⚡ electrical engineering

Uncertainty Quantification for Cardiac Shape Reconstruction with Deep Signed Distance Functions via MCMC methods

This paper proposes a probabilistic framework that combines Deep Signed Distance Functions with Markov Chain Monte Carlo sampling to enable accurate, uncertainty-aware cardiac shape reconstruction from sparse or noisy data, thereby enhancing clinical reliability.

Original authors: Jan Verhülsdonk, Thomas Grandits, Francisco Sahli Costabal, Thomas Beiert, Simone Pezzuto, Alexander Effland

Published 2026-05-11

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jan Verhülsdonk, Thomas Grandits, Francisco Sahli Costabal, Thomas Beiert, Simone Pezzuto, Alexander Effland

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Guessing a Shape from a Few Dots

Imagine you are trying to draw a perfect picture of a human heart, but you only have a few scattered dots to work with. Maybe the dots are from a noisy scan, or maybe you only have data from the left side of the heart and need to guess what the right side looks like.

This is a common problem in medicine. Doctors need a 3D model of a patient's heart to plan surgery, but the data they get is often incomplete or "fuzzy."

The authors of this paper created a new computer method to solve this. Instead of just drawing one best guess, their method draws many possible guesses and tells you how confident it is about each part of the drawing.

The Tools: The "Mental Atlas" and the "Magic Compass"

To understand how they did it, think of two main tools:

1. The DeepSDF (The "Mental Atlas")
Imagine a master artist who has studied thousands of healthy hearts. This artist has built a "Mental Atlas" in their head. They know that hearts generally look a certain way; they know how the left ventricle usually curves and how the right ventricle sits next to it.

In the paper: This is called a Deep Signed Distance Function (DeepSDF). It's a neural network (a type of AI) trained on many heart shapes. It doesn't just store pictures; it stores the rules of what a heart shape looks like. When you give it a code (a "latent code"), it can instantly draw a smooth, perfect heart based on those rules.

2. The MCMC (The "Magic Compass")
Usually, when you try to fit a heart shape to a few dots, the computer finds the single "best" fit and stops there. But what if there are two different shapes that fit the dots almost equally well? A standard computer would just pick one and pretend it's the only answer.

In the paper: The authors use Markov Chain Monte Carlo (MCMC) methods. Think of this as a "Magic Compass" that doesn't just point to one destination. Instead, it explores the entire map of possibilities. It wanders around the "Mental Atlas," finding all the different heart shapes that could possibly fit your dots.
The Result: Instead of one heart, you get a cloud of hundreds of slightly different hearts. If all the hearts in the cloud look the same in one spot, the computer is very confident. If the hearts in the cloud look very different in another spot, the computer knows it's unsure.

How It Works: The "Blindfolded Sculptor"

Here is the step-by-step process the paper describes:

The Training: First, the AI learns from a library of 40 real heart models. It learns the "rules" of heart shapes (the Atlas).
The Input: The user gives the AI a sparse point cloud (a few dots) from a patient's heart.
The Search: The AI asks, "What are all the possible heart shapes that could create these dots?"
- It uses the MCMC method to sample hundreds of different "latent codes" (the secret keys that unlock different heart shapes in the Atlas).
- It treats the difference between the dots and the shape as "noise" (like static on a radio).
The Output: The AI produces:
- The MAP (Maximum A Posteriori): The single "best" heart shape that fits the dots.
- The Uncertainty Map: A visual guide showing where the AI is guessing and where it is sure.

What They Found (The Results)

The authors tested this method on public heart data and found three main things:

More Data = More Confidence: As they added more dots to the input, the "cloud" of possible hearts got tighter, and the reconstruction became more accurate. The uncertainty estimates correctly showed that the model was becoming more sure.
Better than "One Guess": They compared their method to other ways of estimating uncertainty (like the "Laplace approximation," which is a faster but less detailed math trick). Their MCMC method was more accurate at telling the truth about how uncertain the model was, though it took longer to compute.
Cross-Checking Chambers: This is a cool feature. If you only give the AI dots from the Left ventricle, it can still guess what the Right ventricle looks like because it learned they are connected in the "Mental Atlas."
- The Catch: The AI knows it's guessing. It will show a "certainty map" where the Right ventricle is fuzzy (because it wasn't measured), but it will highlight the parts of the Right ventricle that are physically touching the Left ventricle as being more certain.

Why This Matters (According to the Paper)

The paper emphasizes that this is the first time this specific combination (DeepSDF + MCMC) has been used for cardiac shape reconstruction to quantify uncertainty.

Reliability: In medicine, knowing what you don't know is just as important as the diagnosis. If a doctor sees a reconstruction, they need to know if the computer is 99% sure or just guessing.
No "Normal Vectors" Needed: Unlike other methods that require complex data about the direction of the surface (normals), this method works with just simple points, making it easier to use with real-world, messy medical data.

Summary Analogy

Imagine you are trying to guess the shape of a hidden sculpture in a dark room by feeling a few spots on it with your hands.

Old Method: You feel the spots, make your best guess, and draw one picture. You don't tell anyone if you are guessing or sure.
This Paper's Method: You feel the spots, then you ask 100 other experts who have seen thousands of sculptures. They all draw a picture based on what they think the sculpture looks like. You stack all 100 drawings on top of each other. Where the lines overlap perfectly, you know the shape. Where the lines are all over the place, you know that part is a guess.

