Accelerating multidimensional NMR and MRI experiments using iterated maps

Abstract

Techniques that accelerate data acquisition without sacrificing the advantages of fast Fourier transform (FFT) reconstruction could benefit a wide variety of magnetic resonance experiments. Here we discuss an approach for reconstructing multidimensional nuclear magnetic resonance (NMR) spectra and MR images from sparsely-sampled time domain data, by way of iterated maps. This method exploits the computational speed of the FFT algorithm and is done in a deterministic way, by reformulating any a priori knowledge or constraints into projections, and then iterating. In this paper we explain the motivation behind this approach, the formulation of the specific projections, the benefits of using a ‘QUasi-Even Sampling, plus jiTter’ (QUEST) sampling schedule, and various methods for handling noise. Applying the iterated maps method to real 2D NMR and 3D MRI of solids data, we show that it is flexible and robust enough to handle large data sets with significant noise and artifacts.

1. Introduction

For more than 40 years, pulsed Fourier transform spectroscopy [1] has been the dominant approach to nuclear magnetic resonance (NMR), and more recently, to magnetic resonance imaging (MRI). The fast Fourier transform (FFT) algorithm has enabled rapid computation of the spectra obtained from MR experiments, even as they were extended to 2D, 3D, and higher dimensions [2]. Practically speaking, the bottleneck in exploiting the higher information content of multidimensional spectra is the time required to acquire the MR data. For example, in most 2D NMR experiments, only a single 1D ‘row’ of N uniformly-spaced discrete samples of the time-dependent, complex signal is measured per ∝ T₁ (spin-lattice relaxation time). Filling the densely-sampled 2D Cartesian grid of M ×N data points requires M additional experiments, for a total acquisition time T_acq ∝ M ×T₁. One way to accelerate the 2D experiment is to measure only some of the M rows (i.e., to use sparse sampling). Unfortunately, this results in a low quality spectrum, unless non-Fourier reconstruction techniques such as l₁ minimization [3], entropy maximization (e.g., MaxEnt [4, 5], FM [6], or MINT [7]), or multi-dimensional decomposition (e.g., MDD [8]) are used [9, 10, 11, 12].

We became interested in this problem while developing a 3D MRI of solids technique, since the long T₁ of phosphorus-31 in bone meant that one dense data set required T_acq > 45 hours [13]. Taking the first step towards sparse MRI, or compressed-sensing (CS) MRI [14, 15], k⃗ space was sparsely-sampled in a pseudo-random way for T_acq < 1.5 hours. We then Fourier transformed this data, yielding an image of modest quality (Figure S3 in [13]) but in much less time. Typically, the next step to improve upon this image is to adopt non-Fourier reconstruction algorithms like l₁ minimization [16, 17, 18]. However, these methods were not readily applied to our 3D MRI of solids data, since the data sets were sizeable (e.g. 64 × 64 × 64 complex points) and the spectra suffered from both large artifacts and relatively poor signal-to-noise ratios. We sought an easy-to-implement, familiar, and robust reconstruction method that exploited the computational speed of the FFT algorithm.

In this paper we demonstrate the use of iterated maps to accelerate MR data acquisition. In this approach, a speedily-acquired sparse data set is deterministically converted into a high quality approximation of the full, dense data set. We show several examples drawn from 2D NMR of liquids, 2D NMR of solids, and 3D MRI of solids. The approximate output spectra are in excellent agreement with the target, dense spectra over the full dynamic range, for positive and negative signals. This approach is built upon the FFT, so it has the computational speed to handle large multidimensional data sets. As we show below, the time-savings afforded by this method can be used to improve signal-to-noise and frequency resolution compared to spectra obtained from conventional Fourier reconstruction of dense data sets.

Our iterated maps approach was inspired by Veit Elser’s use of the iterated ‘Difference Map’ (DM) algorithm [19, 20] to determine the phase of complex signal when only the magnitude is measured, given a set of a priori constraints that the Fourier reconstruction of the object must satisfy. We realized that Elser’s approach could be adapted to the MR problem by filling in the ‘missing’ complex points in a sparely-sampled MR data set to approximate the dense MR data set. Since both the magnitude and phase are measured in MR, the MR problem may be solved using fewer input data points than Elser’s phase-retrieval problem. While preparing this paper, we became aware of the pioneering work of Herzfeld and Matsuki, who recently developed SIFT (Spectroscopy by Integration of Frequency and Time Domain Information) [21, 22, 23] to accelerate multidimensional NMR experiments. Our work is related to theirs, and both approaches may be traced back along distinct lineages to a common ancestor (see Supporting Information). The conclusions drawn independently by both groups are in great agreement. In the discussion that follows we will point out similarities and differences between SIFT and our iterated maps approach.

2. Results

Our approach offers speed-up advantages when signals occupy a limited region of the frequency domain, as we show using 2D NMR of liquids (Figures 1–3), 2D NMR of solids (Figure 4), and 3D MRI of solids (Figure 5). For each demonstration of the method, we start by describing the data sets that have been densely-sampled in the nD time-domain as a complex vector T(t⃗); the corresponding target ‘dense spectrum’ T̃(f⃗) = Ph(FFT(T(t⃗))) is obtained following nD FFT and a known phase correction Ph, which are reversible operations: IFFT(Ph⁻¹(T̃(f⃗))) = T(t⃗) (see Supporting Information). In practice, our algorithm acts upon T(t⃗) and T̃(f⃗) in their natural form as nD matrices.

Figure 1.

Reconstructing liquid-state 2D NMR data of the ¹⁵N-LuxU sample. (A) Magnitude plot of 50.8% sparsely-sampled (t₂,t₁)-domain NMR data, |S⁰(t⃗)|, (256 rows × 4096 columns). Pink shows the t₁ rows set to zero by P̂₀. (B) Real part of phase-corrected FFT of the sparsely-sampled data shown in A, ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$, which shows the resulting aliasing artifacts along f₂ columns. Red pixels are ≥ 6% of the maximum dense signal (MDS) in T̃_r(f⃗) (not shown). The P̂₁ mask is shown, where blue surrounds the artifact region, black surrounds the positive support regions, and green surrounds the negative support regions. (C) Plot of pixel values along the six columns marked by red arrows in $B({\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})={\stackrel{\sim }{\mathit{F}}}_r^0(\overrightarrow{f}))$ and the blue arrows in $E({\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f}))$ versus corresponding pixels from T̃_r(f⃗). The long dashed line y = (0.508)x shows the poor quality of the fit before reconstruction (red open circles are ${\stackrel{\sim }{F}}_r^0(\overrightarrow{f})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{f})$). The short dashed line y = x shows the excellent agreement after reconstruction (blue open triangles are ${\stackrel{\sim }{F}}_r^{15}(\overrightarrow{f})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{f})$), over the full range of positive and negative pixel values. (D) The resulting time-domain data after 15 iterations of our difference map (DM) algorithm, F¹⁵(t⃗), using a small value of P̂₁ noise-handling (±0.3% of the MDS). (E) Real part of phase corrected FFT, ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$, of the DM reconstructed data shown in D. Red pixels are ≥ 6% of the MDS. (F) A portion of a contour plot showing the close match between the real parts of the dense spectrum (dashed) and the reconstructed spectrum (solid), using the same color scale and contour values (6% → 72% of the MDS, in 6% steps) for both T̃_r(f⃗) and ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$.

Figure 3.

Comparing performance of the difference map algorithm using two different P̂₁ masks for the IGPS data (from Figure 2). (A) The l₂ difference between the reconstructed and dense spectra (inside the positive support) plotted versus the number of positive t₁ rows sampled (N_t1), where N_t1 ≤ 128, for both looser (> 0.14% of MDS, red triangles) and tighter (> 0.70% of MDS, blue triangles) P̂₁ masks. The tighter mask (blue) was used in Figure 2. Both traces have a peak in l₂ at N_t1 = 65, but only the looser mask (red) has a peak at N_t1 = 85. (B) The aliasing of the looser mask due to uniform sampling at N_t1 = 85. (C) Zooming into the region outlined by the black dashed line in B. Multiple, strong (≥ 20% of MDS), aliased peaks (in red and blue) are inside the positive support of the looser P̂₁ (highlighted in yellow), and they result in a worse reconstruction. (D) The same region shown in C, but now using the tighter P̂₁, which makes most aliased peaks fall outside the tighter support region, improving the reconstruction quality and explaining the disappearance of the peak in l₂ at N_t1 = 85 for this mask in A (blue).

Figure 4.

Reconstructing the ¹³C-¹³C 2D MAS correlation spectrum for the ¹³C/¹⁵N enriched GB1 solid state sample. (A) Real part of phase-corrected FFT ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$ of 34.3% sparsely-sampled (t₂,t₁)-domain data, which shows the resulting aliasing artifacts along the f₂ columns (4096 rows × 4096 columns). Red pixels are ≥ 2.7% of the MDS. The P̂₁ mask is shown, with artifact regions surrounded by blue, negative support regions by green, and positive support regions surrounded by black. As described in the main text, a tiny ‘keyhole’ was cut into the mask over the region ((17.7kHz ≥ f₂ ≥ 13.4kHz) ∩ (9.5kHz ≥ f₁ ≥ −1.1kHz)), which significantly reduced the requisite sampling percentage (53.7% → 34.3%). (B) The reconstructed (f₂, f₁)-domain data after 15 iterations of our difference map algorithm, ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$, without using any noise-handling. (C – F) Zoom-in contour plots of various regions comparing the full dense spectrum (dashed) to the reconstructed spectrum (solid), using identical color scales and contour values (1%,2%,3%,4%,8% of the MDS) for both spectra. C shows a high-signal region near the main diagonal; D shows a low-signal region far from the main diagonal; E shows another low-signal region far from the main diagonal; and F shows a low-signal region where there is a rotary resonance (MAS rate ≈ 18.2 kHz). (F, Inset) Plot of pixel values along the two columns marked by blue arrows in B versus corresponding pixels from the full dense spectrum. The short dashed line y = x shows the high-quality fit over a wide dynamic range after reconstruction (blue open triangles are ${\stackrel{\sim }{F}}_r^{15}(\overrightarrow{f})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{f})$).

