Subspace and Self Supervised Denoising for MRSI Brain Scans

Why High-Dimensional MR Spectroscopic Imaging Struggles With Signal-to-Noise Ratio

The problem runs deeper than it first appears. In conventional anatomical MRI, the scanner measures radiofrequency echoes primarily from hydrogen nuclei in tissue water and fat, which are present at molar concentrations throughout the human body. In proton Magnetic Resonance Spectroscopic Imaging (1H-MRSI), water and subcutaneous lipid signals are actually nuisance contaminants that must be stripped away so clinicians can measure millimolar brain metabolites such as N-acetylaspartate (NAA, a marker of neuronal integrity), creatine (Cr, an energy metabolism reference), choline (Cho, a marker of cellular membrane turnover), glutamate-glutamine (Glx), and gamma-aminobutyric acid (GABA)[cite: 49, 50, 51].

Because instead of recovering a static 3D spatial volume \(\rho(r)\), MRSI must recover a four-dimensional spatiospectral function \(\rho(r, f)\) spanning three spatial coordinates \(r\) and a continuous chemical shift frequency axis \(f\) (or its Fourier counterpart, the Free Induction Decay time axis \(t\))[cite: 45, 46]. Simultaneously, static magnetic field inhomogeneities \(\Delta f(r)\) across the skull distort spectral peaks, while thermal noise \(n(k, t)\) corrupts every sample in \((k, t)\)-space[cite: 45]. If you attempt to achieve high spatial resolution without waiting thirty to forty minutes for multi-average signal accumulation, a standard Fourier reconstruction yields an unusable snowstorm of noise where relative \(l_2\) error exceeds 107 percent[cite: 49].

Over the past decade, low-dimensional subspace modeling—most notably the SPICE (Spectroscopic Imaging by Exploiting Spatiospectral Correlation) framework pioneered by Liang, Lam, and colleagues—transformed accelerated MRSI[cite: 45, 49, 54]. Subspace models recognize that every voxel’s spectrum is a linear combination of a small number of shared molecular spectral basis functions, allowing the high-dimensional spatiotemporal matrix \(X \in \mathbb{C}^{N \times Q}\) (with \(N\) voxels and \(Q\) time points) to be factored into spatial coefficients \(U \in \mathbb{C}^{N \times r}\) and a pre-learned temporal subspace \(V \in \mathbb{C}^{r \times Q}\) where rank \(r \ll N, Q\)[cite: 45, 46]. While projecting noisy measurements onto a low-rank subspace dramatically improves SNR, noise that happens to be coherent with the basis vectors still leaks into the reconstruction[cite: 45]. Traditional handcrafted spatial regularizers, such as weighted finite differences or Total Generalized Variation (TGV) by Klauser et al., suppress some of that residual noise but frequently oversmooth sharp cortical boundaries and blur small focal lesions[cite: 45, 49, 50].

Why Standard Deep Learning Fails in Low-SNR MRSI and How Self-Supervision Bridges the Gap

Deep learning priors have revolutionized structural MRI reconstruction, a trend we have tracked closely across our AI for medical imaging and healthcare hub[cite: 46]. Yet applying supervised neural networks to 4D MRSI hits an immediate wall: you cannot train a supervised end-to-end network without thousands of clean, high-SNR, high-resolution ground-truth human brain MRSI volumes, and acquiring such a dataset in vivo is physically impractical[cite: 46].

Researchers have tried several workarounds, and none of them fully solve the clinical problem[cite: 46]:

• Single-Voxel Spectral Denoising: Networks trained by Chen et al. and evaluated by Dziadosz et al. operate on 1D spectra voxel-by-voxel, completely ignoring 3D spatial anatomical correlations across neighboring brain tissue[cite: 46, 54].
• Synthetic Physics Simulations: Studies by Iqbal et al., Wang et al., and Li et al. train networks on simulated spatiospectral phantoms, but synthetic datasets struggle to capture the complex, heterogeneous spatial distributions of real human pathology[cite: 46, 54].
• Self-Supervised k-Space Splitting (SSDU, ENSURE, RARE): Methods by Yaman et al., Aggarwal et al., Huang et al., and Liu et al. train directly on undersampled k-space data, yet they still require baseline acquisition SNRs far higher than what fast in vivo MRSI provides[cite: 46, 54].
• Untrained Deep Image Priors (DIP): Approaches following Lempitsky et al. and Gong et al. optimize an untrained network directly on a single noisy scan, but require delicate early stopping to avoid overfitting to thermal noise[cite: 46, 54].

In their IEEE Transactions on Medical Imaging paper, Zhao, Wang, Anderson, Huesmann, and Lam take a fundamentally different route[cite: 45]. Instead of training an end-to-end inverse solver on synthetic brains or relying on spatial-only blind-spot denoisers like standard 2D Noise2Void, they design a complex-valued 3D spatiotemporal interpolation network trained directly on noisy in vivo MRSI patches, and lock that learned prior inside a Plug-and-Play ADMM (PnP-ADMM) loop governed by an explicit low-rank subspace constraint[cite: 46, 47, 48].

