2.1 Compressive Sensing

Compressive sensing is a mathematical theory describing how a sparse signals can be faithfully recovered after sampling its projections well below the Nyquist sampling rate. Consider a signal x in the n dimensional complex space ℂⁿ that can be sparsely represented in Ψ domain as ρ = Ψx, where Ψ is the n × n sparsifying transform matrix satisfying Φ*Φ = I. The signal ρ is K-sparse, that is, only the K coefficients in ρ are non-zero. A measurement system measures signal y in m dimensional space by taking only m projections of the signal x as (1)

In CS MRI, x ∈ ℂⁿ is the vectorized image with the dimension n = pq for a p × q image, and Φ ∈ ℂ^{m × n} is usually a (partially) randomly under-sampled discrete Fourier transform matrix resulting from encoding process, where m = rq and r < p is the number of phase encoding lines used to acquire y. Given the image size p × q, the under-sampling ratio n/m = p/r is determined by the number of phase encoding lines, r, used in data acquisition, which in turn determines the reduction of data acquisition time [7]. So the under-sampling ratio n/m = p/r is called the acceleration factor in CS MRI.

Eq (1) can be further expressed as (2) where * represents the conjugate transpose operation and the signal x is sparse in the Ψ domain. MR images can be sparsely represented in the wavelet domain using the wavelet transform matrix. Given the measurement y and the matrices Φ and Ψ, there exist many solutions satisfying (1) and recovering x becomes an ill posed problem. The CS theory provides a unique solution to the ill-posed problem by solving the following optimization program: (3) where ‖x‖_l1: = ∑_i∣x_i∣ is the l₁ norm of x, with x_i the ith element of x. Exact reconstruction of the signal x is achievable if certain mathematical conditions hold.

2.2 Restricted Isometry and Incoherence in Compressive Sensing

An important sufficient condition for exact reconstruction of x is the so called restricted isometry property (RIP) [2, 21, 23]. For a normalized measurement matrix Φ with unit column norms, the RIP is given as (4) where δ_K ∈ (0,1) is called the RIP constant. The RIP (4) is equivalent to [2, 38, 39] (5) where Φ_sub(K) is the m × K submatrix formed from K distinct columns of Φ, and σ_min[Φ_sub(K)] and σ_max[Φ_sub(K)] are the minimum and maximum singular values of Φ_sub(K), respectively (In [2, 38, 39], the eigenvalue λ[Φ_sub(K)*Φ_sub(K)] is used for (5), where λ[Φ_sub(K)*Φ_sub(K)] = σ²[Φ_sub(K)]). The RIP constant δ_K is the smallest constant that satisfies the inequality (5) for every m × K sized submatrix of Φ, and it is essentially a bound on the distance between unity and the singular values of all Φ_sub(K)s. It is shown in [23] that if δ_2K < 1, then a K-sparse signal x can be exactly reconstructed from the measurements of Φ.

While δ_K ∈ (0,1) renders the exact reconstruction of x, the value of δ_K determines the stability of reconstruction. In the presence of measurement noise ϵ, y = Φx+ϵ and the reconstructed signal $\widehat{x}$ satisfies (Section 5.2 in [40]) (6) Thus, the smaller the δ_K, the smaller the reconstruction error, and vice versa. Since measurement noise always exists in practice, the value of δ_K is an important performance measure of a measurement matrix Φ for both the reconstruct-ability and reconstruction error. However, the computation of δ_K for a given Φ is NP hard and hence intractable. Since the RIP constant δ_K is essentially a bound on the distance between 1 and the singular values of all Φ_sub(K)s, the value of δ_K can be assessed by the distances from 1 to the σ_min[Φ_sub(K)]s and σ_max[Φ_sub(K)]s. The smaller the distance, the smaller the δ_K and hence the better performance of Φ. Since (5) must hold for all the m × K submatrices of Φ, the statistics of σ_min[Φ_sub(K)] and σ_max[Φ_sub(K)] over randomly sampled Φ_sub(K)s are used in [5, 38, 39] to assess the RIP performance of a given measurement matrix Φ. This method is also adopted in this paper.

