Most real world signals contain certain amount of noise, so, signal denoising algorithms are a topic of interest in many different applications. Wavelet transforms proved to perform very well in noise removal. Their success is based on the fact that, in the transform domain, crucial signal information is contained in a small number of larger magnitude wavelet coefficients. Also, white noise contained in a signal will map to a white noise in the transform domain. It will be represented by a large number of smaller magnitude wavelet coefficients, concentrated about zero. Noise removal can be efficiently carried out by thresholding wavelet coefficients before signal reconstruction. We proposed an adaptive lifting scheme with a goal of improving the transform performance about edges in a signal. The adaptive algorithm is based on the statistical method of intersection of confidence intervals (ICI). It is used on a point-by-point basis, and on each scale, to determine support of the lifting filter P. After deciding the support, filter P is selected from a predefined set of filters. Since the proposed lifting scheme uses filter U defined as: U = P/2, choosing the filter P is equivalent to selecting a wavelet from a predefined set of wavelets. As a final result, longer and smoother wavelets ares used in smooth signal regions, while shorter wavelets are used in higher frequency regions. The approach allows for efficient reconstruction of edges or, in general, higher signal frequencies. We compared the method efficiency to a number of conventional wavelets. It was shown that for the signal classes with prevailing low frequencies, the proposed method is at least comparable to the best performing conventional wavelet. For signal classes characterized by edges or abrupt changes in local signal properties, the method outperforms the conventional wavelets by a significant margin. The shortcoming of the proposed method is its reliance on the ICI gamma parameter, which defines the method sensitivity. Poorly chosen gamma parameter value can have a detrimental effect to a transform performance. As it is not possible to find the optimal Γ value analytically, we proposed a statistical method for its selection. It is based on a distribution of wavelet coefficients at the last decomposition level. Although there is still much room for improvement, it was shown that the method performs reasonably well across a range of signal classes, resolutions and noise levels. The adaptive algorithm was also tested in a real-world application of fluoroscopic image sequences denoising. It is the application in which edge preservation is an essential requirement. Original catheter insertion sequence was examined, as well as the sequence with artificially raised noise level. Set of subsequent images was converted to a 1-D signal and the proposed algorithm applied to it, as for any other native 1-D signal. Denoised 1-D signal is converted back to 3-D and fused with another estimate of denoised images, based on the basic ICI rule. The final result is a high quality denoised image with excellent edge preservation. The proposed adaptive edge preserving lifting scheme and accompanying Γ parameter selection method represent a well performing model of the second generation wavelets, i.e., wavelets which inherit all the benefits and good properties of the classical wavelet transforms, while in the same time introducing additional advantages and features. |