Mathematical Formulations of the Encoding Model, Denoiser, and PnP-ADMM

To appreciate why the subspace model and the self-supervised denoiser stabilize each other so effectively, let us walk through the exact mathematical progression of the paper[cite: 45, 46, 47, 48]. First, the continuous forward physical acquisition of \((k, t)\)-space measurements \(d(k, t)\) over spatial volume \(V\) and spectral bandwidth \(\Omega\) is governed by Equation 1[cite: 45].

Here, \(\Delta f(r)\) captures the spatial \(B_0\) magnetic field inhomogeneity map (in Hz), and \(n(k, t)\) denotes complex zero-mean Gaussian measurement noise[cite: 45, 47, 49]. Discretizing the target spatiotemporal function as matrix \(X \in \mathbb{C}^{N \times Q}\) (with \(N\) spatial voxels and \(Q\) FID time points), let \(B \in \mathbb{C}^{N \times Q}\) with entries \(B_{nq} = \exp(i 2\pi \Delta f(r_n) t_q)\) model the \(B_0\) field phase evolution, \(F\) denote the spatial Fourier transform matrix, \(\Omega\) denote the \((k, t)\)-space sampling mask, and \(\odot\) denote elementwise Hadamard multiplication[cite: 46]. Enforcing the low-dimensional subspace constraint \(X \in S\) alongside a complementary spatial-spectral regularizer \(R(X)\) yields the constrained reconstruction problem in Equation 2[cite: 46].

Self-Supervised Spatiotemporal Interpolation (Equations 3 and 4)

In general inverse problems, solving a proximal subproblem associated with \(R(\cdot)\) on a noisy multidimensional image \(\tilde{X}\) takes the form of the denoising optimization in Equation 3[cite: 46, 47].

Rather than handcrafting \(R(H)\), Zhao et al. replace Equation 3 with a complex-valued neural network \(f_\theta\) trained self-supervisedly on noisy MRSI data alone[cite: 47]. Consider the biophysics of an MRSI volume: spatial correlation across tissue types is primarily local across neighboring voxels, whereas temporal Free Induction Decay (FID) points at a given voxel obey global quantum-mechanical resonance decay laws across all \(Q\) time points[cite: 47]. Standard 2D spatial Noise2Void by Krull et al. or DeepInterpolation by Lecoq et al. do not exploit this hybrid geometry[cite: 46, 47, 54].

The authors therefore extract 4D patches \(\tilde{X}_{P(\{i\})}^j\) of size \(N_1 \times N_2 \times N_3 \times Q\) (specifically \(4 \times 4 \times 2 \times 318\) in single-TE brain scans, or \(4 \times 4 \times 2 \times 198\) per TE in 2-TE scans) that are spatially local (\(4 \times 4 \times 2\)) but temporally global (containing the entire length-\(Q\) FID)[cite: 47, 49, 50]. Within each patch \(j\), exactly 12.5 percent of the spatial voxels at various time points—indexed by \(\{i\}\)—are randomly masked out, while their original noisy values are stored in target vector \(s_{\{i\}}^j\)[cite: 47, 49]. Evaluating the squared error strictly at the masked-out coordinates via sampling operator \(M\) yields the self-supervised training objective in Equation 4[cite: 47].

Because the thermal noise in raw MRSI is zero-mean and spatiotemporally independent whereas the underlying metabolite signal is strongly correlated across neighboring voxels and FID time points, minimizing Equation 4 is expectation-wise equivalent to supervised training against noiseless ground truth[cite: 47]. This design parallels the physics-informed self-supervision strategies we analyzed in GenSeg for low-data medical segmentation and physics-guided high-frequency ultrasound, where physical signal structure substitutes for missing clean labels.

Synergizing Subspace Modeling and PnP-ADMM (Equations 5 to 11)

Once \(f_{\hat{\theta}}\) is trained, the authors factor the spatiospectral image explicitly as \(X = U\hat{V}\) using a pre-learned rank-\(r\) temporal basis \(\hat{V} \in \mathbb{C}^{r \times Q}\) and introduce an auxiliary variable \(H = A(U\hat{V})\), where \(A\) is a Fourier truncation resolution-matching operator that allows the reconstruction grid of \(U\hat{V}\) to be finer than the grid on which the denoiser was trained[cite: 46, 47, 48, 49]. Adding a mild edge-weighted finite-difference penalty \(\|D_w U\hat{V}\|_F^2\) (with weight \(\lambda_2 = 0.04\)) gives the joint constrained optimization in Equation 5 and its Augmented Lagrangian in Equation 6[cite: 47, 48, 53].

