A Strong Linear Baseline for Whole-Heart Cardiac Shape Completion on CT, with an Open Eleven-Structure Statistical Shape Model

Matej Gazda; Jakub Gazda; Juraj Gazda; Peter Drotar

cs.AIAug 20, 2026

A Strong Linear Baseline for Whole-Heart Cardiac Shape Completion on CT, with an Open Eleven-Structure Statistical Shape Model

Authors: Matej Gazda, Jakub Gazda, Juraj Gazda, Peter Drotar

Organizations: Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 1/9, 042 00 Košice, Slovakia · 2nd Department of Internal Medicine, Faculty of Medicine, Pavol Jozef Šafárik University and L. Pasteur University Hospital, Trieda SNP 1, 040 11 Košice, Slovakia · Department of Computers and Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 1/9, 042 00 Košice, Slovakia

Abstract

Public cardiac cohorts annotate different subsets of the heart, so shapes from separate sources cannot be pooled without shared correspondence. Among released cardiac shape resources, none we identified carries the atrial appendage, pulmonary veins, and caval stumps as separate blocks in one mesh. Completion benchmarks also compare deep models against a least-squares projection onto shape modes, not the conditional estimator the same fitted model implies. We release an eleven- structure cardiac computed-tomography (CT) statistical shape model, built from 383 automatically labelled cases in 11 571-vertex correspondence, and compare completion estimators under one frozen internal split and endpoint. On a 76-case internal list held out from fitting, a closed-form conditional-Gaussian estimator reconstructed the missing non-chamber structures at 3.717 mm mean per-vertex error, averaged equally over one, three, five, and nine observed structures. A five-refit mask-conditioned graph variational autoencoder reached 5.248 mm and nearest-neighbour retrieval 8.931 mm. The paired difference was 1.531 mm (95% confidence interval 1.384 to 1.711), and the ordering held in a raw-coordinate sensitivity arm. Expert manual labels exist for 58 external CT cases, but our registered reference is close enough to score only five structures. There the closed-form estimator again had lower average surface distance, 95th-percentile Hausdorff distance, and Chamfer error for both completed atria. On a second public benchmark of 20 cases the reference was close enough for three of four completed structures, and the same ordering held there. Four structures have no expert reference. The released model and its completion operator support cohort-unification research on aligned CT, not clinical use.

Figures & tables

Resource	Anatomy outside the four chamber walls	Public package
Hoogendoorn et al. [ 20 ]	Detailed multiregion whole-heart atlas, including great vessels	Atlas and SSM reported; no reusable model package located
Rodero et al. [ 2 ]	AO and PA walls; PV and caval openings tagged; LAA body omitted	SSM parameters and 1 000 synthetic finite-element meshes
Strocchi et al. [ 3 ]	AO and PA walls; LAA, PV, SVC and IVC at cut rings	24 patient-specific finite-element meshes
This work	AO, PA, LAA, PV, SVC and IVC as separate surface blocks in one shared topology	Template, per-case displacement fields and reconstruction code

TABLE I: Closest multistructure CT geometry resources. “Public package” means a direct public download of a reusable model or mesh cohort, assessed from the cited publications and their linked repositories at the time of writing; boundary tags or cut rings are not counted as separate anatomical surface bodies. None of the prior resources supports missing-label conditioning; this work conditions on any observed structure-block subset.

Fig. 1: End-to-end pipeline and frozen evaluation benchmark. After segmentation QC, joint-label registration, registration/FOV QC, and same-structure R-local repair, the 383 corresponding meshes are divided into a development set used for model selection and a 76-case evaluation set withheld from every fit reported here (the graph architecture predates this partition; Section V ). The orange block carries the conditional-mean equation; all comparators use the same split, panels, endpoint, and evaluation cases. CARE2026 and MM-WHS provide separate expert-surface validation without fitting or tuning.

	Internal (TotalSegmentator)	CARE2026	MM-WHS CT
Role	SSM fit; internal eval.	indep. validation	indep. secondary
Source cases	631	60 (A/B/G)	20
Segmentation QC	$-206$ (vol., extent, centr.) $\rightarrow$ 425	–	–
Affine check	–	$-2$ (non-orthon.)	–
Registration QC	$-26$ ( $>$ 25 mm) $\rightarrow$ 399	–	–
FOV truncation	$-16$	–	–

TABLE II: The three cohorts and the exclusions that produce them. The released internal metadata supplies age, sex, manufacturer, scanner model, tube voltage, pathology class, and study type; “n/a” attributes are absent from the released metadata and are not estimated, and “none” means no fitting or tuning. The 76-case evaluation list is frozen (split file released with the artifacts). Both CARE2026 exclusions fall in subset B; MM-WHS has no exclusions.

