Quaternionic Fourier-Mellin Transform

In this contribution we generalize the classical Fourier Mellin transform [3], which transforms functions f representing, e.g., a gray level image defined over a compact set of R. The quaternionic Fourier Mellin transform (QFMT) applies to functions f : R → H, for which |f | is summable over R+ × S under the measure dθ dr r . R ∗ + is the multiplicative group of positive and non-zero real numbers. We investigate the properties of the QFMT similar to the investigation of the quaternionic Fourier Transform (QFT) in [5, 6]. 1. Quaternions Gauss, Rodrigues and Hamilton introduced the 4D quaternion algebra H over R with three imaginary units: (1) ij = −ji = k, jk = −kj = i, ki = −ik = j, i = j = k = ijk = −1. Every quaternion (2) q = qr + qii+ qjj + qkk ∈ H, qr, qi, qj, qk ∈ R has quaternion conjugate (reversion in Cl 3,0) (3) q̃ = qr − qii− qjj − qkk, This leads to norm of q ∈ H, and an inverse of every non-zero q ∈ H (4) |q| = √ qq̃ = √ q2 r + q 2 i + q 2 j + q 2 k, |pq| = |p||q|, q −1 = q̃ |q|2 = q̃ qq̃ . The scalar part of quaternions is symmetric (5) Sc(q) = qr = 1 2 (q + q̃), Sc(pq) = Sc(qp). The inner product of quaternions defines orthogonality (6) Sc(pq̃) = prqr + piqi + pjqj + pkqk ∈ R. 1Remark: By reading this paper you agree to the terms of the Creative Peace License of page 132. 123 ar X iv :1 30 6. 16 69 v1 [ m at h. R A ] 7 J un 2 01 3