Figure 5.

Reconstruction of sparsely-sampled solid-state MRI data. (A) Isosurface rendering of a portion of the 3D (64 ×64 ×64) T̃_r(r⃗) showing ³¹P density in pork rib in PBS solution. The isosurface value is 33% of the MDS and shows the thick cortical bone ring. The spatial resolution is (1.19mm)³ and T_acq was 35.2h. (B) A 2D slice of T̃_r(r⃗) with thickness of 0.595mm. The support region for this 2D slice (where signal is expected to be positive) is outlined in blue. (C) The same 2D slice from ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{r})$ which would require only 17% of the imaging time used for the dense image. (Inset) Plot of ${\stackrel{\sim }{S}}_r^0(\overrightarrow{r})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{r})$ for voxels within the support region. The thick, dashed line y = x shows the poor-quality fit prior to DM reconstruction. By definition, ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{r})={\stackrel{\sim }{\mathit{F}}}_r^0(\overrightarrow{r})$. (D) The same 2D slice of ${\stackrel{\sim }{\mathit{F}}}_r^{20}(\overrightarrow{r})$ after DM reconstruction with P̂₁ and P̂₂ error handling. (Inset) Plot of ${\stackrel{\sim }{F}}_r^{20}(\overrightarrow{r})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{r})$ for voxels within the support region. The thick, dashed line y = x shows the high-quality fit after reconstruction, where most of the points are within the 10% noise level for the measured data.

As explained in the Supporting Information, even though the causal data set acquired using a ‘States’-like 2D NMR experiment (i.e., with t₁ ≥ 0, t₂ ≥ 0) can be converted into the conventional, purely-real (absorptive) 2D spectrum using ‘States’ processing [24], this cannot be easily reversed, which is a requirement of our iterated maps method. Instead, we followed the MRI model and used the acquired data to construct a T(t⃗) that filled all four quadrants of the time-domain, with Hermitian symmetry about the origin, such that each component of the 2D matrix satisfies T (t⃗) = T^*(−t⃗). A 2D complex Fourier transform (followed by a phase correction) yields a purely-real spectrum, T̃(f⃗). The pseudo-echo transformation in 2D NMR [2] has a similar motivation.

To simulate the effect of skipping particular experiments, we construct an initial ‘sparsely-sampled’ data set S⁰(t⃗) = P̂₀T(t⃗), where P̂₀ is a “projection” that sets to zero all skipped points, leaving the rest alone. In this paper, we follow Elser’s use of the term projection [20], as explained in Supporting Information. For example, Figure 1A shows S⁰(t⃗) if just 50.8% of the required t₁ rows are measured using States acquisition, which halves the normal T_acq. Unfortunately, the real part of the corresponding spectrum ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})=Re(Ph(\mathit{FFT}({\mathit{S}}^0(\overrightarrow{t}))))$ is a poor approximation to T̃_r(f⃗), with pronounced sparse-sampling artifacts that smear along the f₂ columns, as shown in Figure 1B. To quantify the disagreement, Figure 1C (red circles) shows a point-by-point comparison of ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$ to T̃_r(f⃗) along six f₂ columns of Figure 1B, which shows that large signals in T̃_r(f⃗) have only ~ 50.8% of their dense value in ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$, while points that should be zero in the dense spectrum can have large positive or negative values in ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$. To do better than this, we use iterated maps to deterministically convert S̃⁰(f⃗) → F̃ⁿ(f⃗) ≈ T̃(f⃗).

Similar to SIFT, our iterated maps approach uses two projections which enforce a priori knowledge about the dense data in the time and frequency domains. The first projection (P̂₁) is a support constraint in the frequency domain. The simplest version (P̂_1SIFT as is used by SIFT) zeros all points outside of the support region and leaves the points inside the support intact. As we show in Figure S1, we further strengthen this constraint by ensuring proper phasing of the time-domain data, so the spectrum T̃(f⃗) in our stronger version of P̂₁ has no imaginary part, and the real part has a known sign (+ in positive, − in negative, and ± in artifact support regions). The second projection (P̂₂) resets all sampled time-domain points back to their measured values (i.e., the non-zeroed points in S⁰(t⃗)), as in SIFT. Both of these projections are altered slightly to allow for noise-handling with given error thresholds in one or both domains (see Supporting Information).

