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:

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

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

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

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

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

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 [15] and [13]. The coarse scale approximation of at scale can be represented in the Fourier domain as:

where , and . Correspondingly, the two detail images are obtained as:

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

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]: