A Probabilistic Perspective on Wasserstein-Based Evidential Uncertainty for Out-of-Distribution Segmentation

Arnold Brosch; Abdelrahman Eldesokey; Michael Felsberg; Kira Maag

cs.CVOct 7, 2026

A Probabilistic Perspective on Wasserstein-Based Evidential Uncertainty for Out-of-Distribution Segmentation

Authors: Arnold Brosch, Abdelrahman Eldesokey, Michael Felsberg, Kira Maag

Organizations: Heinrich-Heine-University Düsseldorf, Universitätsstraße 1, 40225, Düsseldorf, Germany · Hamad Bin Khalifa University (HBKU), Education City, Gate 8, P.O. Box 5825, Doha, Qatar · Linköping University, SE-581 83, Linköping, Sweden

Abstract

Semantic segmentation networks operate on a fixed set of classes and therefore fail when out-of-distribution (OOD) objects appear during deployment, a critical limitation for safety-critical applications such as autonomous driving. Reliably identifying OOD objects requires well-calibrated epistemic uncertainty, yet common softmax-based confidence scores remain overconfident, while Bayesian alternatives such as Monte Carlo dropout or deep ensembles require costly repeated forward passes. Evidential Deep Learning (EDL) offers an efficient alternative by modeling class probabilities as a Dirichlet distribution learned from a single deterministic forward pass. Existing EDL formulations rely on Euclidean objectives that push predictions towards the simplex vertices, encouraging overconfidence rather than preserving uncertainty for unfamiliar inputs. We instead employ Wasserstein-based objectives, which respect the geometry of the probability simplex, and study the influence of the Wasserstein order on segmentation accuracy and OOD detection within a unified evidential framework. We evaluate this framework on a convolutional (DeepLabV3+) and a transformer-based (SegFormer) architecture on the SegmentMeIfYouCan benchmark, including LostAndFound, RoadObstacle21, RoadAnomaly21, and Fishyscapes. Our results show the optimal Wasserstein order is architecture-dependent: second-order objectives dominate on the convolutional backbone, third-order objectives on the transformer backbone, and our framework surpasses comparable baselines on most metrics, with a single deterministic forward pass.

Figures & tables

Panel of Figure 1 — Figure 1: Top: Semantic segmentation by a deep neural network. Bottom: Evidential uncertainty heatmap obtained by our method. Brighter colors correspond to higher uncertainty.

Panel of Figure 2 — Figure 2: Illustration of probability mass transport for support extension.

Panel of Figure 3 — Figure 3: Illustration of the MSE effect on the probability simplex.

Panel of Figure 4 — Figure 4: Illustration of the Wasserstein effect on the probability simplex.

Panel of Figure 5 — Figure 5: Left: Semantic segmentation prediction. Right: OOD heatmap obtained by our method using DeeplabV3+, $W_{2}$ , $0.45$ MSE and $0.75$ Dice. The top images are from the LostAndFound dataset (a kid playing on the street next to a pile of rubble) and the bottom images from RoadAnomaly21 (a car with an attached caravan).

	LostAndFound test-NoKnown					RoadObstacle21
	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
Ensemble	2.9	82.0	6.7	7.6	2.7	1.1	77.2	8.6	4.7	1.3
MC Dropout	36.8	35.6	17.4	34.7	13.0	4.9	50.3	5.5	5.8	1.1
Maximum Softmax	30.1	33.2	14.2	62.2	10.3	15.7	16.6	19.7	15.9	6.3
Entropy	47.1	21.6	30.7	42.1	30.2	28.4	26.7	14.7	20.5	9.7
PGN	69.3	9.8	50.0	44.8	45.4	16.5	19.7	19.5	14.8	7.4

Table 1: OOD segmentation benchmark results for the LostAndFound and RoadObstacle21 dataset.

	Fishyscapes LostAndFound					RoadAnomaly21
	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
Ensemble	0.3	90.4	3.1	1.1	0.4	17.7	91.1	16.4	20.8	3.4
MC Dropout	14.4	47.8	4.8	18.1	4.3	28.9	69.5	20.5	17.3	4.3
Maximum Softmax	5.6	40.5	3.5	9.5	1.8	28.0	72.1	15.5	15.3	5.4
Entropy	15.8	36.1	7.5	16.3	8.6	30.0	73.0	17.8	15.6	5.1
PGN	26.9	36.6	14.8	29.6	16.5	42.8	56.4	25.8	21.8	9.7

Table 2: OOD segmentation benchmark results for the Fishyscapes LostAndFound and RoadAnomaly21 dataset.

	DeepLabV3+	SegFormer
Baseline	$80.09$	$84.00$
$W_{1}$	$67.67_{-15.55}^{+9.43}$	$80.61_{-0.24}^{+0.27}$
$W_{2}$	$56.14_{-5.01}^{+6.53}$	$80.60_{-0.44}^{+0.38}$
$W_{3}$	$66.88_{-11.04}^{+9.46}$	$80.54_{-0.36}^{+0.20}$

Table 3: In-distribution evaluation on the Cityscapes dataset using mIoU. The rows correspond to the three Wasserstein orders

W_{1}

W_{2}

, and

W_{3}

, and the two columns to the DeepLabV3+ and SegFormer backbones.

