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62 3. Standard destriping techniques and application to MODIS
3.7.6 Destriping with wavelet coefficient thresholding
As exposed in the previous section, the discrete dyadic wavelet transfom offers a sparse
representation of a signal, well suited for many image processing applications. In the case
of denoising, a major limitation not mentionned this far is the translation invariance condition,
not satisfied by the discrete dyadic wavelet transfom. This drawback does not affect
the wavelet decomposition in itself, however, if the wavelet coefficients are modified, the
wavelet reconstruction introduces artifacts along the image discontinuities. These artifacts,
visible as pseudo-Gibbs oscillations come from the decimation of wavelet coefficients between
successive decomposition levels. To avoid these effects, we will rely on the stationary
wavelet transform [Nason and Silverman, 1995]. The multiscale decomposition of an image
provides wavelet coefficients with a magnitude dependent on the local regularity of the
signal in a given resolution. Assuming a moderate amount of noise in the observed signal,
strong wavelet coefficients will be distributed along the edges of an image while weak coefficients
will be located in homogenenous areas. The noise can then be reduced from the
image by considering only specific wavelet coefficients in the reconstruction process. The
seminal work of [Donoho and Johnstone, 1994] describes two denoising techniques based
on the thresholding of wavelet coefficients. A first intuitive approach consists in setting
to zero all wavelet coefficients with an amplitude lower than a fixed threshold λ. This
operation is known as hard thresholding and is represented by a function S hard that takes
the following form :
{ 0 if d
Sλ hard (d k k
j )=
j ≤ λ
d k j if d k j >λ (3.47)
where {d k j } k=1,2,3 are detail coefficients in the horizontal (k = 1), diagonal (k = 2) and
vertical (k = 3) directions, obtained at the resolution 2 j .
An alternative approach in the selection of wavelet coefficients is based on soft thresholding
or wavelet shrinkage ; Coefficients that exceed a threshold are attenuated by the value of
the threshold. The resulting thresholding function S soft
λ
is given by :
{
S soft
0 if d
λ
(d k k
j )=
j ≤ λ
d k j − sign(dk j )λ if dk j >λ (3.48)
The quality of the reconstructed signal highly depends on the choice of the threshold which
can be the same for all resolutions or vary from one resolution to the other. A common
methodology relies on the universal thresholding of [Donoho and Johnstone, 1994] where
the value of the threshold is fixed as :
λ =ˆσ √ 2 log(M × N) (3.49)
where M × N is the number of pixels in the image and ˆσ is an estimation of the noise
variance, determined from the detail coefficients of the highest resolution with :
λ
ˆσ = median{dk 0 } k=1,2,3
0.6745
(3.50)