Estimation of a growth development with partial diffeomorphic mappings

Abstract

In the field of computational anatomy, the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework has proved to be highly efficient for addressing the problem of modeling and analyzing of the variability of populations of shapes, allowing for the direct comparison and quantization of diffeomorphic morphometric changes. However, with the progress achieved in medical imaging analysis, the interest for longitudinal data set has substantially increased in the last years and requires the processing of more complex changes, which especially appear during growth or aging phenomena. The observed organisms are subject to transformations over time that are no longer diffeomorphic, at least in a biological sense. One reason might be a gradual creation of new material. The evolution of the shape can then be described by the joint action of a deformation process and a creation process.In this paper, we extend the LDDMM framework to address the problem of non diffeomorphic structural variations in longitudinal data. We keep the geometric central concept of a group of deformations acting on embedded shapes. The need for partial mappings leads to a timevarying dynamic that modifies the action of the group of deformations. We develop a theoretical framework and two algorithms to estimate realistic individual growth scenarios from a set of observations sparsely distributed in time. We present few numerical experiments on animal horns where the shapes are modeled by oriented varifolds. Each computed scenario is parametrized by low-dimensional variables providing the support for statistical analysis.

Shape spaces: From geometry to biological plausibility

“…A different approach at modeling growth can be found in Kaltenmark [44], Kaltenmark and Trouvé [43]. In this work, a growing shape at a given time t is defined as a transformation q t of a co-dimension-one foliation X, which encodes the full growth process.…”

“…The value of q t (x) remains constant until t is reaches the foliation index of x, so that the function q 0 encodes all future initializations of the growth process. This process can be constructed through an evolution equation in the form ∂ t q t = v(t, q t ), and an example is developed in Kaltenmark and Trouvé [43] to model animal horn growth.…”

Organ registration from partial surface data in augmented surgery from an optimal control perspective

“…Outside the augmented surgery domain, other approaches based on the physical causes of displacement involve an optimal control problem. In [30], the authors control the accretion process leading to the shape of a horn. In [31], the forces driving the motion of a viscoplastic material are estimated.…”

Partial Matching in the Space of Varifolds

“…However, the data attachment metrics proposed so far aim to compare the source and target shapes in their entirety ( [9]), or look for explicit correspondences between subparts of these shapes ( [13]). In [20] the growth model presented introduces a first notion of partial matching incorporated to the LDDMM framework, yet no explicit partial dissimilarity term was proposed.…”