Volumetrically Consistent Implicit Atlas Learning via Neural Diffeomorphic Flow for Placenta MRI
Authors: Athena Taymourtash, S. Mazdak Abulnaga, Esra Abaci Turk, P. Ellen Grant, Polina Golland
Organizations: MIT Computer Science and Artificial Intelligence Laboratory · Massachusetts General Hospital, Harvard Medical School · Boston Children’s Hospital, Harvard Medical School
Establishing dense volumetric correspondences across anatomical shapes is essential for group-level analysis but remains challenging for implicit neural representations. Most existing implicit registration methods rely on supervision near the zero-level set and thus capture only surface correspondences, leaving interior deformations under-constrained. We introduce a volumetrically consistent implicit model that couples reconstruction of signed distance functions (SDFs) with neural diffeomorphic flow to learn a shared canonical template of the placenta. Volumetric regularization, including Jacobian-determinant and biharmonic penalties, suppresses local folding and promotes globally coherent deformations. In the motivating application to placenta MRI, our formulation jointly reconstructs individual placentas, aligns them to a population-derived implicit template, and enables voxel-wise intensity mapping in a unified canonical space. Experiments on in-vivo placenta MRI scans demonstrate improved geometric fidelity and volumetric alignment over surface-based implicit baseline methods, yielding anatomically interpretable and topologically consistent flattening suitable for group analysis.
Figures & tables
Figure 1: Method overview. The warping function transforms point samples of shape i to their canonical positions, which are then mapped to SDF values by the implicit template.
Chamfer-L2 (↓)
EMD (↓)
NC (↑)
Precision (↑)
Recall (↑)
F1(↑)
Model
@0.01
@0.03
@0.05
@0.01
@0.03
@0.05
@0.01
@0.03
@0.05
DeepSDF [ 29 ]
0.27
0.05
0.97
0.24
0.98
0.99
0.24
0.99
1.00
0.24
0.99
0.99
DIT [ 45 ]
0.17
0.05
0.97
0.37
0.98
0.99
0.37
0.99
1.00
0.38
0.99
0.99
DIF-Net [ 12 ]
0.12
0.04
0.98
0.43
0.99
1.00
0.43
0.99
1.00
0.43
0.99
1.00
NDF [ 35 ]
0.22
0.05
0.97
0.28
0.99
1.00
0.28
0.99
1.00
0.28
0.99
1.00
Ours
0.12
0.04
0.98
0.43
0.99
1.00
0.43
0.99
1.00
0.43
0.99
1.00
Table 1: Surface accuracy metrics on the placenta dataset. CD: Chamfer Distance (multiplied by 103 ), EMD: Earth Mover’s Distance, and NC: Normal Consistency. Precision, Recall, and F1 are evaluated at distance thresholds of 0.01, 0.03, and 0.05.
FlipRate (↓)
logDet- L1(↓)
CycleError (↓)
DIF-Net [ 12 ]
15.34
0.81
0.24
DIT [ 45 ]
11.88
0.67
0.29
NDF [ 35 ]
8.78
0.90
0.26
Ours
4.04
0.69
0.20
Table 2: Diffeomorphism metrics computed on tetrahedral embeddings. FlipRate measures the percentage of tetrahedra with negative signed volume, logDet– L1 captures the average deviation of local volume change from unity, and CycleError quantifies the inverse-consistency of the forward and backward flows. All values are averaged across subjects; lower is better.
Figure 2: Distribution of discrete Laplacian energy ∥Lu∥22 of vertex displacements across subjects. Lower values correspond to smoother deformation fields.
Figure 3: Dense pointwise correspondences via the learned implicit template. (a) Two subject instances mapped to the canonical template through the diffeomorphic flow. (b) ACVD‐remeshed surfaces (5 k vertices) with vertex colors propagated from the template. Consistent colors across subjects visually demonstrate stable, smooth correspondences.
Vol (↓)
Areal (↓)
Edge (↓)
SymDir (↓)
Conformal (↓)
DIF-Net [ 12 ]
4.36
2.8
2.43
12.91
11.47
DIT [ 45 ]
2.02
0.99
1.13
8.48
10.89
NDF [ 35 ]
4.17
2.26
2.32
13.75
10.96
Ours
1.94
1.17
1.19
8.02
9.74
Table 3: Maximum distortion per instance, averaged across subjects. Smaller values correspond to more isotropic, low‐distortion mappings. SymDir and Conformal report symmetric Dirichlet and condition‐number–based distortions.
Figure 4: Pose‐invariant flattening and intensity mapping across two maternal positions of the same subject. The placenta in each position is mapped diffeomorphically to the canonical template via Φ(⋅;ci) and Φ(⋅;cj) , producing consistent, topology‐preserving flattened intensity maps.
Figure 5: Volumetric flattening. Given the same tetrahedral placenta mesh (upper left panel), our method produces a coherent, uniformly compressed volumetric flattening. DIT and NDF maintain surface alignment but distort the interior, exhibiting radial compression and anisotropic stretching of tetrahedra toward a mid-surface layer.
Model
CD-L2 ↓
FlipRate ↓
logDet–L1 ↓
SymDir ↓
LapE ↓
Lrec
0.13
13.75
0.69
10.99
26.34
w/ Ld
0.22
10.92
0.61
9.08
22.88
w/ LJac
0.12
14.76
0.80
12.07
33.05
w/ LbiH (ours)
0.12
4.04
0.69
8.02
8.72
Table 4: Ablation of loss terms on the Placenta dataset.
