Tensor in signal processing

I saw a lot of use of tensor in signal processing. What is the intuition behind it? Is it simply a common representation for audio (1D), image (2D) and video (3D) signals?

Answer

This has little to do with intuition. Tensors are rigorously defined mathematical objects, and in general simple arrays don't qualify as tensors. Specifically, signals are not tensors.

In the language of mathematics, and there the field of differential geometry, a tensor is a linear function with several arguments. It is therefore also called a multi-linear form. It is used to describe properties of manifolds at a single point.

A point on a (smooth) manifold comes with a tangent space that contains all possible "direction" at that point, where a direction is really just a tangent vector at that point. This tangent space at a single point is a vector space and it has the same dimension as the manifold.

The tangent space has a natural so called dual space, which contains the linear functions that map tangent space vectors to real numbers. This is the cotangent space, and it's also a vector space of the dimension of the manifold.

A tensor is now (roughly) a linear function that maps any number of dimensions between the tangent space, the cotangent space and the real numbers.

Tensors typically don't come as singles but in so called bundles, which are the disjoint union of the tensors of all points of the manifold. For example if you have a vector at every point of the manifold then the whole structure is called a vector bundle. Similarly, tensor bundles associate a tensor with every point on the manifold, usually in a smooth way.

In signal processing you can encounter tensors, but usually they are called differently. For example in video processing the image is your manifold (or rather, a function on a 2-dimensional manifold giving the brightness or color for each point) and the velocity field that describes the local motion is a rank 1 tensor field (or bundle) on that manifold.

Tensors also show up a lot in volume data processing, where tensors can describe local properties of the volume data. For example if you have a doppler ultrasonic tomography image, then the velocity data in each voxel is a tensor field as would be mechanical stress in tomographic material analysis.

For a 1-dimensional signals tensors often come in the form of derivative operators. For example if you have a signal $s(t)$, then the operator $\frac{\partial}{\partial t}$ is a tensor field of rank 1, or just a vector field. More generally, if you multiply that operator with a smooth function $g(t)$, then all possible vector fields on the manifold the signal lives on are of the form $g(t)\frac{\partial}{\partial t}$

To sum up, tensors are a description of differential properties of manifolds. If you want to understand them, you'll have to study differential geometry.