This section describes our framework, which generates realistic sparse data from LA shapes and completes the full shape using neural networks. The system trains a learning algorithm to reconstruct the left atrium shape from catheter paths. Due to limited real data, we used simulated data with Biosense LA instance generator [10] and a catheter path generator. Figure 2 (left) illustrates data generation, creating LA samples with synthetic catheter paths. The network takes catheter trajectories to reconstruct LA surfaces, see Fig. 2 (right). Training involves generating path volumes resembling real scenarios from landmarks. In clinical cases, acquired catheter point clouds become voxel volumes, indicating catheter traversal. We show atrium instance sampling in “Input data generation: synthetic atria” section and graph-based path generation in “Input data generation: generating synthetic paths” section. Reconstruction using The proposed dense encoder–decoder (DED) is described in Section [3.3 and using V-Net in “Left atrial shape reconstruction using V-Net network” section.

Input data generation: synthetic atria

We represent shape data as a \(N=45^3\) voxel binary 3D volume (\(2.666\,}^3\) per voxel). Each voxel has the value of “1” for interior/boundary, and “0” otherwise-an occupancy volume. Using a predefined model [10], the atrium shape model blends parametric tube-like shapes transformed nonlinearly. Statistical modeling involves features like PV positions, orientations, and ridge locations as a multivariate normal distribution (MVN), learned from CT scans. A generated sample’s statistical score gauges its atrial representation likelihood within the model. To generate a synthetic left atria we sampled from the MVN of the model parameters. Then, we kept samples that scored well within the statistical model. Our resultant generated dataset contains 5006 and 1800 samples for training and testing, respectively.

Input data generation: generating synthetic paths

The algorithm generates paths within LA shapes, emulating clinical catheter movement. With a single mapping sensor catheter, the path starts at the septum and traverses PVs sequentially: left superior, left inferior, right inferior, and right superior–distinct path sections. Path sections are depicted in Fig. 3a, and the full path is shown in b. Our vast LA dataset allows simulating diverse paths.

The process: We locate entry points at each PV ostium and then determine paths between ostia in a predefined order via graph optimization. This balances shortest distance and navigability, solved using Dijkstra’s algorithm [15]. We give an overview of the two main steps. More detailed steps are described in Supplementary material.

Simulated paths creation procedure overview

This algorithm processes input LA shapes as triangular meshes, with an initial mean shape derived from training data. Landmark points on this mean shape, including ostia and septal points, guide new input pose. Voxel-based representation is achieved by grid sampling within or over the mesh surface. Trajectories, originating at the septum and spanning through PVs’ ostia points, are defined by adjacent voxels. These paths usually curve toward the atrium center before reaching the next target. The algorithm then generates a volume where marked voxels indicate the path, while others remain zero.

Synthetic path augmentation

The generated synthetic path is next augmented by adding nearby points which are mostly inside the corresponding atrium. The augmentation procedure is motivated by the catheter’s physical setting and was tested empirically. The catheter vibrates during movement and may exceed the chamber boundary by slightly pushing it. First, for each (grid sampled) path point \( \mathbf \) we sample n points normally distributed around it, \(\mathbf \sim N(\mathbf ,\sigma )\). Then the points are trimmed using a probability factor \(s_}\), for each point. Next, we only consider points that are interior to the ground truth mesh. These undergo a normally distributed translation \(\mathbf }} \sim N(0,1)\cdot \mu _s\), where \(\mu _s\) is the factor that determines the noise level. Figure 3c shows an augmentation result.

Left atrial shape reconstruction using DED network

Next, we propose a reconstruction solution based on an NN recovery of complete LA shapes from sparse catheter paths, which we take as point clouds. We train a network on the generated synthetic paths, producing a probability volume via a final-layer sigmoid function. This signifies voxel likelihood of atrial interior/boundary. A 0.5 threshold binarizes values. We employ the dense encoder–decoder DED model, a multi-layer perceptron with input/output layers as linear voxel arrays. Interim layers are fixed size. Our model uses tied weights (only for input/output), masked input (akin to dropout after input), enhancing outcomes significantly. The Adam optimizer [16] is used. All layers, excluding the last, use RELU activation; the last employs sigmoid. Batch normalization [17] after each layer (but last) stabilizes activations, bridging synthetic clinical gap. The network combines binary cross-entropy loss (BCE) [18] and negative DICE coefficient [13] via linear mix. That is, for a ground truth volume z and a prediction x, the loss is

$$\begin L(x,z) = \alpha }(x,z) - (1 -\alpha ) }(x,z). \end$$

(1)

