Computing the number of votes to be incremented to a given bin of the accumulator array for blades interpreted as straight lines in the 2-dimensional
homogeneous model of geometry. The red spots represent the uncertainty of a given straight line. The distribution is shown as points computed by sampling
the input line according to its Gaussian distributed uncertainty and mapping each sample to P2 (left) and to A2 (center). The envelopes in (left) and (center)
were obtained analytically by our approach. The points Θiface at the center of the bin's face (left) are mapped to the open affine covering for the
Grassmannian G(2;3) as points aiface (center). In turn, the points aiface are mapped to the basis deffined by the orthonormal eigenvectors of the probability
distribution (right). An axis-aligned bounding box is computed for ai face in such a basis. The number of votes is proportional to the weight of the input
blade and the probabilities of an intended p-blade be inside of the box.
Handling Uncertain Data in Subspace Detection
Pattern Recognition.
Volume 47, Number 10, October 2014, pp. 3225-3241. [DOI]
Contents
Experimental data is subject to uncertainty as every measurement apparatus is
inaccurate at some level. However, the design of most computer vision and pattern
recognition techniques (e.g., Hough transform) overlook this fact and treat intensities,
locations and directions as precise values. In order to take imprecisions into
account, entries are often resampled to create input datasets where the uncertainty
of each original entry is characterized by as many exact elements as necessary. Clear
disadvantages of the sampling-based approach are the natural processing penalty
imposed by a larger dataset and the difficulty of estimating the minimum number
of required samples. We present an improved voting scheme for the General Framework
for Subspace Detection (hence to its particular case: the Hough transform)
that allows processing both exact and uncertain data. Our approach is based on
an analytical derivation of the propagation of Gaussian uncertainty from the input
data into the distribution of votes in an auxiliary parameter space. In this parameter
space, the uncertainty is also described by Gaussian distributions. In turn, the
votes are mapped to the actual parameter space as non-Gaussian distributions. Our
results show that resulting accumulators have smoother distributions of votes and
are in accordance with the ones obtained using the conventional sampling process,
thus safely replacing them with signifficant performance gains.
Paper
Full Paper (pre-print)
The final publication is available at ScienceDirect - Elsevier
Supplementary Material
Notation
Citation
Fernandes, Leandro A. F. and Manuel M. Oliveira.
"Handling Uncertain Data in Subspace Detection".
Pattern Recognition. Volume 47, Number 10, October 2014, pp. 3225-3241.
BibTeX
@article {FernadesOliveira2014HUDSD,
author = {Leandro A. F. Fernandes and Manuel M. Oliveira},
title = {Handling Uncertain Data in Subspace Detection},
journal = {Pattern Recognition},
year = {2014},
volume = {47}
number = {10},
pages = {3225--3241},
doi = {http://dx.doi.org/10.1016/j.patcog.2014.04.013},
issn = {0031-3203}
}
Keywords
Hough transform, Uncertain data, Subspace detection, Shape detection, Grassmannian, Geometric algebra, Parameter space.
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CNPq-Brazil fellowships and grants #142627/2007-0 and 308936/2010-8, FAPERGS PQG 10/1322-0, and FAPERJ E-26/112.468/2012. |