Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



GRID macroscopic growth law and its application to image inference

Authors: Nataliya Portman, Ulf Grenander and Edward R. Vrscay
Journal: Quart. Appl. Math. 69 (2011), 227-260
MSC (2000): Primary 62M40, 62F10, 62F15; Secondary 49N45, 65K05, 92C15
DOI: https://doi.org/10.1090/S0033-569X-2011-01192-4
Published electronically: March 3, 2011
MathSciNet review: 2814526
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Abstract | References | Similar Articles | Additional Information

Abstract: In computational anatomy large deformations of anatomical structures observed in images are represented by diffeomorphic flows on the background space of coordinates. They are usually maximum a posteriori (MAP) estimates obtained by minimization of the cost function (posterior energy) consistent with the material properties of an organism. This paper constructs the underlying transformations induced by biological growth according to the Growth as Random Iterated Diffeomorphisms (GRID) model proposed by U. Grenander. They are diffeomorphic flows generated by the GRID macroscopic growth integro-differential equation that emphasizes dependency of the flow on such GRID variables as the Poisson intensity of cell decisions and relative rate of expansion/contraction.

We explore some cost function models that yield biologically meaningful estimates of these growth parameters. Namely, we seek a prior energy that measures cell activities represented by the Poisson intensity function. Using the macroscopic growth law we formulate an optimal control problem where the GRID variables are optimal controls that force an image of the initial organism to be continuously transformed into an image of the grown organism.

We apply the Polak-Ribière conjugate gradient algorithm for direct estimation of the growth parameters from given images. Then the biological mapping is automatically obtained from estimated growth parameters. The accuracy of GRID variable and image estimates obtained by the inference algorithm depends on the value of the weighting coefficient of the prior energy. We propose an experimental evaluation of this coefficient and reveal growth patterns expressed in GRID variables hidden in confocal micrographs of Wingless gene expression patterns in the larval Drosophila wing disc.

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Additional Information

Nataliya Portman
Affiliation: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Email: nataliya.portman@mcgill.ca

Ulf Grenander
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island, 02912
Email: ulf.grenander@gmail.com

Edward R. Vrscay
Affiliation: Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada
Email: ervrscay@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0033-569X-2011-01192-4
Received by editor(s): May 7, 2009
Published electronically: March 3, 2011
Additional Notes: N. Portman was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Postgraduate Scholarship and by the Ontario Ministry of Training, Colleges and Universities in the form of an Ontario Graduate Scholarship.
E. R. Vrscay was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC) in the form of a Discovery Grant.
Article copyright: © Copyright 2011 Brown University

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