From Objective Functions in Computer Vision to Robust Inlier Structures

What I would like to accomplish in these meeting... 1. Representing nonlinear functions in a higher dimensional linear space. (Generalized) total least squares. 2. Nonlinear least squares estimation. The Levenberg-Marquardt method. 3. Mean shift in the Euclidean domain for segmentation and for tracking. 4. Applying nonlinear mean shift to different types of Riemannian manifold. 5. Mean shift clustering in the kernel space. Only very few cannot-link and must-link pairs are known. 6. Robust estimator for multiple inlier structures. Beside the objective function nothing else is provided. ...and will see how much I can really do. 2 Linear vs. nonlinear objective functions in computer vision. The case Ψ ∈ R k×q (q > 1) is beyond the goal of this course. The inlier points at the input, in general, satisfy a nonlinear objective function. The Ψ can be separated into a matrix of the measurements and a new parameter vector, both being derived from the input Ψ (y i , β) = Φ(y i)θ(β) Φ(y) ∈ R k×m θ(β) ∈ R m. This is a higher dimensional linear relation if we write it as a function of carrier vectors, x The intercept α is not pulled out separately here. 3 A carrier contains both elements of the input measurement and pairwise products of these elements. Each of the ζ carriers are different and all these estimates must satisfy the samê θ. The ζ relations concatenated by rows is quasi-equal to 0 ζ and k = ζ. As will be seen later, a difference exist between k and ζ when you use it for robust estimation of the objective function. If ζ is larger that one, will take only a maximum between all ζ carriers and will do the processing in the null space with dimension k = 1.