Image Denoising using Wavelet Thresholding Techniques Submitted by Yang

Introduction Image denoising: Removing unwanted noise in order to restore the original image. Wavelet transform provides us with one of the methods for image denoising. Wavelet transform, due to its excellent localization property, has rapidly become an indispensable signal and image processing tool for a variety of applications, including denoising and compression. Wavelet denoising attempts to remove the noise present in the signal while preserving the signal characteristics, regardless of its frequency content.

Introduction It involves three steps:  a linear forward wavelet transform  nonlinear thresholding step and  a linear inverse wavelet transform Methods Used  Universal Thresholding  Visu Shrink  Sure Shrink  Bayes Shrink

Wavelet Thresholding Wavelet thresholding (first proposed by Donoho) is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. It removes noise by killing coefficients that are insignificant relative to some threshold. Researchers have developed various techniques for choosing denoising parameters and so far there is no “best” universal threshold determination technique. Types  Universal or Global Thresholding Hard Soft  SubBand Adaptive Thresholding

The soft thresholding operator is defined as D(U, λ) = sgn(U)max(0, |U| - λ) Soft thresholding shrinks coefficients above the threshold in absolute value. The transfer function of the same is shown here. The hard thresholding operator is defined as D(U, λ) = U for all |U|> λ Hard threshold is a “keep or kill” procedure and is more intuitively appealing. The transfer function of the same is shown here. Hard & Soft Thresholding

The threshold (N being the signal length, σ being the noise variance) is well known in wavelet literature as the Universal threshold. It is the optimal threshold in the asymptotic sense and minimizes the cost function of the difference between the function and the soft thresholded version of the same in the L2 norm sense. It is useful for obtain a starting value when nothing is known of the signal condition. Universal or Global Thresholding

Results with Hard & Soft Thresholds (Universal thresholding)

VisuShrink is thresholding by applying the Universal threshold proposed by Donoho and Johnstone. This threshold is given by where σ is the noise variance and M is the number of pixels in the image. For denoising images, VisuShrink is found to yield an overly smoothed estimate. VisuShrink

Results with Visu Shrink

SURE Shrink SUREShrink is a thresholding by applying subband adaptive threshold. It is based on Stein’s Unbiased Estimator for Risk (SURE), a method for estimating the loss in an unbiased fashion. Let wavelet coefficients in the jth subband be { Xi : i =1,…,d } For the soft threshold estimator we have Select threshold t S by

Bayes Shrink BayesShrink is an adaptive data-driven threshold for image denoising via wavelet soft-thresholding. We assume generalized Gaussian distribution (GGD) for the wavelet coefficients in each detail subband. We then try to find the threshold T which minimizes the Bayesian Risk.

Results with SURE & Bayes Shrink

Comparison based on minimum MSE