Negative Centerline Loss in Biomedical Segmentation

• Negative Centerline Loss is a topology-aware loss function that supervises only the ground-truth centerline to preserve the connectivity of elongated structures.
• It employs differentiable soft-skeletonization to compute a compact centerline, guiding CNNs (e.g., 3D U-Net) to mitigate false splits in segmentation masks.
• Empirical results show improved precision, reduced Wasserstein distance, and better instance continuity for axon initial segments compared to traditional losses.

Searching arXiv for the specified paper and closely related topology-aware segmentation work to ground the article in current sources. Negative Centerline Loss is a loss function for biomedical image segmentation that is designed to preserve the connectivity of elongated instances, particularly when missing or weak signal causes conventional segmenters to fragment structures that should remain continuous. Introduced by Szustakowski et al. in the context of axon initial segment segmentation, it penalizes the fraction of the ground-truth centerline that is not covered by the prediction, thereby shifting supervision from generic foreground overlap toward missed topological support along the essential path of the object (Szustakowski et al., 3 Sep 2025).

1. Concept and target problem

Negative Centerline Loss was proposed for settings in which voxel-wise accuracy is not the primary objective and preservation of elongated structure continuity and length is more important than voxel-wise accuracy. The motivating examples given are elongated biological structures such as axons and vessels, where weak signal or signal dropout can create artificial discontinuities in predicted masks even when the underlying instance is continuous (Szustakowski et al., 3 Sep 2025).

The key idea is to supervise only the centerline of the ground-truth label. Rather than penalizing all foreground pixels equally, the loss concentrates on centerline pixels that are missed by the model. In this design, the centerline acts as a compact representation of the structure’s topology, and the loss directly encourages the model to reconnect breaks that would otherwise shorten instances or split them into multiple connected components.

This focus distinguishes the method from overlap-driven objectives whose gradients are distributed across the whole foreground. A plausible implication is that the loss is particularly suited to downstream analyses in which the faithful recovery of path length, continuity, or one-to-one instance correspondence matters more than small boundary deviations.

2. Mathematical formulation

Let $L$ denote the ground-truth label mask, $P$ the predicted probability mask, and $\text{soft-skeleton}(L)$ a differentiable skeletonization of the label mask producing the centerline mask $L_{CL}$. With $\circ$ denoting the Hadamard product and $|A|$ the sum over all pixels or voxels of tensor $A$, the loss is defined as follows (Szustakowski et al., 3 Sep 2025):

$$\begin{aligned} L_{CL} &\gets \text{soft-skeleton}(L) \ NI_{CL} &\gets (1 - P) \circ L_{CL} \ \mathcal{L}_{NegativeCenterline} &= \frac{|NI_{CL}|}{|L_{CL}|} \end{aligned}$$

Its computation is described in three steps. First, the label $L$ is skeletonized to obtain its centerline $L_{CL}$. Second, the set of centerline pixels not predicted is computed as $P$0. Third, the missed centerline mass is summed and normalized by the total centerline length. The result is the fraction of the centerline missed by the prediction.

The formulation makes the optimization target explicit: any increase in predicted probability along the ground-truth centerline reduces the loss, while false negatives on that centerline are maximally penalized. The paper states that the largest gradient is applied precisely where the predicted mask fails to cover the label’s centerline, directly driving the model to heal breaks. Conversely, misclassified outer or border pixels have little effect on the loss.

3. Differentiable skeletalization and relation to other topology-aware losses

The “soft-skeleton” operation used in Negative Centerline Loss employs differentiable minpool and maxpool operations to produce a sparse, differentiable skeleton from the label mask, iteratively peeling layers until only the centerline remains. This preserves gradient flow during training and makes the loss compatible with standard backpropagation in CNN pipelines (Szustakowski et al., 3 Sep 2025).

In relation to conventional objectives, the contrast is direct. Dice and BCE supervise all foreground equally and are not sensitive to connectivity breaks. Negative Centerline Loss instead supervises only the label’s centerline and penalizes missed regions that correspond to disconnections. This is the central topological prior embedded in the loss design.

The closest comparator discussed is clDice. In the vessel-segmentation literature, clDice defines topology precision and topology sensitivity on predicted and ground-truth centerlines,

$P$1

and combines them through the harmonic mean

$P$2

This loss is explicitly skeleton-based and operates on both skeletons of the label and the prediction (Rougé et al., 2023).

Szustakowski et al. characterize the difference as operational rather than merely notational. clDice may penalize minor misalignments and boundaries, and can produce gradients in areas not directly relevant for long-range connectivity, whereas Negative Centerline Loss supervises only the label’s centerline and maximally penalizes missed regions. In this sense, Negative Centerline Loss is not a generic topology metric but a targeted false-negative penalty on the topological core of the ground-truth structure (Szustakowski et al., 3 Sep 2025).

A related technical issue arises in 3D. A separate study on vessel segmentation reports that the soft-skeleton algorithm provides suboptimal results on 3D images and makes clDice hardly suitable on 3D images when skeletonization quality is poor (Rougé et al., 2023). This suggests that methods relying on prediction-side skeletonization may be more exposed to skeletonization errors. Negative Centerline Loss avoids prediction skeletonization altogether by skeletonizing only the label; this implication is interpretive, but it is consistent with the design contrast stated in the source material.

