CardioComposer: Leveraging Differentiable Geometry for Compositional Control of Anatomical Diffusion Models
Authors: Karim Kadry
Shoaib A. Goraya
Ajay Manicka
Abdalla Abdelwahed
Naravich Chutisilp
Farhad R. Nezami
Elazer R Edelman
Venue: ICLR 2026 Poster (2026)
Keywords: Diffusion Models
Computational Geometry
Anatomy
Digital Twins
Diffusion Guidance
Primary area: generative models
TL;DR: We present a novel energy-based guidance framework for imposing a variety of geometric constraints on diffusion models that generate multi-component anatomical structures.

Abstract: Generative models of 3D cardiovascular anatomy can synthesize informative structures for clinical research and medical device evaluation, but face a trade-off between geometric controllability and realism. We propose CardioComposer: a programmable, inference time framework for generating multi-class anatomical label maps from interpretable ellipsoidal primitives. These primitives represent geometric attributes such as the size, shape, and position of discrete substructures. We specifically develop differentiable measurement functions based on voxel-wise geometric moments, enabling loss-based gradient guidance during diffusion model sampling. We demonstrate that these losses can constrain individual geometric attributes in a disentangled manner and provide compositional control over multiple substructures. Finally, we show that our method is compatible with a broad range of anatomical systems containing non-convex substructures, spanning cardiac, vascular, and skeletal organs. We release our code at https://github.com/kkadry/CardioComposer.

Published as a conference paper at ICLR 2026
CARDIOCOMPOSER: LEVERAGING DIFFERENTIABLE GEOMETRY FOR COMPOSITIONAL CONTROL OF ANATOMICAL DIFFUSION MODELS
Karim Kadry* Naravich Chutisilp 9
Shoaib Gorayal
Ajay Manicka"
Abdalla Abdelwahed
Farhad R. Nezami
Elazer R. Edelman"
ABSTRACT
Generative models of 3D cardiovascular anatomy synthesize informative struc- tures for clinical research and medical device evaluation, but face trade-off between geometric controllability and realism. We propose CardioCompooer:: programmable, inference-time framework for generating multi-class anatomical label maps from interpretable ellipsoidal primitives. These primitives represent geometric attributes such as the size, shape, and position of discrete substruc- tures. We specifically develop differentiable measurement functions based on voxel-wise geometric moments, enabling bass-based gradient guidance during diffusion model sampling. We demonstrate that these losses can constrain indi- vidual geometric attributes in a disentangled manner and provide compositional control over multiple substructures. Finally, we show that our method is compat- ible with broad range of anatomical systems containing non-convex substruc- tures, spanning cardiac, vascular, and skeletal organs. We release our code at com htttpe///gggcaaaa.ccc//////ttttp
INTRODUCTION
Three-dimensional segmentations of human anatomy power variety of physics-based simulation platforms. For example, virtual cohorts of anatomy be used for virtual clinical trials to evaluate and optimize novel medical devices and imaging systems "Sarrami-Forousiee
et al. 202 Viceconti et al. 2021 Abadi et al. 2020). Additionally, biophysical simulations can generate insights in the context of both computational physiology research (Niederer et al.) 2020 Straughan et al. 2023 Roney et al.) 2020) and surgical training (Yu et al.
2024). Anatomical segmentations can also be used to augment machine-learning workflows through the formation of synthetic images, either through imaging simulators (Gopalakrishnan al.
2024 Gopalakrishnan & Golland 2022), domain
randomization (Dey et al. 2025; Billot et al. 2023). or generative models (Fernandez et al. 2022 2024).
Generative models of anatomy trained on patient-specific data offer advantages for simulation use- cases. For example, conditional generation can augment computational trial cohorts with anatomical variants that are both novel and rare (Kong et al.) 2024). Moreover, generative editing methods, such as inpainting, can precisely modify existing patient geometries to create anatomically plausible variations (Kadry et al. 2024; 2025). These "digital siblings" can be used with biophysical simulators to model counterfactual scenarios that clucidate the relationship between anatomical form and function.
However, unlike generative modeling of 3D shapes for artistic purposes, generating anatomical models for biophysical simulations presents several unique challenges. The first concerns scale- critical features, in which minor geometric variations on the order of millimeters can cause major
fluctuations in physiological behavior (Fabris et al. 2022 Sacco al. 2018 Moore & Dasi 2015).
Massachusetts Institute of Technology
Brigham and Women's Hospital
1 American University in Cairo
8 Ecole Polytechnique Fédérale de Lausanne

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Published as a conference paper at ICLR 2026
Ellipsoid-ben saaeoddddbaaddddde Anatomic Generation
Disentangled Geometric Control
Size Only
Position Only
Shape Only
Multi-part
Compositional Control
F
Substructure
2 Substructures
4 Substructures
Target Ellipsoid
Sample Ellipsoid
We present guidance framework to constrain diffusion models of multi-label anatomical segmentations based on simple geometric features. Such features include size, shape, and position, and can be represented as ellipsoids in 3D space (panel A). Our inference-time approach enables generation based on independent geometric features (panels B-D), and supports multi- component compositional generation (panels E-G). Gray and blue voxels represent components that are unconstrained and constrained, respectively. Purple ellipsoids indicate strong overlap between target and sample ellipsoids
Figure 1:
Second, anatomical structure exerts petribute-specijyy effects, in which geometric features such as size and position play different roles in determining biophysical outcomes (Kadry et al. (021). Third, the geometry of multiple substructures interact compositional manner (Kadry et al. 2021 Bhalodia et al., 2018:) Kong et al. 2024), where simulated outcomes are influenced by the collective arrangement of various substructures. Lastly, to interface with clinicians and device engineers, such generative models should be controllable via primitives that are interpretable and physiologically relevant.
To address these design requirements, we present CardioComposer, an energy-based guidance framework for controlling unconditional diffusion models with geometric attributes regarding size, shape, and position. We visually represent these constraints via interpretable ellipsoidal primitives [ A). Our inference-time framework can independently control individual attributes such as size or position (Figure B-D), and compose geometric constraints for an arbitrary number of anatomical components or substructures (Figure [E-G). Our key insight is that unconditional diffusion models of multi-class anatomy can be constrained in a compositional manner by simple gradients derived from geometric loss functions applied individually to each substructure. We demonstrate this method multi-tissue cardiovascular segmentations that exhibit wide array of substructures such as star-shaper star-shaped chambers and tubular vasculature. Our framework advances the state of the art the following ways:
(Figure
Differentiable Geometry for Anatomical Characterization: We introduce set of differ- entiable geometric measurement functions that compute physiologically relevant anatomical features from substructure label map. We specifically measure voxel-wise geometric moments, computing size via zeroth-order moments, position via first-order moments, and shape via scale-normalized second-order moments.
Inference-time Guidance to Control Substructure Geometry: We demonstrate that simple gradients derived from differenniable geometric loss functions can guide unconditional latent diffusion models of discretized multi-class label maps. This enables independent or joint control of substructure attributes without retraining, where substructures consist of one tissue class or the union of multiple classes.
Complex Compasitional Control: We validate that multiple substructure-specife geometric losses can be composed to enable more complex anatomical constraints. Further, we show that this control extends to non-convex substructures with branching or curved geometry.
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Predict Final Segmentations
Extract Geometric Moments
Compute Geometric Loss
Noised
Predicted Segmentation
Substructure Selection
Moments Mass
Measured g
Target g
Decode D
M
Liion
Centroid
C
Lagos
De(a,;e)
Denolse
Covariance
Larage
Z
X
f2
do SoLase
Aghistupe
Corrected Denoising Step
Da(a,in)
+a an- 1o--V Vigea
Figure 2: Our method involves applying geometric guidance correction step for every denoising iteration. Left: The noised latent 2, passed through the diffusion model and VAE decoder to produce clean voxel space prediction &o (Section 4.2). Middle: The segmentation is parsed for relevant substructures S2, and geometric moments are extracted for each substructure (Section (43).