Another important sufficient condition for exact reconstruction of x is the incoherence [3, 36]. For a pair of measurement matrix Φ and sparsifying transform matrix Ψ, satisfying Φ*Φ = nI and Ψ*Ψ = I, their incoherence is defined as (7) where Φ_k and Ψ_j are respectively the k^th and j^th columns of Φ and Ψ, and $\mu (\Phi ,\Psi )\in [1,\sqrt{n}]$. The value μ(Φ, Ψ) = 1 is termed as maximal incoherence. As shown in [3], if m ≥ C ⋅ μ²(Φ, Ψ) ⋅ K ⋅ log(n), where C is a small constant, then a K-sparse signal x can be exactly reconstructed. Thus, μ(Φ, Ψ) determines the minimum number of measurements needed for exact reconstruction of x. The smaller the μ(Φ, Ψ), the smaller the m (the fewer the measurements) needed for exact reconstruction of x.

It is important to note that both the RIP and the incoherence are sufficient conditions on the measurement matrix. So they are parallel and either or both of them can be used to design, analyze and assess the measurement matrix for exact reconstruction of x.

2.3 Multichannel Compressive Sensing

MRI systems acquire multiple measurements of the desired signal through multiple channels. Given the multiple channels, the data acquisition process can be modeled as y_i = ΦΓ_i x = ΦΓ_iΨ*ρ, i = 1, 2, ⋯, L, where Γ_i = diag[γ_ij]_{j = 1, 2, ⋯, n} is the complex valued sensitivity map matrix of the ith receive channel, with γ_ij being the sensitivity of the ith channel at the jth pixel of the vectorized image, L is the number of receive channels, y_i is the data acquired from the ith receive coil and ρ = Ψx. In matrix form, the y_is can be written as (8) As seen from above, with L receive channels, the measurement matrix for x becomes E, which has a column of L sub-matrices ΦΓ_is and dimension Lm × n. These L sub-matrices, ΦΓ_is, share a common measurement matrix Φ ∈ ℂ^{m × n} resulting from the encoding process, so they measure the same x simultaneously with the same under-sampling pattern and under-sampling ratio n/m. As discussed in Section 2.1, the under-sampling ratio n/m is solely determined by the number of phase encoding lines used in data acquisition, hence it is independent of the number of channels L. It is important to note that Γ_is are complex valued and Γ_i ≠ Γ_j, i ≠ j, in general. Therefore, ΦΓ_i ≠ ΦΓ_j for i ≠ j and can be independent of each other, depending on the specific phase values of Γ_i and Γ_j. As a result, the multichannel measurement matrix E provides more independent measurements than that of the single channel Φ, which may improve RIP. The improved RIP may in turn reduce the number of measurements, m, and hence the number of phase encoding lines needed for exact reconstruction of x. This reasoning is confirmed by the empirical RIP analysis of E in Section 3.4.

In light of the above discussion, the following MCS optimization is considered for reconstructing the desired image x from the multichannel measurements of MRI. (9) where Ψ is the wavelet transform operator and ϵ determines the allowed noise level in the reconstructed image.

MR images are also known to be sparse in the total variation (TV) domain. It is demonstrated in [41] that the TV penalty is critical to the performance of CS-MRI, and that MR images can be recovered more efficiently with the use of TV penalty together with the wavelet penalty. Therefore, most of the CS-MRI work [6, 7, 12, 13, 25, 28, 30, 33, 34] has used both TV and wavelet penalties for better reconstruction performance. To be consistent with this common practice in CS-MRI, the TV penalty is included in the objective function (9) for MCS-MRI reconstruction, together with the wavelet penalty. Eq (9) is a constrained optimization problem which is computationally intensive to solve. To relax the problem, (9) is converted to the unconstrained optimization problem with the inclusion of TV penalty. (10) where TV is a 2D total variation operator and λ₁, λ₂ are regularization parameters for wavelet and TV penalties, respectively.