We combine these projections (and the identity operator 1̂) in a particular form of Elser’s Difference Map algorithm [19, 20] which reproduces Fienup’s hybrid input-output map [25]. Our map is given by D̂ = 1̂ + P̂₁(2P̂₂ − 1̂) − P̂₂ (while SIFT uses D̂_SIFT = P̂₂P̂_1SIFT) and the iterative mapping becomes Sⁿ⁺¹ = D̂Sⁿ. The P̂₁ and P̂₂ projections are defined in the frequency domain and time domain, respectively, and we move from one domain to another using the FFT/IFFT (see Supporting Information). The final output of the algorithm after n iterations is given by Fⁿ = P̂₂Sⁿ. For example, Figure 1D shows F¹⁵(t⃗), and Figure 1E shows ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$, which were calculated in six minutes on an iMac, using code running in IgorPro. Figure 1C,E,F show the excellent agreement between the dense T̃_r(f⃗) and reconstructed ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$ spectra. For example, Figure 1C (blue triangles) shows a point-by-point comparison of ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$ to T̃_r(f⃗) along six f₂ columns of Figure 1E. Clearly, all of these ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$ points (not just the spectral peaks, and not just the positive signals) have reached ≈ 100% of the target value in T̃_r(f⃗). The Supporting Information provides additional details of the algorithm, projections, convergence, and noise-handling.

The successful demonstration in Figure 1 is for liquid-state 2D NMR of LuxU [26], where the dense spectrum T̃_r(f⃗) had relatively little noise, and only a narrow water ‘artifact’. To a first approximation, the P̂₁ mask was constructed by placing a coarse bounding box on the 1D spectrum as a function of either f₁ or f₂, and was then refined by placing negative signal masks at particular locations. A small noise-handling threshold (±0.3% of max signal) was used for P̂₁.

In the time-domain, Figure 1A shows the ‘QUasi-Evenly Spaced, plus jiTter’ (QUEST) sampling schedule used to pick the particular t₁ rows used by P̂₀ (see Supporting Information for more information). In its current form, the iterated maps approach can be used with any on-grid, non-uniform sampling (NUS) schedule (for a review, see [9]), and we studied many variations to assess their performance. However, in our experience, the iterated maps approach converges to essentially the same F̃ⁿ(f⃗) ≈ T̃(f⃗) for all cases, provided that enough samples are included in S⁰(t⃗). Lowering the number of samples (i.e. making the problem under-constrained) for each case revealed a common trend; the final F̃ⁿ(f⃗) started to deviate from T̃(f⃗) whenever the gaps between the sampled points in S⁰(t⃗) grew too large. Large gaps between samples were also observed to lower the quality of forward maximum entropy (FM) reconstruction [6]. We have a qualitative explanation of this effect, at least for our approach, based upon how it seems to work. Iterating the map causes information to diffuse from the sampled points in the time domain to particular unmeasured points, as determined by the point-spread function for a given P̂₁ (as shown in Figure S1). In practice, if the gap between measured points in S⁰(t⃗) is too large, it may never be filled in by this process in the under-constrained case, and the final F̃ⁿ(f⃗) will differ from T̃(f⃗). This picture suggests that QUEST is a good way to distribute a given number of sparse samples across the time domain, since the gaps between measurements are roughly uniform, and so it is used for Figures 1–4.

Figure 2 shows the method applied to another liquid-state 2D NMR data set. Using an Isoleucine, Leucine, Valine (ILV) ¹³C-methyl labeled sample of imidazole glycerol phosphate synthase (IGPS), we were able to very accurately reconstruct the entire spectrum (Figure 2B–C) over a wide dynamic range, starting with just 58.6% of the time-domain data. In this case, no noise-handling is used for either P̂₁ or P̂₂. Note that the artifact domain (surrounded by blue in Figure 2A) is much larger than that in Figure 1B. In addition, this P̂₁ mask has no negative support regions, and the positive support regions (surrounded by black in Figure 2A) are the portions of the dense spectrum with ≥ 0.7% of the maximum dense signal (MDS). Constructing such a tight mask would typically require the full dense spectrum, which is not available in all situations. However, in a serial experiment such as an NMR relaxation rate measurement, one dense data set (at maximum signal amplitude, to optimize the mask) could be followed by many sparsely-sampled (x%) data sets acquired at various relaxation delays, requiring only ~x% of the normal experimental time, as was recently demonstrated using SIFT [23].

Figure 2.