Right: Mcasured geometric moments G are compared to target moments through geometric moment losses. Bottom: The gradient derived from the aggregate loss corrects the denoising step.
BACKGROUND
Traditional Morphometric Modeling for Anatomy. Morphometry involves quantifying anatomical structure through geometric measurements. Traditional morphometric approaches measure intrinsic features such as length, area, volume, and shape, as well as extrinsic factors such as position and orientation. Geometric measurements enable applications such cardiovascular risk stratification (Asheghan et al. 22023] Mahmod et al. 2024) and orthopedic diagnosis (Gatti et al. 2024). However, traditional morphometric approaches face two key challenges: they cannot represent complex relationships between features, and multiple distinct anatomies may map to the same high-level measurements. To address these limitations, propose framework in which unconditional diffusion model controlled by traditional morphometric features (size, shape, and position) to generate variety of realistic anatomical structures, providing approachable anatomical modeling interface for both clinical and engineering workflows involving numerical simulation.
Generative Models for Numerical Simulation. Anatomical models in the form of 3D meshes or label maps serve as a crucial tool for studying form-function relationships through physical simula- tions, enabling both scientific discovery and medical device design. However, current approaches to create such models must trade off between piweeenfideliyy and control. Simple geometrie primitives, such as cylinders for coronary arteries (Dong et al.) (023) and truncated ellipsoids for cardiac chambers (Aróstica al., 2025) offer parametric control but fail to capture anatomical realism. Data-driven approaches such as autoencoders (Dou et al. 2024 Qiao et al. 2025) represent anatomy in terms of global shape vectors, and can generate synthetic data for mechanistic studies of heart disease (Hermida al. 2024 Williams al. 2022 However, such approaches are limited in their ability to model interpretable geometric attributes. Deformation editing methods (Pham et al. 2023 2024) allow for interpretable control of anatomical geometry, but are limited modifying existing structures. Recently, diffusion-basee approaches such as inpainting and partial diffusion have been used to edit patient-specific anatomy create "digital siblings" (Kadry et al. 2024). However, such edits can induce undesirable morphological bias when applied to rare and pathological cases. To this end, recent studies have imposed anatomical features by explicitly providing scalar conditioning features during training. For example, de Wilde et al. (2025) trained a conditional model on thyroid segmenta- tions, and Kadry et al. (2025) introduced morpho-skeletal conditioning and guidance mechanisms for coronary arteries. However, both approaches rely on conditional training and are restricted to size-related variables such as volume or cross-sectional area. Similarly, Du et al. (0025) presented hierarchical conditional diffusion model for generating aortic centerlines and radial profiles, but restricted to fixed centerline connectivity and cannot flexibly accommodate topological changes such varying branching patterns. In contrast, we propose modular inference-time framework that controls unconditional diffusion models across diverse anatomical structures using geometric attributes such size, shape, and position.
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Spatial Control of Generative Models. Spatial control of generative models is achieved through two principal approaches. The first approach conditions models on interpretable mid-level representations (e.g., bounding boxes, ellipsoid parameters, articulation angles) and has been successfully applied to images (Nie et al. 2024), videos (Feng et al.
2025), 3D objects (Hertz et al. 2022 Koo et al. 2023
Mu et al. 2021). and proteins (Stark al. 2025). ). However, these methods cannot accommodate
novel constraints without retraining. The second approach employs energy-based guidance during the reverse diffusion process (Bansal al., 2023; Du et al., 2023), enabling flexible constraint composition at test time, but is typically limited to general localization rather than exact geometric control. Recent works such as self guffdeui use attention- based loss functions enable basic geometric attribute control (size, position) in text-to-image models (Epsttein et al.] (023). but are not designed for multi-label segmentations, nor they control for orientation or aspect ratio. In our work, we extend merggy-ased guidance by introducing differentiable geometric losses for 3D) multi- componet anatomical voxel maps based on substructure-spcciine geometric properties such as the mass, centroid, and covariance, enabling the composition of multiple constraints across several independent substructures.
3 ANATOMICAL DIFFUSION MODELS
Let E RCCHHxWD be 3D segmentation volume with C' tissue channels and (H, W, D) spatial dimensions. We employ a variational autoencoder (VAE) with an encoder that maps x to a lower-dimensional latent representation
Exx), and decoder D that maps back to a voxel- space reconstruction x = D(z). The latent grid z E Rexhxuxd comprises e channels and spatial dimensions (h, w, d) (H/f. W/f, D/ for integer downsampling factor f-
We use an unconditional latent diffusion model (LDM) as a prior over 3D anatomical segmenta- tions, trained on the encoded latent representations z. We specifically use the elucidated diffusion formulation of Karras et al. (2022). In the forward process, data samples z ~ Paam (z) are pro- gressively corrupted by adding Gaussian noise, resulting in perturbed data Z, z + n where ~ (0, The reverse process reconstructs the original data by approximating the score function Vz,, log o) that controls the reverse diffusion process:
dz, -20 V v, log p(z, ;0) dt + 20 dw
where dw is the Wiener process. This score function Vz, pozo σ)) (Do σ) -z o0² can be expressed via a denoising function Dg parametrized by a 3D U-Net Fe through the following relation:
Delzo o) = Cakip 2, + cour(a) Fa (a) Zoi (o))
where ConiipConttGin.nniie are noise-eevel-dependent scaling coefficients. The neural network is trained by minimizing the clean-data prediction objective L ET,,, (o(z;))) =12 with A(a) balancing loss contributions across noise levels.
GEOMETRIC GUIDANCE
4.1 OVERVIEW
Our objective is guide an unconditional diffusion model that synthesizes anatomical segmentations with geometric constraints for size, position, and shape. These attributes are measured on substructures that correspond discrete tissue labels within the 3D voxel map. To do this, we guide the sampling process with a composite geometric loss applied to a subset of labels. This geometric loss is weighted sum of moment-based terms: size via zeroth-order moments (scalar volumes), position via first-order moments (centroid vectors), and shape via scale-inval invariant second-order moments (normalized covariance matrices). Figure[]illusse illlssraates Jillustaessss four main stages. First, at each sampling step we denoise the latent, decode to voxel-space logits, and apply softmax to obtain class probabilities. Second, we select the desired anatomical substructures Ω and extract the geometric moments = C, representing the mass, centroid, and covariance for each substructure. Third, compute the geometric loss Zgeem with respect to target moments G. Lastly, the gradient of this loss with respect to the noisy latents is used to guide the sampling process.
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4.2 SEGMENTATION DENOISING AND GUIDANCE
We formulate loss-based guidance in terms analogous to diffusion posterior sampling (Chung et al. 2023), where the gradient derived from differentiable geometric loss geom guides the sampling process. To guide anatomical generation, the intermediately noised latent 2, is denoised by the diffusion model to produce clean prediction 2. = (z(zo; a) and subsequently decoded into voxcl-space segmentation & = D (. As the decoder outputs are continuous logits, we apply label-wise softmax to ensure that the segmentation values are between 0 and The geometric loss figomm is then computed in differentiable manner to update the denoiser predictions through the gradient with respect to the noised latent Ze. The update step is parameterized with a guidance weight
as follows:
Dg (Zer: = D2(2nia)
w Va, Egoam
(3)
Guided Denoising
Uncond. Denoising
Geometric Guidance
4.3 GEOMETRIC MOMENT LOSS
To isolate guidance to specific substructures representing individual tissues. we map the input segmentation &o E TR. xx xD to a set of substructure voxel maps Ω E BxHHxxxD Here, E specifies the number of relevant substructures. Substructures are determined either through taking subsets of the tissue channels taking the union of multiple tissue channels.
