*The Basic Definition*

Harmonic wavelets are complex wavelets defined in the Fourier domain. It is consists of an even function *He*(*ω*) (see Figure 1a) as the real part and an odd function *Ho*(*ω*) (see Figure 1b) as the imaginary part, which are defined by

**Figure 1.** The harmonic wavelet function. (**a**) the even part *He*(*ω*) ; (**b**) the odd part *Ho*(*ω*); (**c**) the harmonic wavelet function *<sup>H</sup>*(*ω*).

$$H\_{\ell}(\omega) = \begin{cases} 1/4\pi, & \omega \in [-4\pi, -2\pi) \cup [2\pi, 4\pi), \\ 0, & \text{otherwise.} \end{cases} \quad H\_{\bullet}(\omega) = \begin{cases} i/4\pi, & \omega \in [-4\pi, -2\pi), \\ -i/4\pi, & \omega \in [2\pi, 4\pi), \\ 0, & \text{otherwise.} \end{cases} \tag{1}$$

Combining *He* and *Ho*, we ge<sup>t</sup> the harmonic function

$$H(\omega) = H\_{\mathfrak{e}}(\omega) + iH\_{\mathfrak{e}}(\omega). \tag{2}$$

From Label (1), we have

$$H(\omega) = \begin{cases} 1/2\pi, & \omega \in [2\pi, 4\pi), \\ 0, & \text{otherwise}. \end{cases} \tag{3}$$

This is shown in Figure 1c.

The corresponding scaling function *S* is given in the same way, and the even and odd functions are defined as

$$S\_{\varepsilon}(\omega) = \begin{cases} 1/4\pi, & \omega \in [-2\pi, 2\pi), \\ 0, & \text{otherwise.} \end{cases} \quad S\_{o}(\omega) = \begin{cases} \text{i}/4\pi, & \omega \in [-2\pi, 0), \\ \\ -\text{i}/4\pi, & \omega \in [0, 2\pi), \\ 0, & \text{otherwise.} \end{cases} \tag{4}$$

so that, from Label (4),

$$S(\omega) = S\_\mathfrak{e}(\omega) + iS\_\mathfrak{e}(\omega). \tag{5}$$

Therefore, we have

$$S(\omega) = \begin{cases} 1/2\pi, & \omega \in [0, 2\pi), \\ 0, & \text{otherwise}, \end{cases} \tag{6}$$

shown in Figure 2.

**Figure 2.** The harmonic scaling function. (**a**) the even part *Se*(*ω*); (**b**) the odd part *So*(*ω*); (**c**) the scaling function *<sup>S</sup>*(*ω*).

Then, the shifting and scaling of basic functions are denoted as *Sj*,-(*ω*) and *Hj*,-(*ω*), which are given as

$$\begin{aligned} S\_{j,\ell}(\omega) &= 1/2^j S(\omega/2^j - \ell), \\ H\_{j,\ell}(\omega) &= 1/2^j H(\omega/2^j - \ell), \end{aligned} \tag{7}$$

where *j* ∈ *Z* is the scaling parameter, and - ∈ *R* is the shifting parameter. According to Label (7), the harmonic wavelet system constructs a basis of *L*<sup>2</sup>(*R*) in the frequency domain; then, for *f* ∈ *<sup>L</sup>*<sup>2</sup>(*R*), we have

$$f(\omega) = \sum\_{j=-\infty}^{+\infty} \sum\_{\ell=-\infty}^{+\infty} a\_{j,\ell} H(2^j \omega - \ell),\tag{8}$$

$$f(\omega) = \sum\_{\ell=-\infty}^{+\infty} a\_{\ell} S(\omega - \ell) + \sum\_{j=0}^{+\infty} \sum\_{\ell=-\infty}^{+\infty} a\_{j,\ell} H(2^j \omega - \ell),\tag{9}$$

where

$$a\_{\ell} = \int\_{-\infty}^{+\infty} f(\omega) S(\mathbf{x} - \ell) d\mathbf{x}, \quad a\_{\mathbf{j},\ell} = \int\_{-\infty}^{+\infty} f(\omega) H(2^{\mathbf{j}}\mathbf{x} - \ell) d\mathbf{x}. \tag{10}$$
