1 research outputs found
Robust and large-scale quasiconvex programming in structure-from-motion
Structure-from-Motion (SfM) is a cornerstone of computer vision. Briefly speaking,
SfM is the task of simultaneously estimating the poses of the cameras behind a set of images of a
scene, and the 3D coordinates of the points in the scene.
Often, the optimisation problems that underpin SfM do not have closed-form solutions, and finding
solutions via numerical schemes is necessary. An objective function, which measures the discrepancy
of a geometric object (e.g., camera poses, rotations, 3D coordi- nates) with a set of image
measurements, is to be minimised. Each image measurement gives rise to an error function. For
example, the reprojection error, which measures the distance between an observed image point and
the projection of a 3D point onto the image, is a commonly used error function.
An influential optimisation paradigm in SfM is the βββ paradigm, where the objective function takes
the form of the maximum of all individual error functions (e.g. individual reprojection errors of
scene points). The benefit of the βββ paradigm is that the objective function of many SfM
optimisation problems become quasiconvex, hence there is a unique minimum in the objective
function. The task of formulating and minimising quasiconvex objective functions is called
quasiconvex programming.
Although tremendous progress in SfM techniques under the βββ paradigm has been made, there are still
unsatisfactorily solved problems, specifically, problems associated with large-scale input data and
outliers in the data. This thesis describes novel techniques to
tackle these problems.
A major weakness of the βββ paradigm is its susceptibility to outliers. This thesis improves the
robustness of βββ solutions against outliers by employing the least median of squares (LMS)
criterion, which amounts to minimising the median error. In the context of triangulation, this
thesis proposes a locally convergent robust algorithm underpinned by a novel quasiconvex plane
sweep technique. Imposing the LMS criterion achieves significant outlier tolerance, and, at the
same time, some properties of quasiconvexity greatly simplify the process of solving the LMS
problem.
Approximation is a commonly used technique to tackle large-scale input data. This thesis introduces
the coreset technique to quasiconvex programming problems. The coreset technique aims find a
representative subset of the input data, such that solving the same problem on the subset yields a
solution that is within known bound of the optimal solution on the complete input set. In
particular, this thesis develops a coreset approximate algorithm to handle large-scale
triangulation tasks.
Another technique to handle large-scale input data is to break the optimisation into multiple
smaller sub-problems. Such a decomposition usually speeds up the overall optimisation process,
and alleviates the limitation on memory. This thesis develops a large-scale optimisation algorithm
for the known rotation problem (KRot). The proposed method decomposes the original quasiconvex
programming problem with potentially hundreds of thousands of parameters into multiple sub-problems
with only three parameters each. An efficient solver based on a novel minimum enclosing ball
technique is proposed to solve the sub-problems.Thesis (Ph.D.) (Research by Publication) -- University of Adelaide, School of Computer Science, 201