#### Date of Award

3-22-2012

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Department of Mathematics and Statistics

#### First Advisor

Matthew C. Fickus, PhD.

#### Abstract

Suppose we are given a video of a rotating object and suppose we want to determine the rate of rotation solely from the video itself and its known frame rate. In this thesis, we present a new mathematical operator called the Geometric Sum Transform (GST) that can help one determine the angular frequency of the object in question. The GST is a generalization of the discrete Fourier transform (DFT) and as such, the two transforms have much in common. However, whereas the DFT is applied to a sequence of scalars, the GST can be applied to a sequence of vectors. Most importantly, we show that the GST, like the DFT, can (1) be used to estimate frequency and (2) can be computed surprisingly quickly. Indeed, we provide a Fast Geometric Sum Transform (FGST) algorithm that computes the GST in O(N logN) matrix-vector multiplications, where N is the number of images in the video sequence. This is a vast improvement over the O(N2) such multiplications required for a direct computation of the GST. The remainder of this thesis is devoted to proving other properties of the GST and giving proof-of-concept numerical examples.

#### AFIT Designator

AFIT-GAM-ENC-12-02

#### DTIC Accession Number

ADA557236

#### Recommended Citation

Smith, Lindsay N., "Determining Angular Frequency from a video with a Generalized Fast Fourier Transform" (2012). *Theses and Dissertations*. 1024.

https://scholar.afit.edu/etd/1024