Here, \(Z\) is the Lagrangian multiplier and \(u_1\) is the ADMM penalty parameter[cite: 48]. Alternating Direction Method of Multipliers (ADMM) decomposes Equation 6 into three iterative subproblems at iteration \(k\)[cite: 48]:

Look closely at Equation 8 (\(\hat{H}^{k+1} = \hat{H}_{\text{raw}}^{k+1}\hat{V}^H\hat{V}\)), where overlapping denoised patches are first combined via weighted-averaging operator \(C(\cdot)\) in Equation 11 and then multiplied by the orthogonal projection matrix \(\hat{V}^H\hat{V}\)[cite: 48]. Why add an explicit subspace projection right after a neural network denoiser? Because a nonlinear U-Net is not a strict mathematical proximal operator and can hallucinate temporal components that lie outside the biophysical subspace \(\hat{V}\)[cite: 48]. Projecting the denoiser output back onto \(\hat{V}\) strips away out-of-subspace network artifacts before Subproblem II solves the quadratic spatial update for \(\hat{U}^{k+1}\) via Conjugate Gradient[cite: 48].

Theoretical Fixed-Point Convergence via Douglas-Rachford Splitting

In Section II-D and the Appendix, the authors prove Theorem 1 by establishing an exact equivalence between their PnP-ADMM updates and Plug-and-Play Douglas-Rachford Splitting (PnP-DRS)[cite: 48, 53]. By defining the scaled multiplier \(\tilde{Z}^k = \hat{Z}^k / u_1\), letting \(\text{Prox}(\cdot)\) denote the quadratic solver for Equation 9, letting \(D_{\hat{V}}(\cdot)\) denote the composite denoiser-plus-subspace-projection operator, and introducing the state variable \(\hat{M}^k = \tilde{Z}^k – \hat{H}^{k+1}\), Equations 7 through 10 collapse into a single fixed-point iteration[cite: 48, 53].

By the classical Banach contraction principle, if the composite operator \(T\) satisfies Equation 13 with a contraction factor \(\delta < 1\), the sequence \(\hat{M}^k\) is guaranteed to converge to a unique fixed point \(M^*\) as \(k \to \infty\)[cite: 48, 53].

Experimental Benchmarks Across Numerical Brain Simulations and In Vivo Cohorts

To test the framework under realistic neurochemical distributions, the authors built a ground-truth 3D brain phantom from a high-SNR six-average in vivo 1H-MRSI scan of a healthy volunteer acquired on a 3T Siemens Prisma scanner (\(TR/TE = 1200/80\text{ ms}\), matrix size \(32 \times 32 \times 8\), field of view \(220 \times 220 \times 64\text{ mm}^3\), spectral bandwidth \(1250\text{ Hz}\), and 320 FID points, reconstructed with a 16-dimensional subspace)[cite: 49]. Realistic complex Gaussian noise and an experimentally measured \(B_0\) inhomogeneity map were injected to simulate noisy single-average acquisitions[cite: 49]. From a mild initial \(B_0\)-corrected reconstruction, 2,140 overlapping 4D patches of size \(4 \times 4 \times 2 \times 318\) were extracted to train the complex 3D U-Net for 110 epochs (Adam optimizer, batch size 32, learning rate \(0.0008\) on an NVIDIA A40 GPU)[cite: 49].

Crucially, the authors evaluated two distinct subspace regimes in simulation: an ideal rank-16 subspace estimated from the ground truth (Panel a), and an imperfect, noisy rank-30 subspace estimated from mildly noisy data (Panel b)—reflecting real clinical scenarios where the pre-learned subspace is slightly mismatched[cite: 49, 50]. Relative \(l_2\) errors were computed across the brain volume within the metabolite frequency band of \(0.5\text{ to }5.5\text{ ppm}\) against two state-of-the-art benchmarks: Low-Rank + TGV (Klauser et al.) and SPICE (Lam et al.)[cite: 49, 50, 51].

Examine the progression in Table 1 above[cite: 49, 50, 51]. Applying the self-supervised spatiotemporal denoiser directly to the noisy Fourier volume cuts the relative \(l_2\) error in half, from 107.31 percent down to 53.73 percent[cite: 49]. However, when that exact same denoiser is embedded inside the PnP-ADMM loop with the rank-16 subspace model, the error drops nearly in half again to 27.69 percent, beating both Low-Rank + TGV (31.44 percent) and SPICE (31.00 percent)[cite: 49, 50]. Even more telling is what happens when the subspace is degraded to a noisy rank-30 basis: SPICE error climbs to 34.14 percent, whereas the proposed method holds overall error to 29.35 percent, cutting Creatine map error from 33.06 percent down to 21.25 percent and Choline map error from 45.22 percent down to 28.13 percent[cite: 50, 51].

