Quaternion-based Signal Processing Ben Witten* and Jeff Shragge, Stanford University SUMMARY and multiplication as Hypercomlex numbers are primarily used for pattern recognition, offer many useful applications to geophysics. Image disparity es- qp D (q0 C iq1 C jq2 C kq3)(p0 C ip1 C jp2 C kp3) timation is a hypercomplex, phase-based technique, using quater- D (q p − q p − q p − q p ) nions that can find differences between subtly varying images. This 0 0 1 1 2 2 3 3 technique relies on applying a quaternionic Fourier transform, a C i(q0 p1 C q1 p0 C q2 p3 − q3 p2) quaternionic Gabor filter and exploits the symmetries inherent in C j(q0 p2 C q2 p0 − q1 p3 C q3 p1) the quaternion. Two applications of hypercomplex image dispar- C k(q p C q p C q p − q p ). (4) ity estimation examined here are time lapse analysis and boundary 0 3 3 0 1 2 2 1 detection. INTRODUCTION Hypercomplex numbers are multi-dimensional numbers that have Table 1: The multiplication table for quaternion algebra. more than one complex plane. The most common type of hyper- 1 i j k complex numbers have one real and three complex dimensions. These were introduced by Hamilton (1866) who termed them quater- 1 1 i j k nions. The most common application of quaternions has been to- i i −1 k − j wards Maxwell's equations. Recently, however, quaternions have j j −k −1 i been applied to signal processing, most notably pattern recogni- k k j −i −1 tion. They are useful for color image analysis where previous tech- niques have failed because each complex quaternion axis can be associated with an RGB axis (Sanwine and Ell, 2000) which al- Notice that multiplication in equation 4 is not commutative due to lows for color edge detection. In addition, they can be used for the quaternionic algebra rules defined in table 1. Quaternions are image segmentation, finding structure based not only upon color, often separated into two parts, q0 and q D iq1 C jq2 Ckq3, respec- but repeating patters. This has proven useful for finding defects in tively called the scalar and vector part of the quaternion. Using this textiles (Bülow and Sommer, 2001). Another application is image definition, the conjugate of q, q¯, is disparity, which is used by Bülow (1999) to show how a pattern changes between two frames of a movie. q¯ D q0 − q D q0 − iq1 − jq2 − kq3 (5) Image disparity techniques offer many geophysical applications be- cause only organized structures and patterns are detected by image and the norm of a quaternion is defined by disparity. Thus, noise will have minimal effect. Since it is a phase based technique, even low amplitude signal still retain enough in- 2 2 2 2 jj q jjD qq¯ D q0 C q1 C q2 C q3 . (6) formation to be viable for image disparity estimation. The applica- p q tions that will be discussed here are time lapse analysis and edge detection. To this end we show the basics of hypercomplex math- It is useful to formulate a polar representation of the quaternion, ematics define the the quaternionic Fourier transform. We then de- with phase and modulus. For any complex number, z D a C ib, the fine the Gabor filter and extend it to the quaternionic case. This argument or phase-angle is defined as atan2(b,a). If z is written will allow for the image disparity estimation to be calculated based it the form z Dj r j eiγ , then γ is the phase (argument) of z, de- upon multiple complex phases. noted arg(z)=γ . Quaternions contain three complex subfields and, correspondingly, three phases components which are: HYPERCOMPLEX MATHEMATICS φ D atan2(nφ,dφ), (7) The set of hypercomplex numbers is defined as θ D atan2(nθ ,dθ ), (8) n D arcsin(n ), (9) q D q0 C il ql, ql 2 <. (1) XlD1 where nφ,dφ,nθ ,dθ ,n come from the equivalence of two homo- morphic quaternion transformations and are defined as Hypercomplex numbers define a n+1-dimensional complex space with i orthonormal to im, for l 6D m. For all cases presented in this l nφ D 2(q2 ∗ q3 C q0 ∗ q1), (10) paper l will be limited to 3. Such numbers are quaternions, which 2 2 2 2 can be represented as q D q0 C iq1 C jq2 C