The measured data needs to have a domain in which the data is sparsely represented (2.5.2). The particular sparse transform to utilise is an area of much research in the application of CS to MRI (CS-MRI), and is yet unproven. In the application of CS-MRI the sparse domains of the wavelet, Discrete Cosine Transform (DCT) and the identity transforms have been used (Lustig, Donoho et al. 2008; Haifeng, Dong et al. 2009). In previous work on CS-MRSI the wavelet transform has been utilised although this has not been proven as the optimal sparse MRSI domain. The spectra in the MRSI dataset (x,y,f) exhibit sparsity amidst the peaks surrounded by low level noise and for PRESS (section 2.2.1) the data in the image itself is sparse given that the useful data in VOI is much smaller than the whole FOV. It was therefore initially decided to utilise the identity transform. The identity transform is inherently computationally faster and less complex than other sparsifiying transforms. This identity domain was later compared to the wavelet domain.
Wavelets 3.3.1
In this research wavelets were also investigated as a sparse domain representation of the MRSI data. If a signal possesses variation in frequency with time such as sharp spikes or discontinuities, then a Fourier Transform will not appropriately represent the frequencies present. The problem is such that it is not always possible to know the exact time-frequency representation of a signal due to the finite sampling of the signal. The problem results in the exact spectral component at a particular instance of time being undeterminable. What can be determined are the time intervals in which a band of frequencies may exist, a resolution problem. The use of techniques such as the Short Term Fourier Transform (STFT) may be utilised to determine the frequency-time relationship. The STFT segments, or windows the signal into intervals where it is assumed to be stationary, and non-varying in frequency with
time. The window width must be the same as the stationary portion under analysis. The window starts at t=0 and is multiplied by the signal at this location and then Fourier Transformed. The window is then shifted along, the process repeated and the result summed. Unfortunately because of the finite width of the window function, the STFT is limited in its frequency resolution. It does not give the exact frequencies present in the signal, only a band. If a window of infinite length were used to enhance frequency resolution then this is identical to the original Fourier Transform. The use of wavelets is a better approximation than the STFT. Wavelet analysis is achieved in a similar way to the STFT except that the finite window called a wavelet varies in width according to each spectral component, and in that the Fourier Transform for each window is not taken. The wavelet is contracted for low frequency analysis and dilated for high frequency analysis. The degree to which the wavelet contracted or dilated is termed the scale. The initial wavelet used in the analysis is termed the ‘Mother Wavelet’ and is a prototype for each subsequent wavelet. The choice of mother wavelet can vary dependent upon application. There are different wavelet families such as Daubechies, Coiflets and symmlets which vary according to the number of coefficients and by the level of iteration.
The original signal is represented in terms of wavelet expansion using coefficients in linear combination of each wavelet function. This therefore allows truncation of the wavelet coefficients lending itself to sparse representation. Since this study is examining the ability of CS to reconstruct metabolite edges, a step-like wavelet, the Daubechie 1 (Haar wavelet) was utilised as the mother wavelet for this application.
3.4 Image Reconstruction in CS-MRSI
In CS-MRSI a random k-space sampling pattern is defined 3.2. After acquisition of the random data there will be a sub-sample of measured data. In order to fill in the zeros, the
minimisation of the square error of the measured and the desired datasets via the following objective function are achieved:-
‖𝑭𝒖𝒎 − 𝒚‖𝟐< 𝝐 Equation 3-1
Where:
m = The desired reconstruction of the MRSI dataset (x,y,f).
𝐹𝑢 = The Fourier Transform operator of the under-sampling scheme
y = The sampled k-space data.
𝝐 = a parameter to control the reconstruction quality, set to be below the level of expected noise.
This is combined with the L1 norm of the transform sparsity requirement which can be expressed as:-
‖𝚿𝒎‖ 𝟏 Equation 3-2
Where:
m = The desired reconstruction of the MRSI dataset (x,y,f). The sparsifying transform.
The L1-norm is used to maintain the sparsity. This is because the L1 norm will favour fewer larger components and hence sparsity, whereas the L2-norm penalises large components, maintaining larger datasets of smaller values. The L-norm is expressed as:
‖𝒙‖𝒑= (∑𝒏𝒊=𝟏|𝒙𝒊|)𝟏/𝒑 Equation 3-3
𝑚𝑖𝑛𝑖𝑚𝑖𝑠𝑒 ‖Ψ𝑚‖ 1
𝒔. 𝒕. ‖𝑭𝒖𝒎 − 𝒚‖𝟐 < 𝝐 Equation 3-4
It has been shown that it is often useful to extend the problem to also include a term for the Total Variation (Tsaig and Donoho 2006). The finite differences transform may be used as a sparsifying transform. It is referred to as Total Variation, TV(m), and defines the sum of the absolute variation in all dimensions. This transform is defined mathematically for an N point signal x(n), 1≤ n≤ N as:
𝑻𝑽(𝒙) = ∑𝑵𝒏=𝟐(𝐱(𝐧) − 𝐱(𝐧 − 𝟏)) Equation 3-5 This can also be expressed in matrix form as:
𝑻𝑽(𝒙) = ‖𝑫𝒙‖𝟏 Equation 3-6 Where
D= matrix of size (n-1) x n of the form:
( −1 1 0 0 ⋯⋮ ⋱ 0 ⋮ 0 ⋯ 0 0 − 1 1 )
This means that as well as requiring the imaging dataset to be sparse in the specific transform domain it is also to be sparse in finite differences which provides an effective smoothing of the image. The problem can now be expressed as:
𝑚𝑖𝑛𝑖𝑚𝑖𝑠𝑒 ‖Ψ𝑚‖ 1 + TV(m)
𝑠. 𝑡. ‖𝐹𝑢𝑚 − 𝑦‖2 < 𝜖
Where:
= weight which determines the importance of the TV term over the sparsifying transform In Lagrangian form this may be expressed as:
‖𝑭𝒖𝒎 − 𝒚‖𝟐+ 𝝀𝟏 ‖𝚿𝒎‖ 𝟏+𝝀𝟐 𝐓𝐕(𝐦) Equation 3-7
Where:
𝜆1= Sparsyfying transform weighting factor
𝜆2= Weighting factor of TV transform
The weighting factors 𝜆1and 𝜆2 is also a subject of research. They will be application
dependent and balanced by the level of expected noise. In this study the values were empirically chosen to be 0.002 and 0.005 respectively with due consideration of (Geethanath 2012; Heikal, Wachowicz et al. 2013).
The above equation is solved using a non-linear iterative reconstruction technique. The iterative technique utilised in this research was conjugate gradient descent, the method already utilised in the sparse MRI package.