Image Denoising Using Wavelets: Application in Medical Imaging - Advances in Heuristic Signal Processing and Applications - page 291

ered a nonparametric method, i.e., it makes no a priori assumption. It is distinct from
parametric methods in which parameters must be estimated for a particular model
that is assumed a priori. For example, the most popular parametric method is that of
using least squares estimation.

13.3.1 Additive Gaussian White Noise Model

The following is a measurement model:

Y(k) = X(k) + B(k) (13.10)

where Y(k) is the measurement signal of size N , X(k) is the a priori unknown
useful signal, and B(k) is a random noise perturbation (usually assumed to be white
Gaussian with variance σ 2 ).
The elimination or reduction of this additive noise can be achieved nonlinearly by
using multiresolution analysis under the assumption that the appropriate choice of a
decomposition basis allows discrimination of the useful signal from noise. The un-
derlying idea is that the useful signal can be described by a small number of coeffi-
cients of high-amplitude wavelets, and that the noise is spread across all coefficients.
This hypothesis justifies, in part, the traditional use of denoising by thresholding.
If d j (k) represent the wavelet coefficients of the measured signal, the estimation
of the wavelet coefficients of the useful signal, denoted d

j (k) , is generated by two

types of thresholding:

• Hard thresholding

d j (k) if

d j (k)

| | >S ,

d

j (k)

= (13.11)

0 otherwise;

• Soft thresholding

⎧
⎨
⎩

d j (k) S if d j (k)>S ,

d j (k) + S if d j (k)< − S ,
0

−

d

j (k)

= (13.12)

otherwise

σ √ 2ln N where N is the size of the measured signal, and σ represents the
noise standard deviation.
A robust estimator of σ is given by:

=

with S

Med d 1 (k)

σ = 1 . 4826 × (13.13)

d 1 (k) where Med | | designates the median value of the wavelet coefficients, for

d 1 (k) j = 1, in increasing order {| | , 0 ≤ k ≤ N/ 2 } .