Complex Wavelet Transform in Biomedical Image Denoising

The discrete wavelet transform (DWT) has proved very valuable in a large scope of signal processing problems. However, in many applications, it reaches its limitations, such as oscillations of the coefficients at a singularity, lack of directional selectivity in higher dimensions, aliasing and consequent shift variance. To overcome these problems, the complex wavelet transform (CWT) employs analytic filters, i.e. their real and imaginary parts form the Hilbert transform (HT) pair, securing magnitude-phase representation, shift invariance, and no aliasing. The CWT strategy, that we focus on in this paper, is Kingsbury’s and Selesnick’s dual tree CWT (DTCWT). This moderately redundant multiresolution transform with decimated subbands runs in two DWT trees (real and imaginary) of real filters producing the real and the imaginary parts of the coefficients. Due to its shift invariance and improved directional selectivity, the DTCWT outpreformes the critically decimated DWT in a range of applications, such as, motion estimation, image fusion, edge detection, texture discrimination and denoising. In the final part of this paper, we present biomedical CT image denoising by the means of thresholding magnitude of the wavelet coefficients.