HYPERCOMPLEX POLAR FOURIER ANALYSIS FOR COLOR IMAGEA Paper by Zhuo YANG and Sei-ichiro KAMATA

Graduate School of Information, Production and Systems, Waseda University

Abstract

• Fourier transform tool in image processing and pattern recognition • Hypercomplex Fourier transform treats signal

as vector field and generalizes conventional Fourier transform • Hypercomplex polar Fourier analysis • Can handle signals represented in

hypercomplex numbers like image color• reversible that means it can be used to

reconstruct image

Abstract

• The hypercomplex polar Fourier descriptor has rotation invariance property that can be used for feature extraction.

• Noncommutative property of quaternion• Both left-side and right-side hypercomplex polar Fourier

analysis are discussed and their relationships are also established in this paper

• The experimental results on image reconstruction, rotation invariance and color plate test are given to illustrate the usefulness of the proposed method as an image analysis tool.

Background

• Hypercomplex number• Traditional term for an element of an algebra over a field

where the field is the real numbers or the complex numbers

• quaternions, tessarines, coquaternions, biquaternions, and octonions

• Matrix algebra

Background

• Polar Fourier Analysis

Background

• Polar Fourier Analysis

Background

• Polar Fourier Analysis

|Pnm| is the rotation invariant and is called Polar Fourier Descriptor

Background

Quaternion• A type of hypercomplex number and generalization of

complex number, the quaternion was formally discussed by Hamilton in 1843

• One real part and three imaginary parts • Given a, b, c, d ∈ R, a quaternion q ∈ H (H denotes

Hamilton) is defined as q = S(q) + V(q), S(q) =a, V(q) = bi + cj + dk where S(q) is scalar part and V(q) is vector part. i, j, k are imaginary operators obeying the following rules

Background

QuaternionEuler formula holds for hypercomplex numbers,

eμϕ = cos(ϕ) + μ sin(ϕ)

We also have: eμϕ ∥ ∥ = 1. The quaternion• q can be represented in polar form: q = q eμϕ∥ ∥ . Color• image can be represented in pure quaternion form [1]• f(x, y) = fR(x, y)i+fG(x, y)j+fB(x, y)k, where fR(x, y),• fG(x, y) and fB(x, y) are the

Hypercomplex Polar Fourier Analysis and Its Properties

• Hypercomplex Polar Fourier Descriptor (HPFD)• Left-side Hypercomplex Polar Fourier analysis is defined as

Where the coefficient is

Where μ is unit pure quaternion and is defined as

μ = (1/√3)i +(1/√3)j + (1/√3)k

Hypercomplex Polar Fourier Analysis and Its Properties

• Hypercomplex Polar Fourier Descriptor (HPFD)• Right Side Hypercomplex Polar Fourier analysis is defined

as

Where the coefficient is

Where μ is unit pure quaternion and is defined as

μ = (1/√3)i +(1/√3)j + (1/√3)k

Left Side and Right Side Relationship

Experiment 1: Image Reconstruction

Experiment 2: Color Plate