ID	Structure	Abbrev.	Vertices
1	Left ventricle	LV	1 502
2	Myocardium	MYO	2 502
3	Right ventricle	RV	2 002
4	Left atrium	LA	1 002
5	Right atrium	RA	1 002
6	Aorta	AO	1 202

TABLE III: Eleven cardiac structures and template vertex counts. MYO is the left-ventricular wall; the other ten are blood-pool or luminal surfaces. The model therefore carries no right-ventricular or atrial wall and does not represent total myocardial mass.

Panel of Figure 5 — Fig. 2: The eleven-structure whole-heart template mesh (11 571 vertices) in anterior, posterior, left, and right views, each structure a colour-coded surface in fixed vertex correspondence. This single template is warped into every patient’s space to build the SSM.

Structure	Median	p25	Structure	Median	p25
LV	0.981	0.979	AO	0.824	0.787
MYO	0.962	0.944	PA	0.969	0.910
RV	0.974	0.965	LAA †	0.941	0.938
LA	0.978	0.977	PV †	0.918	0.891
RA	0.975	0.971	SVC †	0.725	0.619
			IVC †	0.528	0.461

TABLE IV: Per-structure warp-overlap Dice (SyN-warped patient segmentations vs template), CARE2026 (

n{=}58

): a registration-consistency measure, not accuracy against a manual reference. Structures marked † (LAA, PV, SVC, IVC) are silver-only. Chambers is the mean of the five per-chamber medians. p25 is the 25th percentile.

Fig. 3: The completion operator. A selection matrix

\mathbf{P}_{o}

extracts whichever structure blocks or partial vertex sets are observed. The covariance is fitted once; inference extracts the corresponding covariance blocks and evaluates the conditional mean. Observed coordinates are copied to the output; missing coordinates are predicted. The figure writes the operator in covariance form for readability; the implementation evaluates the low-rank coefficient-space posterior of Eq. (1), whose ridge acts on the posterior precision rather than on

\mathbf{C}_{oo}

Representation	Method	$k=1$	$k=3$	$k=5$	$k=9$	Four- $k$ [95% CI]
Raw	Mean shape	11.218	11.278	11.189	10.909	11.149 [10.310, 12.040]
	Nearest neighbour	10.096	9.031	8.508	8.644	9.070 [8.621, 9.563]
	Cond-G	5.479	3.918	3.193	3.003	3.898 [3.629, 4.206]
	$\beta$ -VAE	6.596	5.238	4.693	4.748	5.319 [4.991, 5.701]
	PPCA ( $K=1$ )	5.524	3.958	3.248	3.064	3.948 [3.683, 4.251]
	Local Cond-G †	5.479	3.918	3.193	3.003	3.898 [3.629, 4.206]

TABLE V: Frozen 76-case internal evaluation completion MPVED (mm). Each

k

entry is the case mean over its fixed panels; the four-

k

column equally weights the four

k

values, with a case-bootstrap 95% CI. PPCA is the selected

K{=}1,q{=}160

limit; local Cond-G reuses the exact global prediction by construction.

Contrast	Difference (mm)	95% CI (mm)	Multiplicity result
R-local $\beta$ -VAE minus Cond-G, four- $k$ primary	1.531	[1.384, 1.711]	Unadjusted $p<0.001$ (floor)
Raw $\beta$ -VAE minus Cond-G, four- $k$	1.421	[1.264, 1.610]	$p_{\rm Holm}<0.002$
R-local $\beta$ -VAE minus Cond-G, $k=1$	1.218	[1.023, 1.439]	$p_{\rm Holm}<0.002$
R-local $\beta$ -VAE minus Cond-G, $k=3$	1.440	[1.298, 1.602]	$p_{\rm Holm}<0.002$
R-local $\beta$ -VAE minus Cond-G, $k=5$	1.613	[1.466, 1.800]	$p_{\rm Holm}<0.002$
R-local $\beta$ -VAE minus Cond-G, $k=9$	1.852	[1.689, 2.047]	$p_{\rm Holm}<0.002$

TABLE VI: Frozen paired family. Differences are challenger minus Cond-G, so positive values favour Cond-G. The R-local four-

k

primary is unadjusted; the next seven rows are the complete Holm family (

m=7

). The final row is an identity check rather than empirical evidence.

