Many algorithms have been proposed during the last decade in order to deal with inverse problems. Of particular
interest are convex optimization approaches that consist of minimizing a criteria generally composed
of two terms: a data fidelity (linked to noise) term and a prior (regularization) term. As image properties
are often easier to extract in a transform domain, frame representations may be fruitful. Potential functions
are then chosen as priors to fit as well as possible empirical coefficient distributions. As a consequence,
the minimization problem can be considered from two viewpoints: a minimization along the coefficients
or along the image pixels directly. Some recently proposed iterative optimization algorithms can be easily
implemented when the frame representation reduces to an orthonormal basis. Furthermore, it can be noticed
that in this particular case, it is equivalent to minimize the criterion in the transform domain or in the image
domain. However, much attention should be paid when an overcomplete representation is considered. In
that case, there is no longer equivalence between coefficient and image domain minimization. This point
will be developed throughout this paper. Moreover, we will discuss how the choice of the transform may
influence parameters and operators necessary to implement algorithms.
We propose in this paper an iterative algorithm for 3D confocal microcopy image restoration. The image quality
is limited by the diffraction-limited nature of the optical system which causes blur and the reduced amount of
light detected by the photomultiplier leading to a noise having Poisson statistics. Wavelets have proved to be
very effective in image processing and have gained much popularity. Indeed, they allow to denoise efficiently
images by applying a thresholding on coefficients. Moreover, they are used in algorithms as a regularization term
and seem to be well adapted to preserve textures and small objects. In this work, we propose a 3D iterative
wavelet-based algorithm and make some comparisons with
state-of-the-art methods for restoration.
Seismic data and their complexity still challenge signal processing algorithms in several applications. The advent
of wavelet transforms has allowed improvements in tackling denoising problems. We propose here coherent noise
filtering in seismic data with the dual-tree <i>M</i>-band wavelet transform. They offer the possibility to decompose
data locally with improved multiscale directions and frequency bands. Denoising is performed in a deterministic
fashion in the directional subbands, depending of the coherent noise properties. Preliminary results show that
they consistently better preserve seismic signal of interest embedded in highly energetic directional noises than
discrete critically sampled and redundant separable wavelet transforms.
Signals and images in industrial applications are often subject to strong disturbances and thus require robust
methods for their analysis. Since these data are often non-stationary, time-scale or time-frequency tools have
demonstrated effectiveness in their handling. More specifically, wavelet transforms and other filter bank generalizations
are particularly suitable, due to their discrete implementation. We have recently investigated a specific
family of filter banks, the <i>M</i>-band dual-tree wavelet, which provides state of the art performance for image
restoration. It generalizes an Hilbert pair based decomposition structure, first proposed by N. Kingsbury and
further investigated by I. Selesnick. In this work, we apply this frame decomposition to the analysis of two
examples of signals and images in an industrial context: detection of structures and noises in geophysical images
and the comparison of direct and indirect measurements resulting from engine combustion.
The objective of this paper is to design a new estimator for multicomponent image denoising in the wavelet transform domain. To this end, we extend the block-based thresholding method initially proposed by Cai and Silverman, which takes advantage of the spatial dependence between the wavelet coefficients. In the case of multispectral images, we develop a more general framework for block-based shrinkage, the blocks being built from various combinations
of the wavelet coefficients of the different image channels at adjacent spatial positions, for a given orientation and resolution level. In the presence of possibly spectrally correlated Gaussian noise, the parameters of the resulting estimator are optimized from the data by exploiting Stein's principle. Simulations show the higher performance of our estimator for denoising multispectral satellite images.