Parametrization of closed surfaces for 3-D shape description
Computer Vision and Image Understanding
Computing and comprehending topology: persistence and hierarchical morse complexes
Computing and comprehending topology: persistence and hierarchical morse complexes
Discrete & Computational Geometry
Stability of Persistence Diagrams
Discrete & Computational Geometry
Homological illusions of persistence and stability
Homological illusions of persistence and stability
Topological Characterization of Signal in Brain Images Using Min-Max Diagrams
MICCAI '09 Proceedings of the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention: Part II
Persistent homology under non-uniform error
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
An output-sensitive algorithm for persistent homology
Proceedings of the twenty-seventh annual symposium on Computational geometry
On persistent homotopy, knotted complexes and the Alexander module
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Linear-size approximations to the vietoris-rips filtration
Proceedings of the twenty-eighth annual symposium on Computational geometry
Spaces and manifolds of shapes in computer vision: An overview
Image and Vision Computing
An output-sensitive algorithm for persistent homology
Computational Geometry: Theory and Applications
Persistent homology: an introduction and a new text representation for natural language processing
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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We present a novel framework for characterizing signals in images using techniques from computational algebraic topology. This technique is general enough for dealing with noisy multivariate data including geometric noise. The main tool is persistent homology which can be encoded in persistence diagrams. These diagrams visually show how the number of connected components of the sublevel sets of the signal changes. The use of local critical values of a function differs from the usual statistical parametric mapping framework, which mainly uses the mean signal in quantifying imaging data. Our proposed method uses all the local critical values in characterizing the signal and by doing so offers a completely new data reduction and analysis framework for quantifying the signal. As an illustration, we apply this method to a 1D simulated signal and 2D cortical thickness data. In case of the latter, extra homological structures are evident in an control group over the autistic group.