30 research outputs found
Local Uniqueness of the Circular Integral Invariant
This article is concerned with the representation of curves by means of
integral invariants. In contrast to the classical differential invariants they
have the advantage of being less sensitive with respect to noise. The integral
invariant most common in use is the circular integral invariant. A major
drawback of this curve descriptor, however, is the absence of any uniqueness
result for this representation. This article serves as a contribution towards
closing this gap by showing that the circular integral invariant is injective
in a neighbourhood of the circle. In addition, we provide a stability estimate
valid on this neighbourhood. The proof is an application of Riesz-Schauder
theory and the implicit function theorem in a Banach space setting
Optical Flow on Moving Manifolds
Optical flow is a powerful tool for the study and analysis of motion in a
sequence of images. In this article we study a Horn-Schunck type
spatio-temporal regularization functional for image sequences that have a
non-Euclidean, time varying image domain. To that end we construct a Riemannian
metric that describes the deformation and structure of this evolving surface.
The resulting functional can be seen as natural geometric generalization of
previous work by Weickert and Schn\"orr (2001) and Lef\`evre and Baillet (2008)
for static image domains. In this work we show the existence and wellposedness
of the corresponding optical flow problem and derive necessary and sufficient
optimality conditions. We demonstrate the functionality of our approach in a
series of experiments using both synthetic and real data.Comment: 26 pages, 6 figure
Multi-parameter Tikhonov Regularisation in Topological Spaces
We study the behaviour of Tikhonov regularisation on topological spaces with
multiple regularisation terms. The main result of the paper shows that
multi-parameter regularisation is well-posed in the sense that the results
depend continuously on the data and converge to a true solution of the equation
to be solved as the noise level decreases to zero. Moreover, we derive
convergence rates in terms of a generalised Bregman distance using the method
of variational inequalities. All the results in the paper, including the
convergence rates, consider not only noise in the data, but also errors in the
operator