Daubechies-4 (db-4) wavelet is usually used in CS-MRI because of its superior performance in sparsifying the MR images. To be consistent with this fact and fair in comparison with the existing CS-MRI results, the unconstrained objective function (10) with the Ψ of the db-4 wavelet operator will be used throughout all the simulations and reconstructions in this work.

2.4 Noiselets

Noiselets are functions which are noise-like in the sense that they are totally incompressible by orthogonal wavelet packet methods [35, 36]. Noiselet basis functions are constructed similar to the wavelet basis functions, through a multi-scale iteration of the mother bases function but with a twist. As wavelets are constructed by translates and dilates of the mother wavelet function, noiselets are constructed by twisting the translates and dilates [3]. The mother bases function χ(x) can be defined as The family of noiselet basis functions are generated in the interval [0,1) as (11) where $i=\sqrt{-1}$ and f_2ⁿ, ……, f_2ⁿ+1 form the unitary basis for the vector space V_n. An example of a 4 × 4 noiselet transform matrix is given below. (12)

Noiselets totally spread out the signal energy in the measurement domain and are known to be maximally incoherent with the Haar wavelet. The mutual incoherence parameter between the noiselet measurement matrix Φ and the sparsifying Haar wavelet transform matrix Ψ has been shown to be equal to 1 [3], which is the minimum value possible for the incoherence. Therefore, theoretically, noiselets are the best suited measurement basis function for CS-MRI when the wavelet is used as sparsifying transform matrix.

2.5 Motivation

The motivations behind using noiselets as a measurement matrix in MCS-MRI are as follows:

• Noiselets completely spreads out the signal energy in the measurement domain and are maximally incoherent with wavelets.
• Noiselet basis function is unitary and hence does not amplify noise as in the case of random encoding [25].
• Unlike random basis, noiselet basis has conjugate symmetry. Thus, this property of symmetry can be exploited by using the partial Fourier like technique.
• Noiselets are derived in the same way as wavelets, therefore it can be modelled as a multi-scale filter-bank and can be applied in O(n ⋅ log(n)).

We proposed to use the noiselet encoding in the phase encode (PE) direction in 2D MR imaging. Therefore, the acquired data is noiselet encoded in the PE direction and Fourier encoded in the frequency encode (FE) direction.

3.1 Pulse Sequence Design for Noiselet Encoding

In conventional 2D MR imaging sequences, a spatially selective RF excitation pulse is used to select the slice and the linear spatial gradients are used to encode the excited slice onto the Fourier transform space. In [31, 32, 42, 43], it is demonstrated that the spatially selective RF excitation pulse can also be used to encode the imaging volume. In [25, 32, 44, 45], the wavelet, SVD and random encoding profiles have been implemented using the spatially selective RF excitation pulses. An analysis using the linear response model described in [42] provides a theoretical framework to design spatially selective RF excitation pulses for implementation of non-Fourier encoding. Under the small flip angle (≤ 30°) regime, the RF pulse envelope can be calculated directly by taking the Fourier transform of the desired excitation profile. However this method of designing an RF excitation pulse requires excellent RF and main field homogeneity.

To excite a noiselet profile during excitation, one can design an RF pulse envelope by directly taking the Fourier transform of the noiselet basis functions. For an image of size 256 × 256, the noiselet measurement matrix has 256 rows and 256 columns (refer to (12) for the low dimensional example). The Fourier transform of each row of the noiselet matrix will result in 256 RF pulse envelops. To achieve a specific flip angle, say 10 degrees, using such noiselet RF pules, the integral of each individual noiselet pulse envelope over time must equal that of the sinc RF pulse used in Fourier encoding for the same flip angle [46]. This equal integral rule can be used to determine the scaling of noiselet pulse envelops for a required flip angle. Compared to a sinc RF pulse of the same energy, a noiselet pulse envelope tends to have a smaller integral since its phase changes are more diverse than that of sinc pulse. Therefore, noiselet pulses generally require more energy than that of sinc RF pulse to attain the same flip angle. But the energy difference is generally marginal and hence the resulting SAR level of noiselet encoding is comparable to that of Fourier encoding. See Section 4.4 for experiment example.