kq3, where i, j,k are dφ D (q0) − (q1) C (q2) − (q3) , (11) imaginary numbers that satisfying the following relations: nθ D 2(q1 ∗ q3 C q0 ∗ q2), (12) d D (q )2 C (q )2 − (q )2 − (q )2, (13) i j D − ji D k, and i2 D j2 D k2 D −1. (2) θ 0 1 2 3 n D 2(q1 ∗ q2 C q0 ∗ q3). (14) The multiplication table for quaternion unit vectors is shown in Table 1. With these definitions, quaternionic addition between two Quaternionic Transform quaternions, q and p, can be defined as Analogous to complex numbers, quaternions can be represented by a magnitude and three phases with q C p D (q0 C iq1 C jq2 C kq3) C (p0 C ip1 C jp2 C kp3) Djj jj iφ jθ k D (q0 C p0) C i(q1 C p1) C j(q2 C p2) C k(q3 C p3), (3) q q e e e . (15) SEG/New Orleans 2006 Annual Meeting 2862 Quaternion-based Signal Processing will be considered to be 0 since only 2-D images are discussed Analogous to equation 18, a quaternionic Gabor filter is defined in this paper. such that the impulse response to the filter is, Ell (1992) introduced the quaternionic Fourier transform (QFT) for hq (x, y;u ,v ,σ,) D g(x, y;σ,)ei2πu0 x e j2πv0 y two-dimensional signals, 0 0 D g(x, y;σ,)ei!1 x e j!2 y (20) Fq (u) D e−i2πux f (x)e− j2πvyd2x, (16) ZR2 where g(x, y;σ,) is defined as in equation 19. The quaternionic Gabor filter can be split into its even and odd symmetries, just as where x D (x, y)T and u D (u,v)T 2 R2 and f is a two-dimensional the QFT and the original image. In that case the filter h can be quaternion signal. Because two-dimensional signals can be decom- written as posed into even and odd components along either the x- or y-axis, q q q q q f can be written h (x, y;u0,v0,σ,) D (hee C ihoe C jheo C khoo). (21) q q q q f D fee C foe C feo C foo, Note that hee, hoe, heo, and hoo are real-valued functions (see Fig- ure 1). with, for example, foe denoting the part of f that is odd with re- spect to x and even with respect to y. The QFT can now be decom- posed to Fq (u) D cos(2πux)cos(2πvy) f (x)d2x ZR2 −i sin(2πux)cos(2πvy) f (x)d2x ZR2 − j cos(2πux)sin(2πvy) f (x)d2x ZR2 2 Ck sin(2πux)sin(2πvy) f (x)d x. (17) Figure 1: The hee, hoe, heo, and hoo symmetries of a quaternionic Z 2 R Gabor filter The QFT is an invertible transform and most standard Fourier the- orems hold for QFTs with minimal variation. These theorems will QUATERNIONIC DISPARITY ESTIMATION not be re-derived here, as complete proofs of the QFT extension of Given two images, f and f , it is possible to find a vector field, Rayleigh's, the shift, the modulation, the derivative, and the convo- 1 2 d(d ,d ), that relates the local displacement between f and f lution theorem exist elsewhere (Ell, 1992; Bülow, 1999). x y 1 2 (i.e. f1(x) D f2(x C d(x) ). Therefore, if the QFT of f1 is, QUATERNIONIC GABOR FILTERS q f1(x, y) D) F (u,v), (22) The three different types of signal phase, as described by Bülow (1999) are global, instantaneous, and local. Global phase is the an- then by the shift theorem, gular phase of the complex Fourier transform of a signal. It gives a real-valued number for each point in the frequency domain that π π f (x, y) D f (x C d , y C d ) D) ei2 udx Fq (u,v)e j2 udy . (23) is indicative of the relative position of the frequency components. 2 1 x y Instead of giving the phase of a certain frequency component, in- Knowing that f and f have local quaternionic phases (φ ,θ , ) stantaneous and local phase give the phase at a certain position in 1 2 1 1 1 and (φ ,θ , ) and assuming that φ varies only in x and θ varies a real signal. The instantaneous and local phases, however, do this 2 2 2 only in y, then the displacement ( ) is given by in different ways. The instantaneous phase is the angular phase of d x the complex value at each signal position. The instantaneous phase φ2(x) − φ1(x) has the drawback that while it provides local information, that in- dx (x) D (24) formation depends on the entire signal. This causes the instanta- uref neous phase at any point to be affected by changes at any other θ2(x) − θ1(x) point in the image, regardless of separation distance. To overcome dy(x) D .