SHARC: Reference point driven Spherical Harmonic Representation for Complex Shapes

Panagiotis Sapoutzoglou; George Terzakis; Maria Pateraki

cs.CVApr 2, 2026

SHARC: Reference point driven Spherical Harmonic Representation for Complex Shapes

Authors: Panagiotis Sapoutzoglou, George Terzakis, Maria Pateraki

Organizations: National Technical University of Athens, Athens, Greece

Abstract

We propose SHARC, a novel framework that synthesizes arbitrary, genus-agnostic shapes by means of a collection of Spherical Harmonic (SH) representations of distance fields. These distance fields are anchored at optimally placed reference points in the interior volume of the surface in a way that maximizes learning of the finer details of the surface. To achieve this, we employ a cost function that jointly maximizes sparsity and centrality in terms of positioning, as well as visibility of the surface from their location. For each selected reference point, we sample the visible distance field to the surface geometry via ray-casting and compute the SH coefficients using the Fast Spherical Harmonic Transform (FSHT). To enhance geometric fidelity, we apply a configurable low-pass filter to the coefficients and refine the output using a local consistency constraint based on proximity. Evaluation of SHARC against state-of-the-art methods demonstrates that the proposed method outperforms existing approaches in both reconstruction accuracy and time efficiency without sacrificing model parsimony. The source code is available at https://github.com/POSE-Lab/SHARC.

Figures & tables

Figure 1: Overview of the SHARC pipeline. From a dense pool of interior candidates, we select a sparse set of reference points that maximize surface visibility. For each point, the local radial distance field is encoded into Spherical Harmonic (SH) coefficients. To reconstruct the shape, we decode these fields and recover the surface points, using a local consistency constraint to filter noisy reconstructed points.

Method	PSB					Stanford					Thingi10k
Method	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$
Raw MA	30k	0.56	2.43	2.25	2.46	83.7k	0.61	3.26	2.26	3.27	53.3k	2.23	7.34	6.84	7.42
CoverageAxis++	310	0.62	2.86	2.20	2.87	510	0.74	3.30	2.46	3.37	654 ∗	1.59	5.53	5.26	5.83
MASH	200	0.31	1.02	1.53	1.64	400	0.38	1.39	1.33	1.49	400	0.52	1.77	2.74	2.99
Ours	46	0.28	0.73	0.62	0.74	77	0.33	1.07	0.89	1.08	118	0.39	1.72	1.28	1.78

Table 1: Quantitative comparison on the PSB , Stanford , and Thingi10k datasets. We report the primitive count

|C|

, Chamfer Distance (

d_{CD}

), and Hausdorff Distance components

\overrightarrow{\epsilon}

\overleftarrow{\epsilon}

\overleftrightarrow{\epsilon}

. All metrics are expressed as a percentage of the object’s diameter. Best results are highlighted in bold . Detailed per-object results are provided in the Suppl. Sec. D.

Figure 2: Qualitative comparison of reconstruction quality. We compare SHARC against CoverageAxis++ [ 38 ] and MASH [ 21 ] , demonstrating our method’s ability to preserve fine geometric detail using significantly fewer geometric primitives.

Category	Raw MA					CoverageAxis++					MASH					Ours
Category	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\|C\|$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$
Airplane	82k	0.30	0.90	0.58	0.91	199	0.49	2.79	1.55	2.80	200	0.27	1.21	1.47	1.63	46	0.23	0.92	0.53	0.97
Chair	83k	0.44	1.11	0.98	1.15	200	0.72	3.49	2.34	3.52	200	0.46	1.47	2.67	2.72	99	0.35	0.83	0.89	1.02
Lamp	71k	0.32	2.20	0.65	2.22	167	0.54	3.79	1.63	3.87	200	0.34	1.51	2.14	2.33	59	0.28	2.37	0.75	2.60
Sofa	85k	0.44	1.65	0.96	1.68	200	0.79	4.41	2.61	4.41	200	0.43	1.63	1.96	2.16	116	0.36	0.86	1.06	1.19

Table 2: Quantitative comparison across four ShapeNet Categories (Airplane, Chair, Lamp, Sofa). We evaluate 100 models per category.

