Disconnected Skeleton: Shape at its Absolute Scale

We present a new skeletal representation along with a matching framework to address the deformable shape recognition problem. The disconnectedness arises as a result of excessive regularization that we use to describe a shape at an attainably coarse scale. Our motivation is to rely on the stable properties of the shape instead of inaccurately measured secondary details. The new representation does not suffer from the common instability problems of traditional connected skeletons, and the matching process gives quite successful results on a diverse database of 2D shapes. An important difference of our approach from the conventional use of the skeleton is that we replace the local coordinate frame with a global Euclidean frame supported by additional mechanisms to handle articulations and local boundary deformations. As a result, we can produce descriptions that are sensitive to any combination of changes in scale, position, orientation and articulation, as well as invariant ones.Comment: The work excluding {\S}V and {\S}VI has first appeared in 2005 ICCV: Aslan, C., Tari, S.: An Axis-Based Representation for Recognition. In ICCV(2005) 1339- 1346.; Aslan, C., : Disconnected Skeletons for Shape Recognition. Masters thesis, Department of Computer Engineering, Middle East Technical University, May 200

Combinatorial and topological phase structure of non-perturbative n-dimensional quantum gravity

We provide a non-perturbative geometrical characterization of the partition function of $n$-dimensional quantum gravity based on a coarse classification of riemannian geometries. We show that, under natural geometrical constraints, the theory admits a continuum limit with a non-trivial phase structure parametrized by the homotopy types of the class of manifolds considered. The results obtained qualitatively coincide, when specialized to dimension two, with those of two-dimensional quantum gravity models based on random triangulations of surfaces.Comment: 13 page

Phase transition of compartmentalized surface models

Two types of surface models have been investigated by Monte Carlo simulations on triangulated spheres with compartmentalized domains. Both models are found to undergo a first-order collapsing transition and a first-order surface fluctuation transition. The first model is a fluid surface one. The vertices can freely diffuse only inside the compartments, and they are prohibited from the free diffusion over the surface due to the domain boundaries. The second is a skeleton model. The surface shape of the skeleton model is maintained only by the domain boundaries, which are linear chains with rigid junctions. Therefore, we can conclude that the first-order transitions occur independent of whether the shape of surface is mechanically maintained by the skeleton (= the domain boundary) or by the surface itself.Comment: 10 pages with 16 figure

The stochastic renormalized mean curvature flow for planar convex sets

We investigate renormalized mean curvature flow (RMCF) and stochastic renormalized mean curvature flow (SRMCF) for convex sets in the plane.RMCF is the inverse gradient flow for logarithm of square of the perimeter divided by the volume. SRMCF is We investigate renormalized mean curvature flow (RMCF) and stochastic renormalized mean curvature flow (SRMCF) for convex sets in the plane. RMCF is the inverse gradient flow for logarithm of $\sigma/\lambda^2$ where $\sigma$ is the perimeter and $\lambda$ is the volume. SRMCF is RMCF perturbated by some Brownian noise and has the remarkable property that it can be intertwined with Brownian motion, yielding a generalization of Pitman "$2M-X$" theorem. We prove that along RMCF, entropy $\mathcal{E}_t$ for curvature as well as $h_t:=\sigma_t/\lambda_t$ are non-increasing. We deduce infinite lifetime and convergence to a disk after normalization. For SRMCF the situation is more complicated. As $h_t$ is always a supermartingale, for $\mathcal{E}_t$ to be a supermartingale, we need that the starting set is invariant by the isometry group $G_n$ generated by the reflection with respect to the vertical line and the rotation of angle $2\pi/n$, for some $n\ge 3$. But for proving infinite lifetime, we need invariance of the starting set by $G_n$ for some $n\ge 7$. We provide the first SRMCF with infinite lifetime which cannot be reduced to a finite dimensional flow. Gage inequality plays a major role in our study of the regularity of flows, as well as a careful investigation of morphological skeletons. We characterize symmetric convex sets with star shaped skeletons in terms of properties of their Gauss map. Finally, we establish a new isoperimetric estimate for these sets, of order $1/n^4$ where $n$ is the number of branches of the skeleton