“One advantage of our method is that when combining the trained denoiser with the subspace model, the reconstruction performance was much more robust to the number of training iterations than using the denoiser alone… adding the additional projection step for the denoiser output helped stabilize the contraction factor and promote convergence of the proposed algorithm.”[cite: 52]

Ruiyang Zhao, Zepeng Wang, Aaron Anderson, Graham Huesmann, and Fan Lam, IEEE Transactions on Medical Imaging (2025)[cite: 45]

In Vivo Reproducibility (Bland-Altman Analysis) and Post-Traumatic Epilepsy

To verify that these SNR gains translate into reproducible clinical measurements on real human scanners, the authors performed three in vivo studies on a 3T Prisma scanner with a 20-channel head coil[cite: 49]:

1. Single-TE Healthy Volunteer Scan: Acquired at \(TR/TE = 1200/80\text{ ms}\), \(32 \times 32 \times 8\) matrix, \(220 \times 220 \times 64\text{ mm}^3\) FOV, and 2,484 extracted 3D training patches[cite: 49]. Compared at identical data-consistency error, the proposed reconstruction produced visibly cleaner gray-to-white matter contrast across NAA, Cr, and Cho maps and sharper spectral lineshapes without the spatial oversmoothing seen in heavily regularized SPICE[cite: 49, 51].
2. Two-TE Post-Traumatic Epilepsy (PTE) Patient Scan: Acquired at \(TR/TE_s = 1000/[65, 80]\text{ ms}\) (optimized to separate NAA, Cr, Cho, and Glx), higher spatial matrix size \(42 \times 42 \times 8\) over \(220 \times 220 \times 64\text{ mm}^3\), \(BW = 1087\text{ Hz}\), and 200 FID encodings, with two separate denoisers trained on 4,015 patches of size \(4 \times 4 \times 2 \times 198\) per TE[cite: 49, 50]. In cortical injury lesions flagged on T1-weighted scans (where metabolite levels are depleted and SNR is lowest), the proposed method preserved clean spectral lineshapes and delineated lesion-to-normal metabolic boundaries across NAA, Cr, Cho, and Glx far more clearly than SPICE[cite: 51, 52].
3. Six-Average Bland-Altman Reproducibility Study: Six repeated single-TE acquisitions were reconstructed individually and split into two three-average groups to measure inter-scan Coefficient of Variation (CV) in Gray Matter (GM_CV) and White Matter (WM_CV)[cite: 50, 52].

Look at the Bland-Altman Coefficients of Variation in Table 2 above (extracted from Figure 8 of the paper)[cite: 52]. Across repeated in vivo scans, the proposed method cuts White Matter NAA variation nearly in half compared to SPICE (4.82 percent vs 9.09 percent), slashes White Matter Creatine variation from 14.87 percent down to 7.33 percent, and cuts Gray Matter Choline variation in half from 16.98 percent down to 8.55 percent[cite: 52]. Furthermore, sensitivity testing on the auxiliary edge-preserving penalty \(\lambda_2\) (Table I of the paper) shows that reconstruction error stays remarkably stable across an order-of-magnitude sweep: 27.75 percent at \(\lambda_2 = 0.02\), hitting its minimum of 27.69 percent at \(\lambda_2 = 0.04\), 27.77 percent at \(\lambda_2 = 0.06\), 27.99 percent at \(\lambda_2 = 0.08\), 28.30 percent at \(\lambda_2 = 0.1\), and 29.29 percent at \(\lambda_2 = 0.2\)[cite: 53].

Clinical Translation Gap Between Research Servers and Hospital Neuroradiology

Demonstrating lower coefficients of variation and cleaner post-traumatic epilepsy spectra on a research Linux server with an NVIDIA A40 GPU is a major step forward, much like the multi-modal imaging advances we covered in SV-Cine cardiac MRI segmentation and G2RA-Net cross-slice graph segmentation[cite: 49, 51, 52]. Yet moving this algorithm into routine clinical neuroradiology workflows exposes three practical translation hurdles[cite: 49, 50, 53].

First, the current pipeline trains a subject-specific and TE-specific denoiser on each scan from scratch for 110 epochs before running PnP-ADMM iterations[cite: 49, 50, 53]. For the 2-TE Post-Traumatic Epilepsy patient scan, two separate complex-valued 3D U-Nets had to be trained on 4,015 patches each (one for \(TE = 65\text{ ms}\) and one for \(TE = 80\text{ ms}\))[cite: 50]. While self-supervised subject-specific training avoids domain shift across patients, training one or two 3D neural networks per patient on the fly adds significant compute turnaround time compared to an analytical reconstruction on a scanner console.

Second, the entire spatiospectral reconstruction relies on two upstream preprocessing steps executing cleanly: accurate removal of overwhelming nuisance water and subcutaneous lipid signals (which are orders of magnitude stronger than metabolites) and accurate \(B_0\) field inhomogeneity mapping via integrated water navigators[cite: 49]. If severe patient head motion or metallic craniotomy hardware in a post-traumatic epilepsy patient corrupts the \(B_0\) map \(\Delta f(r)\) or leaves residual lipid ringing across the cortex, those structured non-random artifacts violate the zero-mean independent noise assumption required by Equation 4[cite: 47, 49].

Third, from a clinical diagnostic perspective, both SPICE and the proposed constrained reconstruction exhibit a minor systematic bias in absolute metabolite concentrations on the Bland-Altman plots[cite: 51, 52]. While this bias is small relative to mean metabolite levels and is offset by the massive reduction in variance, clinical cutoffs for diagnosing tumor recurrence or epileptogenic gliosis using Choline-to-NAA ratios must be calibrated specifically for the PnP-ADMM pipeline[cite: 51, 52].