Reconstructing liquid-state 2D NMR data of the IGPS sample. (A) Real part of phase-corrected FFT ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$ of 58.6% sparsely-sampled (t₂,t₁)-domain data, which shows the resulting aliasing artifacts along the f₂ columns (256 rows × 4096 columns). Red pixels are ≥ 6% of the MDS. The P̂₁ mask is shown, where blue surrounds the artifact region, and black surrounds the positive support regions (≥ 0.7% of the MDS). (B) The reconstructed (f₂, f₁)-domain data after 15 iterations of our difference map algorithm, ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$, without using any noise-handling. (B, Inset) Plot of pixel values along the three columns marked by blue arrows in B versus corresponding pixels from full dense spectrum. The short dashed line y = x shows the excellent agreement after reconstruction (blue open triangles are ${\stackrel{\sim }{F}}_r^{15}(\overrightarrow{f})vs.{\stackrel{\sim }{T}}_r(\overrightarrow{f})$). (C) A contour plot of a region comparing the real parts of the full dense spectrum T̃_r(f⃗) (black) to the reconstructed spectrum ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$ (colored), using identical contour values (3%,8%,17%,25%,34%,67% of the MDS) for both spectra.

While analyzing the IGPS data, we discovered that using the known frequency-space P̂₁ mask along with the QUEST schedule enabled us to make very powerful predictions of just how sparsely we could sample the data. In Figure 3A we plot the Euclidean distance between the reconstructed and dense spectra (l₂[F̃ⁿ(f⃗)− T̃(f⃗)]) for two different P̂₁ masks, plotted versus the number of positive t₁ rows sampled (N_t1). As expected (see Supporting Information), a looser mask (red curve) results in a larger difference between the reconstructed and dense spectra (larger l₂) than a tighter mask (blue curve), at each N_t1.

To understand the striking peaks in Figure 3A at N_t1= 65 and 85, we pick a P̂₁ mask and study the aliasing that results from uniform undersampling (since this nicely approximates the artifacts from QUEST, see Supporting Information). Figure 3B shows the aliasing of the loose mask expected for N_t1 = 85. Zooming into the black dashed line region (in Figure 3C) shows that many strong peaks push into the support region (highlighted in yellow) as they are aliased from above (red) and below (blue). These aliasing artifacts will not be suppressed by our P̂₁ projection when the loose mask is used, resulting in a worse reconstruction (larger l₂). However, when we do the same analysis with the tight mask (see Figure 3D), we see that most aliased peaks miss the support region so our P̂₁ projection will efficiently suppress these artifacts, leading to a better reconstruction and the disappearance of the N_t1= 85 peak for the tighter mask (red) data in Figure 3A. In this picture, if strong aliased peaks push into the support of the central mask (as N_t1 is lowered), then reconstruction quality suffers. The aliased masks are a convenient proxy for the aliased peaks, and they can be used to quickly estimate the minimum N_t1 for excellent reconstruction using QUEST. To do this, one can plot the overlap of the aliased masks with the central mask while lowering N_t1; all overlaps close to zero will yield excellent reconstructions.

Building upon this understanding, Figure 4 shows a very accurate reconstruction of a solid-state 2D NMR spectrum from a NCGB1 sample, despite starting with just 34.3% of the time-domain data and no noise-handling. This is a very large (4096 × 4096 complex points) 2D data set, with sparse spectral features (including artifacts of the magic angle spinning) that span a wide range of amplitudes. For the most-part, the P̂₁ positive support region is a coarse set of blocks, as we used in Figure 1. In a crucial refinement, a ‘keyhole’ was cut into a signal-free region of the mask, to make room for an aliased rotational sideband (see Figure 4A). Adding the tiny keyhole drove the minimum N_t1 from 1100 → 702 (53.7% → 34.3%), while maintaining excellent quality of reconstruction over a wide dynamic range, as seen in Figure 4B–F.

In Figure 5, our reconstruction algorithm is applied to sparsely sampled 3D ³¹P MRI of solids data (see [13] and Supporting Information for more information on how this data was acquired). In this experiment, the image is restricted to a portion of the full field of view (FOV) in order to avoid known artifacts (see Figure S2), which enables the construction of strong constraints in the P̂₁ mask (where k⃗ and r⃗ in MRI map onto t⃗ and f⃗ in NMR). The dense data set for the image in Figure 5A and B took 35.2h to acquire. For Figure 5C, we sparsely sampled the time domain to reduce the required imaging time to 17% of the value for the dense data set (see Figure S3 and Supporting Information), which also lowered the reconstruction quality. To quantify the disagreement between the sparse ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{r})$ and dense T̃_r(r⃗) images, the signal amplitudes inside the positive support region are compared point-by-point in Figure 5C(inset). However, the agreement improves (Figure 5D) if we use our difference map reconstruction algorithm, with noise handling in both the P̂₁ and P̂₂ projections. Figure 5D(inset) shows the point-by-point comparison of signal amplitudes for our reconstruction ${\stackrel{\sim }{\mathit{F}}}_r^{20}(\overrightarrow{r})$ versus the dense T̃_r(r⃗) image, which looks good despite the (≈ 10%) noise level of the dense spectrum. Considering the large noise and artifacts in this 3D MRI of solids data, this provides further evidence of our reconstruction algorithm’s robustness and usefulness for noisy experimental data.

