Image denoising via the redundant wavelet transform  (Page 2/3)

Denoising algorithms based on the rwt

Soft-thresholding

In the traditional method of soft-thresholding, where the universal threshold is used, coefficients below a specified threshold are shrunk to zero while those above the threshold are shrunk by a factor of $\widehat{\sigma }\sqrt{(}2log\left(N\right))$ . On the orthogonal wavelet transforms, it has been shown to exhibit the following property:

Theorem 1 For a sequence of i.i.d. random variables $z_i\sim N(0,1)$ , $P({max}_{i=1..N}\le \sqrt{(}2logN))\to 1$ for $N\to \infty$ .

Bivariate shrinkage

Sendur and Selesnick proposed a bivariate shrinkage estimator by estimating the marginal variance of the wavelet coefficients via small neighborhoods as well as from the the corresponding neighborhoods of the parent coefficients. The developed method maintains the simplicity and intuition of soft-thresholding.

We can write

where ${\mathbf{w}}_k$ are the parent and child wavelet coefficients of the true, noise-free image and ${\mathbf{n}}_k$ is the noise. We have for our variance, then, that

Noting that we will always be working with one coefficient at a time, we will suppress the $k$ .

In , Sendur and Selesnick proposed a bivariate pdf for the wavelet coefficient $w_1$ and the parent $w_2$ to be

where the marginal variance ${\sigma }^2$ is dependent upon the coefficient index $k$ . They derived their MAP estimator to be

To estimate the noise variance ${\sigma }_n^2$ from the noisy wavelet coefficients, they used the median absolute deviance (MAD) estimator

where the estimator uses the wavelet coeffiecients from the finest scale.

The marginal variance ${\sigma }_y^2$ was estimated using neighborhoods around each wavelet coefficient as well as the corresponding neighborhood of the parent wavelet coefficient. For instance, for a 7x7 window, we take the neighborhood around $y_{1,(4,4)}$ to be the wavelet coefficients located in the square (1, 1), (1, 7), (7, 7), (7, 1) as well as the coefficients in the second level located in the same square; this square is denoted $N\left(k\right)$ . The estimate used for ${\sigma }_y^2$ is given by

where $M$ is the size of the neighborhood $N\left(k\right)$ . We can then estimate the standard deviation of the true wavelet coefficients through :

We then have the information we need to use equation .

Bls-gsm

Portilla, et. al. propose the BLS-GSM method for denoising digital images, which may be used with orthogonal and redundant wavelet transforms as well as with pyramidal schemes. They model neighborhoods of coefficients at adjacent positions and scales as the product of a Gaussian vector and a hidden positive scalar multiplier, so that the neighborhoods are defined similarly as in the BiShrink algorithm. The coefficient within each neighborhood around a reference coefficient of a subband are modeled with a Gaussian scale mixture (GSM) model. The chosen prior distribution is the Jeffrey's prior, $p_z\left(z\right)\propto \frac{1}{z}$ .

They assume the image has additive white Gaussian noise, although the algorithm also allows for nonwhite Gaussian noise. For a vector $\mathbf{y}$ corresponding to a neighborhood of $N$ observed coefficients, we have