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Dyadic Wavelet Transform

Mallat and Zhong [15] have generalized the Canny edge detection approach, and have presented a multiscale dyadic wavelet transform for the characterization of 1D and 2D signals. With a wavelet function $ \psi(x) \in L^2(R)$ , a continuous wavelet transform of $ f(x)$ is given by

$\displaystyle W_{a,b}f(x)=<f,\psi_{a,b}> =\int^{+\infty}_{-\infty}f(x)\frac{1}{a}\psi(\frac{x-b}{a})dx$ (4)

where $ a>0$ is the scale number, $ b\in R$ is the translation parameter, and $ \psi_{a,b}(x)=\frac{1}{a}\psi(\frac{x-b}{a})$ . With a differentiable smoothing function $ \theta(x)$ , $ \psi(x)$ is given by

$\displaystyle \psi(x)=\partial \theta(x)/\partial x.
$

For the 2D wavelet transform, the wavelet functions $ \psi^1(x,y)$ and $ \psi^2(x,y)$ are defined as:

$\displaystyle \psi^1(x,y)=\frac{\partial \theta(x,y)}{\partial x}  $and$\displaystyle   \psi^2(x,y)=\frac{\partial \theta(x,y)}{\partial y}.$ (5)

The dyadic wavelet transform of $ f(x,y)\in L^2(R^2)$ at the scale $ 2^j$ (or level $ j$ ) has two components defined by:

$\displaystyle W^d_{j}f(x,y)=f\ast\psi^d_{j}(x,y)       d=1,2.$ (6)

Hence, the wavelet coefficients $ W^1_jf(x,y)$ and $ W^2_jf(x,y)$ are proportional to the gradient of $ f\ast\theta(x,y)$ :

$\displaystyle \left( \begin{array}{c} W^1_{j}f(x,y)\ W^2_{j}f(x,y) \end{array}\right)$ $\displaystyle = 2^j \left( \begin{array}{c} \frac{\partial}{\partial x}(f\ast\t...
...)(x,y)\ \frac{\partial}{\partial y}(f\ast\theta_{j})(x,y)\ \end{array}\right)$    
  $\displaystyle = 2^j\nabla(f\ast\theta_{j})(x,y).$ (7)

The modulus of the wavelet coefficients at scale $ 2^j$ is defined as:

$\displaystyle M_{j}f(x,y)=\sqrt{\vert W^1_{j}f(x,y)\vert^2+\vert W^2_{j}f(x,y)\vert^2},$ (8)

which represents the multiscale edge information obtained by combining the horizontal and vertical wavelet coefficients. With a scaling function $ \phi(x,y)$ , the coarse approximation of $ f(x,y)$ at scale $ 2^j$ is

$\displaystyle S_jf(x,y)=f\ast\phi_j(x,y).$ (9)

A finite-level discrete dyadic wavelet transform of the 2D discrete function $ f\in l^2(Z^2)$ can be represented as:

$\displaystyle W=\bigg\{S_J f,(W^d_{j}f)^{d=1,2}_{1\leq j\leq J} \bigg\},$ (10)

where $ S_Jf$ is a coarse scale approximation of $ f$ at final scale $ 2^J$ , and $ W^d_jf$ represents the detail image at scale $ 2^j$ . We refer to this discrete wavelet transform as the MZ-DWT.

A 2D discrete function $ f$ can be decomposed by a lowpass filter $ H$ and a highpass filter $ G$ , and reconstructed with a lowpass filter $ \tilde{H}$ (the conjugate filter of $ H$ ) and two highpass filters $ K$ and $ L$ . In the Fourier domain, the Fourier transform of five filters are denoted by $ \hat{H}$ , $ \hat{G}$ , $ \hat{\tilde{H}}$ , $ \hat{K}$ and $ \hat{L}$ , respectively. Details about filter construction can be found in [15] and [13]. The coarse scale approximation of $ f(u,v)$ at scale $ 2^{j+1}$ can be represented in the Fourier domain as:

$\displaystyle \hat{S}_{{j+1}}f(u,v)=\hat{H}(2^ju)\hat{H}(2^jv)\hat{S}_{{j}}f(u,v),$ (11)

where $ j\geq0$ , and $ \hat{S}_0 f(u,v)=\hat{f}(u,v)$ . Correspondingly, the two detail images are obtained as:

$\displaystyle \hat{W}^1_{{j+1}}f(u,v)$ $\displaystyle =\hat{G}(2^ju)\hat{S}_{{j}}f(u,v),$ (12)
$\displaystyle \hat{W}^2_{{j+1}}f(u,v)$ $\displaystyle =\hat{G}(2^jv)\hat{S}_{{j}}f(u,v).$ (13)

With the reconstruction filters, the signal is represented recursively as:

$\displaystyle \hat{S}_{j}f(u,v)$ $\displaystyle =\hat{S}_{{j+1}}f(u,v) \hat{\tilde{H}}(2^ju)\hat{\tilde{H}}(2^jv)$    
  $\displaystyle \quad+\hat{W}^1_{{j+1}}f(u,v) \hat{K}(2^ju)\hat{L}(2^jv)$    
  $\displaystyle \quad+\hat{W}^2_{{j+1}}f(u,v) \hat{L}(2^ju)\hat{K}(2^jv).$ (14)

The time domain representation of (11)-(14) can be found in [15,13]. By substituting (11)-(13) into (14), a necessary and sufficient condition for perfect reconstruction is given as [16]:

$\displaystyle \hat{\tilde{H}}(u)\hat{\tilde{H}}(v)\hat{H}(u)\hat{H}(v)$ $\displaystyle + \hat{K}(u)\hat{L}(v)\hat{G}(u)$    
  $\displaystyle + \hat{L}(u)\hat{K}(v)\hat{G}(v)=1.$ (15)


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Next: Wavelet Diffusion Up: Wavelet Diffusion Previous: Nonlinear Diffusion
yyue 2006-04-28