+a

and

-b

describe the size to the largest and smallest deviation respectively.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}1}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	47.5	27.4	21.5	39.2	19.7	4.1	66.0	6.9	8.5	2.0
1.00	0.00	0.00	0.40	50.4	25.8	21.2	38.0	19.2	30.7	42.3	11.6	26.4	8.7
1.00	0.00	0.00	0.45	53.2	23.2	25.3	37.7	20.7	20.2	31.2	11.0	18.3	5.8
1.00	0.00	0.00	0.50	44.4	30.5	25.2	42.8	23.6	12.2	32.6	3.6	34.0	3.4

Appendix

Table A.1: OOD segmentation results for LostAndFound and RoadObstacle21 using DeepLabV3+ with Wasserstein order

W_{1}

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}2}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	43.8	38.9	18.9	53.9	21.6	1.1	44.5	14.1	3.6	1.7
1.00	0.00	0.00	0.40	50.4	36.9	18.2	45.3	20.3	13.7	53.9	8.0	15.2	3.3
1.00	0.00	0.00	0.45	46.1	47.6	22.7	53.2	26.3	13.5	40.9	7.2	20.6	4.4
1.00	0.00	0.00	0.45	48.8	38.9	17.0	44.4	18.4	17.2	49.9	6.4	13.7	3.0

Appendix

Table A.2: OOD segmentation results for LostAndFound and RoadObstacle21using DeepLabV3+ with Wasserstein order

W_{2}

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}3}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	34.0	88.1	13.8	40.1	17.0	12.0	74.1	8.4	18.6	5.2
1.00	0.00	0.00	0.40	10.5	93.2	1.3	12.0	1.2	13.7	50.6	4.7	19.3	3.8
1.00	0.00	0.00	0.45	12.6	89.5	8.4	23.5	8.0	2.8	63.1	5.0	6.4	1.9
1.00	0.00	0.00	0.50	11.7	84.9	2.1	10.6	1.6	5.4	73.7	2.9	6.0	1.4

Appendix

Table A.3: OOD segmentation results for LostAndFound and RoadObstacle21using DeepLabV3+ with Wasserstein order

W_{3}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}1}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	47.5	27.4	21.5	39.2	19.7	4.1	66.0	6.9	8.5	2.0
1.00	0.00	0.00	0.40	50.4	25.8	21.2	38.0	19.2	30.7	42.3	11.6	26.4	8.7
1.00	0.00	0.00	0.45	53.2	23.2	25.3	37.7	20.7	20.2	31.2	11.0	18.3	5.8
1.00	0.00	0.00	0.50	44.4	30.5	25.2	42.8	23.6	12.2	32.6	3.6	34.0	3.4

Appendix

Table A.4: OOD segmentation results for Fishyscapes and RoadAnomaly21 using DeepLabV3+ with Wasserstein order

W_{1}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}2}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	4.6	53.6	7.1	12.6	9.2	33.0	60.3	21.1	19.5	5.0
1.00	0.00	0.00	0.40	10.1	38.4	4.6	20.8	4.2	50.3	57.9	20.0	22.6	5.6
1.00	0.00	0.00	0.45	2.5	42.0	11.0	11.8	7.3	42.9	59.2	18.9	23.5	6.0
1.00	0.00	0.00	0.50	3.9	46.5	8.5	10.4	6.9	35.6	65.2	23.4	20.3	6.8

Appendix

Table A.5: OOD segmentation results for Fishyscapes and RoadAnomaly21 using DeepLabV3+ with Wasserstein order

W_{2}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}3}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	25.7	56.8	25.9	27.7	14.6	1.6	62.1	18.5	5.6	2.0
1.00	0.00	0.00	0.00	7.7	53.3	6.2	23.6	20.7	36.0	85.0	16.5	23.8	6.5
1.00	0.00	0.00	0.40	2.7	68.3	1.6	13.3	2.4	40.6	67.2	17.0	26.8	6.3
1.00	0.00	0.00	0.45	2.2	54.6	8.9	9.7	7.8	42.3	76.8	20.7	25.5	7.4
1.00	0.00	0.00	0.50	2.3	52.5	5.8	10.8	4.7	40.2	76.3	15.2	28.3	6.0

Appendix

Table A.6: OOD segmentation results for Fishyscapes and RoadAnomaly21 using DeepLabV3+ with Wasserstein order

W_{3}

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}1}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	22.4	63.1	7.0	13.1	26.9	21.2	64.7	4.2	55.7	3.9
1.00	0.00	0.00	0.00	16.1	57.7	4.5	11.6	1.3	8.9	66.1	3.0	13.6	0.8
1.00	0.00	0.00	0.40	41.9	43.9	23.3	25.5	13.1	29.1	48.6	13.0	20.9	5.2
1.00	0.00	0.00	0.45	34.1	39.8	22.4	19.2	9.7	31.5	35.8	9.8	28.6	5.8
1.00	0.00	0.00	0.50	34.1	49.8	16.7	23.8	9.2	24.3	39.0	11.0	13.8	3.0

