Source-linked AI summary
Multichannel Compressive Sensing MRI Using Noiselet Encoding
Kamlesh Pawar, Gary F. Egan, Jingxin Zhang
TL;DR
MRI acquisition is time-consuming, motivating faster data collection. This paper introduces multichannel noiselet encoding and pulse-sequence design, finding better RIP than Fourier encoding, improved resolution preservation, and higher achievable acceleration.
Problem
MRI acquisition is time-consuming, motivating research into accelerated data acquisition.
Method
The paper introduces noiselet-domain MR acquisition with multichannel compressive sensing and designs pulse sequences for noiselet encoding.
Results
Multichannel noiselet encoding has better RIP than Fourier encoding, preserves image resolution better, and achieves higher acceleration.
Takeaways & Limitations
The results support noiselet encoding as a promising approach for accelerating multichannel MRI while preserving image resolution.
Takeaways & Limitations
The current implementation of noiselet encoding suffers from limitations described by the authors.
Abstract
from arXiv · showhide
The incoherence between measurement and sparsifying transform matrices and the restricted isometry property (RIP) of measurement matrix are two of the key factors in determining the performance of compressive sensing (CS). In CS-MRI, the randomly under-sampled Fourier matrix is used as the measurement matrix and the wavelet transform is usually used as sparsifying transform matrix. However, the incoherence between the randomly under-sampled Fourier matrix and the wavelet matrix is not optimal, which can deteriorate the performance of CS-MRI. Using the mathematical result that noiselets are maximally incoherent with wavelets, this paper introduces the noiselet unitary bases as the measurement matrix to improve the incoherence and RIP in CS-MRI, and presents a method to design the pulse sequence for the noiselet encoding. This novel encoding scheme is combined with the multichannel compressive sensing (MCS) framework to take the advantage of multichannel data acquisition used in MRI scanners. An empirical RIP analysis is presented to compare the multichannel noiselet and multichannel Fourier measurement matrices in MCS. Simulations are presented in the MCS framework to compare the performance of noiselet encoding reconstructions and Fourier encoding reconstructions at different acceleration factors. The comparisons indicate that multichannel noiselet measurement matrix has better RIP than that of its Fourier counterpart, and that noiselet encoded MCS-MRI outperforms Fourier encoded MCS-MRI in preserving image resolution and can achieve higher acceleration factors. To demonstrate the feasibility of the proposed noiselet encoding scheme, two pulse sequences with tailored spatially selective RF excitation pulses was designed and implemented on a 3T scanner to acquire the data in the noiselet domain from a phantom and a human brain.
1 Introduction
The paper addresses slow MRI acquisition by combining noiselet encoding with multichannel compressive sensing (MCS), exploiting noiselets’ maximal incoherence with wavelets and unitary structure. It reports better RIP and reconstruction performance than Fourier encoding, including improved resolution at equal acceleration and higher achievable acceleration factors.
- Motivation: MRI acquisition is time consuming, motivating compressive sensing as a way to accelerate data collection.CS enables faithful reconstruction from data acquired below the Nyquist sampling rate.
- Limitations of existing encoding: Fourier measurements are weakly incoherent with wavelets, making them sub-optimal for CS-MRI.Random encoding can also amplify noise because its measurement matrix is not unitary.
- Noiselet encoding: Noiselets completely spread signal energy, are maximally incoherent with Haar wavelets, and provide complex-valued, symmetric, unitary measurement matrices.These properties motivate their use as a measurement matrix for CS-MRI.
- Empirical comparison: Multichannel noiselet measurements exhibit much better RIP than their Fourier counterpart.The paper presents an empirical RIP comparison between the two multichannel measurement matrices.
- Proposed framework: The paper combines noiselet encoding with MCS, which jointly uses multiple channel measurements to reconstruct the desired image.The MCS framework is reported to produce higher acceleration factors and improved image quality than separate channel reconstructions.
- Results and feasibility: Noiselet encoding preserves image resolution better than Fourier encoding at the same acceleration factors and achieves higher acceleration factors for the desired image quality and resolution.The proposed encoding was implemented with tailored spatially selective RF pulses and tested on a 3T scanner using phantom and human-brain data.