In the previous results, we aimed for nearly ideal output spectra where F̃ⁿ(f⃗) ≈ T̃(f⃗). In some cases, a reduced quality output spectrum F̃ⁿ(f⃗) may be sufficient for the user’s needs, which means that even less data (and a smaller T_acq) is needed. Of course, in order to determine how much data is required for the task at hand, one first needs to understand how the output spectrum is affected by using less data at the input. As we pushed down to even lower sampling percentages for 2D NMR, we found that QUEST helped us to identify the particular regions of the output spectrum that degrade first, due to aliasing artifacts (see Figures 6–9 and Supporting Information). Using the LuxU data as an example, the same panels A–F found in Figure 1 are shown again for different row numbers: N_t1 = 65 (Figure 6), N_t1 = 50 (Figure 7), N_t1 = 35 (Figure 8), N_t1 = 20 (Figure 9). In addition, panel G shows how the output vector becomes more ‘parallel’ and less ‘perpendicular’ to the target vector when the algorithm works as desired (see Supporting Information). Lastly, panel H shows a plot in the style of Figure 3C, showing the central LuxU mask (dark yellow), the −1×BW₁ mask (dark red) poking in from above, and the +1×BW₁ mask (dark blue) poking in from below, as explained in the Supporting Information.

Figure 6.

The first of four figures illustrating the specific ways in which the LuxU reconstruction quality drops, as fewer time points are sampled (i.e., as N_t1 is systematically lowered over Figures 6–9). (A–F) Identical to the corresponding panels in Figure 1, using the lowest N_t1 = 65 for a very high quality output ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})\approx {\stackrel{\sim }{\mathit{T}}}_r(\overrightarrow{f})$). (G) As explained in Supporting Information, we calculate how ‘parallel’ and ‘perpendicular’ F̃ⁿ(f⃗) is to the target vector, T̃(f⃗). The resulting parametric plot of F̃_||T̃ (n) vs. F̃_⊥T̃ (n), from the n = 0 point (thin black arrow), to n = 15 (thick black arrow), shows the approach to the target (red arrow). (H). A plot like Figure 3C, showing the central LuxU mask (dark yellow), the −1×BW₁ mask (dark red) poking in from above, and the +1×BW₁ mask (dark blue) poking in from below, as explained in the Supporting Information. The black contours are at n_1% = 1% × MDS of T̃_r(f⃗). The few blue pixels are where ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})-{\stackrel{\sim }{\mathit{T}}}_r(\overrightarrow{f})\ge n_{1\%}$, and the few red pixels are where ${\stackrel{\sim }{\mathit{T}}}_r(\overrightarrow{f})-{\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})\ge n_{1\%}$. The blue and red pixel density will grow as N_t1 is lowered in Figures 7,8,9.

Figure 9.

Same as Figure 6, if we drop to N_t1 = 20, which is a lower quality output. The long dashed line in (C) has slope 20/128. Compared to Figure 6, most blue points in (C) fail to reach the short dashed line (with a slope of 1). Note also that this is the first case where many blue points in (C) that should be zero to match the dense image have instead grown larger than the red point values they had at the start of the algorithm (look at points at zero on the horizontal axis, in Figs. 6C–9C). The ‘perpendicular’ component in G grows monotonically for n = 0 to n = 15. In D, dark stripes are noticeable, along with residual aliasing within the mask in E. For this N_t1 = 20, H shows that the dark red and dark blue masks overlap by a lot, and the ±2×BW1 masks (not shown) overlap by a little, and the ±3×BW1 masks (not shown) are starting to poke into the central mask. As a result, even more blue pixels are scattered across the central mask. Red pixels fill even more of the black contours. Still, C shows that the final output is better than ${\stackrel{\sim }{\mathit{S}}}_r^0(\overrightarrow{f})$, and E shows that some T̃_r(f⃗) features are recognizable (F), in just 16% of the normal acquisition time.

Figure 7.

Same as Figure 6, if we drop to N_t1 = 50 (note the increase in pink row density in A). The long dashed line in (C) has slope 50/128. Still a high quality output. (H) shows more blue pixels near the overlap of masks (top and bottom of yellow box), the residual traces of sparse sampling artifacts. The red pixels are concentrated in the black contours.

Figure 8.

