Member-only story

Introduction to Geometric Algebra Computing for Computer Vision

Gus Hayes

·13.8k Followers· Follow

Published in Introduction To Geometric Algebra Computing (Computer Vision)

4 min read

996 View Claps

61 Respond

Save

Listen

Geometric algebra is a mathematical system that combines the concepts of vector algebra and geometric calculus into a single, more powerful framework. Geometric algebra is used in many different applications, including computer vision, robotics, and graphics. In computer vision, geometric algebra can be used to represent and manipulate geometric objects such as points, lines, planes, and rotations. This makes geometric algebra a powerful tool for tasks such as object detection.

Basic Concepts of Geometric Algebra

The basic concepts of geometric algebra are vectors, bivectors, and multivectors. A vector is a directed line segment. A bivector is a geometric object that is equivalent to a parallelogram. A multivector is a sum of vectors and bivectors. Vectors are represented using lowercase italic letters (e.g., a),bivectors are represented using uppercase italic letters (e.g., B),and multivectors are represented using bold symbols (e.g., A). Using multivectors, vectors can be denoted as v = a + b and bivectors can be denoted as B = ab - ba, where a and b are vectors. We can directly compute the dot and cross products of vectors as:

$$ a\cdot b = \frac{1}{2}(ab + ba) $$

Introduction to Geometric Algebra Computing (Computer Vision)

by Franz Kafka

4.3 out of 5

Language	:	English
File size	:	11290 KB
Print length	:	212 pages
Screen Reader	:	Supported

$$ a\times b = \frac{1}{2}(ab - ba) $$

Geometric Algebra for Computer Vision

Geometric algebra is a powerful tool for computer vision. Geometric algebra can be used to represent and manipulate geometric objects such as points, lines, planes, and rotations. This makes geometric algebra a powerful tool for tasks such as object detection, image registration, and motion estimation.

Points, Lines, and Planes

In geometric algebra, points are represented using vectors. Lines are represented using bivectors. Planes are represented using trivectors. The following table shows how to represent geometric objects using geometric algebra.

| Geometric Object | Geometric Algebra Representation | |---|---| | Point | Vector | | Line | Bivector | | Plane | Trivector |

Rotations

In geometric algebra, rotations are represented using rotors. A rotor is a special type of multivector that can be used to represent a rotation about an axis. The following equation shows how to represent a rotation about the z-axis by an angle of θ:

R = cos(θ/2) + sin(θ/2) k

where k is the unit vector in the z-direction.

Applications of Geometric Algebra in Computer Vision

Geometric algebra has been used in a wide range of computer vision applications, including:

Object detection
Image registration
Motion estimation
3D reconstruction
Computer graphics

Geometric algebra is a powerful tool for computer vision. Geometric algebra provides a unified framework for representing and manipulating geometric objects. This makes geometric algebra a valuable tool for a wide range of computer vision applications.

Geometric algebra is a powerful mathematical system that can be used to represent and manipulate geometric objects. Geometric algebra is used in many different applications, including computer vision, robotics, and graphics. In computer vision, geometric algebra can be used to represent and manipulate geometric objects such as points, lines, planes, and rotations. This makes geometric algebra a powerful tool for tasks such as object detection.