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ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulator

T0 review · 4 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read ReMiDi reconstructs axonal microstructure meshes from diffusion MRI signals via gradient-based optimization in a learned latent space.

desk verdict Useful differentiable dMRI simulator and a clever latent-space inversion, but the 'arbitrary microstructure' claim goes well beyond what the three synthetic deformation families in the latent space can support. read the letter →

arxiv 2502.01988 v2 pith:PC3CJL7I submitted 2025-02-04 eess.IV cs.GRcs.LGphysics.med-ph

Prathamesh Pradeep Khole , Zahra Kais Petiwala , Shri Prathaa Magesh , Ehsan Mirafzali , Utkarsh Gupta , Jing-Rebecca Li , Andrada Ianus , Razvan Marinescu This is my paper · ORCID
classification eess.IVcs.GRcs.LGphysics.med-ph
keywords diffusionMRImicrostructurereconstructionBloch-TorreyPDEfiniteelementmethodspectralauto-encoderdifferentiablesimulationaxonalgeometrylatentspaceoptimization
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

ReMiDi aims to solve the inverse problem of diffusion MRI: given a measured signal, recover the 3D shape of the axons that produced it. The method builds a differentiable forward simulator that computes the diffusion signal from a finite-element mesh, then optimizes the mesh so its simulated signal matches the reference. To avoid the ill-posedness of tweaking every vertex, it encodes the mesh into a low-dimensional latent space using a spectral auto-encoder and runs gradient descent on that latent vector. On synthetic axon meshes with bending, beading, and fanning deformations, the recovered meshes match the reference signal and approach the true geometry, with lower Chamfer distance than a neural network baseline on bending and beading.

What carries the argument

The load-bearing mechanism is the coupling of a differentiable diffusion MRI simulator with a spectral auto-encoder (SAE). The simulator solves the Bloch-Torrey partial differential equation on a tetrahedral mesh using the matrix formalism—projecting the solution onto a truncated eigenbasis of the Laplace operator—so that the output signal is differentiable with respect to the mesh vertex coordinates. The SAE encodes the 315-vertex mesh into 300 spectral coefficients and then into a 16-dimensional latent vector $\mathbf{z}$, and its decoder $D$ maps $\mathbf{z}$ back to a mesh; optimization is performed on $\mathbf{z}$ with the loss $\lVert F(D(\mathbf{z})) - S_{\mathrm{ref}}\rVert_2^2$, which confines the search to a learned manifold of plausible axon shapes.

What would settle it

Take a reference mesh that is not a bent, beaded, or fanned cylinder—for example, an axon with an elliptical cross-section, a branch point, or a wavy surface with high-frequency features—generate its dMRI signal with the same simulator, and run ReMiDi from a straight cylinder; if the signal loss reaches near zero while the Chamfer distance to the reference remains large, the claim that ReMiDi can reconstruct arbitrary microstructure shapes is refuted.

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Extended reading notes

Core claim

The paper's central claim is that a diffusion MRI signal contains enough information to recover the geometry of a single axon-like microstructure, provided the search is restricted to a plausible shape manifold. ReMiDi demonstrates this by combining a differentiable finite-element solver of the Bloch-Torrey equation with a spectral auto-encoder: the mesh is mapped to spectral coefficients and then to a 16-dimensional latent vector, and the latent vector is updated by gradient descent on the difference between the simulated and reference signals. The reconstruction converges to the reference signal, and for the deformation families considered—bending and twisting, beading, and fanning—the recovered mesh converges in Chamfer distance to the ground truth, with the most accurate results for beading and the least for fanning at large angles.

Load-bearing premise

The latent space learned by the spectral auto-encoder is assumed to contain a shape close enough to the true microstructure, and that auto-encoder was trained only on three families of synthetic deformations of a single cylinder, so any real or complex shape outside that manifold cannot be recovered.

