Ichnofacies at the Petra Tou Romiou outcrop: the record of sea level

Mallat [Mal89] has proposed an iterative algorithm to compute the discrete wavelet transform. Since then this algorithm is the most known method to apply the wavelet transform. It is based on the multiresolution analysis. It applies two bands subband coding procedure in an iterative fashion and builds the wavelet transform from the bottom up, that is, computing small coefficients for small scales first.

The algorithm by Mallat [Mal89] was proposed for signal analysis. He has proposed a mathematical operator which transforms a signal into an ap- proximation at lower resolution, say 2j_{. Then he showed that the difference}

of information between two approximations at resolutions 2j _{and 2}j+1 _{is ex-}

tracted by the wavelet function.

20 40 60

−10 0 10

The original signal

10 20 30 −10 0 10 Approximation A1 5 10 15 −10 0 10 Approximation A2 2 4 6 8 −10 0 10 Approximation A3 1 2 3 4 −10 0 10 Approximation A4 1 1.5 2 −10 0 10 Approximation A5 10 20 30 −10 0 10 Detail D1 5 10 15 −10 0 10 Detail D2 2 4 6 8 −10 0 10 Detail D3 1 2 3 4 −10 0 10 Detail D4 1 1.5 2 −10 0 10 Detail D5

Figure 4.3: Multilevel decomposition using the wavelet transform. Consider the Fig. 4.3. Every time the function is analysed by the wavelet we go down one level. Certain portions of the function (details) are removed, shown in the right-hand-side plots. Then there are the “approximation” parts, which are further decomposed to give smaller scale representation of the function.

4.2 Wavelet Transform 69

As described in Chapter 3, for a closed approximation vector space Vj, j ∈

Z a scaling function φ(t) is defined such that:

φj,k(t) = 2−j/2φ(2−jt − k), j, k ∈ Z (4.14)

An associated function, the wavelet function, ψ(t), can be deduced from φ(t) for the closed detail vector spaces Wj, j ∈ Z such that:

ψj,k(t) = 2−j/2ψ(2−jt − k), j, k ∈ Z (4.15)

The families {φj,k(t)} and {ψj,k(t)} form orthonormal bases for the closed

approximation vector spaces Vj, j ∈ Z and the closed detail vector spaces

Wj, j ∈ Z, respectively.

The scaling function φ and the wavelet function ψ are related to the low- pass filter H and a high-pass filter G, respectively. The impulse response of H is:

h(n) = hφ(t/2), φ(t − n)i, _{∀n ∈ Z} (4.16)

while the impulse response of G is:

g(n) = (−1)1−nh(1 − n) (4.17)

The filter G is the mirror filter of H. G and H are called quadrature mirror filters [Mal89]. As described more fully in Chapter 3 the relation between the approximation and details is:

Vj = VJ J

M

i=j+1

Wi, ∀j ∈ Z

Based on this, we can go further on the wavelet representation of the signals, which is shown in the following subsections.

The Approximation

First of all the signal is projected on the approximation vector space Vj, j ∈

Z. As mentioned before the scaled and translated versions of the function φ(t) at a certain level j0, i.e., {φj0(t)} = (2

−j0/2_φ(2−j0_{t − k)), ∀k ∈ Z form}

an orthonormal basis of Vj. Then the orthogonal projection on Vj can be

computed by decomposing the signal f (t) on the orthonormal basis φj(t),

that is ∀f (t) ∈ L2_(R): Aj0 = 2 −j0 ∞ X n=−∞ hf (t), φj0,n(t)iφj0,n(t) (4.18)

The inner products

Acj0 = (hf (t), φj0,n(t)i)_n∈Z (4.19)

are called the approximation coefficients of f (t) at the scale j0.

