Heteroscedastic Errors-in-Variables Models in Computer Vision

Ph.D. Thesis Bogdan Matei


Abstract

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 different ways: 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 techniques.

The thesis contains 215 pages. The compressed version has 10M.


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