A pulse sequence for the noiselet encoding of 2D MR imaging is shown in Fig 1. The pulse sequence is designed by tailoring the spin echo sequence. The RF excitation pulse in the conventional spin echo sequence is replaced by the noiselet RF pulse, and the slice select gradient is shifted to phase encoding axis. The 180° refocusing RF pulse is used in conjunction with the slice selection gradient to select the slice that refocuses the spins only in the desired slice. Spoilers are used after the readout gradient to remove any residual signal in the transverse plane. A new RF excitation pulse is used for every new TR to excite a new noiselet profile, and a total of 256 TR are required for excitation of the complete set of noiselet bases. The readout gradient strength determines the FOV in the readout direction, while the phase encoding gradient strength and duration of the RF excitation pulse determines the field of view (FOV) in phase encoding direction. The FOV in phase encoding direction is determined as (13) where G_y is the gradient strength in PE direction, γ is the gyromagnetic ratio, and Δt_p is the dwell time of the RF pulse which is defined as Δt_p = (Duration of RF pulse) / (Number of points in RF pulse). Eq (13) is used to calculate the gradient strength G_y required in the phase encoding direction during execution of RF excitation pulse.

3.2 Under-sampling in noiselet encoding

Noiselet transform is a type of Haar-Walsh transform. The noiselet transform coefficients totally spread out the signal in scale and time (or spatial location) [35]. As a result, each subset of the transform coefficients contains a certain information of the original signal at all the scales and times (spatial locations), and can be used alone with zero padding to reconstruct the original signal at a lower resolution. This important property is demonstrated by the example shown in Fig 2.

Fig 2 shows a brain image of size 256 × 256, and the 3D magnitude map of the noiselet transform of the brain image along the phase encoding direction (all noiselet encodes). Fig 2(c)–2(f) shows the images reconstructed with the first, second, third and fourth 64 noiselet encodes by zero padding the rest. Each of these images are reconstructed using one quarter of the noiselet encodes and has low resolution than the original image. However, each of these images have complementary information about the original image and have approximately the same amount of energy and information because they are reconstructed using the same size of partial matrix from the original coefficient matrix.

Based on the above property of noiselet transform, we propose to under-sample the noiselet encoded data along the phase encoding direction according to the uniform probability distribution function. One sampling mask using this scheme is shown in Fig 3(a) where the white lines represent the sampled data points and the black lines represent the unsampled data points. Fig 3(b) shows the sampling mask for Fourier encoding scheme drawn from a variable density probability distribution function shown in Fig 3(c).

3.3 Coil Sensitivity Estimation

The performance of the MCS-MRI reconstruction depends on the accuracy of the sensitivity matrix estimated. We used the regularized self-calibrated estimation method [47] to estimate the sensitivity maps from the acquired data. This method estimates the sensitivity map ${\widehat{\Gamma }}_i$ of the ith receive coil by using (14) where i ∈ [1, 2, ⋯, L] and R(Γ) is a spatial roughness penalty function with weighting factor β. The reference image I_ref can be obtained using the sum of squares of individual coil images I_i’s. For the experimental results presented below, the sensitivity maps were estimated from fully sampled images using (14). The data was acquired using a 32 channel head coil, but only the data from 14 channels with good SNR was used in the reconstruction.