However, the function φj0,n is a member of the vector space Vj0 which

is included in the vector space Vj0−1. Hence, it can be projected in the

orthonormal basis of Vj0−1 by using Eq. 4.18. Let the filter H, with its

impulse response h be defined by Eq. 4.16 and consider the expansion of φj0,n on Vj0−1. Eq. 4.19 yields then the approximation coefficients of f (t)

that can be re-formulated after Mallat in [Mal89] as follows:

Acj0 = hf (t), φj0,n(t)i = ∞ X k=−∞ ˜ h(2n − k)hf (t), φj0−1,k(t)i (4.20)

where ˜h is the impulse response of the symmetric filter ˜H, ˜h(n) = h(−n). Eq. 4.20 shows that one computes the approximation coefficients Acj for

any level j by convolving Acj−1 with the filter function ˜H and retains every

other sample of the output. Moreover, it shows that the length of the filter is not changed by the change of the scale j, but the resolution of the signal to be filtered is changed by 2j_{. This can be formulated as follows:}

Acj,n = f (t)φj,n(t) = ∞ X k=−∞ ˜ h(2n − k)Acj−1,n (4.21)

All the approximation coefficients Acj, 0 < j ≤ J , for some J ∈ Z∗, of a

4.2 Wavelet Transform 71

Since H is a low-pass filter, this Acj can be interpreted as the result

of a low-pass filtering of f (t) followed by a uniform sampling at the rate 2j _{[Mal89]. Assuming j = 1, then the highest frequency of the signal f (t)}

is suppressed to one half. Then the sampling rate of the approximation coefficients A1 must be halved. This is done by retaining every other sam- ple point of the output. This process is called down sampling. Due to the convolution with low-pass filters the details of f (t) that have a frequency ω > 1₂ kargmax

ω

( ˆf (ω))k are removed.

The Details

To extract the details lost between the approximations of the signal f (t) at the scales j − 1 and j, the original signal must be projected on Wj, which is

the orthogonal complement of Vj in Vj−1.

The construction is similar to the one in the approximation. The detail coefficients, Dcj, are computed at any level j by convolving the signal f (t)

with the wavelet function ψ or with the help of the high-pass filter G: Dcj = hf (t), ψj,n(t)i = ∞ X k=−∞ ˜ g(2n − k)hf (t), φj−1,k(t)i (4.22)

where ˜g is the impulse response of the symmetric filter ˜G, ˜g(n) = g(−n). Eq. 4.22 shows that the details signal Dcj is computed by convolving

Acj−1 with the filter ˜G and retaining every other sample of the output:

Dcj,n = f (t)ψj,n(t) = ∞ X k=−∞ ˜ g(2n − k)Acj−1,n (4.23)

In practice, the signal f (t) is decomposed into two new subband signals using the low-pass filter H and the high-pass filter G yielding the approx- imation coefficients A1 and the detail coefficients D1 at a lower resolution at level j = 1. Each of which has as many sample points as the half of the sample points of f (t). The process is repeated on the new approximation coefficients at level j giving the approximation and details at level j + 1 until a level J is reached. At that level both subbands have only one sample point and the analysis must stop. In fact AJ represents the global approximation

Thus the signal can be represented completely with the sequence:

(AcJ, (Dcj)0<j≤J) (4.24)

which has the same total number of samples as the original signal f (t) = A0. Fig. 4.4 illustrates this process and shows the customary graphical representation.

The original signalf(t) = Ac₀

ApproximationAc₁ DetailDc₁ Ac₂ Dc₂ Ac_J Dc_J j=0 j=1 j=2 j=J

Figure 4.4: Customary representation of 1D dyadic wavelet analysis.

Signal Reconstruction

The reconstruction of the signal can be done by performing the inverse anal- ysis in the inverse direction of the pyramid:

Acj,n = f (t)φj,n(t) = ∞ X k=−∞ h(2n − k)Acj+1,n+ ∞ X k=−∞ g(2n − k)Dcj+1,n (4.25)

The approximation coefficients at a coarser resolution Acj+1 and the sig-

nal details at the same resolution Dcj+1 are combined together to give the

approximation coefficients at level j. The original signal f (t) at resolution j = 0 is reconstructed by repeating this procedure for 0 ≤ j < J . Because the reconstructed signal should have a band width as double as each subband of the approximation or details, it should have double the sampling rate and thus double the number of sample points as each one of them. This is done by up sampling Acj+1 and Dcj+1, where zeros are inserted between each two

4.2 Wavelet Transform 73