The CS proposed by Candès et al. [11] and Donoho [12] is a new theory to do undersampling and reconstruct the signal of interest from limited physically acquired data. They build a theoretical foundation that one can exactly or approximately recover signals from highly incomplete measurements. The two basic tenets to guarantee the performance of CS are sparsity and incoherence.

(a) Sparsity. For the signal x ∈ R^N and a basis dictionary Ψ ∈ R^{S × N} (e.g., identity matrix, FT, discrete cosine transform or wavelet transform matrix), the sparsity is often interpreted as: (6) S = ‖ α ‖ 0 = ‖ Ψ x ‖ 0 ≪ Nwhere ‖ α ‖₀ denotes the ℓ₀ norm that counts the nonzero entries in α, and S is the number of nonzero entries. If x is sparse without transformation (namely sparse in identity matrix I ∈ R^{N × N}), it is called self-sparse since other complicated sparsifying transform, e.g., wavelet transform, is not required.

Candès et al. [11] and Donoho [12] proved that it is possible to recover the original signal x from O(NlogS) measurements. This means the required number of measurements is proportional to the number of nonzero entries in the basis Ψ. The smaller the S is, the less the number of measurements is required.

(b) Incoherence. When a signal x is sampled by a sensing matrix Φ_{M × N}, the measurements y ∈ R^M of x is: (7) f = Φ x

The coherence is defined as [27,28]: (8) μ ( Φ , Ψ ) = max k , j | 〈 ϕ k , ψ j 〉 |where ∅_k is the kth rows of Φ and Ψ_j is the jth column of Ψ. The coherence measures the largest correlation between any row of Φ and column of Ψ. The less the coherence between Φ and Ψ is, the smaller the μ is. The value range of μ is [ 1 , N ]. The minimal coherence μ = 1 occurs when Φ and Ψ is a time-frequency pair [29]. CS requires the coherence to be as small as possible, which means each measurement vector ∅_k must be ‘spread out’ in the Ψ domain [28].

If the signal x satisfies [30]: (9) S = ‖ α ‖ 0 < 1 2 ( 1 + 1 μ ( Φ , Ψ ) )it can be perfectly recovered by solving: (10) α ^ = min α ‖ α ‖ 0 ,       s . t .       y = Φ Ψ αwhere ‖ α ‖₀ denotes the ℓ₀ norm that counts the nonzero entries in α.

The recovered signal is: (11) x ^ = Ψ α ^

Equation (9) implies that if the coherence between Φ and Ψ is small, more non-zeros can be allowed in the sparse representation α. CS suggests Φ to be random enough to guarantee its incoherence with any Ψ. This is also observed that random sampling in time domain can improve the quality of reconstructed spectra [31].

However, ℓ₀ norm is known to be intractable and sensitive to noise [11,12], and ℓ₁ norm convex optimization is commonly used in CS to recover x by solving: (12) α ^ = min α ‖ α ‖ 1 ,       s . t .       y = Φ Ψ α

The accuracy of CS reconstruction using Equation (12) can be guaranteed if ΦΨ satisfies the appropriate restricted isometry properties [32]. A restricted isometry constant σ_s [32] defined as the smallest number such that: (13) ( 1 − σ S ) ‖ α ‖ 2 2 ≤ ‖ Φ Ψ α ‖ ≤ ( 1 + σ S ) ‖ α ‖ 2 2holds for all vectors that have at most S nonzero entries. If σ 2 S < 2 − 1, the solution to the ℓ₁ norm problem is that of the ℓ₀ problem [32].

The number of measurements M should satisfy: (14) M ≥ C ⋅ μ 2 ( Φ , Ψ ) ⋅ S ⋅ log   Nso that the signal x can be exactly recovered from measurements y in overwhelming majority of cases [28]. Equation (14) implies that the number of measurements is proportional to the number of nonzero entries S in α and the square of coherence μ. If both S and μ are small, the required number of measurements M could be small. This means that one can perform fewer measurements to save acquisition time while reconstruct original signal x very well.

Iddo [16] applied CS to remove the aliasing artifacts from incompletely acquired FID data by enforcing the sparsity of 2D NMR spectra in wavelet domain according to: (15) α ^ = min α ‖ α ‖ 1 ,     s . t . ‖ y − Θ F T Ψ T α ‖ ≤ σwhere y is the measurements in time domain, Θ is a random sampling operator defining the FIDs acquired in the indirect dimension, F^T denotes the inverse 2D FT, and Ψ^T is the inverse 2D wavelet transform. According to Equation (11), the recovered spectrum is x̂ = Ψ^Tα̂.

In this paper, we focus on the reconstruction of self-sparse NMR spectra in which significant peaks take up partial locations of the full NMR spectra while the remaining locations have very small or even no peaks. Ideally, if the number of sinusoids J in Equation (1) is very small, and the meaningful peaks are narrow enough relative to the whole 2D frequency coverage, the spectra can be considered to be sparse since the number of non-zeros for the spectra is much smaller than the number of spectrum points in the 2D NMR spectra.