Method	Type	Internal 76	MM-WHS ( $n=20$ )
Cond-G	Closed-form linear	3.717 [3.480, 3.984]	4.823 [4.446, 5.226]
Latent optimisation ‡	Published method	5.044 [4.746, 5.389]	6.248 [5.755, 6.777]
$\beta$ -VAE	Feed-forward deep	5.248 [4.935, 5.613]	6.616 [6.109, 7.152]
SpiralNet++ §	Published operator	6.710 [6.283, 7.218]	8.916 [8.105, 9.741]
Nearest neighbour	Retrieval floor	8.931 [8.503, 9.405]	10.907 [10.145, 11.684]
Mean shape	Population floor	11.090 [10.245, 11.990]	13.174 [12.045, 14.419]

TABLE VII: Additional non-linear baselines, R-local representation, MPVED (mm) with case-bootstrap 95% intervals. The internal column is the frozen 76-case split at four observed-structure counts; the MM-WHS column is exhaustive panels at

k=1,3,5

over the seven expert-overlap structures, so only the ordering within a column is comparable. Both arms lie outside the prespecified Holm family and are descriptive; the latent-optimisation setting is not retuned externally. Cond-G, the

\beta

-VAE and the floors repeat Table V for reference; the same arms on CARE2026 expert surfaces, a different endpoint, are in Table VIII .

Method	Type	LA	RA
Cond-G	Closed-form linear	3.275 / 7.846 / 4.138	3.479 / 9.007 / 4.586
Latent optimisation	Published method	3.902 / 8.941 / 4.822	4.243 / 10.265 / 5.413
$\beta$ -VAE	Feed-forward deep	3.934 / 8.990 / 4.860	4.333 / 10.372 / 5.495
SpiralNet++	Published operator	4.299 / 9.701 / 5.266	4.605 / 10.868 / 5.779

TABLE VIII: Additional non-linear baselines on CARE2026 expert surfaces, R-local, biventricular panel, mean ASSD / HD95 / Chamfer RMS (mm) over 58 cases. The endpoint is distance to expert manual segmentations and is not comparable to Table VII . AO and PA are suppressed for every method by the prespecified reference-closeness rule. Cond-G and the

\beta

-VAE repeat Table IX for reference.

Fig. 4: Completion across six observed-structure panels, one internal TotalSegmentator case (TS, s0365) beside one external CARE2026 case (A_Case1020), each the median case of its cohort by the displayed Cond-G estimator’s mean missing-vertex error over the six panels. Observed structures are grey; the conditional-Gaussian model completes the rest, coloured by per-vertex error (mm). The estimator is the selected configuration of Table V , fitted on the 307 development cases in the R-local domain.

Population	Method	LA	RA
Pooled	Cond-G	3.275 / 7.846 / 4.138	3.479 / 9.007 / 4.586
Pooled	$\beta$ -VAE	3.934 / 8.990 / 4.860	4.333 / 10.372 / 5.495
A	Cond-G	2.634 / 6.643 / 3.353	3.035 / 8.085 / 3.989
A	$\beta$ -VAE	3.103 / 7.553 / 3.911	3.780 / 9.373 / 4.811
B	Cond-G	4.010 / 9.642 / 5.032	4.028 / 10.598 / 5.351
B	$\beta$ -VAE	4.878 / 11.412 / 6.034	4.706 / 11.572 / 6.011

TABLE IX: R-local CARE biventricular native prediction-versus-expert surface metrics, pooled (

n=58

) and by acquisition subset (A/B/G contain 20/18/20 cases). The panel observes LV/MYO/RV; values are mean ASSD / HD95 / Chamfer RMS in mm. AO and PA are omitted after failing the reference-closeness rule.

