Shift invariance associated with good directional selectivity is important for the use of a wavelet transform (WT) in many fields of image processing. Generally, complex wavelet transforms, e.g., the double-tree complex WT (DTCWT), have these useful properties. In this paper, we propose the use of a recently introduced implementation of such a WT, namely, the hyperanalytic WT (HWT), in association with filtering techniques already used with the discrete WT (DWT). The result is a very simple and fast image denoising algorithm. Some simulation results and comparisons prove the performance obtained using the new method.

We here establish and exploit the result that 2D isotropic self-similar fields beget quasi-decorrelated wavelet coefficients and that the resulting localised log sample second moment statistic is asymptotically normal. This leads to the development of a semi-local scaling exponent estimation framework with optimally modified weights. Furthermore, recent interest in penalty methods for least square problems and generalized Lasso for scaling exponent estimation inspires the simultaneous incorporation of both bounding box constraints and total variation smoothing into an iteratively reweighted least-square estimator framework. Numerical results on fractional Brownian fields with global and piecewise constant, semi-local Hurst parameters illustrate the benefits of the new estimators.

The SOund Navigation And Ranging (SONAR) images are perturbed by speckle noise. This paper presents a new denoising method in the wavelet domain, which tends to reduce the speckle, preserving the structural features and the textural information of the scene. Shift invariance associated with good directional selectivity is important for the use of a wavelet transform (WT) in denoising of SONAR images. In this paper, we propose the use of a variant of hyperanalytic WT, which is quasi-shift invariant and has good directional selectivity in association with a maximum a posteriori filter named bishrink. This filter makes a very good treatment of the contours. The corresponding denoising algorithm is simple and fast. Its performance was proved on images perturbed by synthesized speckle noise and on real SONAR images.

Semi-local Hurst estimation is considered for random fields where the regularity varies in a piecewise manner. The recently developed generalised lasso is exploited to propose a spatially regularised Hurst estimator. Dual-tree complex wavelets are used to formulate the usual log-spectrum regression problem and an interlaced penalty matrix is constructed to form a 2-d fused lasso constraint on the double-indexed parameters. We thus extend a regularity-based denoising approach and demonstrate the utility of our method with experiments.

The imperfections of image acquisition systems produce noise. The majority of edge detectors, including gradient-based edge detectors, are sensitive to noise. To reduce this sensitivity, the first step of some edge detectors’ algorithms, such as the Canny’s edge detector, is the filtering of acquired images with a Gaussian filter. We show experimentally that this filtering is not sufficient in case of strong Additive White Gaussian or multiplicative speckle noise, because the remaining grains of noise produce false edges. The aim of this paper is to improve edge detection robustness against Gaussian and speckle noise by preceding the Canny’s edge detector with a new type of denoising system. We propose a two-stage denoising system acting in the Hyperanalytic Wavelet Transform Domain. The results obtained in applying the proposed edge detection method outperform state-of-the-art edge detection results from the literature.

We here harmonise two significant contributions to the field of wavelet analysis in the past two decades, namely the locally stationary wavelet process and the family of dual-tree complex wavelets. By combining these two components, we furnish a statistical model that can simultaneously access benefits from these two constructions. On the one hand, our model borrows the debiased spectrum and auto-covariance estimator from the locally stationary wavelet model. On the other hand, the enhanced directional selectivity is obtained from the dual-tree complex wavelets over the regular lattice. The resulting model allows for the description and identification of wavelet fields with significantly more directional fidelity than was previously possible. The corresponding estimation theory is established for the new model, and some stationarity detection experiments illustrate its practicality.