The sparsifying transform and the coherence between Ψ and Φ = ΘF^T play important roles in the CS, as we have discussed. In the following sections, we will demonstrate that wavelet is not necessary to sparsify or even worsens the self-sparse NMR spectra based on the concept of sparsity and coherence. We will then reconstruct the NMR spectrum by enforcing its sparsity in an identity matrix domain with ℓ_p (p = 0.5) norm optimization algorithm.

To represent the NMR spectra in conventional way [4–7,17], the X and Y coordinate axes are shown with unit of parts per million (ppm) [21] defined as: (16) δ = ω − ω ref ω 0 × 10 6where δ is the chemical shift of a peak with frequency ω, ω_ref is the frequency of a reference peak and ω₀ is the spectrometer carrier frequency.

Figure 4(a) shows a 2D ¹H-¹H correlation spectroscopy (COSY) spectrum where most of the peaks fill partial and very limited regions of the full spectrum. This leads to the sparsity of spectrum because the number of non zeros in the 2D spectrum is much smaller than the number of spectrum points. This phenomenon is also observed by Yoh Matsuki et al. [17].

To test the sparsity of NMR spectra, we can measure the decay of coefficients in a sparsifying transform domain and evaluate the approximation error by retaining the k-term largest coefficients, because the reconstruction error is proportional to the power law decay k^−r, where r is a constant implying the sparsity of signal [29]. Rapid decay of coefficients implies that one can use less non-zero coefficients to approximate a NMR spectrum. If we directly measure the decay of signal without complicated sparsifying transform, e.g., wavelets, it means measure the self-sparsity of signal. Mathematical saying is measuring its sparsity in the identity matrix.

As shown in Figure 4(b), both the spectra and its wavelet coefficients can achieve rapid decay. By retaining 3% largest magnitude coefficients, the spectra can be reconstructed well in Figure 4(c,d). However, the spectrum is sparser than its representation in the wavelet domain. This is demonstrated by the faster decay of spectrum than that of its wavelet coefficients in Figure 4(b). By retaining the 1% largest magnitude coefficients, the wavelet fails to represent some peaks while the spectrum itself can represent these peaks, as marked by the arrows in Figure 4(e,f).

For a 2D ¹H-¹³C COSY spectrum, the spectrum decays faster than its wavelet coefficients (Figure 5(b)). This implies that the identity matrix can provide a sparser representation of spectra than a wavelet does. Peaks are lost or distorted by using the wavelet transform to represent the spectrum (Figure 5(e)), but the spectrum is represented very well with the identity matrix (Figure 5(f)). This phenomenon is consistent with the observation on the 2D ¹H-¹H COSY spectrum discussed above.

As a result, this spectrum is self-sparse, which means spectrum is sparse in the identity matrix. Thus, according to Equations (9) and (14), it is better to use an identity matrix than to use a wavelet to reconstruct the self-sparse spectra from undersampled FIDs since the wavelet cannot provide a sparser representation of the spectrum. In fact, Stern et al. [33] proposed to do iterative soft thresholding on the spectrum directly, not on wavelet coefficients, to recover one dimensional NMR spectra from the truncated FIDs. Although the sparsity of NMR spectra is not explicitly expressed in that work [33], the recovered spectrum is obtained from minimizing ℓ₁ norm of spectrum, which implies enforcing the sparsity of the spectrum. The problem of their method is that truncation violates the random sampling scheme in CS and results in strong Gibbs ringing which is hard to suppress [29]. What is more, truncating the 1D FID is not necessary to save the time to scan a spectrum since scanning a 1D NMR spectrum is fast and only takes on the order of seconds.

Besides the sparsity of signal, another key factor for CS is the coherence between Φ and Ψ According to Equations (9) and (14), fewer measurements are required for signal sampling system Φ if it is less coherent with Ψ and the signal has same sparsity for different Ψ.

Pioneering work on CS has pointed out that the coherence of a time-frequency pair is μ(Φ, I) = μ(ΘF^T, I) = 1 [28]. Thus, we only need to compute the coherence between undersampled Fourier operator Φ and wavelet basis Ψ^T.

The undersampling of Θ in the indirect dimension is carried out by choosing some of the FID points in this dimension. To make this undersampling intuitive, a binary mask which has the same size of 2D FID is shown as the undersampling pattern in Figure 6(a). If the value of mask at location (i, j) is equal to 1 shown as a white pixel, the FID at location (i, j) is acquired.

To avoid the influence of randomness on the coherence calculation, Θ is randomly generated 10 times and the coherence is averaged for each sampling rate. Figure 6(b) shows that the coherence between wavelet and undersampled Fourier operator Φ is larger than the coherence between identity matrix and Φ. So, from the aspect of coherence, it is also better to choose the identity matrix for self-sparse NMR spectra.

In this paper, we propose to reconstruct the self-sparse 2D NMR spectra with identity matrix I as follows: (17) x ^ = min x ‖ x ‖ 1 , s . t .       y = Φ xwhere Φ = ΘF^T.