Figure 6: Flattened intensity maps obtained via volumetric pullback. Our method preserves interior geometry, yielding coherent anatomical texture in the flattened domain, cotyledons and vascular structure are immediately apparent in our flattened images, whereas NDF and DIT cause intensities to be interpolated across collapsed surface to mid-surface layers.
Figure 7: Distribution of gestational ages for our dataset of EPI BOLD placenta scans.
Figure 8: Preprocessing and model outputs for a representative instance with gestational age of 34w+4d (a) Initial surface extracted from the manual segmentation using Marching Cubes (b) Dense signed-distance samples generated using depth-based nearest-surface queries (red: outside; blue: inside)(c) Overlay of the SDF samples with the initial mesh; a random subset of 5k points is used for training (d) Reconstructed implicit surface produced by our template-conditioned SDF decoder; since the representation is continuous, the surface can be extracted at arbitrarily high resolution independently of the input mesh resolution (e) Overlay of the reconstructed surface with the input SDF samples, illustrating geometric consistency (f) Dense pointwise correspondences obtained from the learned diffeomorphic flow, mapping the subject into the template space.
Figure 9: Neural ODE–based deformation module. Top: a velocity field is predicted by a residual MLP from the current point position and latent code ci , and integrated with an ODE solver to produce the updated position. Middle: The full deformation Φi is obtained by concatenating four NODE blocks, where the output of block k serves as the input to block k+1 . Bottom: Visualization of the progressive deformation steps, illustrating the smooth, monotonic trajectory that warps each placenta instance into the learned template space.
Figure 10: Distribution of diffeomorphism and distortion metrics across gestational age.
Spatio-temporal fetal brain atlases are important for characterizing normative neurodevelopment and identifying congenital anomalies. However, existing atlas construction pipelines necessitate days for slice-to-volume reconstruction (SVR) to generate high-resolution 3D brain volumes and several additional days for iterative volume registration, thereby rendering atlas construction from large-scale cohorts prohibitively impractical. We address these limitations with INFANiTE, an Implicit Neural Representation (INR) framework for high-resolution Fetal brain spatio-temporal Atlas learNing from clinical Thick-slicE MRI scans, bypassing both the costly SVR and the iterative non-rigid registration steps entirely, thereby substantially accelerating atlas construction. Extensive experiments demonstrate that INFANiTE outperforms existing baselines in subject consistency, reference fidelity, intrinsic quality and biological plausibility, even under challenging sparse-data settings. Additionally, INFANiTE reduces the end-to-end processing time (i.e., from raw scans to the final atlas) from days to hours compared to the traditional 3D volume-based pipeline (e.g., SyGN), facilitating large-scale population-level fetal brain analysis. Code: https://github.com/hu2274898/INFANiTE
Xiaotian Hu, Mingxuan Liu, Hongjia Yang +10
Tsinghua University · University of Oxford · Sichuan University
Automated analysis of articulated bodies is crucial in medical imaging. Existing surface-based models often ignore internal volumetric structures and rely on deformation methods that lack anatomical consistency guarantees. To address this problem, we introduce a differentiable volumetric body model based on the Skinned Multi-Person Linear (SMPL) formulation, driven by a new Kinematic Tree-based Log-Euclidean PolyRigid (KTPolyRigid) transform. KTPolyRigid resolves Lie algebra ambiguities associated with large, non-local articulated motions, and encourages smooth, bijective volumetric mappings. Evaluated on 53 fetal MRI volumes, KTPolyRigid yields deformation fields with significantly fewer folding artifacts. Furthermore, our framework enables robust groupwise image registration and a label-efficient, template-based segmentation of fetal organs. It provides a robust foundation for standardized volumetric analysis of articulated bodies in medical imaging.
Yingcheng Liu, Athena Taymourtash, Yang Liu +5
Computer Science and Artificial Intelligence Lab, MIT, Cambridge, USA · California Institute of Technology, Pasadena, USA · Boston Children’s Hospital and Harvard Medical School, Boston, USA +1
Diffeomorphic image registration is central to medical image analysis, enabling anatomically consistent alignment across subjects. Most learning-based diffeomorphic methods model autonomous ODEs(ordinary differential equations) by parameterizing a stationary velocity field and recovering deformations via scaling-and-squaring. While non-autonomous ODEs with time-dependent velocities increase expressiveness, existing approaches rely on numerical integration to implicitly enforce flow structure that entangles model expressiveness with discretization accuracy. We propose a framework to directly learn the continuous-time solution of a non-autonomous ODE formulated as a two-parameterflow map. By enforcing cocycle consistency, a fundamental structural property of time-varying flows, we learn the flow maps without time discretization and velocity integration during training. The framework recovers diffeomorphic mappings at inference using a small number of compositions. Our proposed framework seamlessly incorporates standard registration backbones and improves alignment accuracy consistently across nine datasets while preserving diffeomorphic structure. Notably, the proposed method achieves an average Dice improvement of 2.1% on brain MRI benchmarks, a 12% TRE reduction on lung CT, and a 2.6% Dice gain on cardiac MRI and ultrasound datasets.