The Wavelet Transform

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.

Regularization using Significant Structures

Gaussian Noise

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.

Poissonian 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.

Regularization of Van Cittert's Algorithm

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 .

Regularization of the One-Step Gradient Method

The one-step gradient iteration is

with . Regularization by significant structures leads to

Regularization of the Richardson-Lucy Algorithm

Define . Then , and hence . The Richardson-Lucy equation is

and regularization leads to

Convergence

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