3.4 Empirical RIP analysis of measurement matrix

According to the CS theory summarized in Section 2.2, the measurement matrix is crucial to the performance of CS reconstruction, and the performance of a measurement matrix Φ for a K-sparse signal x can be assessed by the statistics of σ_min[Φ_sub(K)] and σ_max[Φ_sub(K)] over the Φ_sub(K)s consisting of k distinct columns of Φ. To understand the behavior and advantage of the noiselet encoding proposed above, we have used this method to assess the noiselet measurement matrix in comparison to the conventional Fourier measurement matrix.

In the assessment, the size of the signal was n = 256, the number of measurements m = 100, the number of channels L = 1,8 and 14, and the sparsity K was varied from 5 to 100 with an increment of 5. For L = 1, the measurement matrices, Φs, were generated for the noiselet and Fourier encodings, respectively. For L = 8 and 14, the measurement matrices Es as given in (8) were generated for the noiselet and Fourier encodings, respectively. For each K, 2000 submatrices Φ_sub(K)s were drawn uniformly randomly from the columns of Φ, then the σ_min[Φ_sub(K)] and σ_max[Φ_sub(K)] of every Φ_sub(K) were calculated. The same procedure is used to obtain the submatrices E_sub(K)s from E and to calculate the σ_min[E_sub(K)] and σ_max[E_sub(K)] of every E_sub(K). The statistics of σ_min[Φ_sub(K)]s, σ_max[Φ_sub(K)]s, σ_min[E_sub(K)]s and σ_max[E_sub(K)]s were accumulated from their respective 2,000 samples.

Fig 4 shows the means and standard deviations of the minimum and maximum singular values of Φ_sub(K)s and E_sub(K)s versus the sparsity K for the Fourier and noiselet measurement matrices. As seen from the figure, in single channel case, the singular values of noiselet measurement matrix are closer to 1 than those of Fourier measurement matrix, but are not significantly different. As the number of channels increases, the singular values of noiselet and Fourier measurement matrices all move towards 1, but those of noiselet measurement matrix move much closer to 1 than those of Fourier measurement matrix. By the CS theory, when the maximum distance from unity to the singular values is less than 1, it equals roughly the RIP constant δ_K. Therefore, the figure actually reveals two facts: 1) For both the noiselet and Fourier measurement matrices, the RIP constant δ_K decreases as the number of channels increases. 2) As the number of channels increases, the RIP constant δ_K of noiselet measurement matrix deceases much more than that of Fourier measurement matrix. According to the CS theory, these imply that the multichannel measurement matrix should generally outperform the single channel measurement matrix, and that the multichannel noiselet measurement matrix should generally outperform the multichannel Fourier measurement matrix.

As a particular example consider the curves in Fig 4(d) for the noiselet measurement matrix. To facilitate discussion, the distances from unity to the singular values of a measurement matrix will be called the δ-distances here. In single channel case, the δ-distances of noiselet measurement matrix are less than 1 for K ≤ 40. By RIP, this implies that the single channel noiselet measurement matrix can guarantee the recovery of the signals with sparsity K ≤ 20. When the number of channels is increased to 14, the δ-distances are less than 1 for K ≤ 85. So the 14 channel noiselet measurement matrix can guarantee the recovery of the signals with sparsity K ≤ 42. The improvement in terms of sparsity is two folds. In contrast, for the 14 channel Fourier measurement matrix shown in Fig 4(c), its δ-distances are less than 1 only for K < 30, so it can only guarantee the recovery of the signals with sparsity K < 15.

From the above assessment, it can be expected that the multichannel CS MRI will outperform the single channel CS MRI and that the noiselet encoding multichannel CS MRI will outperform the Fourier encoding multichannel CS MRI in practice. These are confirmed by the simulation and experiment results presented in the next sections.

4.1 Simulations

Simulations were performed on a (256 × 256) brain image acquired using a spin echo sequence to investigate the performance of noiselet encoded and Fourier encoded MRI. The simulation study was divided into two parts: (i) a simulation study with a single channel using uniform sensitivity and (ii) a simulation study with multiple channels where the sensitivity profiles were estimated from the acquired data.