The paper proves this "group of experts" approach (MCMC) gives a much more honest and reliable picture of the heart than just asking for one opinion.

Technical Summary: Uncertainty Quantification for Cardiac Shape Reconstruction with Deep Signed Distance Functions via MCMC methods

Problem Statement
Accurate reconstruction of cardiac anatomy is essential for understanding cardiovascular disease and planning patient-specific therapies. While atlas-based approaches using Deep Signed Distance Functions (DeepSDFs) enable high-fidelity modeling from sparse or noisy data (e.g., point clouds), current methods are predominantly deterministic. They rely on finding a single Maximum A Posteriori (MAP) latent code, thereby failing to quantify the uncertainty inherent in reconstructions derived from incomplete observations. This lack of uncertainty estimation limits clinical reliability, particularly when data is sparse or ambiguous, as it prevents the assessment of prediction confidence and variability.

Methodology
The authors propose a probabilistic framework that integrates DeepSDFs with Markov Chain Monte Carlo (MCMC) sampling to perform Bayesian inference in the latent space.

DeepSDF Framework: Cardiac geometries (specifically the left and right ventricular endocardial and epicardial surfaces) are modeled implicitly as the zero-level sets of a neural network conditioned on a learned latent code $z$ . The network is trained on a dataset of biventricular shapes using a loss function that combines signed distance reconstruction error with a Lipschitz regularization term and a Gaussian prior on the latent codes.
Bayesian Formulation: The reconstruction task is recast as a Bayesian inference problem. The reconstruction loss is interpreted as the negative log-likelihood of the observed point cloud data given a latent code. A Gaussian prior is placed on the latent space, derived from the training data.
Uncertainty Quantification via MCMC: Instead of optimizing for a single point estimate, the authors employ MCMC methods to sample from the posterior distribution of the latent codes. Specifically, they utilize Hamiltonian Monte Carlo (HMC) and the No-U-Turn Sampler (NUTS) to explore the high-dimensional latent space. This yields a distribution of plausible shapes rather than a single deterministic output.
Evaluation Metrics: The quality of uncertainty estimates is assessed using the Achieved Coverage (AC) and Expected Calibration Error (ECE). These metrics compare the empirical distribution of signed distance values from the posterior samples against the ground truth on a mesh.

Key Contributions

First UQ Framework for DeepSDF Cardiac Reconstruction: To the authors' knowledge, this is the first work to introduce an uncertainty quantification framework for cardiac shape reconstruction based on DeepSDFs.
MCMC-Based Inference: The paper formulates latent shape inference as a posterior sampling problem, proposing MCMC algorithms (HMC and NUTS) to handle uncertainty-aware reconstruction from sparse or noisy data.
Joint Surface Modeling: The approach supports multi-surface reconstruction of biventricular anatomy, allowing for cross-chamber uncertainty analysis (e.g., inferring Right Ventricle uncertainty from Left Ventricle data).
Noise Parameter Inference: The framework is extended to jointly infer the observation noise variance ( $\zeta^2$ ) alongside the latent codes, enabling the model to adapt to unknown noise levels in the input data.

Results
Experiments were conducted on a public cardiac dataset comprising 40 training meshes and 4 test meshes.

Data Sparsity: As the number of input points increases, reconstruction accuracy improves, and the estimated uncertainty (standard deviation of SDF values) decreases, demonstrating the method's sensitivity to data density.
Sampling Comparison: The study compares Laplace approximation, HMC, and NUTS. While Laplace approximation is computationally efficient, MCMC methods (HMC/NUTS) provide well-calibrated uncertainty estimates. The Expected Calibration Error (ECE) for MCMC methods was lower (approx. 0.03–0.04) compared to Laplace approximation (0.041), indicating superior calibration.
Cross-Chamber Analysis: When reconstructing the Right Ventricle (RV) using only Left Ventricle (LV) point clouds, the MAP estimator produced variable results depending on initialization. The probabilistic framework successfully identified regions of high certainty (adjacent to the LV) and quantified ambiguity in other regions.
Comparison with Baselines: The proposed method was compared against Stochastic Poisson Surface Reconstruction (SPSR), Neural Stochastic Poisson (NSPSR), and Bayesian Smooth Signed Distance (BSSD). The proposed method demonstrated competitive or superior calibration (lower ECE) and, crucially, does not require surface normal information in the input, unlike the compared baselines.
Noise Inference: The model successfully inferred relative noise magnitudes when the noise level was unknown, showing a strong correlation between inferred and true noise levels, which improved calibration metrics.

Significance and Claims
The paper claims that reformulating cardiac shape reconstruction within a Bayesian framework allows for the generation of well-calibrated uncertainty estimates, moving beyond conventional point-estimate methods. This enables clinicians and researchers to assess the reliability and variability of reconstructions, particularly in scenarios with sparse data.

The authors note that while iterative MCMC sampling provides the most accurate calibration, it is computationally demanding. Conversely, the Laplace approximation offers a faster, albeit less precise, alternative suitable for time-critical settings. The work highlights that uncertainty-aware reconstruction is vital for identifying regions where the model is confident versus where the data is insufficient, thereby supporting more robust decision-making in patient-specific modeling. The authors acknowledge limitations regarding the dependency on the size and diversity of the training dataset for the shape prior and the need for careful tuning of hyperparameters (noise variance and prior covariance).