Shape models based on elliptic PDES, associated energies, and their applications in 2D and 3D

Gençtav, Aslı
By using an elliptic PDE or its modifications, we develop implicit shape representations and demonstrate their two- and three-dimensional applications. In the first part of the thesis, we present a novel shape characterization field that provides a local measure of roundness at each shape point. The field is computed by comparing the solution of the elliptic PDE on the shape domain and the solution of the same PDE on the reference disk. We demonstrate its potential via illustrative applications including global shape characterization, context-dependent categorization, and shape partitioning. In the second part, we solve the elliptic PDE multiple times varying either the diffusion parameter or the right hand side function and construct high-dimensional feature space. We then apply low-dimensional reduction to assign a distinctness value to each shape point. We use the obtained distinctness values for non-structural representation of two-dimensional shapes and saliency measurement of surfaces of three-dimensional shapes. In the third and the final part, we use the elliptic PDE modifications for bringing a pair of 3D shapes into comparable topology.


Quasi-Cartan companions of elliptic cluster algebras
Velioğlu, Kutlucan; Seven, Ahmet İrfan; Department of Mathematics (2016)
There is an analogy between combinatorial aspects of cluster algebras and diagrams corresponding to skew-symmetrizable matrices. In this thesis, we study quasi-Cartan companions of skew-symmetric matrices in the mutation-class of exceptional elliptic diagrams. In particular, we establish the existence of semipositive admissible quasi-Cartan companions for these matrices and exhibit some other invariant properties.
Source-aggregated-poisson with applications to groupwise shape analysis and mesh segmentation
Gençtav, Murat; Tarı, Zehra Sibel; Department of Computer Engineering (2018)
By computing multiple solutions to Poisson’s equation with varying source functions within the shape and aggregating those solutions, we obtain a novel function for shape analysis, which we call Source-Aggregated-Poisson, or SAP. Despite the local computations, by means of specially designed source functions, our model mimics the part-coding behavior of a previous nonlocal model. We show that SAP is robust under geometric transformations and nuisance factors including topological distortions, pose changes, ...
Painleve classification of coupled Korteweg-de Vries systems
Karasu, Emine Ayşe (1997-07-01)
In this work, we give a classification of coupled Korteweg-de Vries equations. We found new systems of equations that are completely integrable in the sense of Painleve. (C) 1997 American Institute of Physics.
Spherical harmonic-based random fields based on real particle 3D data: Improved numerical algorithm and quantitative comparison to real particles
Liu, X.; Garboczi, E. J.; Grigoriu, M.; Lu, Y.; Erdoğan, Sinan Turhan (2011-02-15)
The shape of particles often plays an important role in how they are used and in the properties of composite systems in which they are incorporated. When building models of systems that include real particles, it is often of interest to generate new, virtual particles whose 3D shape statistics are based on the 3D shape statistics of a collection of real particles. A previous paper showed mathematically how this can be carried out, but only had a small set of real particle shape data to use and only made a l...
Coloring 3D symmetry set: perceptual meaning and significance in 3D
Tarı, Zehra Sibel (1999-12-01)
A computational implementation for assigning perceptual meaning and significance to the points in the symmetry is presented. The coloring scheme allows recovery of the features of interest such as the shape skeletons from the complicated symmetry representation. The method is applicable to arbitrary data including color and multi-modality images. On the computational side, for a 256 × 256 binary image, two minutes on a low-end Pentium machine is enough to compute both the distance function and the colored n...
Citation Formats
A. Gençtav, “Shape models based on elliptic PDES, associated energies, and their applications in 2D and 3D,” Ph.D. - Doctoral Program, Middle East Technical University, 2018.