Noise Filtering Methods in MRI - Advance Image Processing in Magnetic Resonance Imaging - page 168

noiseless image can be reconstructed using the inverse DWT. As a result, noise
is removed or filtered from the observations. Therefore, the key to effective noise
removal in the wavelet domain is to determine which wavelet coefficients do not
have significant signal energy and, hence, can be discarded without detrimental
signal loss. Of course, a similar procedure could be carried out using Fourier-
domain filtering. But the Fourier method is a spatially global operation that cannot
be adjusted to local spatial variations, thus leading to uncontrolled smoothing in
regions with high-frequency content, such as the edges. On the other hand, the
wavelet basis functions enable DWT-based filtering procedures to adapt to such
spatial variations. Only a brief algorithm outline will be given here; details can
be found in [7-9].
There are three fundamental concepts in the wavelet algorithm: wavelet packet
transformation, best-basis selection, and coefficient thresholding.
In wavelet packet transformation, the original 2-D images are split in terms
of shifts and dilations of the low-pass scaling function and band-pass wavelet to
obtain the relevant 2-D coefficients. For special choices of these functions, the
shifts and dilations form an orthonormal basis. The wavelet decomposition can
be implemented iteratively by successive filtering and downsampling (by a factor
of 2 at each iteration) operations, using the so-called quadrature mirrors filters.
At each step four 2-D images of coefficients are obtained (one low-pass subimage
and three subimages corresponding to wavelet orientations that are horizontal,
vertical, and diagonal).
Wavelet bases are bases of nested function spaces, which can be used to
analyze signals at multiple scales. Best-basis selection consists of finding the
mathematical function that not only completely represents the original signal,
but that also concentrates the maximum amount of structured signal into the
minimum number of coefficients. However, in many applications such as MRI,
the key signal features are not well known and the optimal basis functions
cannot be specified in advance. In such cases, a basis with general properties
is preferable.
The thresholding operation in the wavelet domain is the operation that deter-
mines which wavelet coefficients do not have significant signal energy and hence
can be discarded without detrimental signal loss. Ideally, the wavelet-domain
filtering procedure should be adapted to the local SNR in each wavelet coefficient,
so that wavelet coefficients with very low SNR can be suppressed. Methods for
making this determination are called wavelet-domain filters.
When designing a filter in the wavelet domain, we have to account for the nature
of the noise process. For example, if the SNR in a pixel is greater than 15 dB, the
Rician distribution is assumed to become approximately Gaussian. If the SNR is
below 15 dB, then the Rician distribution deviates from the Gaussian.
The goal of wavelet-domain filtering is to obtain a better estimate of the
noise-free image wavelet coefficients by filtering the observed d j o () coefficients k

at any level and wavelet orientation o. In fact, the wavelet

filter should adapt to the local SNR in each wavelet coefficient in order to suppress
wavelet coefficients with very low SNR.

j , spatial position k