Wavelets are mathematical tools for signal processing analysis in the time-frequency do- main [44]. A signal can be represented by wavelets from basis functionsψ(t), called mother wavelets: ψs,l(x) = 1 √ sψ x−l s , (2.16)
where the waveletψis scaled bysand translated bylto generate a family of discrete wavelets, such as a Daubechies wavelet family [69]. Wavelet transforms may be either continuous or discrete. The discrete wavelet transform (DWT) of a signal f is given by:
W f(s,l) =
∑
f(x)ψs,l(x). (2.17)When a signal is represented using a DWT, (2.17) can be calculated using filters [165]. The signal is approximated using low-pass filters while high-pass filters are used to bring out details. This wavelet decomposition is calculated as:
a(n) =
∑
k h(k) f(x−k) = f?h, (2.18a) d(n) =∑
k g(k) f(x−k) = f?g, (2.18b)whereh,gare the low-pass and high-pass filters ofkcoefficients, respectively, andf a discrete signal of sizen. The decompositionaanddare called approximation coefficients and wavelet details, respectively. Equations (2.18a), (2.18b) are the circular discrete convolution (?) of f
with the filtershandg, respectively. As illustrated in Figure 2.13(a), a signalx(n)is passed through each filter, denoted byh andg, where a circular discrete convolution of the signal and the filters is calculated. This wavelet decomposition producesa(n), the approximation coefficients, andd(n), the detail coefficients which are down-sampled at each level to produce half the coefficients, represented by the symbol↓2 in Figure 2.13.
2.6. Wavelet transform 31
(a) (b)
Figure 2.13:1D-DWT: (a) low-passhand high-passgfilter diagram, (b) Mallat’s tree scheme for 2 levels of decomposition. (?stands for circular convolution).
Preserving low and high frequency features is an inherent property of the wavelet trans- forms, as is the ability to compress data. The DWT can be applied recursively to increase the decomposition of the signal, as shown in Figure 2.13(b). The signal is down-sampled at each level to produce half the coefficients. This is called the Mallat algorithm or Mallat-tree decomposition [106]. As the number of wavelet decomposition levels increases, the signal becomes smoother. Figure 2.14 illustrates a 3-level decomposition of a signal representing a pixel vector of 103 features. It is worth observing that a greater level of decomposition implies a smoothing of the original pixel vector. The size of the pixel vector is reduced by a factor of two, from 103 features in the original vector to 13 wavelet coefficients in the third level of decomposition.
Signal denoising is a common task performed by wavelets. This task is known as wavelet thresholding or shrinkage. The smallest high frequency subband coefficients, which are usu- ally considered as noise, might be suppressed without substantially affecting the main features of the signal. These small wavelet coefficients can be removed (hard-thresholding) or attenu- ated (soft-thresholding). For soft thresholding the following nonlinear transform is used [47]:
ηt(y) =sgn(y)(|y| −t)+, (2.19)
(a) (b) (c)
Figure 2.15:Separable 2D-DWT applied to the rows of the image (a) and then, to the columns (b), and the result of 2 levels of decomposition (c).
wheret is the threshold andythe signal to be de-noised. The subscript+in (2.19) indicates that the pixels which are greater than the threshold in absolute value, that is,|y(x)|>t, are attenuated, whereas the rest are removed.
By applying an inverse wavelet transform (IWT) after hard- or soft-thresholding, it is possible to reconstruct the original signal with less amount of noise [84]. In any case, the idea of shrinkage is to preserve only the details that are above a particular threshold.
The DWT is usually applied to one-dimensional signals, although it can be extended to two dimensions. In the case of separable 2D-DWT, the 1D-DWT is extended by applying the wavelet analysis separately to each dimension. This approach simplifies the mathematics and leads to a faster numerical algorithm [106]. In image processing, the two dimensions correspond to the spatial location[x,y], matching the columns and the rows of the image, re- spectively. First, the 1D-DWT decomposition is applied to one of the two dimensions, for example, the rows of an image, resulting in two subbands L and H, with the approximation and detail coefficients of the original image. Figure 2.15(a) shows this step where an im- age is divided into the subband L and the subband H as a result of the convolution by rows with the low-pass and high-pass filters, respectively. Second, the 1D-DWT is applied to the columns of L and H, which results in four subbands LL, HL, LH, HH, corresponding to the low resolution approximation and three subbands of details, as shown in Figure 2.15(b). Like the 1D-DWT, the 2D-DWT can be applied recursively to increase the decomposition of the image, as represented in Figure 2.15(c).
The three subbands HL, LH, HH reveal features related to spatial orientations. Horizontal and vertical structures are highlighted in the LH and HL subbands, respectively, while diago- nal features are represented in the third subband, that is HH. However, one disadvantage of the
2.6. Wavelet transform 33
(a) (b) (c)
Figure 2.16:2D Double-Density DWT: (a) low-pass and high-pass filters diagram, (b) separable transform applied to the rows of an image, (c) separable transform applied to the columns of (b). (?stands for circular convolution).
separable DWT for the application thereof in image processing is that the coefficients reveal only three spatial orientations [59], which results in a poor selectivity of features at different orientations, for example in a curve.
The Double-Density DWT presented in [148] is based on a single scaling function (low- pass filter) and two distinct wavelets (high-pass filters). Although this DWT still suffers from some lack of spatial orientation, with more wavelets than necessary, the Double-Density DWT outperforms the standard 2D-DWT in terms of denoising [148]. Figure 2.16(a) shows the filter decomposition of one dimensional signalx(n)and the resulting three subbandsa(n),d1(n) andd2(n)corresponding to the convolution ofx(n)with the low-pass filterh and the two waveletsg1andg2, respectively. As in the case of the standard 1D-DWT, the result is down- sampled (↓2) at each level to produce half the coefficients.
Applied to an image, the wavelet transform presented in [148] creates nine subbands, instead of four as the general case. The Double-Density DWT is applied to the rows of the image resulting in three subbands, L, H1and H2, as illustrated in Figure 2.16(b), and then it is applied to the columns of L, H1and H2, which results in nine subbands as shown in Figure 2.16(c). One of the subbands is the low resolution approximation (LL subband in Figure 2.16(c)), and the other eight subbands correspond to the wavelet details. The Mallat algorithm can be recursively applied with the Double-Density DWT as explained above for the 2D-DWT. For further details on wavelets, we refer the reader to [106].