## Summary

The mathematical and statistical modeling of biological growth over time is an important problem with a variety of applications ranging from medical diagnostics to evolutionary biology. Here we use 2D and/or 3D images taken across time, species, or specimens to compare or to extract salient differences in anatomical structures, and to analyze and model their variations both within and across biological classes. There is a large body of work on representing differences in imaged objects using deformations of background space. Motivated by Grenander's Growth as Random Iterated Diffeomorphisms (GRID) model, our goal is to solve its inverse problem where we want to decompose large biological growth into smaller biologically-interpretable units.

Time-indexed images and the resulting deformation map.

## Project Description

The parameter estimation problem in Grenander's GRID model [1,3] has been studied by several papers in the past[2,4]. The estimation of growth components was done in two steps [2]: (1) estimate the full deformation between a pair of images that represents biological growth, (2) estimate parameters for growth components, loosely termed seeds, under the GRID model, with a major simplification that different components are spatially local and do not interact with each other.

For step (2) of the estimation process, a common simplifying assumption in the past papers is that different seeds are placed away from each other so that there is no or negligible interaction between the corresponding deformations. In this case, the total displacement field can be written as a superposition of the displacements resulting from individual seeds. Our work has focuses on solving the parameter estimation problem under the original GRID model that advocates sequential composition of arbitrarily interacting components.

The parameter estimation problem is formulated as likelihood maximization or minimization of an energy associated with the negative of a log-likelihood function. It is rather difficult to solve for all the parameters (for all the seeds) simultaneously. Therefore, we take a sequential approach and add one local deformation to the model at a time.

The estimation within the iterative approach is done by using a gradient approach. Even though the complexity of the composed deformation and thus the energy increases with the number of seeds, we still have analytical expressions for the gradients using the chain rule. Note that this process is iterative not incremental. We also demonstrate the superiority of this method over the past additive methods using synthetic data.

Comparison of Parameter Estimation Biases. Top: additive method; Bottom: our method.

## Refereneces

1. Grenander, U., Miller, M.I.: Diffeomorphisms groups and pattern matching in image analysis. Int. J. Comput. Vision 28(3), 213-221 (1998)
2. Grenander, U., Srivastava, A., Saini, S.: A pattern-theoretic characterization of biological growth. IEEE Transactions on Medical Imaging 26(5), 648-659 (2007)
3. Portman, N.: The Modeling of Biological Growth: a Pattern Theoretic Approach. Ph.D. thesis, University of Waterloo, Waterloo, Ontario, Canada (2009)
4. Portman, N., Grenander, U., Vrscay, E.R.: Maximum-likelihood estimation of biological growth variables. In: ICIAR '09 Proceedings of the 6th International Conference on Image Analysis and Recognition. vol. LNCS 5627, pp. 832-843. Springer-Verlag, Berlin (2009)