(14.47) |

and we can compute the set

(14.48) |

and

(14.49) |

where

(14.50) |

The cut-off frequency is reduced by a factor 2 at each step, allowing a reduction of the number of samples by this factor.

The wavelet coefficients at the scale *j*+1 are:

(14.51) |

and they can be computed directly from

(14.52) |

where

(14.53) |

and

(14.54) |

The frequency band is also reduced by a factor 2 at each step. Applying the sampling theorem, we can build a pyramid of elements. For an image analysis the number of elements is . The overdetermination is not very high.

The B-spline functions are compact in this directe space. They
correspond to the autoconvolution of a square function. In
the Fourier space we have:

(14.55) |

(14.56) |

In the direct space we get:

(14.57) |

This function is quite similar to a Gaussian one and converges rapidly to 0. For 2-D the scaling function is defined by , with . It is an isotropic function.

The wavelet transform algorithm with *n*_{p} scales is the following one:

- 1.
- We start with a B3-Spline scaling function and we derive ,
*h*and*g*numerically. - 2.
- We compute the corresponding image FFT. We name
*T*_{0}the resulting complex array; - 3.
- We set
*j*to 0. We iterate: - 4.
- We multiply
*T*_{j}by . We get the complex array*W*_{j+1}. The inverse FFT gives the wavelet coefficients at the scale 2^{j}; - 5.
- We multiply
*T*_{j}by . We get the array*T*_{j+1}. Its inverse FFT gives the image at the scale 2^{j+1}. The frequency band is reduced by a factor 2. - 6.
- We increment
*j* - 7.
- If , we go back to 4.
- 8.
- The set describes the wavelet transform.

If the wavelet is the difference between two resolutions, we have:

(14.58) |

and:

(14.59) |

then the wavelet coefficients can be computed by .