Intrinsic Deformation

Intrinsic deformations are defined as sequences of speed vectors applied to the points of a shape. The length of such a sequence is computed thanks to a Riemannian metric, whose role is to penalize certain transformation. Choosing such a metric roughly corresponds to penalize some distortions of the shape.

Math

In intrinsic deformation is determined by a sequence of velocity vector fields $V=(V^t{)}_{1\le t\le T}$. The shape after deformation is defined as:

$$\text{Morph}(X_i,V)=X_i+\sum _{t=1}^T{V}_i^t.$$

We also define intermediate states $(X^m{)}_{1\le m\le T}$ of the deformation:

$$X_i^m=X_i+\sum _{t=1}^m{V}_i^t$$

The length of the deformation is determined by a Riemannian metric:

$$\text{Length}(X,V^t)=\frac{1}{T}\sum _{t=1}^T\ll V^t,V^t{\gg }_{X^t},$$

where $\ll V^t,V^t{\gg }_{X^t}$ is a Riemannian metric depending on the position of the points and the topology (edges, triangles) of the shape. This term informs about the “distortion” of the shape caused by the transformation:

$$X^t\to X^{t+1}=X^t+V^t$$

Code

Intrinsic Deformation is accessible in scikit-shapes through the class IntrinsicDeformation. The argument n_steps controls the number of time steps $T$, the higher n_steps is, the more flexible is the model. However, the memory impact grows linearly in n_steps and the running time is also impacted. The Riemannian metric is given with the argument metric. Available metrics are:

• as-isometric-as-possible (requires points and edges)

• shell-energy (requires triangles)

import skshapes as sks

loss = ...
metric = sks.ShellEnergyMetric
model = sks.IntrinsicDeformation(n_steps=10, metric=metric)

registration = sks.Registration(loss=loss, model=model)
registration.fit(source=source, target=target)

path = registration.path_
morphed_source = registration.morphed_shape_