Method	PCB					Stanford					Thingi10k
Method	$\rho$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\rho$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$	$\rho$	$d_{CD}$	$\overrightarrow{\epsilon}$	$\overleftarrow{\epsilon}$	$\overleftrightarrow{\epsilon}$
Ours( $L$ =32)	1.94	0.30	0.98	0.97	1.21	153.82	0.36	1.44	1.84	1.96	9.91	0.42	2.14	1.86	2.54
Ours( $L$ =64)	0.51	0.28	0.73	0.62	0.74	40.65	0.33	1.07	0.89	1.08	2.64	0.39	1.72	1.28	1.78
Ours( $L$ =128)	0.13	0.27	0.59	0.57	0.61	9.85	0.32	0.83	0.78	0.87	0.66	0.38	1.37	1.17	1.43
Raw Ma	$10^{-3}$	0.56	2.43	2.25	2.46	0.08	0.61	3.26	2.26	3.27	0.01	2.23	7.34	6.84	7.42
CoverageAxis++	0.17	0.62	2.86	2.20	2.87	32.99	0.74	3.30	2.46	3.37	2.09	1.59	5.53	5.26	5.83

Table 3: Storage Efficiency vs. Fidelity. Mean Compression Ratio (

\rho

, higher is better), Chamfer Distance (

d_{CD}

), and Hausdorff Distances (

\overrightarrow{\epsilon}

\overleftarrow{\epsilon}

\overleftrightarrow{\epsilon}

) across datasets. Best results are highlighted in bold .

Figure 3: Runtime analysis. Mean execution time (in seconds) for CoverageAxis++ (CA++), Raw Medial Axis (RMA), MASH, and our method across four datasets. The total time is decomposed into Preprocessing (gray), Encoding (blue), and Decoding/Reconstruction (orange).

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Visual Impact of Proximity Threshold ( $\tau_{prox}$ ). We evaluate reconstruction quality on the Multi-Face Bust and Armadillo models. Without a threshold ( $\tau_{prox}=\text{None}$ ), the selection is driven purely by visibility, resulting in a sparse representation ( $|\mathcal{C}|=31$ for Armadillo) that fails to capture fine details like the scales on the leg or facial creases. Our default setting ( $\tau_{prox}=0.2$ ) enforces locality, recovering these high-frequency features with a compact set of anchors ( $|\mathcal{C}|=60$ ). A strict threshold ( $\tau_{prox}=0.05$ ) results in an excessive number of reference points ( $|\mathcal{C}|=830$ ) without significant visual improvement.

Figure 5: Visual Impact of Spherical Harmonic Bandwidth ( $L$ ). We compare reconstructions of the Stanford Dragon , Thai Statue , Earth , and City models at varying bandwidths. Lower bandwidths ( $L=16,32$ ) result in overly smoothed geometry, failing to capture high-frequency features. Our default setting ( $L=64$ ) successfully recovers fine details such as the dragon’s scales and the sharp corners of the skyscrapers. Doubling the bandwidth to $L=128$ offers minor visual improvement while increasing the storage footprint (see Sec.4.2 of main paper).

Figure 6: Effect of Lanczos Smoothing. Visual comparison on the Thingi10k Dragon model. The No smoothing result exhibits significant ringing artifacts, visible as high-frequency ripples on the neck and body scales. Applying Lanczos smoothing effectively suppresses this noise, restoring a clean surface while preserving the underlying geometric structure.

Figure 7: Runtime Scalability Analysis. Comparison of computation time (seconds) vs. number of primitives ( $|C|$ ) across three datasets. SHARC (blue) is consistently fast with minor overhead as $|C|$ increases. MASH (red) exhibits steep linear scaling, while Coverage Axis (green) is fast (for the simplified version of the models; see Sec.B) but offers lower reconstruction fidelity. Shaded areas represent standard deviation.

Figure 8: Reconstruction Error vs. Primitive Count. Chamfer Distance ( $10^{-3}$ ) for varying $|C|$ . SHARC achieves the lowest error even at the lowest primitive counts, confirming that its speed (shown in Fig. 7 ) does not compromise quality.