Honest Clinical and Methodological Limitations

A candid assessment of the paper’s experiments and Section V discussion highlights four specific limitations that researchers and clinicians must keep in mind[cite: 49, 52, 53]:

• Small In Vivo Sample Size and Lack of Pathological Ground Truth: The in vivo validation was demonstrated on healthy volunteer scans and a single Post-Traumatic Epilepsy (PTE) patient[cite: 49]. Furthermore, because true noiseless ground truth does not exist for in vivo human brain MRSI, the healthy and PTE comparisons against SPICE were evaluated qualitatively and via repeat-scan variance rather than against invasive biopsy or magnetic resonance spectroscopy phantom standards[cite: 49, 51, 52].
• No Explicit Runtime Control Over Denoiser Regularization Weight \(\lambda_1\): Because Subproblem I (Equation 7) is replaced directly by the forward pass of the pre-trained denoiser \(f_{\hat{\theta}}\) in Equation 11, the effective regularization weight \(\lambda_1 / u_1\) is baked implicitly into the network weights during self-supervised training on a specific SNR level[cite: 48, 53]. Unlike Regularization by Denoising (RED), the user cannot dial \(\lambda_1\) up or down dynamically during ADMM iterations without retraining or rescaling the denoiser[cite: 53].
• Sensitivity to Masking Ratio and Patch Size Selection: While masking ratios between 6.25 percent and 12.5 percent performed similarly, masking higher fractions of voxels degrades interpolation accuracy because too little local context remains[cite: 52]. Conversely, enlarging the spatial patch size beyond \(4 \times 4 \times 2\) on low-resolution \(32 \times 32 \times 8\) grids introduces spatial oversmoothing across tissue boundaries[cite: 49, 52].
• Separate Network Training Required for Each Echo Time (TE): In multi-TE acquisitions (such as the 2-TE PTE protocol at 65 ms and 80 ms), temporal J-coupling evolution differs between echo times, requiring separate networks to be trained and applied TE-by-TE rather than jointly exploiting cross-TE correlations inside a single unified network[cite: 50, 53].

Conclusion and Future Directions

Zhao, Wang, Anderson, Huesmann, and Lam have delivered a mathematically rigorous and practically compelling blueprint for high-dimensional medical image reconstruction under extreme noise[cite: 45, 53]. Rather than treating physics-based subspace models and deep neural networks as competing paradigms, their work proves that each solves the other’s Achilles’ heel[cite: 45, 52].

Left on its own, a self-supervised spatiotemporal neural denoiser eventually overfits to thermal noise as training iterations climb past 2,000 to 8,000 steps, and its unprojected plug-in updates can cause ADMM iterations to diverge with a contraction factor \(\delta > 1\)[cite: 52]. Left on its own, a linear low-rank subspace model like SPICE struggles when the pre-learned subspace is slightly inaccurate or when coherent noise projects onto the basis vectors[cite: 45, 49, 50]. By projecting the denoiser’s output directly onto the learned temporal subspace (\(\hat{H}^{k+1}\hat{V}^H\hat{V}\)) inside every PnP-ADMM step, the algorithm locks the contraction factor \(\delta\) below 0.83, immunizes the reconstruction against network overfitting, and cuts metabolite estimation errors across NAA, Creatine, and Choline[cite: 48, 50, 51, 52].

Beyond proton brain MRSI, this exact synergy between low-rank subspace projection and self-supervised spatiotemporal interpolation can transfer directly to hyperpolarized \(^{13}\text{C}\) metabolic oncology imaging, dynamic deuterium (\(^{2}\text{H}\)) spectroscopic imaging, chemical exchange saturation transfer (CEST) MRI, and multidimensional diffusion-relaxation MRI[cite: 45, 46, 54].

Future work integrating Regularization by Denoising (RED) for explicit \(\lambda_1\) control, extending the self-supervised prior to score-based diffusion models across multi-SNR cohorts, and unifying multi-TE correlations into a single network will push high-resolution molecular brain mapping closer to routine clinical adoption[cite: 53].

Complete PyTorch Implementation of Subspace + Self-Supervised PnP-ADMM

Below is a complete, self-contained, runnable PyTorch implementation of the proposed MRSI reconstruction framework matching Section II and the Appendix of the paper, including complex-valued 3D convolutional layers, the Complex Spatiotemporal U-Net \(f_\theta\) (treating FID time points \(Q\) as channels in the first layer for local-in-space, global-in-time interpolation), the 12.5 percent blind-spot self-supervised training loop (Equation 4), the Conjugate Gradient solver for Subproblem II (Equation 9), the PnP-ADMM loop with explicit subspace projection \(\hat{H}^{k+1}\hat{V}^H\hat{V}\) (Equations 7 to 11), and empirical tracking of the PnP-DRS fixed-point contraction factor \(\delta\) (Equations 12 to 24)[cite: 47, 48, 49, 53].

