Computational Intelligence - August 2015 - 11

Abstract-Missing data is a frequently encountered problem for structure-from-motion (SFM)
where the 3D structure of an object is estimated based on
2D images. In this paper, an effective approach is proposed to deal
with the missing-data estimation problem for small-size image
sequences. In the proposed method, a set of sub-sequences is first
extracted. Each sub-sequence is composed of the frame to be estimated and
a part of the original sequence. In order to obtain diversified estimations, multiple weaker estimators are constructed by means of the column-space-fitting
(CSF) algorithm. The various sub-sequences are in turn used as the inputs to the
algorithm. As the non-missing entries are known, the estimation errors of these
entries are computed so as to select weaker estimators with better estimation performances. Furthermore, a linear programming based weighting model is established to compute the weights for the selected weaker estimators. After the
weighting coefficients are obtained, a linear weighting estimation which is used
as the final estimation of the missing entries is computed based on the outputs of the weaker estimators. By applying the strategies of weaker-estimator selection and the linear programming weighting model, the proposed missing entry estimation method is more accurate and robust
than the existing algorithms. Experimental results on several
widely used image sequences demonstrate the effectiveness and feasibility of the proposed algorithm.

In [8], the missing data was handled by the Grassmannian rank-one update subspace estimation under a rigid SFM model. A subspace method was proposed in
[9] for projective reconstruction from multiple images with missing data. In [10],
the low-rank approximation was formulated as a minimization problem on subspaces, and two implementations of the Levenberg-Marquardt algorithm were carried out to solve the optimization problem. In [11], [12], a robust weighted lowrank matrix-approximation method was proposed for missing-data estimation. By
incorporating a damping factor into the Wiberg method, a low-rank matrixapproximation method was proposed in [13] to estimate the missing entries.
Instead of rigid deformation, a column-space-fitting (CSF) method was
recently proposed in [14] to estimate the missing entries based on a non-rigid
SFM (NRSFM) model. Simulations on multiple sequences have demonstrated that the CSF algorithm can achieve a very good estimation performance for the missing data of deformed objects. In real situations, the data
available for SFM may come from a few images only, i.e. a small-size image
sequence. Under such a condition, the performance of existing methods may
not be satisfactory when they are applied directly to estimate the missing data.
In this paper, a sub-sequence-based integrated CSF approach is proposed
to deal with the missing-data estimation problem for small-size image
sequences. In the proposed method, sub-sequences are first extracted from an
original sequence to form the training data. In order to obtain accurate and
diversified estimations, multiple weaker estimators are constructed by applying
different sub-sequences to the CSF algorithm. Furthermore, the strategies of
using weaker-estimator selection and a linear weighting model are devised to
integrate the outputs of the weaker estimators. Experimental results on several
widely used image sequences demonstrate the effectiveness and feasibility of
the proposed algorithm.

august 2015 | IEEE ComputatIonal IntEllIgEnCE magazInE

Table of Contents for the Digital Edition of Computational Intelligence - August 2015