To extract geometric features, we compute the set of geometric moments G= [M,C, where M E REx1 represents the masses volumes for each substructure, C E REx3 represents the centroids, and S E IR Ex3x3 represents the covariances. Specifically, for each individual substructure index k we define Q4 E (R D) as the flattened substructure voxel grid and p E R as the normalized voxel coordinates between 0 and 1. We compute the geometric moments as
2 M₁
1 p diag(Ra)) C Ck. M M
Mr = 1T. Ss and C₁
and Se =
(4)
where 1 is the all-ones vector, and diag(-) refers to a diagonal matrix embedding. To enable independent control over size and shape characteristics, we compute a normalized representation of the covariance matrix. The ceale-nommalized covariance matrix is defined as S = S2/ tr(A) where is the eigenvalue matrix obtained from the eigendecomposition of Sg. Intuitively, the normalized covariance matrix represents the aspect ratio and orientation of the substructure.
Following the computation of geometric moments, we calculate individual loss terms by comparing each moment to its corresponding target moment G [M, C, S".. For each geometric feature, we compute the mean squared error (MSE) between the measured and target values. These individual loss terms are defined as:
Laize LASE (M, M), Lps = LMSE(C,C), Lsttpee = LS )
Using prescribed weight factors λo Å1 A2, we compute the aggregate geometric loss as Leeom Sazz + A Coos+ + Aagfahapp- The weighted sum of each weight X; allows us to control the contribution of each individual loss to the guidance process, enabling easy disentangled control by zeroing out the associated weighting factor.
5
EXPERIMENTS
5.1
UNCONDITIONAL MODEL TRAINING
For diffusion training, we use the label maps provided in the TotalSegmentator dataset (Wasserthal et al., 2023). We extract heart-related labels, which include aorta (Ao), pulmonary artery (PA). pulmonary veins (PV), inferior vena cava (IVC), superior vena cava (SVC), left atrium (LA), right atrium (RA), left ventricle (LV), right ventricle (RV), and left ventricular myocardium (Myo). We manually filter out low low-quality label maps, resulting in 596 3D cardiac segmentations with 11 channels and isotropic voxel edge length of 2 mm (Section 8.2 We split the dataset into training and validation sets with an 80/20 split. All target moments and evaluation metrics are computed on the validation set.
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Weigttd
Guidance Weghtias
Guidance Whight-1
Conditiona Fidelity Siee (41
Morphological Metrics Fracsion: !T]
Pointcloud Metrics
MMD (4)
50605
Attn :CFG
0.2
0.000
Pcslibn (1)
Recall 11:
Coverage (7)
0.00
0005
0.2
0.3
.DFG
0.22
Stupe (1)
Frachal O Staree 11:
1.4hh 14thAccarady
0.000
0.7
0.00
82
0.5-
0.5
-Oars
0.10
032
25
1.0
0.5
10
Weigf
Megg
Phigh
Tirget Fligaord
Sample Ellpaoid
.OF3
.ANG
Figure Geometric guidanee can generate synthetic anatomy with geometric constraints. Grid shows example synthetic label maps where constraints are applied to the myocardium voxels Rows: baseline conditioning and guidance methods (CFG classifier- free guidance, ANG adaptive null guidance).
Figure 4: Geometric guidance can enforce conditional fidelity while maintaining realism. Line plots compare conditioning and guidance mechanisms based on the geometric properties of myocardium. MMD values are multiplied by 10
We train unconditional diffusion model on cardiac label maps similarly to Kadry al. (2024) (further details in Section 8.3p. To compute our geometric guidance loss, we use the weighted sum of the individual geometric moment losses, where the guidance weights are tuned experimentally (Section
5.2 BASELINES
We compare our approach (unconditional diffusion combined with geometric guidance) to conditional training approaches. Given the target geometric moments representing the size M, centroid C, and covariance S each cardiac substructure, we condition the model in the following ways:
Explicit Concatenation: We directly encode geometric attributes as scalar values in the conditioning signal (Kadry et al. 2025). Here, we adapt this method to positional and shape- based features. We flatten and stack all geometric moments into a 13-dimensional vector for all substructures. We then expand this vector into voxel grid Geep (1xx which is concatenated to the latents along the channel dimension.
Implicit Concatenation: We indirectly encode geometric attributes the conditioning signal through 3D heatmaps (Kadry et al. (025). Here, embed geometric moments 3D Gaussians in voxel space. For each substructure, we create a voxel map Gump E R Exhxwxd where the voxel values encode the Mahalanobis distance.
Cross-attention We express the conditioning signal sequence of tokens where each token represents substructure geometry. The dimension of each token corresponds to the embedded geometric moments Geross I Fx 256 To enable sequence conditioning for the denoising U-Net, we convert the self-attention layers to cross-attention layers, similar to Rombach al. (2022

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We implement guidance mechanisms such as adaptive null guidance (ANG) (Kadry et al. 2025
for explicit concatenation, and classifier-fres guidance (CFG) (Ho & Salimans 2022) for implicit concatenation and cross- aosss-ttention. Further details can be found in Section 8.5
5.3 EVALUATION METRICS
We evaluate pairwise conditional fidelity for size, shape, and position by taking the L1 -o-morm between the target and sample moments, averaging over all relevant substructures. We measure morphological quality metrics by comparing the distribution of real and synthetic anatomy in morphological feature space (Kadry et al., 2024). To embed each label map, consider all 10 tissues substructures and concatenate, over all substructures, the masses, centroids, and eigenvalues of the normalized covariance matrices. We specifically use morphological variants of improved precision and recall, as well as the Fréchet distance (FD) (Kynkäänniemi et al. 2019; Kadry et al. 2024). Lastly, we leverage pointcloud- based metrics to assess 3D shape Yangg et al. 2019), such as minimum matching distance (MMD), coverage (COV), and I-nearest neighbor accuracy (I-NNA). Distances between pointclouds are computed with Earth Mover's Distance (EMD). Further details can found Section 8.6
Samples (Overlald)
2
Maes
Guidance
Guidance
Guidance
Cenooix Masi Only
Eipsood
Ellppotd
Real
Comtined
Cento Only
Staps Only
Figure 5: Geometric guidance enables inde- pendent control of size, shape, and position. Columns show synthetic label maps generated by geometric guidance applied to the right ven- tricle voxels using various geometric losses. Rows represent which geometric feature is being independently controlled.
Figure 6: Geometric guidance enables inde- pendent control of size, shape, and position. Pair plot shows kernel density estimate plots (di- agonals) and pairwise scatterplots (off-diagonals) of morphological metrics. Guidance is applied to control the right ventricle geometry.
