Deep neural networks are vulnerable to input deformations in the form of vector fields pixel displacements and other parameterized geometric e.g. translations, rotations, etc. Current deformation certification methods either (i) do not scale deep on large datasets, or (ii) can only certify a specific class deformations, rotations. We reformulate randomized smoothing setting for both general fie...

Prediction of respiratory motion has the potential to substantially improve cancer radiation therapy. A nonlinear finite element (FE) model of respiratory motion during full breathing cycle has been developed based on patient specific pressure-volume relationship and 4D Computed Tomography (CT) data. For geometric modeling of lungs and ribcage we have constructed intermediate CAD surface which ...

This paper evaluates a nonlinear registration method for warping a 3D histological atlas of the basal ganglia into patient data for deep brain stimulation (DBS) planning. The power of the method is the possibility to combine iconic registration with geometric constraints under a unified diffeomorphic framework. This combination aims to ensure robust and accurate atlas-to-patient warping and ana...

In the paper we present a novel algorithm for collision detection between complex geometric objects represented by polygonal models and undergoing rigid motions and deformations. Most algorithms described in the literature deal with rigid bodies and are based on some kind of hierarchical representations. We present the alternative approach. The algorithm relies on the idea of "sensor particles"...

Caffarelli-Friedman [7] proved a constant rank theorem for convex solutions of semilinear elliptic equations in R2, a similar result was also discovered by Yau [28] at the same time. The result in [7] was generalized to R by Korevaar-Lewis [27] shortly after. This type of constant rank theorem is called microscopic convexity principle. It is a powerful tool in the study of geometric properties ...

This paper studies functionals defined by multiple integrals associated to differential forms on the jet bundle of first order corresponding to some Riemannian manifolds; the domain of these functionals consists in submanifold maps satisfying certain conditions of integrability. Our idea is to give geometric properties to the domain of a functional allowing us to properly define convexity. The ...

The dynamic theory of deformable ellipsoidal inclusions in slow viscous flows was worked out by J.D. Eshelby in the 1950s, and further developed and applied by various authors. We describe three approaches to computing Eshelby’s ellipsoid dynamics and other homogeneous deformations. The most sophisticated of our methods uses differential-geometric techniques on Lie groups. This Lie group method...

This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclide...

Patient-specific biomechanical models implemented using specialized nonlinear (i.e., taking into account material and geometric nonlinearities) finite element procedures were applied to predict the deformation field within the brain for five cases of craniotomy-induced brain shift. The procedures utilize the Total Lagrangian formulation with explicit time stepping. The loading was defined by pr...

Wave propagation in elastomeric materials undergoing large deformations is relevant in numerous application areas, including nondestructive testing of materials and ultrasound techniques, where finite deformations and corresponding stress states can influence wave propagation and hence interpretation of data. In the case of periodically structured hyperelastic solids, the effect of deformation ...