Editorial extensions

If this is right

  • dMRI signals can be inverted into explicit 3D mesh reconstructions of axonal geometry, surfacing features such as undulation, focal beading, and fanning that parametric models only summarize as scalar metrics.
  • Because the forward simulator is differentiable, the same pipeline can be turned to other gradient-based tasks, such as estimating diffusion coefficients or optimizing acquisition parameters jointly with the mesh.
  • The success of latent-space optimization in this setting supports the view that learned priors can regularize PDE-constrained inverse problems, making vertex-level reconstruction unnecessary.
  • The ablation results imply that acquisition design matters: using at least two diffusion times and 15 or more gradient directions substantially improves the spatial fidelity of the recovered mesh.
  • The comparison with a neural network suggests that physics-based simulation contributes the most when deformations are large; for shapes close to the straight cylinder, a direct regressor can be more accurate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • I interpret the claim of reconstructing 'arbitrary shapes' as an aspiration: the experiments cover only three deformation families on a single 315-vertex cylinder, so the method's generalizability to other topologies is untested. A stronger test would use meshes with heterogeneous cross-sections or branching.
  • Because the diffusion signal is invariant under certain rotations and reflections—the authors even adjust their evaluation metric accordingly—the objective function may have multiple equivalent optima; the latent prior then determines which orientation or mirror image is recovered, which may or may not match the true mesh.
  • A practical extension would be to train the SAE on a broader corpus of realistic neuron geometries and then test ReMiDi on signals from those shapes, which would reveal whether the latent space is the bottleneck or whether the simulator is the limiting factor.
  • The paper's finding that two diffusion times improve reconstruction hints at what information the signal carries about length scales; combining more diffusion times with variable b-values could push the resolution limit further, but the computational cost grows with the number of gradient evaluations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper presents ReMiDi, a PyTorch-based differentiable dMRI simulator that solves the Bloch-Torrey PDE via finite elements and matrix formalism, and uses it to solve a mesh-reconstruction inverse problem. The reconstruction is performed by optimizing a 16-dimensional latent code of a spectral auto-encoder (SAE) so that the simulated signal matches a reference signal S_ref. Experiments on synthetic single-axon meshes with bending/twisting, beading, and fanning deformations show that the optimizer can reduce signal loss and, for some deformation families, reduce spatial error relative to the reference mesh. The paper also compares ReMiDi to an MLP baseline and reports ablation studies on the number of diffusion directions and diffusion times. The core forward simulator is validated against SpinDoctor, and the code is made available.

Significance. The differentiable BTPDE simulator is a useful contribution: it enables gradient-based optimization through a physically grounded dMRI forward model, and the matrix-formalism implementation addresses computational cost. The SAE latent-space optimization is a sensible way to regularize the ill-posed inverse problem. However, the significance is limited by three issues: the inverse validation is a self-consistency check using the same simulator to generate the reference signals; the claimed arbitrary-shape reconstruction is tested only within three narrow synthetic deformation families; and the fanning results are substantially worse than the MLP baseline. These issues do not invalidate the forward simulator or the general optimization framework, but they do mean the paper's central claim, as stated in the abstract, is not supported by the evidence.

major comments (4)
  1. [Abstract, Section 1, Section 4, Section 5.1] The claim that ReMiDi can 'reconstruct microstructures of arbitrary shapes represented by finite-element meshes' is not supported. The SAE is trained on three synthetic deformation families applied to a single 315-vertex cylinder (bending/twisting, beading, fanning), and the test meshes are held-out parameter values from the same families (e.g., odd versus even bead counts, even versus odd fanning angles). Because the decoder D(z) can only produce meshes within the learned manifold, any reference mesh with different topology, connectivity, or deformation type is unreachable by construction. The evaluation therefore demonstrates interpolation within the training manifold, not coverage of arbitrary shapes. The paper should either restrict the claim to the studied deformation families or demonstrate reconstruction of out-of-family geometries.
  2. [Section 5, Appendix E.1, Appendix E.2, Algorithm 1] The optimization objective in Algorithm 1, ||F(D(z)) - S_ref||^2, is shown to be insufficient to determine geometry. The signal loss converges around iteration 250 (Fig. 4), while the Chamfer distance and volume continue changing until roughly iterations 400-700 (Figs. 11 and 12). This is direct evidence that low signal loss does not uniquely identify the mesh, which is consistent with the paper's own discussion of ill-posedness. The authors should either incorporate additional regularizers or multi-sequence information that can reduce non-identifiability, or explicitly frame the result as one member of a signal-equivalence class rather than a unique reconstruction.
  3. [Section 5.1.3, Table 1] The fanning experiments contradict the general claim of reconstruction capability. For fanning, ReMiDi achieves a modified Chamfer distance of 7.433 +/- 4.844, which is an order of magnitude worse than the MLP baseline (0.743 +/- 0.24), and Figure 7 shows significant errors at 60 degrees and non-smooth curvature. The text states that 'ReMiDi can capture fanning to some extent,' but this is a much weaker statement than the abstract's claim of reconstructing fanning fibers. Either the method needs to be improved for fanning, or fanning should be explicitly excluded from the headline claims.
  4. [Section 3.1, Figure 2] The inverse-problem validation is a self-consistency check: the reference signals S_ref are generated by the same differentiable simulator F used for reconstruction. Figure 2 validates the forward simulator against SpinDoctor, but the inverse procedure is not tested against an independent forward model. Thus the experiments show that the optimizer can invert the exact forward model on a training manifold, but do not demonstrate that the method is robust to model error or to signals generated by a different simulator. The authors should validate the inverse approach using an independent forward model or an analytic signal expression, or clearly state this as a limitation.
minor comments (6)
  1. [Section 1] There is a typo: 'unidimentional' should be 'unidimensional'.
  2. [Figure 7 caption] The caption reports negative modified Chamfer distances at 60 degrees due to KDE-based smoothing; plotting negative distances for a distance metric is confusing and should be replaced with raw values or an explicit explanation of the smoothing artifact.
  3. [Algorithm 1, Section 4] Algorithm 1 hardcodes the input mesh as M in R^{315x3}, while the text uses a general V; this inconsistency should be fixed.
  4. [Section 4, Appendix D] The description of the spectral auto-encoder should clarify whether the 300 spectral coefficients are computed per Cartesian coordinate, and how the graph Laplacian and its eigenvectors are defined for the deforming mesh; the appendix says connectivity is consistent but the relationship between geometry changes and the fixed spectral basis should be stated explicitly.
  5. [Section 5.2] The MLP baseline comparison lacks details on the MLP's input representation, training set, and loss function; without these details, the numbers in Table 1 are difficult to interpret.
  6. [Section 5, Appendix E.2] The main text says the mesh volume converges around iteration 400, while Appendix E.2 says it stabilizes around iteration 500; please reconcile these numbers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse optimization is a genuine fit to a reference signal through a differentiable forward model, and the SAE prior is a learned constraint, not a relabeling of the output.