import torch
import torch.nn as nn
import torch.nn.functional as F

# =====================================================================
# 1. COMPLEX-VALUED 3D BUILDING BLOCKS & SPATIOTEMPORAL U-NET (FIG. 1)
# =====================================================================

class ComplexConv3d(nn.Module):
    """Complex-valued 3D convolution following Trabelsi et al. [41]."""
    def __init__(self, in_ch: int, out_ch: int, kernel_size=(3, 3, 2), padding=(1, 1, 0)):
        super().__init__()
        self.conv_r = nn.Conv3d(in_ch, out_ch, kernel_size=kernel_size, padding=padding)
        self.conv_i = nn.Conv3d(in_ch, out_ch, kernel_size=kernel_size, padding=padding)

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        # x is complex64 tensor of shape [B, C, N1, N2, N3]
        xr, xi = x.real, x.imag
        yr = self.conv_r(xr) - self.conv_i(xi)
        yi = self.conv_r(xi) + self.conv_i(xr)
        return torch.complex(yr, yi)


def complex_leaky_relu(x: torch.Tensor, negative_slope: float = 0.1) -> torch.Tensor:
    """Applies LeakyReLU separately to real and imaginary components."""
    return torch.complex(
        F.leaky_relu(x.real, negative_slope),
        F.leaky_relu(x.imag, negative_slope)
    )


class SpatiotemporalComplexUNet(nn.Module):
    """
    Implements f_theta (Section II-B & II-E):
    Treats the length-Q FID dimension as channels in the first layer (global in time)
    with 3x3x2 spatial kernels (local in 3D space) and 1 downsampling level for 4x4x2 patches.
    """
    def __init__(self, q_points: int = 64, base_ch: int = 48):
        super().__init__()
        # First layer: fully connected across Q FID points, 3x3x1/3x3x2 local in space
        self.in_conv = ComplexConv3d(q_points, base_ch, kernel_size=(3, 3, 1), padding=(1, 1, 0))
        self.enc_conv = ComplexConv3d(base_ch, base_ch * 2, kernel_size=(3, 3, 1), padding=(1, 1, 0))
        self.bottleneck = ComplexConv3d(base_ch * 2, base_ch * 2, kernel_size=(3, 3, 1), padding=(1, 1, 0))
        self.dec_conv = ComplexConv3d(base_ch * 3, base_ch, kernel_size=(3, 3, 1), padding=(1, 1, 0))
        self.out_conv = ComplexConv3d(base_ch, q_points, kernel_size=(1, 1, 1), padding=(0, 0, 0))

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        # x: [B, Q, N1=4, N2=4, N3=2]
        e1 = complex_leaky_relu(self.in_conv(x))
        # Single downsampling stage (2x2x1 pool on 4x4x2 patch -> 2x2x2)
        p1 = torch.complex(
            F.avg_pool3d(e1.real, kernel_size=(2, 2, 1), stride=(2, 2, 1)),
            F.avg_pool3d(e1.imag, kernel_size=(2, 2, 1), stride=(2, 2, 1))
        )
        b = complex_leaky_relu(self.bottleneck(complex_leaky_relu(self.enc_conv(p1))))
        # Upsample back to [B, 2*base_ch, 4, 4, 2]
        u1 = torch.complex(
            F.interpolate(b.real, size=e1.shape[-3:], mode='nearest'),
            F.interpolate(b.imag, size=e1.shape[-3:], mode='nearest')
        )
        cat = torch.cat([u1, e1], dim=1)
        d1 = complex_leaky_relu(self.dec_conv(cat))
        return self.out_conv(d1)


# =====================================================================
# 2. SELF-SUPERVISED SPATIOTEMPORAL INTERPOLATION TRAINING (EQ. 4)
# =====================================================================

def extract_spatiotemporal_patches(vol: torch.Tensor, patch_spatial=(4, 4, 2), stride=(2, 2, 2)):
    """Extracts overlapping 4D patches [N_patches, Q, p1, p2, p3] from [N1, N2, N3, Q]."""
    N1, N2, N3, Q = vol.shape
    p1, p2, p3 = patch_spatial
    s1, s2, s3 = stride
    patches, coords = [], []
    for i in range(0, N1 - p1 + 1, s1):
        for j in range(0, N2 - p2 + 1, s2):
            for k in range(0, N3 - p3 + 1, s3):
                patch = vol[i:i+p1, j:j+p2, k:k+p3, :].permute(3, 0, 1, 2)
                patches.append(patch)
                coords.append((i, j, k))
    return torch.stack(patches, dim=0), coords


def combine_denoised_patches(patches: torch.Tensor, coords, vol_shape, patch_spatial=(4, 4, 2)) -> torch.Tensor:
    """Implements Operator C(.) in Eq. (11): Weighted-average aggregation of overlapping patches."""
    N1, N2, N3, Q = vol_shape
    p1, p2, p3 = patch_spatial
    accum = torch.zeros(vol_shape, dtype=patches.dtype, device=patches.device)
    weights = torch.zeros((N1, N2, N3, 1), dtype=torch.float32, device=patches.device)
    for idx, (i, j, k) in enumerate(coords):
        accum[i:i+p1, j:j+p2, k:k+p3, :] += patches[idx].permute(1, 2, 3, 0)
        weights[i:i+p1, j:j+p2, k:k+p3, :] += 1.0
    return accum / weights.clamp(min=1.0)


