Image Denoising Using Wavelets: Application in Medical Imaging - Advances in Heuristic Signal Processing and Applications - page 297

• The signal x and the noise b are uncorrelated, namely, R bx = 0 , ∀ k ;

• The noise, b , is an additive white Gaussian process, with zero-mean and vari-

ance σ b ;

• The useful signal, x , is a random Gaussian process with zero-mean and vari-

ance σ x .
Consequently, the observed signal is a zero-mean process with variance σ y =
σ x + σ y I where I denotes the identity

matrix. The intercorrelation between the observed and the noiseless signal is then
R yx (k)

σ b . Its autocorrelation is reduced to R y (k) =

σ y δ(k),

= ∀ k where δ(k) is defined by

1 f k

= 0,

=

δ(k)

0 otherwise.

In these conditions, the solution of Eq. ( 13.20 ) is then:

σ x
σ x +

h 0 = , and h 1 = h 2 =···= h N = 0 .

σ b

The denoised signal can be put in the following form:

σ x
σ x + σ b

SNR

x = y =

SNR y

(13.21)

1 +

σ X
σ b

where SNR ; it represents the signal-to-noise ratio of the observed signal. In

the case of an image, the variance of the useful signal, σ X , can vary in space and
must be estimated locally.
The relations deduced for time (or space)-domain signals are also available in
the wavelet domain, under the same hypotheses, the estimated coefficients being
computed with:

=

σ X
σ X + σ b

d

d j .

j =

(13.22)

In the context of medical imaging (see Sect. 13.4 ), we compare the performance
of this denoising method to shrinkage wavelet denoising both visually an in terms
of PSNR.

13.3.4 Suppression of Correlated Noise

Images captured by digital devices often contain noise. Various methods of wavelet-
based image exist, but their performance is limited in the presence of correlated
noise in the image or signal of interest.