Uncertainty Quantification for Cardiac Shape Reconstruction with Deep Signed Distance Functions via MCMC methods
Authors: Jan Verhülsdonk, Thomas Grandits, Francisco Sahli Costabal, Thomas Beiert, Simone Pezzuto, Alexander Effland
Abstract
Atlas-based approaches allow high-quality, patient-specific shape reconstructions of cardiac anatomy from sparse and/or noisy data such as point clouds. However, these methods are mainly prior-driven, so the impact of uncertainty can be large, limiting their clinical reliability. We propose a probabilistic framework for uncertainty-aware cardiac shape reconstruction that combines Deep Signed Distance Functions (DeepSDFs) with Markov Chain Monte Carlo (MCMC) sampling. Cardiac geometries are modeled implicitly as zero-level sets of a neural network conditioned on learned latent codes, enabling multi-surface reconstruction of the left and right ventricles. By interpreting the reconstruction loss as a log-likelihood, we perform Bayesian inference in the latent space to obtain both maximum a posteriori (MAP) and posterior-sampled reconstructions. Experiments on a public cardiac dataset show that our approach produces accurate reconstructions and well-calibrated uncertainty estimates.
We propose Bi-PT, a pipeline for reconstructing 3D four-chamber human heart meshes from clinical sparsely sampled cardiac magnetic resonance imaging (CMR) data. This work addresses the error-prone generation of 3D cardiac shape from a sparse point cloud (SPC) extracted from 2D long-axis and short-axis views used in routine clinical CMR protocols. Bi-PT enables accurate inference of the four-chamber heart mesh from the SPC by learning robust point features via bidirectional point cross-attention between an atlas and the SPC, together with per-point semantic labels that improve correspondence estimation. We formulate the deformation field as a Neural Ordinary Differential Equation (NODE) parameterized by a per-point affine transformation and translation to deform the atlas toward the target heart shape. By learning such a NODE, we can guarantee the deformation field to be a locally affine diffeomorphic deformation. We also integrate a semantic label loss into the Chamfer distance to encourage label-consistent correspondences and add a smoothness regularization to stabilize and improve the learning of the deformation field. Extensive experiments demonstrate that Bi-PT achieves accurate and robust performance compared to baselines.
Reconstructing a three-dimensional left-ventricular (LV) endocardial surface from cardiac magnetic resonance (CMR) data is challenging when supervision is available on only a small number of axial slices. Through-plane geometry is weakly constrained, and automatically generated two-dimensional masks can propagate segmentation errors into the recovered shape. We present MR-RS-SDFR, a per-case implicit signed distance field (SDF) framework that reconstructs a continuous LV surface from a CMR volume and sparse axial weak masks. The method first builds a cross-slice SDF initialization from axial and longitudinal geometric cues and then refines the field using two complementary signals: MRI edge-field normal alignment, which provides an image-derived boundary cue independent of the weak masks, and differentiable reslice Dice and contour consistency, which preserve agreement with the observed planes. We evaluate three weak-mask generators -- LOO TransUNet, LOO nnU-Net, and an off-the-shelf Medical SAM3 model used without MM-WHS-specific training or fine-tuning -- and five sparsity levels from 4 to 64 axial planes. In the sparse-16 setting, final MR-RS-SDFR reconstruction reaches 0.928 Dice and 3.80mm HD95 with Medical SAM3 masks. The upstream generators do not exhibit a single common ranking across 2D and dense 3D segmentation, and nnU-Net- and Medical-SAM3-driven sparse reconstruction achieve the same mean final Dice despite different upstream error profiles. Across all three sparse-16 mask sources, MR-RS-SDFR is numerically better than protocol-matched full GHD+DVS in both Dice and HD95. Final Dice improves markedly from sparse-4 to sparse-16 and then saturates at the reported precision through sparse-64. These results support MRI-guided per-case SDF refinement as a reconstruction strategy that remains effective across weak-mask generators and supervision densities.
Reconstructing 4D (3D+t) cardiac geometry from sparse 2D echocardiography is highly desirable yet fundamentally challenged by geometric ambiguity and temporal discontinuity. To tackle these issues, we propose Echo4DIR, a novel test-time 4D implicit reconstruction framework. Specifically, we learn robust 3D shape priors from statistical shape models (SSMs) via a cardiac conditional SDF, constructing an Epipolar Mask Encoder module with epipolar cross attention to effectively fuse multi-view features. To bridge the synthetic-to-real domain gap, we introduce a self-supervised SDF-tailored differentiable rendering strategy for patient-specific 3D shape adaptation using uncalibrated clinical masks without requiring 3D ground truth. Crucially, the inherent continuity of implicit representation overcomes sparse observations, enabling anatomically reliable geometry at arbitrary resolutions. Furthermore, to empower our framework with physically continuous 4D extension, we introduce a Radial SDF Alignment strategy that strictly locks shape evolution to the predicted velocity field, fundamentally eliminating mesh drift. Extensive experiments on synthetic benchmarks and real clinical datasets demonstrate that Echo4DIR achieves state-of-the-art 4D cardiac mesh reconstruction, notably yielding an impressive clinical overlap of up to 98.35% Dice and 96.75% IoU.