model for cameras and multi-camera networks us-
ing fuzzy sets is presented. The model captures the vague-
ness inherent in the concept of visual coverage. At present,
the model can be used to evaluate, given a scene model and
an objective, the coverage performance of a given camera or
multi-camera network configuration, as a single numerical
metric. Plans to use the model for optimal camera place-
ment and other problems involving coverage are discussed.
Examples of qualitative experimental validation of the cov-
erage model are presented.
1. INTRODUCTION
In computer vision, the property of coverage of a given
point in space by a camera or multi-camera network is vague.
A point in space is normally considered covered, for a specific
task, if useful information can be obtained for that task from
the point, assuming such information exists in the scene. In
most applications, visibility is not enough for a point to be
covered; we require a certain level of “quality” in the imag-
ing, which is generally imprecise. High-level computer vision
applications are complex. Even when performance metrics
for the final results are well-defined, it is usually impractical
or impossible to trace these analytically, through the myriad
factors imparted by various low-level algorithms and scene
conditions, back to a set of requirements and/or optima for
the parameters of the imaging system. As a result, these
parameters are often designed by educated guessing, some-
times partially informed by application performance metrics,
often involving much trial and error. There is no analytic
indicator function for coverage.
A fuzzy representation, we contend, lends itself well to
modeling coverage. The membership degree of a scene point
in a fuzzy set representing a camera’s volume of coverage
can concisely encapsulate the various underlying factors into
a single metric, yielding a useful model for evaluating the
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coverage of such points. A number of these sets can then be
transformed and combined in appropriate ways to yield an
equivalent coverage model for a multi-camera network of a
given type.
We present the fuzzy coverage model, a comprehensive
three-dimensional coverage model for cameras and multi-
camera networks, based on fuzzy sets, which is derived from
well-studied parameters of the standard imaging model (ob-
tained from internal and external calibration), and can be
tuned for specific tasks via a small number of intuitive ap-
plication parameters. The single-camera model can be seen
as an adaptation of work on task-oriented, constraint-based
modeling of camera coverage by Cowan and Kovesi [5], Sta-
mos and Allen [17], Reed and Allen [16] and others to a more
powerful and flexible representation with fuzzy sets, which
itself is reminiscent of the attenuated disk coverage models
found in the sensor network literature [19]. The extension
to the multi-camera context has parallels to work by Mittal
and Davis [14].
By providing a model of the desired coverage volume, a re-
alistic and quantitative performance metric can be obtained.
Besides evaluating the coverage of existing or proposed cam-
era network deployments, the obvious application is as an
objective model for optimal camera placement (sensor plan-
ning for multi-camera networks). Previous models used for
this purpose by Erdem and Sclaroff [6], Ho¨rster and Lien-
hart [9], Angella et al. [1], Zhao et al. [22], and others have
used bivalent models (i.e., points are either covered or not
covered) and do not account for the direction of observa-
tion. Our ambition is to develop optimization tools capable
of taking advantage of the richer information in the fuzzy
coverage model to achieve various goals. Since the model is
rather complex and has not been explicitly designed for use
in any specific optimization method, an analytic approach
will involve reducing the model to various simpler forms fo-
cusing on the variables of interest. The scalar performance
metric we propose would already allow for a generate-and-
test approach similar to Yi et al. [20].
The primary contributions of this work are the concept
of a coverage model based on fuzzy subsets of directional
Euclidean space, a full parameterization of this model using
well-studied models from computer vision and intuitive ap-
plication parameters, and the use of the model to obtain a
scalar performance metric of coverage in multi-camera net-
works. It is intended as a stepping stone toward the solution
of a number of specific multi-camera network problems, as
well as a better theoretical understanding of camera net-
works.