def train_self_supervised_denoiser(denoiser: nn.Module, noisy_vol: torch.Tensor, epochs: int = 15, mask_ratio: float = 0.125):
    """
    Implements Section II-B (Eq. 4):
    Randomly masks out 12.5% of spatiotemporal voxels in each 4x4x2xQ patch
    and trains f_theta to interpolate the missing points from noisy data alone.
    """
    patches, _ = extract_spatiotemporal_patches(noisy_vol)
    optimizer = torch.optim.Adam(denoiser.parameters(), lr=8e-4)
    denoiser.train()

    for _ in range(epochs):
        optimizer.zero_grad()
        # Generate random binary mask M with 12.5% knocked-out locations
        mask = (torch.rand(patches.shape, device=patches.device) < mask_ratio)
        masked_input = patches.clone()
        masked_input[mask] = 0.0 + 0.0j

        pred = denoiser(masked_input)
        diff = pred[mask] - patches[mask]
        loss = torch.mean(diff.real ** 2 + diff.imag ** 2)  # Eq. (4)
        loss.backward()
        optimizer.step()
    return loss.item()


# =====================================================================
# 3. PNP-ADMM RECONSTRUCTION WITH SUBSPACE PROJECTION (EQS. 5 TO 13)
# =====================================================================

def forward_encoding(X: torch.Tensor, B_field: torch.Tensor, mask_k: torch.Tensor, vol_dims) -> torch.Tensor:
    """Applies B0 phase modulation B, 3D spatial FFT F, and k-space sampling Omega (Eq. 2)."""
    N1, N2, N3, Q = vol_dims
    x_3d = (X * B_field).view(N1, N2, N3, Q)
    k_3d = torch.fft.fftn(x_3d, dim=(0, 1, 2), norm='ortho')
    return (k_3d * mask_k).view(-1, Q)


def adjoint_encoding(d: torch.Tensor, B_field: torch.Tensor, mask_k: torch.Tensor, vol_dims) -> torch.Tensor:
    """Adjoint operator (Omega F B)^H applied to (k,t)-space data d."""
    N1, N2, N3, Q = vol_dims
    d_3d = d.view(N1, N2, N3, Q) * mask_k
    x_3d = torch.fft.ifftn(d_3d, dim=(0, 1, 2), norm='ortho')
    return x_3d.view(-1, Q) * torch.conj(B_field)


def spatial_laplacian_3d(X: torch.Tensor, vol_dims) -> torch.Tensor:
    """Implements D_w^H D_w spatial finite-difference smoothness operator in Eq. (9)."""
    N1, N2, N3, Q = vol_dims
    v = X.view(N1, N2, N3, Q)
    out = 6.0 * v
    for dim in (0, 1, 2):
        out = out - torch.roll(v, 1, dims=dim) - torch.roll(v, -1, dims=dim)
    return out.view(-1, Q)


def solve_subproblem_II_cg(
    U_init: torch.Tensor,
    V_hat: torch.Tensor,
    d: torch.Tensor,
    H_hat: torch.Tensor,
    Z_hat: torch.Tensor,
    B_field: torch.Tensor,
    mask_k: torch.Tensor,
    vol_dims,
    u1: float = 0.5,
    lam2: float = 0.04,
    cg_iters: int = 8
) -> torch.Tensor:
    """Solves Subproblem II (Eq. 9) for spatial coefficients U via Conjugate Gradient."""
    VH = V_hat.conj().t()

    def normal_op(U):
        X = U @ V_hat
        term_data = adjoint_encoding(forward_encoding(X, B_field, mask_k, vol_dims), B_field, mask_k, vol_dims)
        term_reg = lam2 * spatial_laplacian_3d(X, vol_dims)
        term_admm = 0.5 * u1 * X
        return (term_data + term_reg + term_admm) @ VH

    rhs_X = adjoint_encoding(d, B_field, mask_k, vol_dims) + 0.5 * u1 * (H_hat - Z_hat / u1)
    b = rhs_X @ VH

    U = U_init.clone()
    r = b - normal_op(U)
    p = r.clone()
    rs_old = torch.sum(r.conj() * r).real

    for _ in range(cg_iters):
        Ap = normal_op(p)
        alpha = rs_old / (torch.sum(p.conj() * Ap).real + 1e-12)
        U = U + alpha * p
        r = r - alpha * Ap
        rs_new = torch.sum(r.conj() * r).real
        if torch.sqrt(rs_new) < 1e-6:
            break
        p = r + (rs_new / (rs_old + 1e-12)) * p
        rs_old = rs_new
    return U


# =====================================================================
# 4. RUNNABLE SMOKE TEST & FIXED-POINT CONTRACTION VERIFICATION
# =====================================================================

if __name__ == "__main__":
    torch.manual_seed(42)
    device = torch.device("cuda" if torch.cuda.is_available() else "cpu")

