# Klas Nordberg

### Linköping University, Sweden

# Tensors in Computer Vision and Image Processing

The concept of tensors has been around in image processing and computer vision a few decades, with two main applications areas: as descriptors of local features in image data, mainly in the context of local orientation, and in geometry where they are used for representing various types of matching constraints or mappings between geometric objects. The tutorial consists of three parts. (1) A mathematical background to what tensors are and why it is reasonable that such a rather abstract mathematical construction should be useful in different fields of physics as well as in image processing and computer vision.Notation issues are also discussed.(2) An overview of tensors for orientation representation, with applications to motion estimation, interest point detection, and image de-noising.Recent developments in this field are extensions of the basic orientation tensors to more complex descriptors, e.g., of multiple orientations or multiple line segments, as well as novel methods for estimating orientation tensors. (3) An overview of tensors in multiple-view geometry and multiple-point geometry.Some recent developments in this field are presented, such as a general framework for constructing both constraint tensors for multiple views/points and mappings for the reconstructing of point/views based on multiple views/points, and minimal parameterizations of such tensors.

### Content

- Introduction to tensors
- What are tensors?
- Why do we need them?
- Indices or no indices?
- Operations on tensors

- Tensors in image processing
- Orientation tensors
- Orientation tensor processing
- Extensions of orientation tensors
- Applications

- Tensors in geometry
- Matching constraints, is there a general principle?
- Reconstruction, not only of points
- Minimal representations