   Next: Wavelet Diffusion Up: Wavelet Diffusion Previous: Nonlinear Diffusion

Mallat and Zhong  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 , a continuous wavelet transform of is given by (4)

where is the scale number, is the translation parameter, and . With a differentiable smoothing function , is given by For the 2D wavelet transform, the wavelet functions and are defined as: and (5)

The dyadic wavelet transform of at the scale (or level ) has two components defined by: (6)

Hence, the wavelet coefficients and are proportional to the gradient of :   (7)

The modulus of the wavelet coefficients at scale is defined as: (8)

which represents the multiscale edge information obtained by combining the horizontal and vertical wavelet coefficients. With a scaling function , the coarse approximation of at scale is (9)

A finite-level discrete dyadic wavelet transform of the 2D discrete function can be represented as: (10)

where is a coarse scale approximation of at final scale , and represents the detail image at scale . We refer to this discrete wavelet transform as the MZ-DWT.

A 2D discrete function can be decomposed by a lowpass filter and a highpass filter , and reconstructed with a lowpass filter (the conjugate filter of ) and two highpass filters and . In the Fourier domain, the Fourier transform of five filters are denoted by , , , and , respectively. Details about filter construction can be found in  and . The coarse scale approximation of at scale can be represented in the Fourier domain as: (11)

where , and . Correspondingly, the two detail images are obtained as:  (12)  (13)

With the reconstruction filters, the signal is represented recursively as:    (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 :   (15)   Next: Wavelet Diffusion Up: Wavelet Diffusion Previous: Nonlinear Diffusion
yyue 2006-04-28