5.4 EVALUATING ANATOMICAL GENERATION QUALITY
We first aim to compare and evalu- ate geometric control methods both conditional fidelity and syn- thetic anatomy quality. We sam- ple target moments for single substructure (myocardium) from the validation set and generate 200 anatomical segmentations per method. We sweep over guidance weights w E [0, 2]. In Figure 4 we show that geometric guidance enhances conditional fidelity, espe- cially at higher guidance weights. We observe that our method main- tains generation quality, retaining
Table 1: Comparative analysis of various approaches for multi-
substructures compositional generation. The number of substructures indicates the number of tissues actively constrained during sampling. MMD values multiplied by 10
Morph. Metrics
Pointeloud Metrics
Constraints
Method
FD
(t)
MMD
COV
I-NNA
Implicit
1622
0.00
0.99
55.7
0.288
0.915
Ours
34.6
0.70
0.87
9.40
0.53
0.55
Implicit
227
17.1
0.40
0.79
Ours
38.5
0.60
0.83
9.39
0.52
0.57
Implicit
29,8
0.80
0.81
9.21
0.48
0.58
Ours
32.7
8.60
0.52
Implicit
31.1
8.11
0,50
Ours
35.5
0.80
0.94
8.50
0.58
0.50

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similar levels of morphological and pointcloud evaluation metries with increasing guidance. Figure shows example label maps generated through varying guidance values for methods; only our method and implicit conditioning align the target and sample ellipsoids under guidance. Further information on which features were plotted can be found in Section 8.13
5.5 EVALUATING GEOMETRIC DISENTANGLEMENT
We next show that our guidance framework uniquely enables disentangled control of geometric attributes. We use 100 target moments for myocardial labels from the validation set using no losses (Uncond.), combination of all losses (Cegoom) or each individual moment loss (L-ize: Epsss and L.shppe). Figure 9 shows that each individual loss improves its corresponding conditional-fideln metric while leaving the others approximately unchanged. The main exception is the interaction between shape and mass, where adding guidance weight for the shape Ioss enhances mass fidelity, This phenomenon is likely due to correlations between size and shape in the dataset. Qualitative results can be seen in Figures and where the right ventricle is constrained independently by mass, position, or shape. For example, mass-only guidance produces a narrow peak in the mass marginal while the other morphology metrics remain broad, whereas applying all geometric losses collapses all marginals to narrow peaks at their target values.
Size
Aspect Ratio
Implisit
Ours
Exgenvatve Perus
Uncond.
Component
Expenvatio Ratto
E
Mads
Mass
Ps Mass
Position
Orientation
Comenax CernesX Cornoe
Regle
2 Components
Aogle
Made
Components
Cartrodd
Cantrec
Carrrodd
Angle
Targel Elipsoid
Sample Elipeoid
Real
Substructure
Substructures
3 Substructures
Figure 7: Geometric guidance ex- hibits enhanced multi-part com- positional generation compared to a conditional drop-out baseline, Columns: Baseline vs. method. Rows: Synthetic label maps with a varying number of voxel labels
Figure 8: Our guidance framework enables multi-part compositional generation. Pair plot shows kernel density estimate plots (diagonals) and pairwise scatterplots (off- diagonals) for various morphological metrics. Guidance is applied control the geometry for varying number of substructures.
5.6 EVALUATING MULTI- PART COMPOSITIONALITY
We evaluate the ability of our method to achieve multi-part control under arbitrary constraints. We sample 100 target moment sets from the validation set and constrain generation based on: (a) only the myocardium (I substructure), (b) the right heart labels (3 substructures), and (c) both right and left heart labels (6 substructures). For geometric guidance, we use an unconditional model and select the appropriate substructures Ω during guidance. For baseline, we retrain the best.
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conditional diffusion model (implicit) with 6 substructures using dropout (further details can be found in Section 8.6 Results are shown Table I and Figure 7 which show that with a small number of constrained substructures, implicit conditioning with dropout fails to generate high-quality anatomy as measured by morphological and pointeloud metrics. Because the implicit conditional baseline trained with independent dropout over six ellipsoidal conditioning channels, the fully conditioned case (all channels present) is vastly more frequent than the unconditional case (all channels empty). As result, unconditional sampling corresponds the rarest training configuration and yields degraded anatomical quality in Figure 7 and Table
We further show in Figure that controlling multiple substructures via geometric guidance can effectively sample from lower pweer-dieensioal slices of the original morphological distribution. For instance, when guidance applied around the target value for the right ventricle, while the remaining structures retain broad distributions. When three substructures are guided simultaneously, the corresponding morphological marginals all collapse to narrow peaks their target geometric values. Finally, Figure I0 shows that guidance framework applies to complex, non-star-shaped geometries, including curved and branching substructures, as well Boolean unions involving multiple tissue classes considered single substructure (e.g., both vena cavae or all chambers).
single substructure, the pair plots show a sharp concentration
5.7 GEOMETRIC INPAINTING AND BIOPHYSICAL SIMULATIONS
We demonstrate that our geometric guidance framework can controllably edit patient-specific anatomy for simulation experiments. We consider an example involving biventricular pressurization in which we edit label map to enlarge or shrink the RV. As shown in Figure 11 we define the RV target geometry by doubling or halving the mass measured from the original label map (left column insets). We apply iissue-base inpainting (Kadry et al. 2024) with geometric guidance to edit the RV (left column) and convert the label map tetrahedral mesh (middle column). We simulate biventricular pressurization for the baseline patient and edited variants, showing how RV volume modulates wall displacement (right column). Further details can be found in Section 8.7
Vai Ralelry
Ceemros Ruedly 113
Chalaton Hoiethe
Label Map
Crop-Mesh
Blomechanical Sinulstion
Inpaint Large
Oucula Aegs
Gustance Metrod
Utan Only Cerrrd Only
Shaps
Figure 9: Geometric guidance enables indepen- dent control of size, shape, and position. Line plots compare conditional fidelity for individual losses (mass, position, shape) and losses (full).
Palent
Inpoint
Losses were computed for myocardial tissue.
Small
Pulmonary
Vena Cavas
Chambers & Myocardium
Aorta
Artery
Original Ellipoid
Echted Eliipoid
Figure Il: Geometric guidance can control- lably inpaint anatomical features for coun- terfactual biomechanical simulations. Left column: baseline patient edited to vary right- ventricle size, while maintaining all other sub- structures Middle column: cropped biventricu- mesh from each scenario. Right column: edit- ing vintt-ventrille size while retaining the left ventricle modulates biomechanical outcomes.
Target Ellizooid
Sample Ellppoid
Figure 10: Geometric guidance is compatible with complex substructures. Qualitative results showing geometric control of substructures with non-convex or branched features, as well as sub- structures comprising multiple tissues.

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Unconditional Sample
Cardlocomposer Sample
Sample
Sample
Sample
Sample
Spine
Geometric guidance can control the generation of wide variety anatomical systems. We present label maps that were generated from guided and unguided latent diffusion models. For the aortic dataset, we control the main trunk, for the spinal dataset, we control the sixth, seventh, and eighth thoracic vertebrae, while for the knee dataset, we control the femur.
Figure 12:
5.8 GENERALITY OVER ANATOMICAL SYSTEMS AND STRUCTURES
We aim to show that geometric guidance applies to wide variety of anatomical diffusion models. We construct three datasets of 3D multi-class patient-specific label maps corresponding to the branched ascending aorta, spinal vertebral column, and distal femur. Further details can found in Section 8.2 In Figure 12 we show typical unconditional samples, as well as samples from geometric guidance, which constrains substructure geometry to high degree of fidelity.
LIMITATIONS
Our method has several limitations. First, the relative weights of the geometric moments should be obtained through experimental tuning, similar to all guidance frameworks. However, we found that the same set of loss weightings transfer well entirely different anatomical systems such as the aorta, spinal column, and knee, indicating that only minimal additional tuning is required. Second, substructures are currently defined based on label map class, and cannot represent localized features such as cross sections. Lastly, anatomical diffusion models can generate topologically incorrect substructures, such disconnected aortas or several left atria. making the resulting simulation physies inaccurate. This can be addressed by filtering out topologically incorrect anatomies, at the cost of some wasted computation.
CONCLUSIONS
We present a flexible method to impose geometric constraints diffusion models of 3D multi-class anatomical label maps. By measuring geometric moments relating to size, shape, and position of various substructures during inference, we enable energy- energy-based based guidance without conditional training. show that our framework can independently control geometric attributes such as size, position, shape, and constrain multiple anatomical substructures in compositional manner. We also demonstrate geometric guidance across a wide range of anatomical systems and structures, spanning cardiac, vaseular. and skeletal systems. Our framework enables custom-tailorie realistic anatomy for computational simulation experiments, elucidating the causal relationships between form and simulated function.
