New computational methods for the construction of "darcyan" biological coordinate systems

Authors:
Nataliya Portman;Ulf Grenander;Edward R. Vrscay
Affiliations:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada;L. Herbert Balliou University Professor Emeritus, Brown University, Providence, RI;Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
Venue:
ICIAR'07 Proceedings of the 4th international conference on Image Analysis and Recognition
Year:
2007

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Abstract

In this paper we pursue the goal of constructing a biologically meaningful coordinate system that (i) carries information about the spatial distribution of gene activity regions and (ii) captures its effect on the internal structure and shape of an organism during its growth. Geometrically, this "Darcyan" coordinate system is curvilinear and comprised of an interior grid surrounded by a closed curve, namely the boundary of the organism. We explore two computational methods of constructing it, one based on potential theory (Poisson equation) and the other based on level set methods, with particular emphasis placed on the latter. We propose a novel algorithm that uses image processing tools for the extraction of the boundary, from which is produced the interior Darcyan coordinate system by means of level set evolution. Examples show the ability of the proposed algorithm to handle the complex geometry of the initial boundary such as significant oscillations, corners and cusps.