To further improve the reconstruction, a ℓ_p (0 < p < 1) norm is incorporated which has been demonstrated to give better reconstruction of MR images with fewer measurements than ℓ₁ norm does [34–37]: (18) x ^ = min x ‖ x ‖ p p , s . t .       y = Φ xwhere ‖ x ‖ p p = ∑ n = 1 N | x n | p and x_n is the nth entry of vector x. For the function f(x) = |x|^p, with p → 0, f(x) gets closer to the ℓ₀ norm of x, as shown in Figure 7.

Theoretically, the required number of measurements [38] by enforcing the sparsity with a ℓ_p (0 < p < 1) norm is: (19) M ≥ C 1   ( p )   K + p C 2   ( p )   K   log ( N / K )where C₁ and C₂ are determined explicitly and bounded in p and the recommend p is 0.5 [34].

In this paper, the ℓ_p norm minimization is solved via the p-shrinkage operator [39] with continuation algorithm [40] because of its fast computation. This algorithm is abbreviated as PSOCA and summarized in Algorithm 1.

For a given continuation parameter β, PSOCA is implemented to solve two sub-problems:

(1) p-shrinkage operator (20) α j = S ɛ p ( x j ) = max { x j − ɛ | x j | p − 1 , 0 } x j | x j |where ɛ = β 1 p − 2 and β is a parameter to be updated in the continuation scheme, x_j and α_j are the jth entry of column vectors x and α, respectively.

(2) solve the linear equation: (21) min x β 2 ‖ α − x ‖ 2 2 + λ 2 ‖ y − Φ x ‖ 2 2which can be simplified to: (22) ( β I + λ P ) F T x = β F T α + λ Θ T ywhere the term P = Θ^TΘ is a diagonal matrix consisting of ones and zeros. The diagonal entries of P correspond to the location of FID data and the entry value is 1 if a corresponding FID data point is sampled, otherwise the entry value is 0. Equation (22) can be solved fast since only a discrete Fourier transform and entry-wise division are required.

The improvement by using the proposed method is verified from the less crowed ¹H-¹H COSY spectrum and more crowded ¹H-¹³C COSY spectrum. The sampling patterns of the two spectra are shown in Figure 8.

Figure 9(c–h) show the reconstructed ¹H-¹H COSY spectra corresponding to the sampling pattern in Figure 9(a) with a sampling rate of 0.20. With the ℓ₁ norm minimization, all the peaks are recovered successfully by using identity matrix (Figure 9(d,f)), while some peaks are lost by using wavelets (Figure 9(c,e)).

Since the contours for the marked peaks look faint, we also plot the 1D slices along the indirect dimension in Figure 10. The height of one peak in the wavelet-based reconstruction in Figure 10(a,b) are much lower than those in the fully sampled spectrum, leading to the peak lost in the contour plots in Figure 9(c,e).

Furthermore, the nonlinear operation on wavelet coefficients induces the artifacts labeled in Figure 9(c,e). This phenomenon is also observed in the 1D slices shown in Figure 10(a,b), where wavelet reconstruction generates illusive peaks. With the ℓ_0.5 norm minimization, the errors caused from wavelet and identity matrix reconstruction are reduced, as shown in Table 1. One can still observe the reduced peak height and artifacts in wavelet-based reconstruction, but identity matrix performs very well (Figure 10(d)). The advantage of ℓ_0.5 norm over ℓ₁ norm is obvious in the crowded ¹H-¹³C COSY spectra, as will be shown in the following discussion.

Figure 11 shows the reconstructed ¹H-¹³C COSY spectra corresponding to the sampling pattern in Figure 8(b) with a sampling rate of 0.25. Some peaks are obviously lost in the reconstructed spectra using wavelets with both ℓ₁ norm and ℓ_0.5 norm minimization (Figure 11(c,e,g)). These lost peaks are found in the identity matrix-based reconstruction spectra (Figure 11(d,f,h)). With the ℓ_0.5 norm minimization, the intensities of the peaks marked with arrow in Figure 11(h) are more consistent to the fully sampled spectra in Figure 11(b) than those in the reconstructed spectra with the ℓ₁ norm minimization (Figure 11(d,f)). The smallest reconstruction error is achieved with the proposed identity matrix-based ℓ_0.5 norm minimization method (Table 2).

All above simulation results demonstrate that wavelet-based reconstruction obviously induces the loss of some peaks in the crowded ¹H-¹³C COSY spectrum and loss of some weak peaks in the less crowded ¹H-¹H COSY spectrum. The wavelet may even worsen the reconstructed spectra. Thus, it is not a good choice to use wavelets for the self-sparse spectra discussed in this paper.

Our simulation is run on a dual core 2.2 GHz CPU laptop with 3 GB RAM. The computational time for the algorithms using wavelet is two times that using the identity matrix, as shown in Table 3.

In the simulation, with the gradual increase of continuation parameter β, the previous solution was used as a ‘warm start’ for the next alternating optimization in the PSOCA. For a given β, with the increase of iterations in inner loop, the difference between reconstructed spectra decreases (see Figure 12(a)), so does the error between the reconstructed spectrum and the fully sampled spectrum (see Figure 12(b)). The reconstruction error decreases when β becomes large in the outer loop. The computational time of ℓ_0.5 norm minimization in PSOCA is nearly four times as that of ℓ₁ norm minimization, as shown in Table 3.