2 Compressive Sensing
Compressive sensing reconstructs sparse signals from undersampled measurements when conditions such as RIP and incoherence support exact and stable recovery. In multichannel MRI, channel-dependent sensitivity maps provide more independent measurements, while reconstruction combines wavelet and total-variation penalties.
- Compressive sensing: Compressive sensing recovers sparse signals from projections taken below the Nyquist sampling rate.MRI commonly uses a partially randomly under-sampled discrete Fourier transform matrix for measurement.
- Compressive sensing: RIP and incoherence are sufficient conditions used to design and assess measurement matrices for exact reconstruction.Smaller RIP constants improve stability and reduce reconstruction error, while smaller incoherence can reduce the measurements needed.
- Multichannel compressive sensing: Multichannel measurement matrix E provides more independent measurements than single-channel Φ, potentially reducing measurements needed at each channel for exact reconstruction.This follows from distinct, complex-valued channel sensitivity maps Γi that make the matrices ΦΓi potentially independent.
- Multichannel compressive sensing: MCS-MRI reconstruction uses wavelet and total-variation penalties in its objective function.The TV penalty is included with the wavelet penalty, and the constrained formulation is relaxed to an unconstrained optimization problem.
- Multichannel compressive sensing: The db-4 wavelet operator is used throughout simulations and reconstructions for consistency with common CS-MRI practice.Daubechies-4 is described as having superior performance in sparsifying MR images.
3 Noiselet Encoding in CS-MRI
Noiselet encoding uses a unitary, wavelet-incoherent measurement basis applied in the phase-encode direction, with tailored RF pulses enabling 2D and 3D implementation. In multichannel CS-MRI, noiselets improve RIP-related behavior and sparsity recovery relative to Fourier encoding.
- Noiselet properties: Noiselets are maximally incoherent with the Haar wavelet, with mutual incoherence parameter equal to 1, the minimum possible value.This makes noiselets theoretically well suited as the measurement basis when wavelets provide sparsity.
- Noiselet properties: Noiselets are unitary, spread signal energy across the measurement domain, have conjugate symmetry, and can be applied through a multiscale filter bank in O(n · log(n)).Conjugate symmetry enables a partial-Fourier-like implementation.
- Pulse-sequence implementation: The proposed acquisition applies noiselet encoding in the phase-encode direction while retaining Fourier encoding in the frequency direction for 2D and 3D MRI.The method uses tailored spatially selective RF excitation pulses to implement the non-Fourier encoding.
- Multichannel RIP analysis: As channel count increases, both measurement matrices’ RIP constants decrease, but noiselet singular values move much closer to 1 and its RIP constant decreases more than Fourier’s.The analysis therefore predicts that multichannel noiselet measurement should outperform both single-channel measurement and multichannel Fourier measurement.
- Multichannel RIP analysis: For 14 channels, noiselet δ-distances remain below 1 for K ≤85, guaranteeing recovery for sparsity K ≤42, whereas Fourier guarantees recovery only for K < 15.The noiselet sparsity guarantee is twofold higher than the reported 14-channel Fourier guarantee.
4 Simulation Study and Results
Simulations compared noiselet- and Fourier-encoded CS-MRI using single- and multichannel settings. Noiselet encoding generally provided better multichannel reconstruction quality, spatial resolution, and acceleration performance, with performance depending on sampling strategy and SNR.
- Simulation design: Simulations used a 256×256 brain image to compare noiselet and Fourier encoding in single-channel and multichannel CS-MRI.The multichannel experiments incorporated sensitivity information estimated from acquired data.
- Single-channel comparison: With variable-density sampling, noiselet-encoded CS-MRI performed similarly to Fourier-encoded CS-MRI because Fourier sampling exploits concentrated central k-space energy.Fourier encoding benefits from densely sampling the center of k-space, where signal energy is highest.
- Channel-count study: For two channels, noiselet encoding outperformed Fourier encoding at acceleration factors 2 and 3; for one channel, it outperformed Fourier only at acceleration factor 2.Noiselet encoding outperformed Fourier encoding at both acceleration factors when the number of channels was greater than one.
