Shape Representation using Gaussian Process mixture models

Panagiotis Sapoutzoglou; George Terzakis; Georgios Floros; Maria Pateraki

cs.CVApr 1, 2026

Shape Representation using Gaussian Process mixture models

Authors: Panagiotis Sapoutzoglou, George Terzakis, Georgios Floros, Maria Pateraki

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

Abstract

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.

Figures & tables

Figure 1: We partition a sparse point cloud using distance-based clustering and extract reference points (Sec. 3.1 ). Each 3D point $P_{i}$ is parameterized as a bearing vector in spherical coordinates $U_{i}$ relative to its reference point $C_{i}$ . The directions, with distances as targets, serve as inputs to a Gaussian Process ( $GP_{i}$ ) (Sec. 3.2 ). The GP mixture model forms our shape representation.

Figure 2: Spherical directional distance field (blue rays) centered at reference point $\boldsymbol{C}$ (red).

Panel of Figure 3 — Figure 3: Ray-casting to the objects’s surface (blue) from reference points (red) by retaining the first intersection. Denser coverage achieved with increasing number of reference points (left-to-right).

Panel of Figure 4 — Figure 4: Assignment of training points to clusters. (a)-(b) Ground truth - reconstructed without overlap. (c)-(d) Ground truth - reconstructed with overlap (blue).

	Planes				Chairs				Sofas
	$d_{C}\downarrow$	P $\uparrow$	R $\uparrow$	F $\uparrow$	$d_{C}\downarrow$	P $\uparrow$	R $\uparrow$	F $\uparrow$	$d_{C}\downarrow$	P $\uparrow$	R $\uparrow$	F $\uparrow$
DeepSDF [ 24 ]	0.79	90.2	85.5	87.6	0.81	84.2	88.2	86.0	0.57	89.0	90.0	89.4
DeepSDF (10k)	0.80	90.3	84.5	87.2	0.90	84.5	88.2	86.1	0.56	65.0	60.8	62.5
NKSR [ 16 ]	0.65	78.4	81.3	79.5	0.41	84.6	86.3	85.0	0.45	79.6	82.5	80.8
Ours	0.14	87.4	93.6	90.3	0.21	92.3	86.5	89.2	0.26	92.4	89.2	90.7

Table 1: Comparison with NKSR [ 16 ] and DeepSDF [ 24 ] trained with 500K and 10K points per object across shapes of varying complexity for the planes, chairs, and sofas datasets from ShapeNetCore (120 models each). For the NKSR we used the pretrained model on ShapenetCore. We report

d_{C}

(Chamfer distance),

P

(Precision),

R

(Recall),

F

(F-score) computed using a sample of 30k points.

	Planes				Chairs				Screwdriver
	$d_{C}$ ↓	P ↑	R ↑	F ↑	$d_{C}$ ↓	P ↑	R ↑	F ↑	$d_{C}$ ↓	P ↑	R ↑	F ↑
Polynomial (p=3)	1.56	35.3	58.0	43.7	3.35	26.7	40.4	31.9	1.69	32.6	32.8	32.7
Periodic	0.38	70.4	86.4	77.5	0.62	60.8	73.8	66.2	3.47	26.6	26.8	26.7
Linear	4.37	13.5	36.2	19.5	9.09	10.9	31.8	16.0	9.02	10.5	14.6	12.2
RBF	0.31	73.0	88.9	80.1	0.47	60.7	69.2	66.4	0.20	86.0	88.4	87.2
RQ	0.19	82.6	93.8	87.9	0.51	68.4	82.9	74.5	0.12	95.1	96.5	95.8

Table 2: Ablation study of different kernels and their impact on reconstruction accuracy across various shapes. Chamfer distance is multiplied by

10^{3}

Figure 5: Qualitative comparison of shape representation on the ShapeNetCore dataset. The heatmap quantifies the distance from the ground truth point-cloud to the reconstructed; superimposed into the ground truth point cloud. We demonstrate two examples from the airplanes, chairs and sofas categories with each method. From left to right ours, DeepSDF [ 24 ] , DeepSDF (10k), NKSR [ 16 ]

Figure 6: Qualitative results of our method for the models of the IndustryShape dataset.The heatmap quantifies the distance from the ground truth point-cloud to the reconstructed; superimposed into the ground truth point cloud.

Panel of Figure 7 — Figure 7: (a) Ground truth, (b) Results obtained from automatically estimating reference point locations (16) with k-means, (c) Results based on manually defined reference points (32).

Figure 8: Ablation study for various number of reference points vs shape accuracy based on CD for the planes and sofas object classes of the ShapeNetCore dataset and the screwdriver from the IndustryShapes dataset.

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