Dual-tree Complex Wavelet Transform based denoising for Random Spray image enahcement methods

This work introduces a novel way to reduce point-wise noise introduced or exacerbated by image enhancement methods leveraging the Random Spray sampling approach. Due to the nature of the spray, the sampling structure used, output images for such methods tend to exhibit noise with unknown distribution. The proposed noise reduction method is based on the assumption that the non-enhanced image is either free of noise or contaminated by non-perceivable levels of noise. The dual-tree complex wavelet transform is applied to the luma channel of both the nonenhanced and enhanced image. The standard deviation of the energy for the non-ehanced image across the six orientations is computed and normalized. The normalized map obtained is used to shrink the real coefficients of the enhanced image decomposition. A noise reduced version of the enhanced version can then be computed via the inverse transform. A thorough numerical analysis of the results has been performed in order to confirm the validity of the proposed approach. Introduction The field of image enhancement has been one of the most active even before digital imagery achieved a consumer status, but despite its age it has never stopped evolving. The present work introduces a novel denoising method, tailored to address a specific image quality problem expressed by Random Spray based image enhancement algorithms. Random sprays are two-dimensional collection of points (coordinates) with a given spatial distribution around the origin. Such structures can be scaled and translated to sample an image’s support in a way similar to that employed by the Human Visual System (HVS). Yet, the peaked nature of the sprays introduces unwanted noise in the output image. The amount and statistical characteristics of the noise so introduced depend on several factors, among which are image content and spray properties. Thus, common noise reduction methods tailored to deal only with one particular kind of noise (e.g. Gaussian noise) would not find the expected conditions. The method here described approaches the problem via wavelet coefficient shrinkage. Algorithms based on wavelet shrinkage have a long history, nonetheless this work presents a novel view on the subject. This article was particularly inspired by the works on the Dual-tree Complex Wavelet Transform by Kingsbury [6], the work on the Steerable Pyramid Transform by Simoncelli et al. [15], and the work on Wavelet Coefficient Shrinkage by Donoho and Johnstone [3]. Dual-Tree Complex Wavelet Transform Kingsbury developed the Complex Wavelet Transform (CWT) in order to solve certain problems that arise with the traditional Discrete Wavelet Transform (DWT), as well as other more advanced methods such as the Steerable Pyramid Transform (SPT) [5]. Similarly to the SPT, in order to retain the whole Figure 1: Quasi-Hilbert pairs wavelets used in the Dual Tree Complex Wavelet Transform Fourier spectrum, the CWT needs to be overcomplete by a factor 4, i.e. there are 3 complex coefficients for each real one. The CWT is also efficient, as it can be computed through separable filters, yet it lacks the Perfect Reconstruction property. Kingsbury also introduced the concept of Dual-tree Complex Wavelet Transform (DTCWT), which has the added characteristic of Perfect Reconstruction at the cost of only approximate shift-invariance [6]. Since the a full discussion on the Dual-Tree Complex Wavelet Transform would be too cumbersome, only a brief introduction the 2D variant of the DTCWT is given. The reader is referred to the the work by Selesnick et al. [14] for a very comprehensive coverage on the DTCWT and the relationship it shares with other transforms. The 2D Dual Tree Complex Wavelet Transform can be implemented by using two distinct sets of separable 2D wavelet bases, as shown below. ψ1,1(x,y) = φh(x)ψh(y), ψ2,1(x,y) = φg(x)ψg(y), ψ1,2(x,y) = ψh(x)φh(y), ψ2,2(x,y) = ψg(x)φg(y), ψ1,3(x,y) = ψh(x)ψh(y) ψ2,3(x,y) = ψg(x)ψg(y) (1) ψ3,1(x,y) = φg(x)ψh(y), ψ4,1(x,y) = φh(x)ψg(y), ψ3,2(x,y) = ψg(x)φh(y), ψ4,2(x,y) = ψh(x)φg(y), ψ3,3(x,y) = ψg(x)ψh(y) ψ4,3(x,y) = ψh(x)ψg(y) (2) The following equations shows the relationship between wavelet filters h and g g0(n)≈ h0(n−1), for j = 1 (3) g0(n)≈ h0(n−0.5), for j > 1 (4) where j is the decomposition level. When combined, the bases give rise to two sets of real, two dimensional, oriented wavelets (see Fig. 1). ψi(x,y) = 1 √ 2 ( ψ1,i(x,y)−ψ2,i(x,y) )