Search

We describe and provide code and examples for a polygonal edge matching method., Comment: Contains Matlab code and 4 figures

We show that the recently introduced L1TV functional can be used to explicitly compute the flat norm for co-dimension one boundaries. While this observation alone is very useful, other important implications for image analysis and shape statistics include a method for denoising sets which are not boundaries or which have higher co-dimension and the fact that using the flat norm to compute distances not only gives a distance, but also an informative decomposition of the distance. This decomposition is made to depend on scale using the "flat norm with scale" which we define in direct analogy to the L1TV functional. We illustrate the results and implications with examples and figures.

Regularization functionals that lower level set boundary length when used with L^1 fidelity functionals on signal de-noising on images create artifacts. These are (i) rounding of corners, (ii) shrinking of radii, (iii) shrinking of cusps, and (iv) non-smoothing of staircasing. Regularity functionals based upon total curvature of level set boundaries do not create artifacts (i) and (ii). An adjusted fidelity term based on the flat norm on the current (a distributional graph) representing the density of curvature of level sets boundaries can minimize (iii) by weighting the position of a cusp. A regularity term to eliminate staircasing can be based upon the mass of the current representing the graph of an image function or its second derivatives. Densities on the Grassmann bundle of the Grassmann bundle of the ambient space of the graph can be used to identify patterns, textures, occlusion and lines., Comment: 9 pages, no figures. This informal discussion paper will be updated periodically

The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topology, with a union of discs and straight line segments. Generalization to higher numbers of boundary components shows that minimal surfaces union straight line segments can still be achieved, and that the configuration of straight line segments depends on the direction of approach of domain conformal classes to the boundary point of domain moduli space. Bubbling can occur in variable ways according to the metric representatives of the conformal classes of domains. A method is given whereby bubbling can be eliminated yielding a point-wise limit of maps with image limit containing a surface union straight line segments., Comment: 17 pages, 3 tables, 6 figures. Page breaks repositioned in version 2

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a point. The n-1 dimensional space of outward pointing vectors can be used for such a measure. It represents all C2 rectifiable sets of codimension at least one of Rn as rectifiable sets in RnXSn-1 with n-1 dimensional Hausdorff measure. When viewed as (n-1)-rectifiable varifolds or currents in RnXSn-1 they come equipped with compactness theorems. The projection of their limits to Rn recovers the rectifiable sets of mixed dimensions giving the limits for the desired topology on subsets of Rn. Both varifold and current compactness are required as there are sequences, such as honeycombs, that converge as varifolds but not as currents. Conversely sequences such as lifts of polyhedral approximations converge as currents but not as varifolds., Comment: 20 pages, 10 figures, unpublished preprint version

An ODE variational calculation shows that an image principle curvature ratio factor can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into Rn. In certain situations, including all radially symmetry harmonic maps, equality is achieved., Comment: v1: 8 pages, 1 figure. (Thm 2 proof wrong & counter example found to Cor 3) v2=v3: 6 pages 1 figure

Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and vector space, structure of Euclidean space as possible. This enables structures, such as foliations and calibrations, to generalize over to pseudo-manifolds. The main result is that the cone of any compact topological surfaces can be given a piecewise flat metric structure that admits a pseudo-foliation by minimal surfaces. In some situations orientation reversing holonomy is an obstruction and in others it is just the opposite, a sufficient condition. Also foliations by minimal surfaces can extend across connect sum operations., Comment: 18 pages, 15 figures