4.2 Single Channel Simulation with a Uniform Sensitivity Profile

Fourier encoded CS-MRI A Fourier transform of the image was taken in the PE direction to simulate Fourier encoding. A variable density random sampling pattern as shown in Fig 3(a) where samples were taken in the PE direction according to a Gaussian distribution function was used in this simulation. A non-linear program of (10) was solved to reconstruct the final image for acceleration factors of 2 and 3. In this case the encoding matrix E does not have any sensitivity information (i.e. E = Φ).

Noiselet encoded CS-MRI A noiselet transform of the image was taken in the PE direction to simulate noiselet encoding. A completely random sampling pattern was used to sample the noiselet encoded data in the PE direction and the non-linear program of (10) was solved to reconstruct the final image for acceleration factors of 2 and 3. In this case the encoding matrix E does not have any sensitivity information (i.e. E = Φ).

Fig 5 show the images reconstructed with Fourier encoding and noiselet encoding using variable density random under-sampling and completely random under-sampling pattern respectively. The noiselet encoded CS-MRI performs similar to that of the Fourier encoded CS-MRI. This is due to the fact that in the case of variable density random under-sampling, the Fourier encoding judiciously exploits extra information about the data, namely the structure of k-space. The center of the k-space data has maximum energy and hence, by densely sampling the center of k-space, the Fourier encoding captures most of the signal energy and results in better performance.

In practice, the MR data is collected through the use of multiple channels, and data in each channel is slightly different from the other channels. The actual k-space data is convolved with the Fourier transform of the sensitivity profiles of the individual channel, making the data from each channel different from others. This sensitivity information can also be taken into consideration while performing the CS reconstruction, by applying the multichannel CS frame. Therefore, to further study the effect of sensitivity information on noiselet encoding and Fourier encoding, MCS-MRI simulations were performed. To quantitatively compare the performance of both the encoding schemes, we used the relative error defined in (15) as a metric: (15)

First we investigated the effect of the number of channels on the reconstruction quality using the MCS framework. For a fixed number of measurements, the number of channels was varied and the mean of the relative error for 1000 such simulations was calculated. Fig 6 shows the plot of the mean relative error versus the number of channels for the acceleration factors of 2 and 3. When the number of channels was two, the noiselet encoding scheme outperformed the Fourier encoding scheme for both the acceleration factors of 2 and 3. However, when number of channels was equal to one, the noiselet encoding outperformed the Fourier encoding for an acceleration factor of 2, but not for an acceleration factor of 3. It is interesting to note that noiselet encoding outperformed Fourier encoding for both acceleration factors when the number of channels was greater than one. These simulations suggest that noiselet encoding should take into account the sensitivity information while performing CS, and therefore noiselet encoding is potentially a better encoding scheme for MCS-MRI. Based on the fact that noiselet encoding performs better than Fourier encoding for multi-channel data, we investigated the performance of both the encoding schemes using muti-channel data.

4.3 Multiple Channel Simulation

A (256 × 256) brain image was used to compare the performance of noiselet encoding and Fourier encoding in MCS-MRI for different acceleration factors. Eight complex sensitivity maps (Fig 7) obtained from the head coil of a Siemens Skyra 3T scanner were used to perform the simulations. For solving the minimization program in (10), we used the nonlinear conjugate gradient with the backtracking line search method [7]. The measurement matrix (Φ) was the discrete Fourier transform matrix while the daubechies-4 wavelet transform matrix (Ψ) and TV were used as sparsifying transforms.

Fourier encoded MCS-MRI The reference brain image was multiplied by the sensitivity function to generate eight sensitivity encoded images. The Fourier transform of each these images was taken in the PE direction; only a few PEs were taken according to the Gaussian probability distribution function. MCS-MRI reconstruction of (10) was solved using the nonlinear conjugate gradient on this data. An example sampling scheme for the Fourier encoded MCS-MRI is shown in Fig 3(a).

