Historical Constraint via Wavelet-Packets

Given space constraints, and to avoid excessive notation, we illustrate wavelet-packets in a heuristic fashion but note that full rigor can be found in Percival and Walden (2000) and Misiti et al. (2007). Consider a generic 1-dimensional function, x(t). The DWT de-

(a) Discrete Wavelet Transformation (b) Discrete Wavelet-packet Transformation

Figure 2.2: (a) Decomposition of a function x(t) into three levels using DWT, x(t) = A3+ D3+

D2+ D1. (b) Graphical representation of the decomposition of a function into three levels using

DWPT, x(t) = AAA3+ AAD3+ ADA3+ ADD3+ DAA3+ DAD3+ DDA3+ DDD3.

composes x(t) into an approximation and successive levels of detail Figure 2.2a. For in- stance, a 3-level decomposition of the row vector x(t) using the DWT would start with a decomposition into an approximation A1 and a detail component D1. The second level

of decomposition takes the approximation piece and further decomposes that into ap- proximation and detail components, so that x(t) = A2 + D2 + D1. For the third level of

decomposition the approximation A2 is split, giving x(t) = A3+ D3+ D2+ D1.

Wavelet packets are found in a similar manner as the DWT except at each stage both the approximation and the detail components are further decomposed, Figure 2.2b. The first stage of the DWPT looks the same as above for the DWT, as the function can be repre- sented as x(t) = A1 + D1. For the second stage both the detail and approximation are

decomposed, yielding: x(t) = AA2 + AD2 + DA2+ DD2. The third level of decomposi-

tion gives the final representation in Figure 2.2b. The wavelet coefficients therefore relate to a single level, yielding x(t) = AAA3 + AAD3 + ADA3 + ADD3 + DAA3 + DAD3 +

DDA3+ DDD3. There are 2Lgroupings of wavelet coefficients at the level L decomposi-

tion. Ordinarily, wavelet packets have been used to find an optimal decomposition of a function based on different detail/approximation combinations from the wavelet packet

tree (Misiti et al., 2007). For our purposes we are not interested in any optimal represen- tation using the packets but in the final decomposition at a given level, L. Rather, when using the 2-dimensional DWPT for the regression surface β, preservation of the constraint in Model (2.3) now follows directly because each s and j combination represents nodes at the same level of decomposition so that ` and k are associated with their corresponding time intervals v and t, respectively. Therefore β(v, t) = 0 for v > t can be better approxi- mated using the DWPT by setting βW

(s`,jk) = 0if ` > k.

To illustrate proof of concept, consider the images in Figure 2.3 as an example of a hypo- thetical β(v, t) function. It is a 256×256 pixel image defined for v and t = 1, · · · , 256 where v runs along the horizontal axis and t the vertical axis. The top left figure of Figure 2.3 displays the original image which is true to the historical constraint, β(v, t) = 0 for v > t where blue corresponds to β(v, t) = 0. The top right image in Figure 2.3 shows the results from the na¨ıve approximate DWT restriction using a 3-level 2-dimensional DWT with the Haar wavelet family. After restricting the coefficients with the constraint 1(vs` ≤ tjk),

the coefficients were transformed via 2-dimensional IDWT back into the time (v and t) domain. Here we see considerable distortion along the edge of the constraint. Further distortion can be seen in the upper triangle particularly as the coefficients decrease. A closer examination of diagonal shows an example of “ghosting” where coefficients that should be set to zero are not. Thus the constraint in the wavelet pace does not properly enforce the constraint.

Conversely, the bottom left figure in Figure 2.3 shows the constraint and reconstruction using wavelet packets, again using a 3-level decomposition with the Haar wavelet family. In the wavelet-packet space, we apply the given restriction, βW

s`,jk = 0if ` > k. The result is

an image that is essentially identical to the original image. Use of other wavelet families, such as Daubechies for two or more vanishing moments, results in greater distortion along the edge. This likely due in part to the padding inherent to other wavelet families. We suggest the use of the Haar wavelets as they maintain the edge better. Similar, a dyadic signal is required as padding can lead to distortions along the constraint. The

v t Original Surface 0 50 100 150 200 250 250 200 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 v t DWT with Constraint 0 50 100 150 200 250 250 200 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 v t

DWPT with Constraint (Haar)

0 50 100 150 200 250 250 200 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12 v t

DWPT with Constraint (Daubechies, 4 VM)

0 50 100 150 200 250 250 200 150 100 50 0 0 0.02 0.04 0.06 0.08 0.1 0.12

Figure 2.3: Proof of concept of the historical constraint. Top left: original image. Top right: de- composed and reconstructed original image with constraint in wavelet space. Bottom left: de- composed and reconstructed original image with constraint in wavelet-packet space using Haar wavelets. Bottom right: wavelet-packet space proof of concept using Daubechies wavelets with 4 vanishing moments.

point here is that the DWPT faithfully retains the features of the regression surface while enforcing the upper triangular constraint. But if we use Daubechies wavelets, as depicted in the bottom right corner of Figure 2.3, we see ghosting to either side of the constraint. Thus only the Haar wavelets maintain the constraint.