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Extensive literature exists on the wavelet transform and its applications
(Meyer 1991, Daubechies 1988, Chui 1992, Meyer 1990, Meyer 1992, Ruskai et
al. 1992). In this paper, our discrete approach of the wavelet transform is
implemented with a version of the so-called à trous (with holes)
algorithm (Holschneider and Tchamitchian 1990, Shenasa 1992, Starck and Bijaoui
1993). One assumes that the sampled data are the scalar products
at pixels k of the function
with a scaling function
which
corresponds to a low pass filter:
The scaling function satisfies the dilation equation,
with being a discrete low pass filter.
Consequently the scalar product
is obtained by:
and the signal difference by
or
when the wavelet function is defined by
The B-spline of degree 3 was used in our calculations.
A reconstruction algorithm uses the
last smoothed array added to all the differences
:
The above à trous algorithm is easily extendible to two-dimensional
space. This leads to a convolution with a mask of 55 pixels for the
wavelet related to a
-spline. At each scale
we obtain a set
(we will call this a ``wavelet plane'' throughout the following
discussion), which has the same number of pixels as the image.
If we use an iterative deconvolution algorithm such as Van Cittert or
Richardson-Lucy, we define , the error at iteration
:
By using the à trous wavelet transform algorithm,
can be defined by the sum of its
wavelet planes and the last smooth
plane (see Eq. 12):
The wavelet coefficients provide a mechanism to extract from the residuals at each iteration only the significant structures. A large part of these residuals are generally statistically insignificant. The significant residual is
is the standard deviation of the noise at scale
, and
is
a function which is defined by
The standard deviation of the noise is estimated from the
standard deviation of the noise in the image. This
is done from the study of noise variation in the wavelet space,
with the hypothesis of a white Gaussian noise.
If the noise in the data is Poissonian, the transform
acts as if the data arose from the
Gaussian white noise model (Donoho 1992) with .
The noise in
(with
)
can be suppressed in the same way as previously.
is decomposed into
wavelet coefficients, and only the significant coefficients are kept.
The reconstruction gives
.
The residual
can be expressed as a function of
(see
Appendix) by
and is obtained by
We show now how the iterative deconvolution algorithms can be modified in order to take into account only the significant structure at each scale.
Van Cittert's (1931) iteration is
with .
Regularization using significant structures leads to:
The basic idea of our method consists of detecting, at each scale, structures
of a given size in the residual and putting them in the restored
image
. The process finishes when no more structures are
detected. Then, we have separated the image
into two images
and
.
is the restored image, which does not contain
any noise, and
is the final residual which does not contain any
structure.
is our estimation of the noise
.
The one-step gradient iteration is
with .
Regularization by significant structures leads to
Define . Then
, and
hence
.
The Richardson-Lucy equation is
and regularization leads to
The standard deviation of the residual decreases until no more significant structures are found. Convergence can be estimated from the residual. The algorithm stops when