Abstract
In modern shape analysis deformation is quantified in different ways depending on algorithms used and on the scale at which it is evaluated. While global affine and non-affine deformation components can be decoupled and computed using a variety of methods, the very local deformation can be considered, infinitesimally, as an affine deformation. The deformation gradient tensor F can be computed locally using a direct calculation by exploiting triangulation or tetrahedralization structures or by evaluating locally the first derivative of an appropriate interpolation function mapping the global deformation from the undeformed to the deformed state. A suitable function is represented by the Thin Plate Spline (TPS) that separates affine from non affine deformation components. F, also known as Jacobian, encodes both the locally affine deformation and local rotation. This implies that it should be used for visualizing Primary Strain Directions (PSD) and deformation ellipses and ellipsoids on the target configuration. Using C=FTF allows, instead, to compute PSD and to visualize them on the source configuration. Moreover, C allows the computation of the strain energy that can be evaluated and mapped locally at any point of a body using an interpolation function. In addition, it is possible, by exploiting the second order Jacobian, to calculate the amount of the non affine deformation in the neighborhood of the evaluation point by computing the body bending energy enclosed in the deformation. In this contribution, we present i) the main computational methods for evaluating local deformation metrics, ii) a number of different strategies to visualize them on both undeformed and deformed configurations and iii) the potential pitfalls in ignoring the actual three-dimensional nature of F when it is evaluated along a surface identified by a triangulation in three dimensions.
Highlights
Modern shape analysis exploits the potential of specific computational algorithms applied to phenomena where the deformation and/or the variation of shapes are under investigation
Shapes are represented by vectors of point coordinates (=landmarks) that can be compared by means of different mathematical formalisms
The term “shape” is referred to forms that have been standardized at unit size that can be quantified in various ways
Summary
Modern shape analysis exploits the potential of specific computational algorithms applied to phenomena where the deformation and/or the variation of shapes are under investigation. As for the first case, while most applications, from biology (Adams et al, 2013) to paleontology (Piras et al, 2010) to medicine (Piras et al, 2019), usually analyze shapes and forms by using homologous anatomical landmarks, the use of continuous surfaces without points correspondence is faced by exploiting the potential of a plethora of diffeomorphic techniques not treated in detail here (see Trouvé, 1998; Durrleman et al, 2012) When using these diffeomorphic techniques, shapes are considered as images (2D) or surfaces (3D) that are registered using different algorithms (Ceritoglu et al, 2013): diffeomorphic approaches are used for this purpose such as large diffeomorphic deformation metric mapping (LDDMM; Miller et al, 2014, 2015) that represents, today, one of the most used (among many others) approaches for estimating shape differences, surface matching, and Parallel Transport of deformations (Charlier et al, 2017). Size is more frequently quantified using m-Volume
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