Figure 9: Additional Qualitative Comparisons on PSB models. Visual reconstruction results for the models of Table 4 )

Figure 10: Additional Qualitative Comparisons on Thingi10k models. Visual reconstruction results for the models of Table 4 )

Figure 11: Additional Qualitative Comparisons on ShapeNet models.

3D Shape Generation

3D shape generation creates and edits 3D objects from learned geometric priors and text, images, sketches, or structural constraints.

See all 106 papers

Jun 8, 2026

Beyond Spherical Harmonics: Rethinking Appearance Models for Radiance Reconstruction

View-dependent appearance modeling remains a challenging problem in novel-view synthesis and reconstruction. Accurately representing complex angular effects often requires substantial memory and computational resources. For new learning-based methods, a common approach is to rely on SH. However, capturing high-frequency phenomena such as specular reflections demands high-order expansions, which increase memory usage and computational cost. Consequently, most methods employ low-order SH, which limits the ability to model complex view-dependent effects, resulting in overly smooth or diffuse representations. To address these limitations, we systematically evaluate a wide range of spherical functions in the context of scene reconstruction. Some of them are introduced to graphics and computer vision for the first time in this paper. Based on the insights from the experiment, we develop a novel spherical formulation, the Normalized Anisotropic Spherical Gabor function that enables efficient modeling and learning of high-frequency appearance effects while maintaining compact representation. Compared to existing approaches, our function achieves higher-quality reconstruction of view-dependent phenomena such as glints, while being up to five times more memory-efficient and more efficient to evaluate. We validate its performance in radiance-field reconstruction tasks.

Ewa Miazga, Jorge Condor, Piotr Didyk

1École Polytechnique Fédérale de Lausanne, Switzerland · 2Università della Svizzera Italiana, Lugano, Switzerland

Novel View Synthesis 3D Scene Representation Spherical Harmonics Neural Radiance Fields

Ask PDF

Mar 24, 2026

Patchwork: A compact representation for 3D polygonal shapes

We introduce Patchwork, a new general-purpose shape representation capable of modeling 2D and 3D geometry with a small number of parameters. Patchwork is grounded in a rigorous mathematical framework, providing provable complexity bounds and the ability to approximate arbitrary shapes with arbitrary precision in any dimension. We propose an efficient gradient-based optimization scheme to fit Patchwork representations to 2D and 3D data, along with a novel regularization loss that progressively prunes redundant elements, yielding high compactness after convergence. Our approach offers fast fitting performance, a fraction of the required parameters compared to existing alternatives, and native support for inside-outside classification, making it a versatile and compact representation for geometric learning and reconstruction tasks, with future potential for 3D generation. Our implementation is available at: https://github.com/Ankbzpx/patchwork-experiment.

Ruichen Zheng, Biao Zhang, Michael Birsak +2

King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia

Geometric Representation Learning 3D Representation Learning 3D Reconstruction

Ask PDF

Apr 1, 2026

Shape Representation using Gaussian Process mixture models

Traditional explicit 3D representations, such as point clouds and meshes, demand significant storage to capture fine geometric details and require complex indexing systems for surface lookups, making functional representations an efficient, compact, and continuous alternative. In this work, we propose a novel, object-specific functional shape representation that models surface geometry with Gaussian Process (GP) mixture models. Rather than relying on computationally heavy neural architectures, our method is lightweight, leveraging GPs to learn continuous directional distance fields from sparsely sampled point clouds. We capture complex topologies by anchoring local GP priors at strategic reference points, which can be flexibly extracted using any structural decomposition method (e.g. skeletonization, distance-based clustering). Extensive evaluations on the ShapeNetCore and IndustryShapes datasets demonstrate that our method can efficiently and accurately represent complex geometries.

Panagiotis Sapoutzoglou, George Terzakis, Georgios Floros +1

Laboratory of Photogrammetry, School of Rural, Surveying and Geoinformatics Engineering, National Technical University of Athens, Greece

3D Representation Learning Statistical Shape Modeling Gaussian Processes

Ask PDF

Abstract

3D Shape Generation

Explore similar work

Beyond Spherical Harmonics: Rethinking Appearance Models for Radiance Reconstruction

Patchwork: A compact representation for 3D polygonal shapes

Shape Representation using Gaussian Process mixture models