16-08-2012, 02:22 PM
A Bayesian framework for 3D surface estimation
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Abstract
We develop an evidence-combining framework for extracting locally consistent di!erential structure from curved
surfaces. Existing approaches are restricted by their sequential multi-stage philosophy, since important information
concerning the salient features of surfaces may be discarded as necessarily condensed information is passed from stage to
stage. Furthermore, since data representations are invariably unaccompanied by any index of evidential signi"cance, the
scope for subsequently re"ning them is limited. One way of attaching evidential support is to propagate covariances
through the processing chain. However, severe problems arise in the presence of data non-linearities, such as outliers or
discontinuities. If linear processing techniques are employed covariances may be readily computed, but will be unreliable.
On the other hand, if more powerful non-linear processing techniques are applied, there are severe technical problems in
computing the covariances themselves. We sidestep this dilemma by decoupling the identi"cation of non-linearities in the
data from the "tting process itself.
Introduction
Since the advent of sensor technology capable of volumetric
imaging, three-dimensional scene analysis has
become a critical area of investigation in computer vision.
The main types of data under study are range
images, which consist of an array of sensed depth values,
and density slice data generated by techniques such as
magnetic resonance imaging, or X-Ray computed tomography.
Conventionally, researchers have adopted
a four-stage strategy to scene interpretation [1}6]. Firstly,
surface points are estimated.
Global MAP estimation
The basis of our approach is to formulate the recovery
of consistent di!erential structure as a global MAP estimate.
In this section we introduce our MAP estimation
scheme and show how it leads to separate processes for
surface parameter estimation, and chart re"nement. Further,
we show that optimisation can be realised by the
iterative re-assignment of single charts. This has the desirable
e!ect of permitting a manifestly global process to
be realised in terms of local computations.
Our starting assumption is that we are given the estimated
locations and associated covariances of putative
surface points, as returned by some 3D feature detection
process. Let the estimated locations and covariances,
Surface parameter estimation
In this section we commence development of our
weighted, least-squares technique for estimating best-"t
parameter values for a patch model of a local surface. By
using separate probability density functions to characterise
the measurement process of points which obey and
violate smoothness assumptions, we obtain a robust
technique. Further, by basing estimates of violation on
previous estimates of best-"t parameters and di!erential
structure, we arrive at a recursive "t process which may
be coupled to the surface re"nement stage.
Surface re5nement
Recently, Sander and Zucker have introduced an iterative
minimisation procedure for local surface re"nement
[6]. The basis for their algorithm is to optimise the local
consistency of di!erential structure. The basic idea is
that, since the surface at each point is represented by a
parabolic quadric patch, it is possible to extrapolate
outward from a point j, say, to its neighbour i, in order to
get an idea of what the di!erential surface at i looks like
according to j. If the di!erential structure extrapolated to
i is close to the actual di!erential structure there, then
chart i is said to be consistent with the chart at j (note
that i consistent with j, does not necessarily imply j consistent
with i). Now, in imposing consistency, Sander and
Zucker perform the extrapolation procedure for a set of
neighbours to each point and update each chart via a
least-means-squares estimate.
Discussion
In this paper we have formulated a technique for
extracting locally consistent di!erential structure for surfaces,
as a step towards our goal in establishing a framework
for 3D scene interpretation. Our approach incorporated
standard models of local (smooth) surfaces at the
levels of surface "t and surface re"nement, namely the
parabolic quadric patch and consistency of di!erential
structure, respectively. Its novelty stemmed from (1) its
formulation as a global MAP estimate within a Bayesian
framework, and (2) the decoupling of the irregularity
estimation process from the "ts themselves. This led to
iterative, robust techniques wherein evidential support
for data representations was readily computed and readjustable.
As processing proceeded, more reliable
evidential support led to better surface estimates, and
vice versa.