full rationale

ReMiDi's pipeline is not circular. The optimization objective z* = argmin_z ||F(D(z)) - S_ref||_2^2 (Section 3.2) is a genuine inverse problem: the reference signal S_ref enters only through the loss, not through the mesh geometry or the SAE weights. The SAE decoder D is trained on synthetic deformation families, so the reachable set is restricted to those families; this limits generalization and undercuts the "arbitrary shapes" claim, but it is a validation/generalization limitation, not a definitional reduction. Reference signals are produced by the same forward simulator, which is a standard synthetic-validation setup and not circular reasoning: the forward model is independently checked against SpinDoctor's MATLAB implementation (Figure 2). The paper's own convergence curves (signal loss plateaus around iteration 250 while Chamfer distance and volume converge around iteration 500) show the inverse problem is ill-posed, but the paper explicitly acknowledges this in Appendix E.1. The citation to SpinDoctor (Li et al. 2019) involves a co-author, but the cited toolbox is prior external work, and the present paper reimplements and validates it; no uniqueness theorem or unsupported self-citation is used to force the result. Overall, the derivation chain is self-contained: simulator -> forward model -> signal loss -> gradient update to latent -> decoded mesh. No fitted parameter is renamed as a prediction, and no equation reduces to its input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical entities. It relies on standard FEM/BTPDE mathematics, a pre-existing spectral auto-encoder, and a set of hand-chosen hyperparameters (latent dimension, spectral truncation, loss multiplier). The main domain assumptions are that the BTPDE model and the SAE latent space faithfully represent neural tissue and axon geometry.

free parameters (3)
  • latent dimension z = 16
    Chosen by hand as a balance between smoothness and flexibility; the paper states 'We found that a dimension of 16 for z ensures that the reconstructed meshes are both smooth and flexible' (Section 3.2).
  • number of spectral coefficients = 300
    Set to 300 for the 315-vertex meshes; this truncation is a manual architectural choice that determines the mesh representation capacity (Section 4).
  • loss multiplier k = not specified
    Scales the signal loss to prevent early stopping of latent updates; mentioned in Algorithm 1 and Appendix E.3, but its value is not given and it directly affects convergence.
assumptions (4)
  • domain assumption The Bloch-Torrey PDE with given boundary conditions accurately models the diffusion MRI signal from neural tissue.
    The simulator is built on this PDE; no validation against real tissue data is provided, only against the SpinDoctor MATLAB implementation (Section 2.1, Section 3.1).
  • standard math The matrix formalism with a truncated eigenbasis provides a sufficiently accurate approximation of the BTPDE solution.
    The paper relies on the eigendecomposition of the Laplace operator to approximate the magnetization; the truncation error is not quantified (Section 2.2).
  • domain assumption The spectral auto-encoder's latent space covers the space of plausible axon geometries relevant to the inverse problem.
    The reconstruction can only produce meshes that the SAE can decode; the paper only tests three synthetic deformation families (bending, beading, fanning), so the latent space is not shown to be general (Section 3.2, Section 5.1).
  • domain assumption A single-compartment mesh with 315 vertices is sufficient to represent the axon sections of interest.
    All experiments use single-compartment meshes with 315 vertices, and the paper notes that 8 beads cannot be captured at this resolution (Section 5.1.2, Section 6).