Noiselet encoded MCS-MRI A Noiselet transform of the sensitivity encoded images was taken in the PE direction, with only a few PE selected according to the uniform probability distribution function. MCS-MRI reconstruction of (10) was solved using the nonlinear conjugate gradient on this data. An example of the sampling scheme for noiselet encoded MCS-MRI is shown in Fig 3(b).

For a noiseless simulation, the reconstructed images for different acceleration factors (4, 8 and 16) are shown in Fig 8. The difference images in Fig 8(d)–8(f) and 8(j)–8(l) demonstrate that the error in noiselet encoding is always less than in Fourier encoding, and that the noiselet encoded MCS-MRI reconstruction preserves spatial resolution better than the Fourier encoded MCS-MRI. Fig 8(m) and 8(n) show the zoomed images reconstructed with Fourier encoding for acceleration factors of 8 and 16 respectively, while Fig 8(o) and 8(p) show the zoomed images reconstructed with noiselet encoding for an acceleration factors of 8 and 16 respectively. The zoomed images highlight that the spatial resolution of the noiselet encoded reconstructions outperforms the Fourier encoded reconstructions. Moreover, the spatial resolution provided by the noiselet encoding at an acceleration factor of 16 is comparable to that of the Fourier encoding at an acceleration factor of 8, suggesting that noiselet encoding performs approximately twice as good as Fourier encoding.

To further compare MCS-MRI under Fourier encoding and noiselet encoding, we also reconstructed images using only the wavelet penalty in (9). As shown in Fig 9, the reconstructed images of both encoding schemes are degraded as compared to those reconstructed using both the wavelet and TV penalties. However, the superior image quality of noiselet encoding over Fourier encoding still holds true.

To measure the relative error, simulations were performed on the brain image for 1000 times by randomly generating a sampling mask each time. The mean of the relative errors was calculated after 1000 such reconstructions at every acceleration factor. The mean relative error versus the number of measurements is plotted in Fig 10 and highlights that noiselet encoding outperforms Fourier encoding for all acceleration factors. The relative error for noiselet encoding at an acceleration factor of 16 was the same as the relative error for Fourier encoding at an acceleration factor of 8 indicating that higher acceleration factors are achievable with noiselet encoding compared to Fourier encoding.

In practice, MR data always has some noise and the level of noise depends upon many factors including the FOV, resolution, type of imaging sequence, magnetic field inhomogeneity and RF inhomogeneity. Therefore, simulations were carried out to evaluate the performance of both the noiselet encoding and Fourier encoding schemes in the presence of variable levels of noise. Different levels of random Gaussian noise were added to the measured k-space data, and MCS-MRI reconstruction was performed for noiselet and Fourier encoding schemes. For every level of noise, 1000 simulations were performed and the mean of the relative error was calculated. Fig 11 shows the mean relative error as a function of the Signal to Noise Ratio (SNR), demonstrating the comparative performance of noiselet encoding reconstructions and Fourier encoding reconstructions in the presence of noise. The plots demonstrate that noiselet encoding outperforms Fourier encoding for SNR above 20 dB for all acceleration factors, but does a poor job at extremely low SNR of 10 dB. The poor performance of noiselet encoding at 10 dB SNR can be attributed to the fact that at extremely low SNR, most of the noiselet coefficients are severely corrupted by the noise since their magnitudes are approximately uniform. In contrast, the Fourier coefficients at the center of k-space have much larger magnitudes and hence are less affected by the noise at low SNR. These large magnitude coefficients are fully utilized in reconstruction because of the centralized variable density sampling scheme, hence Fourier encoding is less affected by the noise and performs better at low SNR.