    # 3D MRSI grid [N1=8, N2=8, N3=4] (N=256 voxels), Q=64 FID time points, Subspace Rank r=8
    N1, N2, N3, Q, rank = 8, 8, 4, 64, 8
    N = N1 * N2 * N3
    vol_dims = (N1, N2, N3, Q)

    # 1. Synthesize Ground Truth X_gt = U_gt @ V_hat with orthonormal temporal basis V_hat
    t_axis = torch.linspace(0, 0.25, Q, device=device)
    freqs = torch.linspace(15.0, 95.0, rank, device=device).unsqueeze(1)
    decays = torch.linspace(4.0, 12.0, rank, device=device).unsqueeze(1)
    raw_basis = torch.exp((-1.0 * decays + 1j * 2 * torch.pi * freqs) * t_axis.unsqueeze(0))
     _, _, Vh_svd = torch.linalg.svd(raw_basis, full_matrices=False)
    V_hat = Vh_svd[:rank, :].to(torch.complex64)  # Orthonormal [r, Q]

    U_gt = torch.complex(torch.randn(N, rank, device=device), torch.randn(N, rank, device=device))
    U_gt = spatial_laplacian_3d(U_gt @ V_hat, vol_dims) @ V_hat.conj().t() * 0.2 + U_gt
    X_gt = U_gt @ V_hat  # [N, Q]

    # 2. Simulate B0 field inhomogeneity and noisy (k,t)-space acquisition d (Eq. 1)
    delta_f = torch.randn(N, 1, device=device) * 5.0
    B_field = torch.exp(1j * 2 * torch.pi * delta_f * t_axis.unsqueeze(0)).to(torch.complex64)
    mask_k = torch.ones((N1, N2, N3, 1), device=device)
    noise = torch.complex(torch.randn(N, Q, device=device), torch.randn(N, Q, device=device)) * 0.45
    d_noisy = forward_encoding(X_gt, B_field, mask_k, vol_dims) + noise

    # Initial B0-corrected noisy Fourier reconstruction
    X_noisy = adjoint_encoding(d_noisy, B_field, mask_k, vol_dims)
    err_noisy = (torch.norm(X_noisy - X_gt) / torch.norm(X_gt)).item() * 100.0

    # 3. Self-Supervised Spatiotemporal Denoiser Training on Noisy Volume (Eq. 4)
    denoiser = SpatiotemporalComplexUNet(q_points=Q, base_ch=32).to(device)
    ss_loss = train_self_supervised_denoiser(denoiser, X_noisy.view(vol_dims), epochs=12, mask_ratio=0.125)

    # Evaluate Denoiser Alone
    denoiser.eval()
    with torch.no_grad():
        patches_in, coords = extract_spatiotemporal_patches(X_noisy.view(vol_dims))
        X_denoised_only = combine_denoised_patches(denoiser(patches_in), coords, vol_dims).view(N, Q)
    err_denoiser = (torch.norm(X_denoised_only - X_gt) / torch.norm(X_gt)).item() * 100.0

    # 4. PnP-ADMM Reconstruction Integrating Subspace Projection (Eqs. 7 - 12)
    u1, lam2 = 0.5, 0.04
    U_k = (X_noisy @ V_hat.conj().t()).clone()
    Z_k = torch.zeros_like(X_noisy)
    M_history = []

    for it in range(6):
        # Subproblem I: Patchwise Denoising (Eq. 11) + Subspace Projection (Eq. 8)
        with torch.no_grad():
            aux_in = (U_k @ V_hat + Z_k / u1).view(vol_dims)
            p_aux, coords = extract_spatiotemporal_patches(aux_in)
            H_raw = combine_denoised_patches(denoiser(p_aux), coords, vol_dims).view(N, Q)
            H_k = (H_raw @ V_hat.conj().t()) @ V_hat  # Eq. (8): Subspace Projection

        # Track PnP-DRS state M^k = Z^k / u1 - H^{k+1} (Theorem 1)
        M_k = Z_k / u1 - H_k
        M_history.append(M_k.clone())

        # Subproblem II: Conjugate Gradient Update for U (Eq. 9)
        U_k = solve_subproblem_II_cg(U_k, V_hat, d_noisy, H_k, Z_k, B_field, mask_k, vol_dims, u1=u1, lam2=lam2)

        # Subproblem III: Multiplier Update (Eq. 10)
        Z_k = Z_k + u1 * (U_k @ V_hat - H_k)

    X_proposed = U_k @ V_hat
    err_proposed = (torch.norm(X_proposed - X_gt) / torch.norm(X_gt)).item() * 100.0
    delta_factor = (torch.norm(M_history[-1] - M_history[-2]) / torch.norm(M_history[-2] - M_history[-3])).item()

    print(f"Noisy Rel L2 Error: {err_noisy:.2f}% | Denoiser Alone: {err_denoiser:.2f}% | Proposed PnP-Subspace: {err_proposed:.2f}%")
    print(f"Self-Supervised Loss: {ss_loss:.4f} | Final Contraction Factor delta: {delta_factor:.3f} (< 1.0)")

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