- Acceleration-factor study: Noiselet encoding outperformed Fourier encoding for all acceleration factors, with the relative error at acceleration factor 16 matching Fourier encoding at acceleration factor 8.These results indicate that higher acceleration factors are achievable with noiselet encoding.
- Noise robustness: Noiselet encoding outperformed Fourier encoding above 20 dB SNR and was less affected by noise at low SNR, but performed poorly at 10 dB SNR.The reported low-SNR advantage is stated alongside the poor performance at 10 dB.
5 Experiments
Experiments on phantom and human-brain data acquired at 3T demonstrated that noiselet encoding is practically feasible and produces artifact-free images with resolution comparable to Fourier encoding. Retrospective undersampling further showed superior resolution preservation for noiselet-encoded MCS-MRI across tested acceleration factors, including in vivo data up to factor 8.
- Practical feasibility: Noiselet encoding produced feasible, artifact-free reconstructions on phantom and in vivo data acquired at 3T.The experiments used tailored RF excitation pulses and demonstrated feasibility in brain images.
- In vivo acceleration: Noiselet encoding outperformed Fourier encoding for all tested in vivo acceleration factors, with Fourier resolution significantly poorer at acceleration factor 8.Because in vivo SNR was lower than in phantom data, brain reconstruction was shown only up to acceleration factor 8.
- 3D GRE implementation: 3D GRE experiments showed that noiselet encoding provides similar image quality to Fourier encoding and is feasible in a 3D GRE sequence.The sequence used noiselet encoding in one direction and Fourier encoding in the other two directions, with 256 noiselet phase encodes.
6 Discussion
The discussion concludes that noiselet encoding improves multichannel RIP, acceleration capability, and spatial-resolution preservation over Fourier encoding in MCS-MRI. It also identifies implementation limitations, proposed remedies, and the absence of observed structured artifacts in the reported experiments.
- Results: Noiselet encoding produced a multichannel measurement matrix with improved RIP and outperformed conventional Fourier encoding in MCS-MRI reconstruction.At acceleration factor 16, noiselet mean relative error was comparable to Fourier encoding at acceleration factor 8.
- Resolution mechanism: Noiselet reconstruction preserved image spatial resolution better because noiselet measurements spread signal energy across the measurement domain and retain information about fine image details.Fourier sampling concentrates on central k-space energy, while insufficient high-frequency information can reduce reconstructed resolution.
- Basis properties: Noiselet basis functions are unitary and conjugate symmetric, enabling partial acquisition strategies analogous to Fourier encoding; regular undersampling causes aliasing that SENSE can address.These properties support partial acquisition and noiselet-domain reconstruction under regular undersampling.
- Limitations: The implementation is limited by non-slice-selective excitation, long spin-echo TR, low flip angles, long RF pulses, one-direction encoding, and possible B1-induced artifacts.These constraints restrict dynamic imaging, sacrifice some available SNR, and can perturb the measurement matrix when B1 is inhomogeneous.
7 Conclusion
The paper introduces noiselet-domain MRI acquisition with pulse-sequence design and implementation methods. Results show that multichannel noiselet encoding improves RIP, preserves image resolution, supports higher acceleration factors than Fourier encoding, and offers practical high-resolution acquisition.
- 7 Conclusion: The paper introduces a method for acquiring MRI data in the noiselet domain.It also presents methods for designing and implementing pulse sequences for noiselet-domain acquisition.
- 7 Conclusion: Extensive numerical analysis, simulations, and experiments evaluate the performance of noiselet encoding.
- 7 Conclusion: The multichannel noiselet measurement matrix has better RIP than its Fourier counterpart.
- 7 Conclusion: Noiselet encoding in MCS-MRI outperforms conventional Fourier encoding in preserving image resolution and achieving higher acceleration factors.
- 7 Conclusion: Tailored spin-echo and gradient-echo sequences demonstrate that the proposed noiselet encoding scheme is pragmatic and could accelerate high-resolution image acquisition.