Same as Figure 6, if we drop to N_t1 = 35. The long dashed line in (C) has slope 35/128. This is approximately the lower end of the N_t1 range for high quality outputs. In (H), we see that the dark red and dark blue masks just touch, and so blue pixels are scattered across the central mask. Red pixels fill more of the black contours. This ${\stackrel{\sim }{\mathit{F}}}_r^{15}(\overrightarrow{f})$ requires only 27% of the normal acquisition time of T̃_r(f⃗).

Comparing Figure 6–9, we can summarize how the algorithm seems to work, and what happens as the input data is reduced. The red points in panel C show that most large features start at only ≈ x% of their dense value, where x% is the sparse sampling percentage; at the same time, pixels that should be zero in the dense signal are non-zero, due to the artifacts of sparse sampling. As the algorithm iterates, the artifacts are driven towards zero, while the true signals push up toward their dense values. The total area under the signal is a conserved quantity, since we always include the data point at t⃗ = 0. Panel H shows that artifacts (blue features) that survive to the end of the algorithm are located in the portions of the frequency domain where the aliased and central masks overlap. The consequence of artifact survival is a poorer output quality, since it is also correlated with additional undershoot (red features) of the true signal amplitude in panel H. For clarity, panel H only shows the location of the M = 1 aliases at specific N_t1 values, but the formulas in the Supporting Information indicate when M = 2 and M = 3 aliases will start to matter. For example, at N_t1 = (65/2) ≈ 33 the M = 2 aliases will just start to poke into the central mask (as shown in Figure 6H for M = 1). On the other hand, at N_t1 = (50/2) = 25 the M = 2 aliases will just touch in the central mask (as shown in Figure 7H for M = 1). Finally, at N_t1 = (65/3) ≈ 22 the M = 3 aliases will just start to poke into the central mask (as shown in Figure 6H for M = 1). These ‘special’ numbers are consistent with the trends in Figure 6–9, with reasonably high quality fits from N_t1 = 65 down to N_t1 = 35, and a noticeably lower quality fit for N_t1 = 20.

3. Discussion

SIFT can handle even ‘unphased’ input data, which is a clear advantage whenever the experimental parameters are difficult to change. On the other hand, whenever input data can be ‘phased’ (see Supporting Information), this additional information about the signal can be used to strengthen the projections, enabling our iterated maps approach to push down to even sparser sampling. A few 1D experiments should be sufficient to quickly set up the proper conditions to run an nD experiment that can be accelerated using iterated maps.

For the 2D NMR experiments described here, we used the stronger P̂₁ projection that makes use of the known sign of the real part of the signal (+ amplitude in positive, − amplitude in negative support regions). In cases where the signal sign is not known in advance, our so-called artifact mask should be used instead (with ±real signal amplitude allowed in the artifact support regions). This will also work, but since less a priori information is fed into the algorithm, a larger number of N_t1 rows will typically be required to reach the same high quality output (e.g., see Figures S4–S5). The precise increase in the required number of rows will depend upon the details of the signal, the mask, the sampling schedule, and the noise-handling method. In general, the user should try to take advantage of all known characteristics of the dense data set.

In Figures 1, 2, 4, and 5, we aimed for nearly ideal output spectra F̃ⁿ(f⃗) ≈ T̃(f⃗), since results of that quality can be used for any application. In the case of 2D NMR, we found that QUEST helped us to determine the minimum number of samples consistent with that goal. At first glance, our sparse sampling percentages may not seem that low, but in fact they appear to be quite close to the minimum necessary for a constrained linear system. For example, the corresponding sparse sampling percentages used in Figures 1,2,4 are: (50.8%,58.6%,34.3%), which are similar to the largest percentage of positive or negative support pixels along the f₂ columns for each mask: (55.1%,44.5%,32.6%). Since our current 2D NMR experiments use sparse sampling in the single indirect dimension (by ‘skipping’ some t₁ values), the required sparse sampling percentages should drop quickly as this method is applied in 3D NMR, 4D NMR, etc.

In addition to the Difference Map D̂ shown here, we tried several other iterated maps, including D̂_{f lip} which flips P̂₁ ↔ P̂₂, D̂_{AltPro js} = P̂₁ P̂₂, and Elser’s ‘Divide and Concur’ [27] which accommodates more than two projections. All of the maps worked, and they were relatively easy to implement. The output spectra F̃ⁿ(f⃗) have better signal-to-noise ratios than T̃(f⃗), since fewer noisy samples are used at the input. The ability to sample at very long times, without requiring the acquisition of all intermediate times, can be used to achieve higher spectral resolution in less acquisition time. The time savings offered by iterated maps could be leveraged to allow practical acquisition of higher dimensional (5D and 6D) NMR experiments, which are currently not feasible in all but the most ideal circumstances (for a current review, see [28]). Moreover, iterated maps offer another approach to further accelerate [29] ultrafast 2D NMR [30].