Spatial weight smoothing regularization (SWR)

In order to reconstruct a realistic atrial volume, we added a Spatial Weight Smoothing Regularization (SWR) term to the loss function. The loss including SWR is defined as:

$$\begin L(x,z) + \lambda \sum _^k \sum _^N\left\| \nabla _ } }^_i \right\| ^2 \end$$

(2)

where N is the number of voxels, k is the hidden layer size, \(}_\) denotes the layer weights. The SWR loss term is applied to the weights of the input and output layers only, for which each weight corresponds to a voxel. Specifically, each vector \(\textbf(j) = }_i^\) assigns a weight for each voxel in the input volume if we flatten the volume to a vector. The gradient vector \( \nabla _}\) is computed for each element (voxel) in the \(\textbf(j)\) vector based on the original spatial locations of the volume. The spatial dimension is \(\textbf\), that is, the position within the volume for the three spatial axes, \(\textbf\in (},},})\) in which the input and output reside. \(\lambda \) represents the level of regularization.

The spatial derivatives are computed using finite differences. Figure S1 depicts the relationship between the voxels in space and the neurons in the layer and shows which weight difference is added to the cost.

Boundary enhancement mask

The most significant area in the volume is the surface boundary. In order for the model to have larger loss gradients around the surface boundary, we used a weighted DICE cost with a weighting mask \(\Omega \). The mask assigns a weight for each voxel such that the majority falls over the boundary, and decreases for voxels further away from it, as seen in Figure S2. The weight of a voxel v is given by

$$\begin \Omega (v) = (1+ \beta )/(1+PN(D(v))), \end$$

(3)

where D(v) is the distance from the shape boundary and PN is the probability density function of the normal distribution with zero mean and \(\sigma = 1.5\). The \(\beta \) parameter was experimentally set to 14.

DED parameter selection

Following empirical experimentation, we report the results of the best-performing DED variant. To select parameters, we explored different scales; we then performed finer sampling around values that performed well over the test set. In our earlier work [8], we tested different combinations of depths and widths of the network architecture. The chosen variant includes two hidden layers with 350 neurons each. The cross-entropy and (weighted) DICE were combined in the loss function using \(\alpha =\frac\). The variants are named according to the SWR parameter \(\lambda \): “No SWR” with \(\lambda =0\), “SWR005” \(\lambda =0.05\), “SWR75” \(\lambda =75\). “No Aug” net has the same parameters as “SWR005” but with no path augmentation and no boundary enhancement mask.

Fig. 4

Input examples for the clinical cases. The acquired path is in red, the synthetic template path is in blue, and the CT is in gray. Tagged PV points are color coded

Left atrial shape reconstruction using V-Net network

The V-Net [13] is composed of convolution layers followed by max-pooling, repeated over four stages to halve volume size. Subsequent steps involve four-stage up-sampling using learned filters, with residual connections concatenating input and earlier stage info of same size. Regularization involves batch normalization and dropout.

Figure 2 shows the architecture integrated with our system. Like the DED, the input is the voxelized catheter path and the output is LA reconstruction as occupancy volumes. Network parameters include filter count, down/upsampling layers, and concatenation. Notably, larger receptive field filters in the first and last layers enhance network performance.

Mean shape as a baseline solution

To determine the effectiveness of any reconstruction algorithm, it needs to be compared to a common solution. In our case, we compared our results to the results of a mean shape solution. The mean shape was generated by taking a voxel-wise average over all the ground truth shapes in the training set. The mean location of a PV ostia over all the available atria data should converge to that of the mean shape. We defined a base coordinate frame for the reconstruction using the mean shape four PV ostia points. The input paths from the clinical cases were registered and transformed to have their PVs ostia match that of the mean shape (in the least squares sense) using rigid point set registration [19].