4π Z Z S eφθdΩ (3.3) = 1 2π Z π θ=0 Z π φ=0 eφθ| sin 2θ| dθdφ, (3.4)
where dΩ = |2 sin 2θdφdθ|1 corresponds to the differential solid angle on sphere S. The
mean can then be approximated using the following summation
m(t) ≈ π 2IJ I X i=1 J X j=1 eφjθi|sin 2θi| (3.5)
The weighting factor |sin 2θi| compensates for the higher density of the point set
by giving smaller weights to envelopes near poles; this scheme, however, is not stable for smaller values of I and J.
3.4
The Trivariate EMD Algorithm
In trivariate EMD (TEMD) algorithm, the input signal is represented as a pure quaternion signal for convenience of representation and the computational gains which it offers as compared to R2. Moreover, the rotation property of quaternions is used to generate
projections along multiple directions in 3D space to estimate the local mean.
As discussed earlier, a major challenge in extending standard EMD to trivariate signals is to find an accurate method for calculating the local mean, since the concept of extrema cannot always be rigorously defined for multivariate signals. For this cause, in TEMD algorithm, signal projections in multiple directions in 3D spaces are obtained, whose extrema are then interpolated using component-wise spline interpolation to yield pure quaternion-valued envelope curves. The use of component-wise spline interpolation
1For θ ∈ [0,π
2], 2θ is the colatitude, hence the differential solid angle corresponding to (θ, φ) is dΩ = 2 sin 2θdφdθ; for θ ∈ [π
2, π], 2θ, 2π − 2θ is the colatitude, hence the differential solid angle is dΩ = −2 sin 2θdφdθ. This formula is not the standard one because the angle θ in this thesis is not the standard colatitude or latitude.
2See Appendix A for further details about the quaternion algebra and benefits of using the quaternion representation as compared to R4.
3.4 The Trivariate EMD Algorithm 63 −1 0 1 −1 0 1 −1 0 1 X Y Z O A
(a) A direction vector in 3D space
Figure 3.3: Proposed set of direction vectors in 3D space for local mean estimation in TEMD: (a) The direction vector OA in 3D space, which has unit norm, can also be represented by a point on the surface of a unit sphere. (b) Set of direction vectors used for calculating the local mean in TEMD, with V = 64 directions. .
for pure quaternion signals represents an extension of the concept of ‘complex splines’, employed in RI-EMD and bivariate EMD. The resulting ‘quaternion-valued envelopes’ are then averaged to obtain an estimate of the local mean of a trivariate signal.
3.4 The Trivariate EMD Algorithm 64
To obtain signal projections along different directions, multiple direction vectors are chosen in 3D space which can be represented by points on the surface of a unit sphere (Figure 3.3(a)). In TEMD, these direction vectors are taken along equidistant points on multiple longitudinal lines on the sphere, obtaining so-called ‘equi-longitudinal lines’. The projections of an input signal along points (direction vectors) on an equi-longitudinal line can then be obtained by rotating the input signal along a rotation axis in the xy plane and mapping it along the z-axis.
Multiple points (direction vectors) on a single longitudinal line, corresponding to rotation axis q, along which signal projections can be taken, are shown in Figure 3.4(b). Every rotation in three dimensions can be treated as a rotation about an axis by a given angle. Thus, a unit quaternion, which gives an efficient and convenient mathematical notation for rotation using an angle-axis representation, is a natural choice for performing 3D rotations.
To take projections along direction vectors spanning the whole 3D space, multiple rotation axes along the xy plane are employed. By rotating the input signal along these axes, signal projections along directions corresponding to multiple equi-longitudinal lines on the surface of a sphere are taken, as shown in Figure 3.5. Since rotation axes are 3D vectors, they can also be represented by a set of unit quaternions q in the xy plane, under an angle φ to the x-axis as shown in Figure 3.4(a). Rotation axes, represented by a vector of quaternions q, can therefore be expressed as
q = 0 + cos(φ)ı + sin(φ) + 0κ. (3.6) Since a trivariate signal can also be represented as a pure quaternion x(t), its projections along multiple direction vectors on a sphere can be calculated by rotating it ¡
by using equation (A.7)¢ about a set of vectors q, by an angle 2θ, and then taking its projection along the z-axis (κ), using
3.4 The Trivariate EMD Algorithm 65
(a) Choice of a set of rotation axes q for obtaining multiple signal projections
y z x k j i q f
Figure 3.4: Generating multiple direction vectors on a 3D sphere. (a) Choices of rota- tion axes to obtain projections along multiple directions in 3D space. For projections along longitudinal lines on a sphere, multiple axes represented by a set of vectors q are chosen in the xy plane, with angle φ taken with respect to +x-axis. (b) Multiple direction vectors represented by points on a longitude line. Projections of the input signal are taken by rotation about the vector qφ. To encompass the whole 3D space,
3.4 The Trivariate EMD Algorithm 66
Figure 3.5: Points on multiple longitudinal lines on a sphere, representing directions along which projections of the input signal can be taken by rotating the input signal along rotation axes represented by a set of unit quaternions q
.
pφθ = eqθx(t)(eqθ)∗· κ (3.7)
where symbol (·) denotes the dot product. To calculate the envelopes in multiple direc- tions, angles φ and θ can be selected to have respectively J and I values between 0 to π. The range of π is necessary since both q and −q give projections in the same direction and also because the application of a unit quaternion q represents rotation by an angle 2θ (See Appendix A for details).
While projections of the input signal on points (direction vectors) along equi- longitudinal lines on a sphere provide a reasonable solution, it can be noticed from Fig- ure 3.5 that there is greater concentration of points close to the North and South pole of the sphere, and therefore, the points close to the poles are given more ‘weight’ in the calculation of the local mean signal. As a result, the method would not be strictly in-
3.4 The Trivariate EMD Algorithm 67 −150 −75 0 75 150 −75 −50 −25 0 25 50 75 −75 −50 −25 0 25 50 75 Z Y X
Figure 3.6: Local mean (dotted line) calculated for a trivariate signal (solid line) using equation (3.2)
.
variant to 3D rotations for a relatively small number of directions. However, for a large number of directions, the whole 3D space is spanned and the proposed method effectively becomes invariant to 3D rotations. Alternatively, one can adopt the weighting scheme given in equation (3.5). The extension of EMD for pure quaternion signals is summarized in Algorithm 3.
The stopping criterion used with the TEMD algorithm is similar to the one em- ployed for the standard EMD algorithm as given in section 2.4.2, with the same parameter values ([θ1 = 0.05, θ2 = 0.5, α = 0.05] and S = 1). The only difference is that the con-
ditions for the stopping criteria are imposed on V = IJ number of multiple projections each and the sifting process is stopped when the stopping conditions are fulfilled for all those projections 3. Figure 3.6 shows the results of employing the mean envelope, given in equation (3.2), for calculation of the local mean of a trivariate signal, using I = 6 and
J = 6. Notice that the local mean correctly tracks the dynamics of the signal.
3A detailed illustration of the stopping criterion used for general multivariate extensions of EMD will be given in Section 4.4