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8 APPENDIX
8.1
OVERVIEW
In Section
8.2
provide details dataset curation and processing.
In Section
we provide implementation details for our autoencoder and diffusion model.
In Section 8.4
provide implementation details for our guidance algorithm.
In Section
we provide implementation details for our conditional generation baselines.
In Section
8.6
we provide further experimental details for evaluation and inference.
In Section
8.
we provide implementation details for our biomechanical simulations.
In Section
8.8
we provide a dataset scaling analysis for our latent diffusion model.
In Section
8.9
we provide qualitative results for procedurally generated ellipsoid dataset.
In Section 8. 10 we provide an autoencoder reconstruction error analysis.
In Section 8. we provide scale-factor sweep analysis for target modification.
In Section 8. we provide quantitative results demonstrating disentangled generation for
alternative geometric features derived from the second-orde moment.
we present additional morphological distribution plots that examine the effect guidance weight as well as the choice of control technique.
In Section 8. 13
8.2 DATASETS
For study, construct four separate datasets of anatomical segmentations to qualitatively demonstrate the flexibility of geometric guidance. These datasets represent 1) whole-heart cardiac segmentations with great vessels, 2) the branched ascending aorta, 3) multi-vertebral spinal column, and 4) the femoral condyle and articular cartilage. We primarily use the cardiac dataset for our experiments.
For the cardiac dataset, we utilize TotalSegmentator v2 (Wasserthal et al. 2023). with 596 cases manually selected based on segmentation quality assessment. Cardiac structures including the myocardium (Myo), left and right atria (LA & RA), left and right ventricles (LV & RV), aorta (Ao), and pulmonary artery (PA) were segmented using a specialized TotalSegmentator model trained on sub-millimeter resolution data. For the inferior vena cava (IVC), superior vena cava (SVC), and pulmonary veins (PV), we retain the labels from the original dataset. To ensure anatomical validity, we perform topological filtration all structures except the pulmonary veins, where filtration involves extracting only the largest connected component. The resulting segmentations are standardized by resampling uniform voxel resolution of 2mm and subsequently cropped fixed range. The crop center is determined from the union of all four chamber segmentations, and the crop size is 128² voxels.
For the aorta dataset, we extract labels directly from the original TotalSegmentator v2 (Wasserthal al. 2026) segmentations, without applying specialized model, resulting in 450 3D segmentations manually selected based on segmentation quality assessment. The labels include the main aortic trunk and the ascending branches, which comprise the brachiocephalic trunk (BCT), left common carotid artery (LCCA), right common carotid artery (RCCA), left subclavian artery (LSCA), and right subclavian artery (RSCA), for total of channels per segmentation. All segmentations are resampled to an isotropic voxel size of mm and cropped to a spatial size of 1283 using a crop center determined from the center all combined tissues.
For the spinal dataset, we utilize the CTSpine K dataset (Deng et al. 2021 and extract all vertebral body segmentations, resulting in 784 3D segmentations. The segmentations include 7 cervical vertebrae (C1-C7), 12 thoracic vertebrac (TI-TI2), and 5 lumbar vertebrae (L1-L5), for total of 25 channels per segmentation. To ensure spatial consistency and anatomical completeness, all segmentations first resampled an isotropic voxel spacing of 1 mm. The center of the crop box determined from the union (voxelwise sum) of all vertebral structures in each scan, and a fixed crop of 1283 voxels is applied for each patient.
For the knee dataset, we utilize the ShapeMedKnee dataset (Gatti et al. 2024) and extract 2000 3D segmentations of the left knee. The segmentations include the femur (Fe) and articular cartilage (Ca),
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resulting in channels per segmentation. To ensure spatial consistency and anatomical completeness, all segmentations are first resampled to an isotropic voxel spacing mm. A fixed crop size of 1283 voxels is applied for each patient.
8.3 LATENT DIFFUSION MODEL IMPLEMENTATION
For this study, we adapt the VAE and LDM architectures specified by Kadry al. (2024). The VAE input and output channel counts are set to 11, corresponding to 10 distinct cardiac labels along with an additional channel for the background. The number of input channels for the LDM is set 3 for unconditional sampling. The hyperparameters and training configuration for the VAE and LDM are listed in Table 2 and Table respecillyyy
Table 2: Autoencoder hyperparameters
Hyperparameter
Value
x 10
Epochs Batch Size
40
1
Num. Channels
[16,32,64]
Num. Res. Blocks
2
Downscaling Factor
4
Recon. Loss Weight
1
KI. Weight
10-6
Table 3: Diffusion model hyperparameters
Hyperparameter
Value
Training
Ir Epochs
2.5 x 10-5
50
Batch Size
I
Num. Channels
[64, 128, 196]
Num. Res. Blocks
2
Num. Attn Heads
1
Attn. Res.
8, 4, 2
data
1
p(a) mean
1
p(a) std
1.2
Sampling
x 10
O min
80
P
3
8.4 GEOMETRIC GUIDANCE IMPLEMENTATION
8.4.1 GEOMETRIC MOMENT COMPUTATION
To ensure that the extracted components yield interpretable moments, we require the voxel grid values to be softly binarized, with one tissue channel approaching while the others are close to 0. To achieve this, we apply softmax function with a temperature 1. During the computation of geometric moments, we observed that segmentations that are empty nearly empty, particularly those with small components, lead to unstable gradients that significantly degrade the quality of generation. This instability arises because the centroid and covariance loss calculations utilize mass in the denominator. To mitigate this issue, we introduce a small amount of noise to the mass term whenever it appears in the denominator, thereby stabilizing the overall process. After computing all moments, we normalize the mass term by the total number of voxels N HW such that the term represents volume fraction. Unless stated otherwise, use 50 denoising steps.
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8.4.2 GUIDANCE WEIGHT TUNING
We determine the weight factors À [Ao A1. A2] for our geometric loss through tuning each loss in isolation. We tune for conditional fidelity while retaining reasonable generation quality metrics. The final weight values can be seen in Table
Table 4: Geometric moment losses and their corresponding weight factors.
Guidance Loss
Weight Factor A
Sii
10?
Cpos
105
Shhppe
10
8.5 BASELINE METHODS IMPLEMENTATION
Explicit Conditioning: To ensure that the elements of Geep are roughly between 0 and I, we min-max normalize the masses M, centroids C, and normalized covariances S,, with values calculated from the real dataset (Table[5. The LDM input channel count is increased to accommodate the concatenated input. This method does not readily permit the use of dropout to train diffuusin model in an unconditional manner because the null condition is defined as zero equivaleenn to the minimum moment values. We include explicit conditioning results for guidance weights smaller than 0 in Figure for completeness.
Cross-Attention Conditioning: Our initial tokens consist of 13-dimensional vectors rep- resenting the concatenation of mass M, centroids C, and normalized covariances S,,. The tokens are then min-max normalized similar to explicit conditioning and embedded into a 256 dimensional vector for cross-attention. To embed the component index, we use linear embedding layer. To embed the geometric moments, we use an MLP with three linear layers and apply a ReLU operation after the first and second layers. Both embeddings are added together and used to condition the U-Net with cross-attention, where we use 8 attention heads. To enable unconditional generation, we randomly drop each channel of Geross with a probability of 0.1.
To compute the ellipsoidal distance map, we use the centroids C. and non-normalized covariances & for each component to compute the Mahalanobis distance (De Maesschalck et al. 2000) for each voxel position. We then apply a shifted sigmoid transform with slope of -0.5 and a bias of to constrain the outputs between 0 and 1, and subsequently concatenate the resulting grid to the latents. To enable unconditional generation, we randomly drop each channel of Gimp with a probability of 0. One limitation of this approach is that the target mass can only be targeted indirectly through the non- normalized covariance term, which can be seen in the conditional fidelity plot for size in Figure 4
Implicit Conditioning:
Table 5: Normalizing constants for geometric moments during explicit and cross-attention based conditioning.
