4.2.1
Sparsity of Variations
MR images are usually piecewise smooth because of the structures of human organs and thus they are approximately sparse in the wavelet transform domain [LDP07, LV09, LLAV11] which has been widely used to compress natural images. Wavelet transform are based on small waves, called wavelets, the choice of which depends on the properties of a signal we wish to analyse. It decomposes a signal with sets of scaled and translated waves, and its discrete version (i.e. discrete wavelet transform (DWT)) is normally used in practice where the wavelets are discretely sampled. The scale factors reflect the fre- quency information of the signal: small scale factors measure the details and noise (i.e. high frequency components) of the signal; in contrary, large scale factors characterise the coarse structure (i.e. low frequency components) of the signal.
Figure 4.1: Discrete Wavelet Transform (DWT) of a One Dimensional Signal.
one dimensional signal S into coefficients A and D by using a low-pass(LP) and a high- pass(HP) filter respectively, where A refer to the coefficients of the low frequency com- ponents while D refer to the coefficients of the high frequency components. As an image
(a) DWT image (b) Daubechies-4 wavelet
Figure 4.2: One-level Daubechies-4 DWT for an Functional MR Image.
is a two dimensional signal, it requires a 2D DWT which is implemented by applying the one dimensional DWT to the rows of the image and then applying the one dimensional DWT to the columns of the already horizontal transformed image. Figure 4.2(a) shows an example of one-level Daubechies-4 discrete wavelet transform (DWT) of an functional MR image. Daubechies-v is a common wavelet used in signal processing and MR imag- ing technique [LDP07, LLAV11]. It is easy to implement fast wavelet transform so that it is practical in real applications. A Daubechies wavelet can be decomposed into two func- tions: wavelet function and scaling function which represent the high pass and the low pass filter respectively. The smoothness of both functions are controlled by the parameter v which is called vanishing moment: a larger value of v results in more smooth wavelet and scaling functions. In this work, I empirically choose the Daubechies-4 wavelet which is normally used in the MR imaging technique [LDP07, LLAV11], and its functions are shown in Figure 4.2(b). The 2D DWT decomposes the original image into four sub im- ages which characterise different components of the image. The sub image in the upper-
left contains the low frequency components that produces the final approximation image. It represents approximated version of the original at half the resolution. On the other hand, the details of the original image in vertical, horizontal and diagonal dimensions are represented by the sub images in upper-right, bottom-left and bottom-right respectively. Furthermore, the low frequency components can be further decomposed and therefore a two-level DWT is produced (as shown in Figure 4.3).
Figure 4.3: Two-level Daubechies-4 DWT for an Functional MR Image.
Rather than assuming the functional MR images are sparsely represented in the wavelet domain, in my work I assume that variations of functional MR images are sparse over time in the wavelet domain. I demonstrated this for a fMRI sequence [LLAV11] in Figure 4.4. The sparsity level is determined by |nd\nt|, where ntrefers to the number of two-
level Daubechies-4 2D DWT coefficients which support 96% energy of the functional MR image at time t, and nd = |nt\nt−1| refers to the number of DWT coefficients changes
with respect to the previous frame. In most cases, the number of variations is less than 10% of the signal size, while in the worst case it is less than 13%
Figure 4.4: Example of Sparse Variations. The variations are calculated with 96% energy supports of DWT coefficients of images
4.2.2
Linear Dynamic Sparse Model
Linear dynamic model [KMK12] is a state-space model that describes the probabilistic dependence of a latent state and its corresponding observed measurements. It is charac- terised by a pair of equations: system equation and measurement equation (as shown in Figure 4.5).
Figure 4.5: Linear Dynamic Model.
The system equation defines transition rules from the previous latent state to the current one, while the measurement equation maps the latent state space to the measurement space. My introduced linear dynamic sparse model for functional MR imaging is a special case of the linear dynamic model, where the system equation is modified to meet the sparsity constraint. Both the equations in my model are detailed below.
System Equation
Based on the assumption that variations of functional MR images are sparse, we can model an fMRI sequence as a linear equation with an identity transition matrix:
wt = wt−1+ κt, (4.1)
where random variable wt∈ Rndenotes the DWT coefficients of a functional MR image
at time t. For simplicity, I call wt image in the rest of this chapter. Random variable
κt∈ Rndenotes its sparse variations with respect to the previous image wt−1. To meet the
sparsity constraint, a hierarchical sparseness prior is placed on κt. Each element κtiof the
variation κtis randomly sampled from a zero-mean Gaussian distribution N (κti|0, α−1i ),
the variance αiof which is randomly sampled from a Gamma Γ(αi|a, b). That is,
p(κt|a, b) = n Y i=1 Z ∞ 0 N (κti|0, α−1i )Γ(αi|a, b)dαi. (4.2)
After marginalising the hyperparameter αi, the prior of κt corresponds to a product of
independent Student’s t distribution. Tipping et al. [Tip01] demonstrates a strong sparse property of this hierarchical distribution.
Measurement Equation
The fMRI technique measures a subset of discrete Fourier Transform (DFT) coefficients of MR images. At each time t, the measurement process can be modelled as:
yt= Θtwt+ ηt, (4.3)
where random variable yt ∈ Rm , here called measurements, is a subset of DFT co-
measurement noise. The design matrix Θt is formed by a subset of m vectors selected
from a given projection matrix Θ ∈ Rn×n, which, in my case, is constructed by the DFT
matrix with the inverse DWT matrix (detailed in Section 4.4.1). The budget m is a given positive integer. It determines the number of frequencies to be measured.