‎In this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎To this end‎, ‎we aim to replace the field of scalars $ mathbb{R} $ by self-adjoint elements of a commutative $ C^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎First‎, ‎we put forward the concept of operator-valued tensors and extend semi-Riemannian...

‎in this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎to this end‎, ‎we aim to replace the field of scalars $ mathbb{r} $ by self-adjoint elements of a commutative $ c^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎first‎, ‎we put forward the concept of operator-valued tensors and extend semi-riemannian...

Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame-indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor-valued orientation distribution functions ODFs . Two characterizations of the tensor-valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respective...

Current data processing tasks often involve manipulation of multi-dimensional objects tensors. In many real world applications such as gait recognition, document analysis or graph mining (with graphs represented by adjacency tensors), the tensors can be constrained to binary values only. To the best of our knowledge at present there is no principled systematic framework for decomposition of bin...

In this article, we introduce nonlinear versions of the popular structure tensor, also known as second moment matrix. These nonlinear structure tensors replace the Gaussian smoothing of the classical structure tensor by discontinuity-preserving nonlinear diffusions. While nonlinear diffusion is a well-established tool for scalar and vector-valued data, it has not often been used for tensor imag...

The intrinsic volumes, recalled in the previous chapter, provide an array of size measurements for a convex body, one for each integer degree of homogeneity from 0 to n. For measurements and descriptions of other aspects, such as position, moments of the volume and of other size functionals, or anisotropy, tensor-valued functionals on convex bodies are useful. The classical approach leading to ...

We prove a complete set of integral geometric formulas of Crofton type (involving integrations over affine Grassmannians) for the Minkowski tensors of convex bodies. Minkowski tensors are the natural tensor valued valuations generalizing the intrinsic volumes (or Minkowski functionals) of convex bodies. By Hadwiger’s general integral geometric theorem, the Crofton formulas yield also kinematic ...

In this paper we discuss the notion of singular vector tuples of a complex valued d-mode tensor of dimension m1 × . . . × md. We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous re...

We present a tensor field interpolation method based on tensor-valued Bézier patches. The control points of the patch are determined by imposing physical constraints on the interpolated field by constraining the divergence and curl of the tensor field. The method generalizes to Cartesian tensors of all orders. Solving for the control points requires the solution of a sparse linear system. Resul...

We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on real-valued decompositions, both unconstrained and nonnegative. We discuss when solutions exist and how to formulate the mathematical program. Numerical results sh...