Heteroscedastic Errors-in-Variables Models in Computer Vision
Ph.D. Thesis Bogdan Matei
We have witnessed in the last ten--fifteen years a significant
improvement of 3D
computer vision applications such as scene reconstruction, tracking,
mosaicing. The deeper understanding of the geometry underlining
these tasks is one of the main reasons behind this progress.
The geometry of the scene imposes various constraints on the
projection of 3D objects in the image plane. For example,
in uncalibrated stereo the features
must lie on corresponding epipolar lines.
These constraints are rigorously satisfied only in the
absence of measurement errors.
However, there are multiple sources of errors in
the real data, ranging from quantization noise to violations
of the embedded assumptions at earlier processing stages.
A large number of
geometric constraints encountered in computer vision are linear in the
parameter of interest and depend on the measurements through
relative simple nonlinear functions. For such models, an incorrect
way of handling the measurement noise during the estimation process
may yield parameter estimates with poor accuracy.
We argue that the proper analysis of the geometric constraints
when all the measurements are affected by noise is by employing the
errors-in-variables (EIV) statistical model. We perform the analysis
of the EIV model under the most general assumption of anisotropic and
inhomogeneous, i.e. heteroscedastic, noise.
The main contribution of the thesis is
a novel estimation technique for the EIV model with
heteroscedastic errors, the HEIV algorithm. The HEIV algorithm was
successfully applied to a variety of computer vision applications:
3D rigid motion estimation, conic fitting, fundamental matrix and
trifocal tensor estimation.
Most often, we are interested not only in finding the
parameter estimates, but also in assessing how
accurate these estimates are, given a particular set of
measurements. We address the issue of performance assessment in two
by deriving analytical expressions for the covariance and
bias of the HEIV parameters, or by doing bootstrap simulation.
In the last part of the thesis we present a method for
measuring the uncertainty in correlation based feature matching
between images. The shape of the correlation surface
models the uncertainty associated with a match and is encoded in
covariance matrices. These covariance matrices are then employed in the
outlier rejection and
ensuing parameter estimation using either HEIV, or other optimization
The thesis contains 215 pages. The compressed version has 10M.
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