Missing structure	Population	Cond-G	$\beta$ -VAE
LV	Pooled	2.362 / 6.407 / 3.119	3.899 / 9.623 / 4.942
	A	1.818 / 4.744 / 2.358	3.367 / 8.177 / 4.254
	B	3.697 / 10.361 / 4.969	4.841 / 12.575 / 6.258
	G	1.704 / 4.510 / 2.215	3.582 / 8.411 / 4.446
MYO	Pooled	2.056 / 5.667 / 2.729	2.943 / 7.583 / 3.769
	A	1.695 / 4.541 / 2.212	2.526 / 6.257 / 3.171

TABLE X: R-local CARE leave-one-expert-structure-out completion, pooled (

n=58

) and by acquisition subset (A/B/G contain 20/18/20 cases). Values are mean ASSD / HD95 / Chamfer RMS in mm. AO and PA are omitted because their references are too far.

Structures	Internal benchmark	CARE2026 expert check	MM-WHS expert check
LV, MYO, RV	never a completion target	reference close enough (leave-one-out panel)	expert label, but observed as input
LA, RA	never a completion target	reference close enough (both panels)	reference close enough; Cond-G better
AO	Cond-G better at every $k$	reference too far	reference too far
PA	Cond-G better at every $k$	reference too far	reference close enough under R-local; Cond-G better
LAA, PV, SVC, IVC	Cond-G better at every $k$	no expert label	no expert label

TABLE XI: The evidence behind each structure; structures with identical status share a row. “Reference close enough” means the registered reference lies within the fixed distance limits of the expert surface, so a native comparison is meaningful; the internal benchmark scores only structures that are completion targets there. “Cond-G better” is the lower descriptive mean error against the

\beta

-VAE at every

k

. AO’s reference is too far because the expert annotations cover a shorter aortic extent than the whole-aorta reference, not because of misregistration (Section V-D ).

	Internal ( $n{=}76$ )			CARE2026 ( $n{=}58$ )
Structure	MAE (mL)	medAPE	$\Delta$ [95% CI]	MAE (mL), medAPE
LA	7.99	8.3%	$-6.84$ [ $-9.43$ , $-4.43$ ]	18.47, 18.5%
RA	9.61	8.9%	$-7.29$ [ $-10.36$ , $-4.35$ ]	20.89, 25.7%
AO	13.81	5.6%	$-7.99$ [ $-12.75$ , $-3.33$ ]	23.08, 13.1%
PA	6.74	8.5%	$-4.51$ [ $-6.35$ , $-2.68$ ]	13.39, 17.5%
LAA	0.93	10.7%	$-0.49$ [ $-0.71$ , $-0.27$ ]	1.41, 14.8%

TABLE XII: Morphometric retention from the biventricular panel: per-structure volume agreement between the completion and the case’s own registered mesh. MAE is mean absolute error; medAPE is median absolute percentage error;

\Delta

is the paired difference in absolute error against the development-only regression on the three observed chamber volumes, with its 95% case-bootstrap interval (negative favours completion). Vessel and appendage rows are segment volumes under the fixed template crop.

	Cond-G		$\beta$ -VAE
Observed panel	typical	atypical	typical	atypical
LV	1.000	0.946 [0.868, 0.998]	0.128	0.105 [0.071, 0.138]
LV, MYO, RV	1.000	0.944 [0.878, 0.991]	0.129	0.140 [0.104, 0.174]
Five chambers	1.000	0.974 [0.953, 0.993]	0.141	0.139 [0.103, 0.175]
Nine structures	1.000	0.999 [0.996, 1.000]	0.205	0.194 [0.110, 0.281]

TABLE XIII: Nominal 95% ellipsoid coverage by out-of-support stratum, R-local, 76 evaluation cases (64 typical, 12 atypical). Strata come from the frozen development 95th-percentile Mahalanobis threshold. Exploratory and post-hoc; brackets are descriptive case-bootstrap intervals.

3D Shape Completion

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Bi-PT: Bidirectional Cross-Attention Point Transformers for Four-Chamber Heart Reconstruction from Sparse Cardiac MRI Data

Chenchuhui Hu, Shaoming Pan, Leon Axel, Meng Ye

Department of Computer Science and Engineering, University of Texas at Arlington, Arlington, USA · School of Medicine, New York University, New York, NY, USA

Transformer 3D Medical Imaging Sparse-View 3D Reconstruction Point Cloud Registration Cardiac MRI Point Transformer

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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.

Abstract

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