The wavelet transform is well suited for approximation of two dimensional functions with certain smoothness characteristics. Also point singularities, e.g. texture-like structures, can be compactly represented by wavelet methods. However, when representing line singularities following a smooth curve in the domain -- and should therefore be characterizes by a few parameters -- the number of needed wavelet coefficients rises dramatically since fine scale tensor product wavelets, catching these steep transitions, have small local support. Nonetheless, for images consisting of smoothly colored regions separated by smooth contours most of the information is comprised in line singularities (e.g. sketches). For this class of images, wavelet methods have a suboptimal approximation rate due to their inability to take advantage of the way those point singularities are placed to form up the smooth line singularity.
To compensate for the shortcomings of tensor product wavelets there have already been developed several schemes like curvelets, ridgelets, bandelets and so on. This paper proposes a nonlinear normal offset decomposition method which partitions the domain such that line singularities are approximated by piecewise curves made up of borders of the subdomains resulting from the domain partitioning. Although more general domain partitions are possible, we chose for a triangulation of the domain which approximates the contours by polylines formed by triangle edges. The nonlinearity lies in the fact that the normal offset method searches from the midpoint of the edges of a coarse mesh along the normal direction until it pierces the image. These piercing points have the property of being attracted towards steep color value transitions. As a consequence triangular edges are attracted to line up against the contours.
Wavelet threshold algorithms replace wavelet coefficients with small magnitude by zero and keep or shrink the other coefficients. This is basically a local procedure, since wavelet coefficients characterize the local regularity of a function. Although a wavelet transform has decorrelating properties, structures in images, like edges, are never decorrelated completely, and these structures appear in the wavelet coefficients. We therefore introduce geometrical prior model for configurations of large wavelet coefficients and combine this with the local characterization of a classical threshold procedure into a Bayesian framework. The threshold procedure selects the large coefficients in the actual image. This observed configuration enters the prior model, which, by itself, only describes configurations, not coefficient values. In this way, we can compute for each coefficient the probability of being `sufficiently clean'.
WAILI is a wavelet transform library, written in C++. It includes some basic image processing operations based on the use of wavelets and forms the backbone of more complex image processing operations. We use the Cohen-Daubechies- Feauveau biorthogonal wavelets. The wavelet transforms are integer transforms, calculated using the integer version of the Lifting Scheme. WAILI is available in source form for research purposes.
KEYWORDS: Wavelets, Wavelet transforms, Signal to noise ratio, Convolution, Fast wavelet transforms, Denoising, Smoothing, Image denoising, Data analysis, Image compression
De-noising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold. The optimal threshold minimizes the error of the result as compared to the unknown, exact data. To estimate this optimal threshold, we use generalized cross validation. This procedure does not require an estimation for the noise energy. Originally, this method assumes uncorrelated noise, and an orthogonal wavelet transform. In this paper we investigate the possibilities of this method for less restrictive conditions.
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