Dual-tree complex wavelet coefficient magnitude modelling using the bivariate Cauchy–Rayleigh distribution for image denoising

• Accurately model the univariate complex magnitude statistics of the DT-CWT transform.
• Accurately model the bivariate complex magnitude statistics of the DT-CWT transform.
• Produce a denoising application based on the above.

Wavelet shrinkage is a standard technique for denoising natural images. Originally proposed for univariate shrinkage within the Discrete Wavelet Transform (DWT) domain it has been made more effective through the use of (approximately) translationally invariant wavelet decompositions such as the Dual-Tree Complex Wavelet Transform (DT-CWT) and bivariate shrinkage. Techniques using the DT-CWT benefit from both (approximate) translational invariance and improved directionality. However, these techniques have used the real and imaginary components of the complex valued subband coefficients separately. The proposed work instead uses coefficient magnitudes to produce a novel bivariate shrinkage technique based on a heavy tailed bivariate distribution (of magnitudes) to provide a quantitative improvement in image denoising.The results were compared to state of the art non-local means denoising technique BM3D. The PSNR results for small amounts of noise were comparable and within a small range for larger amounts of noise. However, when using the perceptually based structural similarity metric (SSIM) our developed technique offers improved results across the range of noise inputs when compared to BM3D in many cases. Perceptually, the developed technique is able to retain a greater quantity of the high frequency elements of the input image compared to BM3D.