The theory of compressed sensing is often considered a mature field by non-specialists. While this belief proves to be quite accurate for Gaussian measurements, many impor- tant questions are still open when dealing with structured measurements and structured signal recovery, as met in MRI. In this section, we review a few major theoretical results and open questions to motivate our experimental study.
2.2.1 The case of unstructured measurements
Let
σs(z) = min
z0∈Cn, s-sparsekz−z
0k1 (2.1)
denote the `1-tail of a vector z ∈ Cn. This function is often used to characterize the compressibility of a signal z.
The following theorem ([)Theorem 9.13]foucart2013mathematical provides an accu- rate description of the recovery guarantees in the case of unstructured Gaussian mea- surements.
Theorem 1 Assume that A∈ Cm×nis a matrix with i.i.d. random Gaussian components. There exist universal constants C1, C2, D1and D2such that, for any 1≤s ≤n, for any e∈ (0, 1), if
2.2. The theory of CS in MRI and its limitations 39
then, with probability at least 1−e, for all vectors z ∈Cn, given the measurements y =Az+
e∈Cm, where e is a measurement error satisfyingkek2
2 ≤mσ2, we get: kz− ˆzk2≤ D1 σs(z) √ s +D2 √ mσ, (2.3) where ˆz= arg min kAz0−yk2≤√mσk z0k1. (2.4)
The value of this theorem lies in the fact that it provides a good understanding of the reconstruction quality with respect to the signal’s compressibility (captured by σs(z))
and the input SNR (captured by σ). In particular, it shows that if s 7→ σ√s(z)
s decreases
sufficiently fast with s, a small number of measurements will be sufficient to reconstruct the true signal, up to an error proportional to kek2. The`1-tail σs explains the role of
resolution in the theory of CS. Fig. 2.1 shows the evolution of the normalized σs(z)with
respect to the normalized sparsity s/n for the phantom image Fig. 2.3A) at different resolutions (n=128×128, n=512×512 and n= 2048×2048). Two observations can be drawn from this graph: on the one hand, the larger the relative sparsity of z, the smaller its `1-tail σs(z). On the other hand and more importantly, the higher the resolution,
the faster the `1-tail decay. Hence, CS will allow using higher sub-sampling factors at
higher resolution for an equal error. Similar phenomena are explained in more details in
(Roman et al., 2014). 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 s/n σs ( z ) k z k1 n=128×128 n=512×512 n=2048×2048
FIGURE2.1: Normalized `1-tail σs(z) = minz0∈Cn, s-sparsekz−z0k1 with
respect to the relative sparsity s/n for different resolutions of the brain phantom image in Fig. 2.3A): n=128×128, n=512×512 and n=2048×
2048. The`1-tail decays faster at higher resolutions, thus allowing the use of larger undersampling factors for higher resolutions in the context of CS.
Let us now describe the limitations of this theorem. First, it does not capture the de- noising capabilities of`1reconstructions. The term√mσ in (2.3) coincides with the amount of noise in the data and increases with the number of measurements m. In practice it is often observed that`1minimization not only allows to recover missing information, but also serves as a regularizer able to denoise the data. Second, the constants C1, C2, D1and
D2are not tight and can be huge in some variants of this theorem, meaning that the lower
bound on the number of measurements m may actually be quite large, or equivalently that the undersampling factor R=n/m may be close to 1 or even smaller than 1.
This last criticism has to be moderated by the theoretical analysis of phase tran- sitions in (Amelunxen et al., 2014). It is now well known that in the case of noise- less measurements, perfect recovery will occur with high probability whenever m ≥ 2s(log(n/s) +1) +eand will fail with high probability when m ≤2s(log(n/s) +1)−e,
where e is a small margin. This shows that the minimum number of measurements for good recovery is very well understood in this case.
2.2.2 The case of structured measurements and structured signals
In MRI, signals are highly structured: all brain images share strong similarities and can be modeled much more precisely than arbitrary s-sparse signals. A simple model to describe this structure is the sparsity by levels in wavelet bases(Adcock et al., 2013): each wavelet sub-band of the image contains a number of nonzero coefficients bounded by a known quantity at each scale. (Fig. 2.2) illustrates a brain phantom image and its wavelet decompositions for four sub-bands. Moreover, the traditional way of acquiring data in MRI is far from independent Gaussian measurements: Fourier transform values of the image are probed along continuous trajectories.
(a)Brain phantom (b)Wavelet representation FIGURE2.2: A brain phantom image of size N =256 (a) and its wavelet representation (b) for a Symmlet wavelet basis and 4 leves of decomposi-
tion.
The current sampling theory in this challenging setting can safely be described as sig- nificantly less comprehensive than the case of Gaussian measurements. Let us however remark that significant advances were proposed recently (Adcock et al., 2013; Roman
et al., 2014; Krahmer and Ward, 2014; Bigot et al., 2016; Boyer et al., 2017). Without intro-
ducing the theorems, let us detail the main conclusions and limitations of these studies. First, the use of incoherent sampling and reconstruction bases is not necessary if the sparsity pattern is adapted to the sampling scheme (Boyer et al., 2016). This is the case in particular for MRI, where most images are compressible in the wavelet domain. Second, one important result argues that low frequencies should be probed more often than high frequencies. The reason is subtle and not just due to the fact that there is more informa- tion in the low frequencies (Puy et al., 2011). To understand the reconstruction limits, one needs to precisely describe the links between wavelet bases, Fourier bases and sparse by levels signals. The current major limitations can be listed as follow: