The Proposed Theory
Bias in Linear Estimation
The constraints underlying visual processes often can be formulated as equations linear in the unknowns. There is a number of measurements available from which some unknown parameters must be estimated. Thus, the computational problem amounts to finding a solution to an over-determined equation system of the form
A x = b,where A an n × k matrix, and b an n-dimensional vector denoting measurements, that is the observations, and x a k-dimensional vector denoting the unknowns.
The observations are noisy, that is, they are corrupted by errors. We can say that the observations are composed of the true values (A', b') plus the errors (δA, δb) , i.e. A = A' + δA and b = b' + δb. In addition the constraints are not completely true, they are only approximations; in other words there is system error, ε. The constraints for the true value, x', amount to
A' x' = b' + ε.
We are dealing with what is called the errors-in-variable model in statistics. We have to use an estimator, that is a procedure, to solve the equation system. The most common choice is by means of least squares (LS) estimation. However, it is well known, that LS estimation is biased.
Under some simplifying assumptions (identical and independent random variables δA and δb with zero mean and variance σ2 ) the LS estimate converges to
Large variance in δA , an ill-conditioned A', or an x' which is oriented close to the eigenvector of the smallest singular value of A' all could increase the bias and push the LS solution away from the real solution. Generally it leads to an underestimation of the parameters.
There are other, more elaborate estimators that could be used. None, however will perform better if the errors cannot be obtained with high accuracy.
Examples of visual computations which amount to linear equation systems are the estimation of image motion or optical flow, the estimation of the intersections of lines, and the estimation of shape from various cues, such as motion, stereo, texture, or patterns.