The Morlet wavelet is a complex function; the modulus of the truncated Morlet wavelet (without the correction term) is a Gaussian, elongated in the $x$ direction if $\epsilon >1$, and its phase is constant along the direction orthogonal to $k\u21920$, and linear in $x\u2192$, mod($2\pi /|k0|$), along the direction of $k\u21920$. As compared to the 1-D case, the additional feature of the 2-D Morlet (or Gabor) wavelet function is its inherent directivity, entirely contained in its phase. This turns to be a crucial advantage for studying objects with directional properties. Indeed, since the wavelet transform [Eq. (6)] is a convolution product of the fringe pattern with the dilated wavelet, we see that the wavelet transform smoothes the image in all directions, but detects the sharp transitions in the direction perpendicular to $k\u21920$. In Fourier space, the effective support (footprint) of the function $\Psi ^M$ is an ellipse centered at $k\u21920$ and elongated in the $ky$ direction. In Figs. 2(b) and 2(c) we show two Morlet wavelets computed for $k\u21920=(5.6,0)$, and $\epsilon =1$ and $\epsilon =10$, respectively; their Fourier transforms are shown in Figs. 2(e) and 2(f). Since the ratio of the axes is equal to $\epsilon $, the cone of the wavelet in Fourier space elongates along $ky$ direction as $\epsilon $ increases. This wavelet preferentially detects edges perpendicular to the $y$-direction (i.e., parallel to $k\u21920$), and its angular selectivity increases with $k\u21920$ and with the anisotropy $\epsilon $. For the optical image shown in Fig. 2(a) recorded with the QPM from a glass coverslip coated with a scratched polymer layer, the best selectivity is achieved with $k\u21920$ perpendicular to the long axis of the ellipse in $k\u2192\u2212space$, that is $k\u21920=(k0,0)$. We show in Fig. 2(d) the modulus of the Fourier transform of the fringe image shown in Fig. 2(a). The Morlet wavelet selects the right part of this Fourier transform by performing a band-pass filtering around the grating frequency. The advantage of taking a smooth wavelet and not a simply circular window in Fourier space^{31} is not only to avoid the introduction of artificial oscillations produced by the sharp boundary of such a window, but also to have the ability to use the mathematical formalism of wavelet analysis, for instance, the ridge detection method.^{52} The Morlet wavelet $\Psi M$ is then written as Display Formula
$\Psi M(x\u2192)=exp[\u221212(x2+\epsilon y2)][exp(ik0x)\u2212exp(\u2212k02/2)].$(11)