4.4 Experiments

Experiments were carried out on Siemens Skyra 3T MRI scanner with a maximum gradient strength of 40 mT/m and a maximum slew rate of 200 mT/m/sec. Informed consent was taken from healthy volunteers in accordance with the Institution’s ethics policy. To validate the practical implementation of noiselet encoding, the pulse sequence shown in Fig 1 was used to acquire noiselet encoded data. An RF excitation pulse with 256 points was used. The flip angle was 10° calculated by the equal integral rule described in Section 3.1, with the SAR level checked by Siemens’ RF pulse programming software IDEA to be well below the safety limit and about 5% higher than that of Fourier encoding RF pulse. We also acquired the Fourier encoded data using the spin echo (SE) sequence to compare the quality of the reconstructed image from the data acquired by the noiselet encoding sequence. An apodized slice selective sinc RF excitation pulse was used in the spin echo sequence with a flip angle of 10°. The protocol parameters for the noiselet encoding sequence and the Fourier encoding SE sequence were (i) Phantom experiments FOV = 200 mm, TE/TR = 26/750 ms, image matrix = 256 × 256; and (ii) In vivo experiments FOV = 240 mm, TE/TR = 26/750 ms, image matrix = 256 × 256.

Non-Fourier encoding in general is sensitive to field inhomogeneities, but careful design of the sequence and good shimming can result in high quality images. To reconstruct the noiselet encoded data the inverse Fourier transform was taken along the frequency encoding axis and the inverse noiselet transform was taken along the PE axis. To reconstruct the Fourier encoded data, an inverse Fourier transform was taken along both axes. Fig 12 shows the images reconstructed from the fully sampled noiselet encoded data and Fourier encoded data sets. These images demonstrate that the noiselet encoding reconstructions are practically feasible and produce artifact free images. Fig 12(c) shows a zoomed portion of the noiselet encoded image, while Fig 12(f) shows a zoomed portion of the Fourier encoded image. The zoomed images reveal that the resolution of the image from noiselet encoding with 256 noiselet excitation is the same as that of the image from Fourier encoding with 256 phase encodes. Fig 12(g) and 12(i) show the T2 weighted images for the brain with noiselet encoding and Fourier encoding, respectively. Fig 12(h) and 12(j) show the T1 weighted images for the brain with noiselet encoding and Fourier encoding, respectively. It is evident from the in vivo images that the proposed noiselet encoding is feasible in practice.

To validate the feasibility of the proposed reconstruction method, we performed retrospective under-sampling on the acquired noiselet encoded data and Fourier encoded data to simulate accelerated data acquisition. After retrospective under-sampling, the unconstrained optimization program (10) was solved using the non-linear conjugate gradient method to reconstruct the desired image for different acceleration factors. Fig 13(a)–13(c) shows the reconstructed images for the acceleration factors of 4, 8 and 16 on the Fourier encoded data while Fig 13(d)–13(f) shows the corresponding difference images. Similarly, Fig 13(g)–13(i) shows the reconstructed images for the acceleration factors of 4, 8 and 16 on the noiselet encoded data, and Fig 13(j)–13(l) shows the corresponding difference images for noiselet encoded MCS-MRI. These results on the acquired data are consistent with the simulation results and indicate that the noiselet encoding is superior to the Fourier encoding in preserving resolution.

Fig 13(A)–13(H) shows the zoomed portion of the reconstructed images with Fourier encoding and noiselet encoding. One can distinguish between the small dots in the zoomed images reconstructed with noiselet encoding while it is difficult to distinguish these dots in the images reconstructed with Fourier encoding. This demonstrates that noiselet encoding is able to preserve resolution better than the Fourier encoding. Fig 14 show the images reconstructed with Fourier encoding and noiselet encoding for various acceleration factors on the data acquired for one axial slice of the brain. Since the SNR of the in vivo images is less than in the phantom images, reconstruction is shown only up to an acceleration factor of 8. The difference images demonstrate that noiselet encoding outperforms Fourier encoding for all acceleration factors. In particular, at the acceleration factor of 8 the image reconstructed with Fourier encoded data has significantly poorer resolution compared to the image reconstructed with noiselet encoded data.