Geometric Moment
Normalizing Minimum
Normalizing Maximum
M
3.19 10-3
1.3 1
C
0
1
S
-1 x 10
x 10-2
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8.6 ADDITIONAL EXPERIMENTAL DETAILS
Morphological evaluation metries:
To compute the morphological metrics, the features are normalized by the mean and standard deviation of the real data. To calculate precision and recall, we use 5 neighbors.
Pointcloud evaluation metrics To compute the point cloud metrics, we calculate MMD, COV, and NNA for every tissue label using 256 points sampled using farthest point sampling. The metrics are then averaged over the number components. To compute the pointeloud distances, we approximate Earth Mover's Distance (EMD) through the Sinkhorn divergence (Feydy et al. 2019).
Disentangled Generation: Disentangled generation is done by zeroing out the inactive loss weights, Exact configuration details are shown in Table 6 We use 50 denoising steps for all generated samples.
Compositional Generation: Our compositional generation experiments vary the number of constrained substructures. The exact labels used for each experiment are detailed in Table We use 100 denoising steps for all generated samples.
Table 6: Configuration details for the disentangled generation ablation study. Checkmarks / indicate the associated weight factor A is active while x indicates the weighting factor is zeroed out.
Guidance Loss
A1
X
Uncond.
X
X
X
Czize
ppps
X
V
X
Lihhpe
x
x
r
Lgoom
Table 7: Configuration details for the compositional generation study.
Substructures
Labels
None
RV
RV. RA RV, RA, PA RV, RA PA, LV, LA, Ao
8.7 BIOMECHANICAL SIMULATION DETAILS
Biventricular Cropping: As only myocardial tissue is available for the left ventricle, we approximate an RV myocardial wall by dilating the RV cavity mask constant thickness of 4 mm (2 voxels) corresponding to the clinical literature (Ho & Nihoyannopoulos (006).. crop the left and right ventricles at the base of the heart, we define vector from the LV centroid to the LA centroid, and crop the ventricles by adjusting the position threshold along the defined direction.
Tetrahedral Meshing and Processing: The segmentation is then converted into surface mesh using marching cubes, with a voxel size of mm. Tetrahedral mesh generation is performed using the open-source software Gmsh and MeshLab. The three anatomical models, large RV, baseline patient, and small RV (see Figure are discretized into 39,780, 42,768, and 47,347 linear tetrahedral elements, respectively, with an average edge length of 2 mm.
Pressurization Simulation: An in-house finite element method (FEM) solver, implemented in Fortran with MP MPI, is used for the simulations. The solver based the variational multiscale method, providing stabilized FEM formulations (Goraya et al. 2024 Kang et al.
2022). Simulation results visualized using the open-source package ParaView.
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The myocardium is modeled as standard neo-Hookean material with Young's modulus of 25 kPa and a Poisson's ratio of 0.4. Physiological pressure loads of 12 mmHg and 6 mmHg were applied to the LV and RV endocardium, respectively, corresponding to normal diastolic blood pressure. To constrain rigid body motion, zero-displacement Dirichlet boundary conditions are imposed the base of the heart, while stress-free Neumann boundary condition is applied on the pericardium.
The nonlinear dentradefoonttinn elasticity problem is then solved using the New- ton-Raphson (NR) method. A direct solver (MUMPS) is employed to solve the discretized algebraic system at each NR iteration, with convergence tolerance set 10-20 20 for the initial residual. Simulations are carried out on a cluster using 128 processors.
8.8 DATASET SCALING ANALYSIS
We aim to understand the effect of dataset size on both generation quality and conditional fidelity under guidance. To this end, train four additional autoencoder-diffusion model pairs different split sizes, using 20%, 40%, 60%, and 80% of the original training set, while keeping the validation set fixed across all models. As shown in Figure 13 generation-quality metrics improve as the training dataset size increases up to 40%. In contrast, conditional fidelity under geometric guidance remains approximately invariant across dataset sizes.
8.9 PARAMETRIC ELLIPSOID DATASET ANALYSIS
To further characterize our geometric guidance procedure isolation from complex anatomical variation, we construct a toy dataset of 3D two-channel ellipsoidal label maps with varying sizes, shapes, and positions. To generate each voxel map, we sample the ellipsoidal radii uniformly from to 0.5, where 1 corresponds to the full length of the voxel map. We additionally sample Fuler angles uniformly from 0 to 2x, and choose the centroid to anywhere within the voxel map such that the ellipsoid is not cropped by the voxel boundaries. We generate 800 training and 200 validation label maps.
We then train latent diffusion model with double the number of base channels to accommodate the large geometric variation in the dataset. We apply our geometric guidance method with centroid and covariance loss weights multiplied by a factor of 10. As shown in Figure [1]] our geometric guidance framework can enforce precise geometric constraints on parametric ellipsoid geometries in disentangled manner.
8.10 AUTOENCODER RECONSTRUCTION FIDELITY ANALYSIS
We aim to determine whether the conditional fidelity metrics and topological quality lower- bounded by the VAE reconstruction error. We first auto-encode 24 seed label maps and measure conditional fidelity for size, position, and shape, well as the Betti error for each anatomical structure. We then sample 24 label maps over 50 diffusion steps with and without right-ventricular geometric guidance. As summarized in Table geometric guidance substantially improves conditional fidelity relative to unconditional sampling, while the resulting errors for position and shape remain above the VAE reconstruction error.
In terms of topology, we quantify quality using the Betti error, defined as the number of extra connected components relative the expected topology (e.g., if the aorta is expected be a single connected component but two are measured, the Betti error is 1). We observe that the VAE introduces only a small number of topological defects, whereas the unconditional diffusion model produces more frequent errors, especially for the aorta (Ao) and pulmonary artery (PA) labels. Finally, we find that geometric guidance can further increase the Betti error, particularly for the PA and inferior vena cava (IVC) labels.
8.11 EDITING SCALE FACTOR ANALYSIS
We investigate how far the target right- ventrieen mass can be scaled while still producing plausible samples. For each editing factor (0.1, 0.5, 1.0, 2.0, 4.0}. we take 64 seed label maps, compute the RV mass, multiply it by the editing factor, and use the scaled mass the conditioning target
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Conditional Fidelity
Morphological Metrics Precision (4)
Pointcloud Metrics
Size 4
MMD ( (4)
0.020
0.8
0.6
0.096
0.010
0.088
0.005
0.2
0.080
Position ( (4)
Recall ( (4)
Coverage (f)
1.0
1.0
0.20
0.16
0.6
0.6
0.4
0.4
0.08
0.2
0.2
0.04
0.0
0.0
Shape ( )
FMD ( (L)
1-NN Accuracy
210
0.5
0.8
180
0.4
0.6
0.4
120
0.2
0.2
90
0.1
20
40
60
80
100
20
40
60
80
100
20
40
60
80
100
Training Set Split %
Training Set Split %
Training Set Split %
Guidance
Figure 13: Conditional fidelity is invariant training set size, while generation quality metrics benefit from training set size up to limit. Line plots show conditional fidelity and generation quality for latent diffusion models trained on different-sized datasets. this plot, the right ventricle constrained.
for mass-only geometric guidance. As summarized in Table decreasing the target mass (factors 1) yields samples whose size error and distributional metrics remain close the unedited (factor = 1): size еттог increases moderately, and FMD and 1-NNA remain within the same order of magnitude as the baseline. In contrast, increasing the target mass beyond a factor
leads to clear degradation: at factor of 4, both the size error and FMD increase by more than one order of magnitude, and 1-NNA worsens, indicating that strong mass upscaling produces distorted label maps.