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Cite this review

Pith. "Pith review of ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulator." pith.science (2026). https://pith.science/paper/PC3CJL7I

@misc{pith2026250201988,
  author       = {Pith},
  title        = {Pith review of: ReMiDi: Reconstruction of Microstructure Using a Differentiable Diffusion MRI Simulator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PC3CJL7I}},
  note         = {Machine review of arXiv:2502.01988}
}
abstract

We propose ReMiDi, a novel method for inferring neuronal microstructure as arbitrary 3D meshes using a differentiable diffusion Magnetic Resonance Imaging (dMRI) simulator. We first implemented in PyTorch a differentiable dMRI simulator that simulates the forward diffusion process using a finite-element method on an input 3D microstructure mesh. To achieve significantly faster simulations, we solve the differential equation semi-analytically using a matrix formalism approach. Given a reference dMRI signal $S_{ref}$, we use the differentiable simulator to iteratively update the input mesh such that it matches $S_{ref}$ using gradient-based learning. Since directly optimizing the 3D coordinates of the vertices is challenging, particularly due to ill-posedness of the inverse problem, we instead optimize a lower-dimensional latent space representation of the mesh. The mesh is first encoded into spectral coefficients, which are further encoded into a latent $\textbf{z}$ using an auto-encoder, and are then decoded back into the true mesh. We present an end-to-end differentiable pipeline that simulates signals that can be tuned to match a reference signal by iteratively updating the latent representation $\textbf{z}$. We demonstrate the ability to reconstruct microstructures of arbitrary shapes represented by finite-element meshes, with a focus on axonal geometries found in the brain white matter, including bending, fanning and beading fibers. Our source code is available online.

Figures

Figures reproduced from arXiv: 2502.01988 by the authors.

Figure 1
Figure 1. Overview of the ReMiDi-based dMRI reconstruction pipeline. A 3D mesh decoded by a Spectral Auto-Encoder (SAE) is given as input to the differentiable dMRI simulator. The final loss function between the simulated dMRI signal and a reference signal is backpropagated to the latent of the SAE, and the mesh is iteratively updated with gradient-based learning. where ϕ(⃗x) = [ϕ1(⃗x), . . . , ϕNeig (⃗x)]T is the vector of e… view at source ↗
Figure 2
Figure 2. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Overview of iterative reconstruction of a bent axon by ReMiDi at different gradient descent iterations. The ground-truth mesh is shown to the left. Top row: Evolution of the triangulated mesh surface. Bottom row: Corresponding point cloud representations of mesh vertices. The color scale shows Chamfer distance (lower values are better). 5. Results [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: dMRI signal loss over iterations for reconstruction of a bent axon using ReMiDi validation meshes, and 4,500 testing meshes. The bending coefficient (β) controls the deformation intensity, where points experience horizontal displacement proportional to their squared he…
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Modified Chamfer distance as error between reference meshes (middle) and reconstructed meshes (bottom) for increasing fan angles. While Chamfer distances are always non-negative, the visualization shows negative values at 60◦ due to applying KDE-based smoothing. 5.1.4.…
Figure 8
Figure 8. Figure 8: Modified Chamfer distance as error between reference meshes (middle) and reconstructed meshes (bottom) for increasing number of diffusion directions. Multiple diffusion sequences with varying diffusion times are sensitive to varying amounts of displacements of wa￾ter m…
Figure 10
Figure 10. Figure 10: Cylindrical section of a bent region of neuron (University of California, San Diego, 2021) In the brain, the neurons are connected to each other by a thin elongated process called the axon. Axons transmit electrical signals from the neuron to the dendrites of other ne…
Figure 11
Figure 11. Figure 11: Chamfer distance (measuring spatial error) with respect to the reference mesh over gradient descent iterations, while recon￾structing a bent axon using ReMiDi The above figure 11 shows the change in spatial error over the iterative reconstruction process by ReMiDi, th…
Figure 12
Figure 12. Figure 12: Evolution of the mesh volume over gradient descent iterations during reconstruction by ReMiDi. Figures 4, 11, 12 further highlight the inherent ill-posedness of the inverse reconstruction problem. While signal loss converges much earlier, around iteration 250, spatial…
Figure 13
Figure 13. Figure 13: Updating of 16x1 latent space vector representing encoded meshes over iteration by ReMiDi by backpropagation through SAE to latents [PITH_FULL_IMAGE:figures/full_fig_p015_13.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.