The principle behind the wavelet
transform, as described by Grossman & Morlet (1984), Daubechies (1988) and
Mallat (1989), is to hierarchically decompose an input signal into a series
of successively lower resolution reference signals and their associated detail
signals. At each decomposition level, *L*, the reference signal has a
resolution reduced by a
factor of 2^{L} with respect to the original signal.
Together with its respective detail signal, each scale contains the
information needed to reconstruct the reference signal at the next higher
resolution level.
Wavelet analysis can therefore be considered as a series of bandpass filters
and be viewed as the decomposition of the signal into a set of independent,
spatially oriented frequency channels. Using the
orthogonality properties, a function in this decomposition can be completely
characterised by the wavelet basis and the wavelet coefficients of the
decomposition.

The multi-level wavelet transform (analysis stage) decomposes the signal into
sets of different
frequency localisations. It is performed by iterative application of a pair of
Quadrature Mirror Filters (QMF). A scaling
function and a wavelet function are associated with this analysis filter bank.
The continuous scaling
function satisfies the following two-scale equation:

(1) |

(2) |

(3) |

(4) |

(5) |

The system is said to be bi-orthogonal if the following conditions are satisfied:

(6) | ||

(7) | ||

(8) |

Cohen et al. (1990) and Vetterli & Herley (1992) give a complete treatment of the relationship between the filter coefficients and the scaling functions.

- 1.
- A "dyadic'' decomposition refers to a transform in which only the reference
sub-band (low-pass part of the signal) is decomposed at each level. In this
case, the analysis stage is
applied in both directions of the image at each decomposition level. The total
number of sub-bands after
*L*levels of decomposition is then 3*L*+1 (Fig. 1, upper panel). - 2.
- A "pyramidal'' decomposition is similar to a "dyadic'' decomposition in the
sense that only the reference sub-band is decomposed at each level, but it
refers here to a transform that is performed separately in the two directions
of the image. The total number of sub-bands after
*L*levels of decomposition is then (*L*+1)^{2}(Fig. 1, lower panel). - 3.
- A "uniform'' decomposition refers to one in which all sub-bands are
transformed at each level. The total number of sub-bands after
*L*levels of decomposition is then 4^{L}.

The wavelet functions are localised in space and, simultaneously, they are also localised in frequency. Therefore, this approach is an elegant and powerful tool for image analysis because the features of interest in an image are present at different characteristic scales. Moreover, if the input field is Gaussian distributed, the output is distributed the same way, regardless of the power spectrum. This arises from the linear transformation properties of Gaussian variables. The distribution of the wavelet coefficients of a Gaussian process is thus a Gaussian. Conversely, we expect that any non-Gaussian signal will exhibit a non-Gaussian distribution of its wavelet coefficients.

In our study, we have used bi-orthogonal wavelets, which are mainly used in data compression, because of their better performance than orthogonal wavelets, in compacting the energy into fewer significant coefficients. There exist bi-orthogonal wavelet bases of compact support that are symmetric or antisymmetric. Antisymmetric wavelets are proportional to, or almost proportional to, a first derivative operator (e.g. the 2/6 tap filter (filter #5) of Villasenor et al. (1995), or the famous Haar transform which is an orthogonal wavelet). Symmetric wavelets are proportional to, or almost proportional to, a second derivative operator (e.g. the 9/3 tap filter of Antonini et al. 1992). In the frame of detecting non-Gaussian signatures, the choice of the wavelet basis is critical because non-Gaussian features exhibit point sources or step edges. The wavelet must have a very good impulse response and a low shift variance, i.e. it better preserve the amplitude and the location of the details. Villasenor et al. (1995) have tested a set of bi-orthogonal filter banks, within this context, to determine the best ones for image compression. They conclude that even length filters have significantly less shift variance than odd length filters, and that their performance in term of impulse response is superior. In these filters, the high pass QMF is antisymmetric which is also a desirable property in the sense that we will also be interested in the statistical properties of the multi-scale gradients. Consequently, in our study, we have chosen the 6/10 tap filter (filter #3) of Villasenor et al. (1995) (Fig. 2) which represents the best compromise between all the criteria and energy compaction. Using this filter, we have chosen to perform a four level dyadic decomposition of our data. This particular wavelet and decomposition method have already been used for source detection by Aghanim et al. (1998).

Figure 2:
Scaling function (top) and wavelet function (bottom)
corresponding to the filter #3 of Villasenor et al. (1995). Note that the
wavelet function is antisymmetric |

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