Abstract. We propose using Partial Differential Equation (PDE) techniques in wavelet based image processing to remove noise and reduce edge artifacts generated by wavelet thresholding. We employ a variational framework, in particular the minimization of total variation (TV), to select and modify the retained wavelet coefficients so that the reconstructed images have fewer oscillations near edges while noise is smoothed. Numerical experiments show that this approach improves the reconstructed image quality in wavelet compression and in denoising. 1. Introduction. In this paper, we consider the noise removal and reducing edge artifacts generated by wavelet thresholdings in digital image denoising and compression. The standard wavelet thresholding techniques, including linear (i.e. truncating the high frequency coefficients) and nonlinear thresholdings (i.e. retaining large wavelet coefficients), such as hard, soft wavelet thresholding, wavelet shrinkage, are well understood and have been widely used in many areas. The most remarkable applications are in digital image denoising and compression. There is a large literature on this subject, we simply refer to several books [34] , [45] , [6] and references therein. However, it is also well known that wavelet thresholdings may generate oscillations near discontinuities, which is called Gibbs' phenomenon in math literature and ringing effects in digital image processing. Many methods have been proposed to reduce the severity of this problem. One approach is to introduce geometry features into wavelet constructions (the standard wavelet transforms are not designed to handle the geometries explicitly), so that wavelet representations can express discontinuous functions more efficiently. The popular way along this line is to construct special wavelet-like bases for discontinuous functions. Typical examples include a series of efforts by Donoho and his collaborators, such as wedgelets [23] , ridgelets [24] , curvelets [7] and beamlets [25] , and Le Pennec-Mallat's bandelets [40] . A different approach is to modify the standard wavelet transforms so that fewer large high frequency coefficients are generated near discontinuities. This results in less edge related information being lost in the thresholding processes. Therefore, the edge artifacts are reduced in the reconstructions. Along this direction, Harten proposed an original and general framework for multiresolution representations [29, 30] . Claypoole, Davis, Sweldens and Baraniuk [20] have proposed an adaptive lifting scheme which lowers the length of the wavelet filters near jumps, thus reducing the severity of the Gibbs' effects. We have proposed ENO-wavelet transforms, which apply the idea of Essentially Non-Oscillatory (ENO) schemes invented for shock capturing in computational fluid dynamics (CFD), to design adaptive wavelet transforms such that no large high frequency coefficients are generated through filtering across discontinuities [16] . The ENO-wavelets can essentially eliminate oscillations in the reconstructed functions if they are piecewise smooth.
Tony and Zhou proposed a technique for noise removal and image compression in wavelet domain thresholding which is based on Partial Differential Equation (PDE) and it takes the advantage of variations in framework REF .
Total Variation Wavelet Thresholding
Journal of Scientific Computing
Mathematics Computer Science