8.12 GEOMETRIC GUIDANCE WITH ALTERNATIVE MOMENT- SANEENT-FEATUES
In our main study, we demonstrated guidance by targeting the normalized second moment to control shape and orientation independently from size. We aim in this section to preliminarily demonstrate
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Sample
Sample
Sample
Sample
Sample
Sample
Uncond.
Mass Guidance
Centroid Guldance
Shape
Guidance
Combined Guidance
G
Figure 14: Geometric guidance can control unconditional diffusion models of procedurally generated ellipsoids. We generate ellipsoidal label maps with varying loss-function combinations to achieve disentangled control.
Table 8: Conditional fidelity and Betti error rates for reconstructed or synthetic label maps. Betti error computed the mean number of connected components minus 1. Values for size, position, and shape fidelity were multiplied by le5, le4, 1e4 respectively.
Cond. Metrics
Connected Component Betti Error
Method
Size
Pos.
Shape
Ao
PA
IVC
SVC
LA
RA
LV
RV
Myo
VAE Recon.
4.74
2.51
21.53
0.125
0.0
0.125
0.042
0.042
0.083
0.0
0.083
0.0
Unconditional
106.03
168.75
512.42
1.208
0.583
0.125
0.0
0.083
0.083
0.0
0.0
0.0
Guided
9.34
25.61
81.77
14458
1.012
0.417
0.042
0.333
0.042
0.042
that achieve fine-grained disentangled control of second-moment derived attributes such as extent, stretch, and orientation. We first decompose the covariance matrix as follows:
S -UA"UT
(6)
where define the extent R the trace of the eigenvalue matrix A R3x3 and normalize by v to obtain the anisotropic stretch A" A/v. Finally, orientation is represented by the eigenvectors U 3x3 derived from the decomposition.
We then define three new geometric losses, which consist of MSE losses for extent and stretch, as well as dot product loss for orientation.
Lextem LMSE 7), Latretch LASE (A" N²), Lurien Las U).
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Table We generate label maps with mass-only geometric guidance applied to the right ventricle and artificially changing the target mass derived from the seed label map.
Cond. Metrics
Morph. Metrics
Pointcloud Metrics
Editing Factor
Size
FMD (L)
Pr. (f)
Re. (1)
MMD ())
COV (t)
I-NNA
0.1
84.06
126.0
0.14
0.34
13.51
0.295
0.826
0.5
19.95
59.6
0.55
0.50
10.51
0.406
0.713
1.0
12.60
40.3
0.59
0.73
9.26
0.520
0.567
2.0
18.55
167.5
0.16
0.84
11.08
0.430
0.745
4.0
732.10
1690.0
0.00
0.97
26.81
0.273
0.926
The dot product loss Ldot is computed as the mean misalignment between corresponding cigenvectors from U and U,
Lavi U, Ú)
5 (1 i url?))
(8)
1=1
where u, and u, denote the i-th columns of U and U, respectively, and the absolute value enforces sign-invariance of eigenvector alignment.
With these losses, we conduct a disentangled generation experiment where we sample 32 label maps for each loss ablation setting, with the loss weightings detailed in Table TO] Conditional fidelity for extent and stretch is quantified using the mean absolute error, while conditional fidelity for orientation quantified using the dot-product loss directly. As shown in Table IT geometric guidance based second-order derived features can be applied in disentangled manner. For example, orientation- guidance achieves smaller orientation error while maintaining extent and stretch fidelity comparable to unconditional sampling.
Table 10: Second order moment losses and their corresponding weight factors.
Guidance Loss
Weight Factor X
Lextent
105
Satech
10
Cerient
102
Table 11: We enable disentangled control over geometric features derived from decomposing the second moment into extent, stretch, and orientation. Conditional fidelity metrics for extent and stretch, as well as MMD values were multiplied by le3.
Cond. Metries
Morph. Metries
Pointcloud Metries
Method
Extent
Stretch
Orient.
FMD (L)
Pr. (*)
(1)
MMD (L)
COV (t)
I-NNA
None
2.14
42,30
0.21
67.63
0.44
0.80
10.22
0.484
0.563
Extent Only
1.42
41.85
0.20
61.99
0.50
0.78
10.12
0.459
0.566
Stretch Only
2.26
3.60
0.23
60.53
0.53
0.80
10.08
0.491
0.559
Orient Only
1.94
41.29
0.0064
60.81
0.59
0.77
10.08
0.525
0.564
8.13 MORPHOLOGICALL ANALYSIS
We represent size the mass of each substructure. Position is represented by the centroid coordinate. To characterize shape, we extract the largest eigenvalue and its associated eigenvector from the covariance matrix. Orientation is represented by the polar angle of the principal axis (in spherical coordinates), while elongation is defined as the ratio between the largest eigenvalue and the second-largest eigenvalue.
Additional morphological plots are presented below. Figure 16 we show that geometric guidance better aligns the distribution of geometric features when comparing real and synthetic anatomies.
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Figure 13 shows that all geometric-contro methods can recapitulate the morphological distribution exhibited by the real data.
0.015
Myo Mass
0.010
0.005
0.425
Myo Centroid X
0.400
0.375
0.350
0.325
1.000
Myo Eigenvalue
0.900
Radio
0.800
0.600
Myo Spherical
80.00
60.00
Angle
40.00
0.000
0.005
0.010
0.015
0.30
0.35
0.40
0.45
0.6
0.8
1.0
50
100
Myo Mass
Myo Centroid x
Myo Eigenvalue Ratio
Myo Spherical Angle
Real
CrossAttn
Implicit
Explicit
Ours
Figure 15: Geometric guidance can help recapitulate morphological distributions. Pair plot shows kernel density estimate plots (diagonals) and pairwise scatterplots (off-diagonals) for various morphological metries. We plot metries for anatomies generated through conditional baselines and geometric guidance (ours), In this plot, the myocardium is being constrained.
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Published as a conference paper at ICLR 2026
Aspect Ratio
Size
0.015
0.9
LV Eigenval Ratio
0.0125
0.8
0.01
0.7
LV
0.0075
0.005
0.5
0.0025
0.0
RV Eigenval
Ratio 1
0.4
RV
0.005
0.2
0.0075
0.5
0.01
0.015
LV Eigenval Ratio
RV Eigenval
LV Mass
RV Mass
Ratio
Orientation
Position
0.44
LV Centroid
0.42
LV Spherical Angle
60
0.38
40
20
0.32
RV Spherical Angle
RV Centroid
0.55
0.525
0.475
60
0.45
50
100
LV Spherical Angle
RV Spherical Angle
LV Centroid X
RV Centroid
Real
Guidance Weight-0.00
Guidance Weight= 0.50
Guidance Weight=1.00
Guidance Weight-1.50
Guidance Weight=
Figure 16: Geometric guidance improves morphological distribution similarity between real and synthetic anatomy. Pair plot shows morphological relationships for mass (top left panel), centroid (bottom left panel), normalized axis lengths (top right panel), and orientation (bottom right panel), where the myocardium labels are being constrained. Diagonal plots show kernel density estimates (LV RV), off-diagonal plots show pairwise scatterplots.
