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Singularities in Gaussian scale space that are relevant for changes in its hierarchical structure
Author(s) -
Kuijper A.
Publication year - 2008
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200810935
Subject(s) - hessian matrix , space (punctuation) , scale space , scale (ratio) , gravitational singularity , intensity (physics) , mathematics , extension (predicate logic) , matching (statistics) , image (mathematics) , order (exchange) , gaussian , topology (electrical circuits) , geometry , combinatorics , pure mathematics , computer science , physics , mathematical analysis , artificial intelligence , statistics , image processing , optics , quantum mechanics , operating system , finance , economics , programming language
The Gaussian scale space for an n–dimensional image L ( x ) is defined as its n +1 dimensional extension L ( x , t ) being the solution of ∂ t L =Δ L , L t =0 = L ( x ). A hierarchical structure in L ( x , t ) is derived by combining the critical curves (∇ x L ( x , t )=0) and special points on them, viz. catastrophe points (where det H =0, H being the Hessian matrix) and scale space saddles (where Δ L =0), with iso–intensity manifolds through the scale space saddles. Until now, this structure has only been used for topological segmentation. In order to perform image matching and retrieval tasks based on the hierarchical structure, one needs to know which transitions are allowed when the structure is changed under influence of one control parameter. In this way, the gradual change of one structure into another is described. In this work we describe such relevant possible transitions for the hierarchical structure. These transitions describe the creations and annihilations of catastrophe points and scale space saddles, as well as their interaction with iso–intensity manifolds. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)