Noise covariance estimation in low-level computer vision

In image processing literature, thus far researchers have assumed the perturbation in the data to be white (or uncorrelated) having a covariance matrix σ2I, i.e., assumption of equal variance for all the data samples and that no correlation exists between the data samples. However, there has been very little attempt to estimate noise characteristics under the assumption that there is correlation between data samples. We develop a new and novel approach for the estimation of the unknown colored noise covariance matrix. We use the facet model to describe the noise free image, because of its simple, yet elegant mathematical formulation.; Original contributions of this dissertation include: (1) Development of a new and novel approach for the simultaneous estimation of the unknown colored (or correlated) noise covariance matrix and the hyperparameters of the covariance model using the facet model . We also estimate, simultaneously, coefficients of the facet model. (2) Introduction of the Generalized Inverted Wishart ($GJW$) distribution to the image processing and computer vision community. (3) Formulation of the problem solution in a Bayesian framework using an improper uniform prior distribution for the facet model coefficients (i.e. a flat prior) and a Generalized Inverted Wishart ($GJW$) prior distribution for the unknown noise covariance matrix that is to be estimated. (4) Placing a structure on the hypercovariance matrix of $GJW$ distribution, such that its elements are a function of the coefficients of a correlation filter. These filter coefficients in addition to the number of degrees of freedom parameter of the $GJW$ distribution are called the hyperparameters. (5) Hyperparameters have constraints placed on them so that the resulting hypercovariance matrix remains positive definite. Therefore, we designed a new extension of the expectation maximization algorithm called the generalized constrained expectation maximization (GCEM) algorithm for the estimation of the hyperparameters using the sequential unconstrained minimization technique (SUMT) via barrier functions to incorporate the constraints. (6) Development of a new ridge operator called the integrated second directional derivative ridge operator (ISDDRO) based on the facet model. Our main focus here is the optimal estimation of ridge orientation. The orientation bias and orientation standard deviation are the measures of performance. The latter measures the noise sensitivity. (7) Comparison of ISDDRO using the noise covariance matrix estimation (ISDDRO-CN) with the same ridge operator under white noise assumption (ISDDRO-WN) and also with the most competing ridge operator multilocal level set extrinsic curvature (MLSEC) [5]. ISDDRO-CN has superior noise sensitivity characteristics compared to both ISDDRO-WN and MLSEC.