24


Reviews:
[{"review_id": "AiLx7GHAhb", "reviewer": "Reviewer_fX4P", "summary": "CardioComposer adds a geometric control mechanism to an unconditional diffusion model at inference time, using differentiable moments—volume (0th), centroid (1st), and scale-normalized covariance (2nd)—to control the size, position, and shape of cardiac parts.\nNo retraining is required: each denoising step decodes a segmentation, measures geometry, computes a loss to target values, and backpropagates its gradient to adjust the latent, enabling disentangled and composable control.\nTrained on 596 multi-label cardiac segmentations, the method achieves stable single- and multi-part control, geometric inpainting, and biomechanical simulation compatibility.", "questions": "1. The generation quality appears to depend on the amount of training data available. How large a dataset is required for the diffusion model to achieve stable and realistic shape synthesis?\n\n2. Apart from the 0th- and 1st-order moments, are there other geometric features that could serve as control signals?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. **Clinical Significance**\nThe paper addresses a clinically relevant issue—how to create anatomically realistic yet controllable 3D cardiac models. \n\n2. **Flexible Post-training Optimization**\nThe proposed inference-time optimization framework is elegant and practical. By introducing differentiable geometric moments as guidance losses, the method allows users to customize geometric targets on demand without retraining or modifying network weights. This design provides an interpretable and lightweight mechanism for controllable shape generation.\n\n3. **Comprehensive Experimental Validation**\nThe experiments are thorough and convincing. They demonstrate geometric controllability across multiple substructures, evaluate both single- and multi-part guidance, and further validate the realism of generated shapes using distributional metrics and biomechanical simulations. Together, these results strongly support the effectiveness and robustness of the proposed approach.\n\n4. **Clarity and Completeness of Presentation**\nThe paper is clearly written and technically detailed.", "weaknesses": "1. **Lack of comparison with existing conditional shape generation methods.**\n\nWhile the paper clearly articulates the benefits of inference-time guidance, it does not compare against established conditional generative models trained for geometric control (e.g., de Wilde et al., 2025 (https://arxiv.org/abs/2504.03313); Kadry et al., 2025 (https://www.nature.com/articles/s41746-024-01332-0), A-SDF (https://arxiv.org/abs/2104.07645). Including such baselines would better quantify the extent to which performance or controllability is improved by the proposed guidance relative to conditional training.\n\n2. **Absence of simulated validation data.**\n\nAlthough the authors compute geometric attributes (mass, centroid, covariance) from real cardiac segmentations, the evaluation relies solely on real anatomies. A complementary experiment on simulated or procedurally generated geometries—with known ground-truth attributes—would allow denser and more diverse target sampling, providing clearer evidence of control accuracy, disentanglement, and robustness.\n\n3. **Limited explanation for Figures 6 and 8.**\n\nThe pair-plot visualizations are intriguing but under-explained: it remains unclear which visual trends (e.g., sharper peaks, denser clusters) indicate successful guidance.\n\n\n4. **Code is not available**", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b5QQmDk2yZ", "reviewer": "Reviewer_aon8", "summary": "This paper introduces \"CardioComposer,\" a novel inference-time guidance framework designed to control unconditional diffusion models for the generation of 3D multi-class anatomical segmentations. The primary problem it addresses is the trade-off between geometric controllability and realism in generative models for cardiovascular anatomy, which is a critical component for clinical research and in silico medical device evaluation.\n\nThe core contribution is the development of differentiable measurement functions based on voxel-wise geometric moments. These functions compute interpretable geometric attributes—specifically size (from zeroth-order moments), position (from first-order moments), and shape (from scale-normalized second-order moments)—from the generated label maps. The gradients from a loss function comparing these measured moments to target moments are then used to guide the diffusion sampling process at inference time.", "questions": "1) How sensitive is the method to VAE reconstruction error?\n\n2) Can you control orientation independently of shape?\n\n3) What happens under extreme geometric targets (e.g., 10× mass)?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1) The core contribution—a flexible, inference-time guidance mechanism based on differentiable geometric moments for 3D multi-class segmentations—is somewhat novel. This method directly tackles a significant bottleneck in the use of generative models for scientific simulations, where interpretable control over specific geometric properties is paramount. The ability to generate \"digital siblings\" for counterfactual simulations is interesting.\n\n2) The paper is well-structured and clearly written. Figures are informative (outstanding), though some are dense. The related work section is comprehensive. The writing is precise.\n\n3) Extensive evaluation: 200 samples per method, 5+ baselines, 3 metric families (morphological, pointcloud, conditional fidelity). Strong ablations: disentanglement, compositional control, guidance weight tuning. Real-world validation: Used to edit patient anatomy for biophysical simulation (RV volume vs. wall displacement), showing causal form-function modeling.\n\n4) The method is technically sound, with a clear formulation of the geometric moment losses (size, position, and scale-normalized shape). The experimental validation is rigorous.", "weaknesses": "The authors are commendable for their transparent limitations section. Some weaknesses:\n\n1) The framework relies on weights $\\lambda_i$ for the aggregate loss and a guidance weight $w$. The paper notes $\\lambda_i$ require \"experimental tuning\" and provides the final values in the appendix37. However, the sensitivity to these weights is not explored. How robust is the method to changes in these hyperparameters?\n\n2) The authors rightly admit that the underlying diffusion model can generate topologically incorrect anatomies, which is a critical failure for biophysical simulations. The proposed solution is post-hoc filtering. It is unclear if the proposed geometric guidance affects the rate of these topological errors\n\n3) While framed as general, all experiments are on cardiac data.\n\n4) In Figure 4, while the proposed method  generally maintains high-quality metrics, there appears to be a slight degradation in realism at high guidance weights ($w > 1.5$)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lx6jrlqxFx", "reviewer": "Reviewer_WPU8", "summary": "The paper proposes CardioComposer, a 3D method enabling differentiable, geometry-aware control of anatomical diffusion models. The main contribution is to introduce geometric moment-based losses that are differentiable, which can guide sampling to produce anatomically realistic and geometrically controllable multi-class label maps of the heart.", "questions": "Will the code be released to the public?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. Using geometric moments as differentiable constraints for guiding diffusion models of anatomy is novel.\n2. The paper presents clear mathematical formulation of 0th, 1st and 2nd order geometric moments.\n3. The paper includes baselines such as explicit and implicit concatenation, cross-attention.\n4. The paper evaluates different combinations of proposed loss as the ablation study.", "weaknesses": "1. The experiment is focusing on cardiac structures with limited samples (less than 600 in total). The generalizability and scalability remain concerns.\n2. Are the unconditional latent diffusion models (3D VAE and 3D UNet) trained from scratch?\n3. In the Figure 7, for unconditional generation, why does the implicit fail but the proposed one seems to be able to generate a much more realistic map? Further explanation or more visualization would strengthen this observation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "tnhNphuDWs", "reviewer": "Reviewer_gDx6", "summary": "This paper presents a novel framework for imposing geometric constraints models without model retraining. This is achieved by introducing a differentiable geometric loss during inference. The loss aims to match the geometric moments (size, position, and shape) between the predictions and targets. The framework was evaluated on the Totalsegmentator dataset and compared to several baseline methods that integrate the geometric information either explicitly or implicitly.", "questions": "Good paper with clear practical relevance and a well-motivated clinical application. While it does not introduce major technical innovations, it effectively addresses a real-world problem and demonstrates solid empirical value in the medical field.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The paper presents a novel framework for anatomical structure generation that enables flexible control over geometric attributes during inference without requiring retraining, a notable advantage over conventional conditional generative approaches.\n2. The proposed geometric loss formulation is both interpretable and modular, leveraging moment-based constraints to disentangle control over size, shape, and position. The method further supports compositional multi-part control and demonstrates utility in simulation-based downstream tasks.\n3. The authors have shown model's robustness against hyperparameters and non-convex shapes.", "weaknesses": "The method introduces a novel formulation in Eq. (4) to define and normalize geometric moments, offering a clear way to separate size, shape, and orientation characteristics. However, the overall idea of combining a Latent Diffusion Model with test-time constraints has been explored in prior works, so the main novelty lies in the specific geometric moment formulation rather than the general framework.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)