4. Integration into segmentation pipelines

Negative Centerline Loss was implemented as an auxiliary loss in 3D U-Net architectures generated with nnU-Net. In the reported study it was added on top of standard BCE and Dice losses, with a weight; in the study, $P$3 (Szustakowski et al., 3 Sep 2025).

The main application scenario was a 3D light-sheet fluorescence microscopy dataset of axon initial segments in mouse brains. This task is prone to discontinuity due to signal dropout, and manual annotation is challenging. The paper also discusses characteristics of experiment design, such as downscaling and spacing correction, that help obtain continuous segmentation masks. These design choices are presented as part of a broader continuity-oriented workflow rather than as a property of the loss alone.

The per-sample or per-batch computation is straightforward. The label is skeletonized, the centerline mask is multiplied by $P$4, and the missed centerline pixels are summed and normalized by the total centerline length. Because the skeletonization and multiplication use soft operations, the loss is differentiable and can be incorporated into standard CNN training without changing the network architecture.

Although the paper’s main demonstrations are biomedical, it also identifies other application domains in which preserving connectivity of thin, elongated structures is critical: vessels, nerves, root systems, roads, and pipelines.

5. Empirical behavior on axon initial segment segmentation

The reported evaluation compares Negative Centerline Loss with a baseline nnU-Net, clDice, topology loss, and Simplified Topology Loss on metrics that reflect both overlap and structural fidelity. The metrics include Dice coefficient, Wasserstein Distance between length distributions of AIS in ground truth and prediction, Strictly Standardized Mean Difference, Precision and Recall at the instance level, and the average number of prediction instances overlapping a label (Szustakowski et al., 3 Sep 2025).

For the main experiment with $P$5 downscaling, the paper reports the following values:

Metric	Negative Centerline	Comparison
Dice	0.818	baseline 0.826; clDice 0.821; topology loss 0.824; Simplified Topology 0.822
Wasserstein	42.9	baseline 55.5; clDice 54.0; topology loss 54.4; Simplified Topology 49.3
SSMD	0.172	baseline 0.197; clDice 0.194; topology loss 0.220; Simplified Topology 0.178
Precision	0.862	baseline 0.822; clDice 0.830; topology loss 0.844; Simplified Topology 0.857
Recall	0.879	baseline 0.892; clDice 0.902; topology loss 0.896; Simplified Topology 0.888
Overlapping inst.	1.042	baseline 1.038; clDice 1.049; topology loss 1.045; Simplified Topology 1.054

These numbers support several conclusions stated in the paper. Negative Centerline Loss yields the lowest Wasserstein distance, the best SSMD values, and the highest precision for detecting connected objects. Its Dice score remains on par with the other methods rather than collapsing under the stronger continuity prior. The paper also states that instances are rarely split and that the method reduces the average number of predicted instances per label, bringing the mapping closer to the optimal one-to-one correspondence.

The same pattern is reported without downscaling. Negative Centerline Loss attains Wasserstein distance $P$6, compared with $P$7 for clDice and $P$8 for the baseline, and precision $P$9, compared with $\text{soft-skeleton}(L)$0 for clDice and $\text{soft-skeleton}(L)$1 for the baseline. Visual inspection in Figure 1 is reported to show fewer disconnected AIS segments than clDice or baseline losses, preserving more of the true structure’s length and continuity.

6. Interpretation, use cases, and limitations

The principal downstream significance of Negative Centerline Loss is improved instance length calculation. Because it directly penalizes disconnected centerlines, it reduces false splits and improves length estimation in downstream applications. In the AIS setting this is particularly important because discontinuities induced by signal dropout can otherwise bias measurements of elongated neuronal structures (Szustakowski et al., 3 Sep 2025).

A common misconception is that topology-aware segmentation losses must always symmetrically compare label and prediction skeletons. The Negative Centerline formulation contradicts that assumption: it supervises only the label’s centerline and uses the prediction only to determine whether centerline regions are covered. The practical consequence is that missed centerline support dominates the loss, while peripheral boundary disagreements have limited influence.

Another misconception is that connectivity-focused losses necessarily trade away overlap quality. In the reported experiments, Negative Centerline Loss maintains Dice score on par with other methods while improving length-distribution matching and connected-object precision. The improvement is therefore not presented as a replacement of overlap metrics, but as a reweighting of learning pressure toward functionally relevant structural errors.

The paper further states that the method works better than other topology-aware losses in data-poor regimes in combination with the Simplified Topology Loss. It also reports that the model can learn to “interpolate” connections in regions of missing input signal if the label indicates continuity. This should not be read as unrestricted shape completion; the concrete claim is narrower, namely that continuity supervision helps recover connected predictions in the presence of dropout.

As a practical recommendation derived directly from the reported application scenarios, Negative Centerline Loss is suited to image segmentation problems in which structure continuity is critical and signal interruptions may otherwise lead to undesired splits or length underestimation. The evidence presently documented is strongest for AIS segmentation in 3D light-sheet fluorescence microscopy, with the paper explicitly identifying analogous relevance for vessels, nerves, root systems, roads, and pipelines (Szustakowski et al., 3 Sep 2025).