Appendix

Table A.7: OOD segmentation results for LostAndFound and RoadObstacle21 using SegFormer with Wasserstein order

W_{1}

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}2}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	22.4	63.1	7.0	13.1	26.9	21.2	64.7	4.2	55.7	3.9
1.00	0.00	0.00	0.00	23.2	65.7	10.3	12.8	3.7	15.1	46.5	3.3	24.3	1.5
1.00	0.00	0.00	0.40	21.0	62.9	7.3	14.6	2.9	18.9	67.9	7.7	11.6	2.2
1.00	0.00	0.00	0.45	8.8	79.9	3.2	8.1	0.9	9.4	81.0	3.0	40.9	2.7
1.00	0.00	0.00	0.50	34.8	62.9	17.8	20.5	9.1	27.7	54.7	10.0	24.6	4.7

Appendix

Table A.8: OOD segmentation results for LostAndFound and RoadObstacle21 using SegFormer with Wasserstein order

W_{2}

				LostAndFound test-NoKnown					RoadObstacle21
$\lambda_{\mathcal{W}3}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	22.4	63.1	7.0	13.1	26.9	21.2	64.7	4.2	55.7	3.9
1.00	0.00	0.00	0.00	33.0	63.0	13.2	21.6	6.9	20.5	54.1	6.1	27.9	3.4
1.00	0.00	0.00	0.40	38.8	59.7	20.0	30.7	14.1	42.1	57.8	23.3	26.7	11.8
1.00	0.00	0.00	0.45	21.1	64.2	10.1	14.5	4.1	20.9	64.6	10.0	14.8	3.1
1.00	0.00	0.00	0.50	35.5	47.5	18.7	19.3	8.9	29.5	34.6	14.5	24.1	6.8

Appendix

Table A.9: OOD segmentation results for LostAndFound and RoadObstacle21 using SegFormer with Wasserstein order

W_{3}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}1}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	32.6	70.1	14.4	49.7	14.4	48.2	96.4	30.1	29.5	8.9
1.00	0.00	0.00	0.00	28.9	67.1	8.9	40.3	11.7	49.4	86.6	32.2	27.9	9.9
1.00	0.00	0.00	0.40	50.4	25.8	21.2	38.0	19.2	30.7	42.3	11.6	26.4	8.7
1.00	0.00	0.00	0.45	53.2	23.2	25.3	37.7	20.7	20.2	31.2	11.0	18.3	5.8
1.00	0.00	0.00	0.50	44.4	30.5	25.2	42.8	23.6	12.2	32.6	3.6	34.0	3.4

Appendix

Table A.10: OOD segmentation results for Fishyscapes and RoadAnomaly21 using SegFormer with Wasserstein order

W_{1}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}2}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	32.6	70.1	14.4	49.7	14.4	48.2	96.4	30.1	29.5	8.9
1.00	0.00	0.00	0.00	28.2	72.2	6.8	32.0	8.1	48.9	85.1	33.5	24.4	8.1
1.00	0.00	0.00	0.40	35.6	55.0	6.9	63.3	11.5	50.3	87.3	31.0	26.9	8.6
1.00	0.00	0.00	0.45	16.2	73.0	2.7	31.0	3.5	44.6	75.0	35.3	26.0	10.6
1.00	0.00	0.00	0.50	36.5	70.5	8.2	53.7	11.9	59.6	79.4	37.8	28.4	12.0

Appendix

Table A.11: OOD segmentation results for Fishyscapes and RoadAnomaly21 using SegFormer with Wasserstein order

W_{2}

				Fishyscapes LostAndFound					RoadAnomaly21
$\lambda_{\mathcal{W}3}$	$\lambda_{\text{Dice}}$	$\lambda_{\text{KL}}$	$\lambda_{\text{MSE}}$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$	AuPRC $\uparrow$	FPR 95 $\downarrow$	$\overline{\text{sIoU}}$ $\uparrow$	$\overline{\text{PPV}}$ $\uparrow$	$\overline{F_{1}}$ $\uparrow$
0.00	0.00	0.00	1.00	32.6	70.1	14.4	49.7	14.4	48.2	96.4	30.1	29.5	8.9
1.00	0.00	0.00	0.00	28.4	76.0	9.1	49.1	12.2	57.5	84.4	34.3	28.9	10.6
1.00	0.00	0.00	0.40	37.3	71.2	12.7	54.4	18.8	64.5	72.4	35.7	31.7	13.6
1.00	0.00	0.00	0.45	22.4	69.0	8.4	58.5	13.6	42.4	90.1	30.3	26.3	8.3
1.00	0.00	0.00	0.50	38.0	60.3	10.3	66.7	15.5	56.8	85.0	36.6	28.6	11.4

Appendix

Table A.12: OOD segmentation results for Fishyscapes and RoadAnomaly21 using SegFormer with Wasserstein order

W_{3}

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