The iterative maps approach has many characteristics that should be familiar to magnetic resonance practitioners, such as its use of on-grid sampling, the FFT/IFFT, and final outputs which look the same as dense data sets in both the time and frequency domains. It complements existing methods to reconstruct spectra from sparsely-sampled data, and should find further applications in NMR and MRI of solids. More generally, any data acquisition and image modalities which make use of two reciprocal spaces related by a transformation can use this technique to harness a priori knowledge to fill in under-sampled data. The speed, simplicity, and robustness to error makes this an ideal technique for fast analysis of noisy experimental data.

Highlights.

• An iterated maps algorithm is applied to sparsely-sampled time domain data.

• Used to reconstruct spectra from noisy 2D NMR and 3D MRI of solids data.

• High quality results achieved with sparse sampling approaching theoretical minimum.

• We use the QUEST sampling schedule and discuss its benefits for 2D NMR data.

• FFT-based method is computationally fast, simple to implement, and robust.

Appendix A. Materials

The spectral acquisition parameters used throughout this paper were determined by the goals of each experiment, and they were not adjusted to optimize the performance of the algorithm.

The ¹⁵N HSQC of LuxU in Fig. 1 was collected at 14.1 T and 20°C with a ¹H spectral width of 12000 Hz and a ¹⁵N spectral width of 2500 Hz [26]. For Figs. 2–3, the ²H, ¹³C-methyl ILV labeled IGPS (t. maritima) NMR sample was prepared as previously described [31]. The zero time point of a ¹³CH₃ multiple quantum CPMG relaxation dispersion experiment was collected at 14.1 T and 30°C with 120 t₁ increments using a previously published sequence [32]. The ¹H carrier was centered at 4.70 ppm with a spectral width of 8500 Hz, while the ¹³C carrier was centered at 19.5 ppm with a spectral width of 3200 Hz.

The ¹³C-¹³C correlation spectrum for a microcrystalline sample of uniformly ¹³C/¹⁵N enriched GB1 shown in Fig. 4 was taken using a RAD mixing sequence [33] on a Varian VNMRS spectrometer operating at a ¹H frequency of 798.89 MHz, a MAS rate of 18.2 kHz, and 4 scans per t₁ point. A spectral width of 100 kHz was used in both the direct and indirect dimensions. The sample was expressed and purified according to a previously reported protocol [34], using the pET-11a plasmid incorporating the gene for the T2Q mutant of GB1 (kindly provided by Dr. Angela M. Gronenborn, University of Pittsburgh) into BL21 E. coli cells. For solid-state NMR, microcrystalline GB1 was precipitated from a solution containing 25 mg/mL GB1 in 50 mM Na₂HPO₄ at pH 5.5, and 50% (v/v) Methyl-2,4-pentanediol (MPD), 25% (v/v) Isopropyl alcohol (IPA) as reported by Rienstra and co-workers [35].

The sample used in Fig. 5 is a marrow-filled section of a pork rib bone, which was cut from a whole fresh pork rib. The rib bone section was placed in a cryotube vial, which was then filled to volume with phosphate buffered saline (PBS) solution to keep the sample hydrated. The ³¹P MRI data was acquired at the Yale University Medical School’s Magnetic Resonance Research Center, using the Bruker Avance 4.0 Tesla / 31 cm animal system, running ParaVision 3.0.1. See reference [13] for more information on how this data was acquired.

Appendix B. Supporting Information

In addition to the figures and details referred to in the main text, the Supporting Information includes several figures referred to in that section. Figure S6 shows the mapping of 2D NMR ‘States’-like data into the four quadrants of T(t⃗). Figure S7 is a schematic depiction of the behavior of the difference map near a gap between the subspaces defined by the projections. Figure S8 shows how different types of noise-handling affect a slice of the output, and the metrics we can use to follow the iterations of the algorithm.

We also include seven Quicktime movies, which show the algorithm iterating from the beginning to the end (Movie S1), how the final output compares to the dense target (Movie S2), and a slice-by-slice comparison of the output before and after the difference map to the dense target spectrum (Movie S3), all for the case of the 2D NMR of liquids LuxU data (from Figure 1).

Movie S4 is a slice-by-slice comparison of the output before and after the difference map to the dense target spectrum, for the case of the 2D NMR of liquids IGPS data (from Figure 2). Movie S5 is a slice-by-slice comparison of the output before and after the difference map to the dense target spectrum, for the case of the 2D NMR of solids ¹³C/¹⁵N enriched GB1 data (from Figure 4).

Movies S6 and S7 show the algorithm iterating from beginning to end for the case of the 2D NMR of liquids LuxU data, using two different sampling schedules. Movie S6 uses N_t1 = 65 and the QUEST schedule (just as in Figure 6), while Movie S7 uses a larger N_t1= 71 and a very different sampling schedule that skips contiguous rows in the intermediate t₁ range.