cs.CVOct 2, 2026
SaveA Theory of Shape Reconstruction from Heat Conduction and Shading
Abstract
Shape from shading using a single image of a Lam- bertian surface is inherently ambiguous. When the light source direction is known, the surface normal estimation has a cone- ambiguity, which worsens when the source is unknown. Recently, shape from heat conduction has emerged as an approach that leverages heat transport equations to estimate the Shape Lapla- cian operator, an intrinsic measure of shape. However, deriving surface normals from the Laplacian operator encounters a local binary convex/concave ambiguity. Our contribution introduces a novel theory to resolve these local shape ambiguities (excluding a few degeneracies) without relying on priors like smoothness, by combining the cues from shading and heat conduction. Our method ensures the mathematical constraints of both shading and the Laplacian are satisfied simultaneously, even with an unknown light source. We validate our theory through simulations of complex shapes and analyze its performance in the presence of noise, as well as on a noisy single thermal video of real-world objects with complex shapes and material properties, including varying albedo.
Figures & tables
Fig. 1: Reconstructed shapes using estimation techniques based on Laplacian cues, shading cues, or both. While Laplacian cues constrain the local curvature around a pixel, it is invariant to convex/concave deformations. On the other hand, shading cues constrain the normal at each pixel to lie on a cone around the source direction, but do not constrain neighboring normals. This paper combines these complementary cues to resolve ambiguities while enabling faithful 3D reconstructions.
Fig. 2: Left: A regular grid triangular mesh whose cotangent weights for any vertex depend on its neighboring six vertices. Right: Illustrate the local surface patch around vertex , as an embedding of the triangular mesh in 3D space.
Fig. 3: Left: Illustrates ambiguity in shape-from-shading, where a cone of normals yields identical shading, and a local binary ambiguity in shape-from-Laplacian, where different shapes with the same cotangent weights appear as mirror reflections in normal space. The true object shape is uniquely determined at the intersection of the shading cone and Laplacian normals. Right: Demonstrates degenerate cases where the proposed shape-from-Laplacian-and-shading method fails.
Fig. 4: Shape estimation results for a simulated object using shading and Laplacian as inputs. Figures (a) and (b) show the inputs cues: shading and Laplacian, respectively. The Laplacian is visualized as the magnitude of Gaussian curvature, which is an intrinsic geometric property of shape. Figure (c) presents the rendering of the estimated lighting and reconstructed surface according to the algorithm described in Sec. V . Note that this rendering is derived entirely from the unknown lighting and unknown shape. Figure (d) shows the depth map recovered from the reconstructed surface, and it can be seen that it aligns with the ground truth in (e) . The estimated shape and ground truth are both scaled, and the mean distance between them was at most 0.45%. Detailed quantitative results are summarized in Table II .
Fig. 5: Shape estimation results using only Laplacian or shading cues, and joint optimization over lighting and shape using an analysis-by-synthesis approach. Due to the non-convex nature of the underlying optimization, the results exhibit ambiguities—locally binary ambiguities from Laplacian constraints and conic ambiguities from shading cues—reflecting the presence of multiple global minima. The light source estimation errors with the analysis-by-synthesis approach were , , , , and from left to right object, compared to our approach shown in Table I .
Fig. 6: Shape estimation with varying levels of additive gaussian noise. (a) and (b) are the input shading and Laplacian, (c) the rendered shading image estimated from lighting and shape, and (d) the estimated depth-map. From the top to bottom, results are shown for no noise, and for Gaussian noise with standard deviations , , and , respectively. Although greater noise levels introduce more local distortion in the reconstructed shape, our proposed method still yields a reasonable estimation overall. Table III presents the corresponding quantitative numbers.
| Droplet Wave | Wineglass | Cow | Igea | Armadillo | |
| Initial Estimate | 27.02 | 2.026 | 1.076 | 48.94 | 25.03 |
| After Refinement | 0.088 | 0.029 | 0.066 | 0.038 | 0.049 |
TABLE I: Angular error (in degrees) between estimated and true lighting direction, evaluated after a single iteration of our refinement strategy and after convergence.
| Droplet Wave | Wineglass | Cow | Igea | Armadillo | |
| Shape from Laplacian | 2.64% | 12.80% | 3.23% | 4.89% | 5.56% |
| Shape from Shading | 3.52% | 12.97% | 6.89% | 12.21% | 71.94% |
| Analysis by Synthesis | 2.23% | 13.79% | 13.65% | 3.29% | 28.08% |
| SFLS (Ours) | 0.02% | 0.06% | 0.08% | 0.06% | 0.45% |
TABLE II: Average percentage distance error with the ground truth for shapes estimated using shape from Laplacian [ 9 ] , shape from shading [ 13 ] (Fig. 5 ), and proposed shape from Laplacian and shading (Fig. 4 ), computed after scaling each shape to fit within a unit bounding box.
| Added Gaussian Noise | |||
| Avg. distance error | |||
| % Distance error | 0.49% | 0.79% | 1.12% |
TABLE III: Quantitative results for a droplet wave object with increasing Gaussian noise applied to the inputs of both the shading and Laplacian weights.
| Gaussian | Gaussian + Fixed Pattern | |
| SFLS with propagation [ 1 ] | 0.81% | 1.20% |
| SFLS (Ours) | 0.36% | 0.42% |
TABLE IV: Average percentage distance error with the droplet wave object reconstructed from simulated noisy thermal videos with Gaussian noise and Gaussian + fixed pattern noise that is commonly seen in thermal cameras.
Fig. 7: We simulated and reconstructed shape from the noisy thermal videos with Gaussian noise (G) and the Gaussian + fixed pattern noises (G+F). We used the algorithm from [ 9 ] to estimate Shape Laplacian and we apply median filtering and total variation denoising before using them as input to our shape estimation. We compare our method with the propagation-based approach proposed in prior work [ 1 ] . Table IV presents the corresponding quantitative numbers.
Fig. 8: Shape estimation results for a simulated object under known lighting conditions, using Laplacian and shading as inputs for two cases: (1) shading includes interreflections with the Lambertian BRDF and (2) shading includes interreflections with a non-lambertian BRDF with roughness parameter . Figures (a) , (b) and (c) show pairs of the input shading image rendered using Mitsuba and the estimated shape. Similarly, Figures (d) , (e) and (f) show the corresponding results for the non-Lambertian BRDF case. Below the visual results, we also show the corresponding depth profiles comparing the estimated shapes with the ground truth along the vertical center line of the image. The mean distance between them was at most 0.48%. Detailed quantitative results are in Table V .
Fig. 9: Shape estimation results using shading images with interreflections under different roughness parameters as input. As the surface becomes smoother (i.e., as decreases), the specular highlights in the shading introduces rapid changes in surface normals. The depth profiles comparing the estimated shapes with the ground truth along the vertical center line of the image. Quantitative results are summarized in Table VI .
Fig. 10: Shape estimation results using shading images with spatially varying albedo modeled using a checkerboard pattern. As the relative albedo contrast increases, high-contrast intensity variations increasingly confound shading cues and affect normal estimation. The depth profiles comparing the estimated shapes with the ground truth along the vertical center line of the image. Quantitative results are summarized in Table VII .
| Interreflections | 0.02% | 0.02% | 0.43% |
| BRDF with interreflections | 0.04% | 0.02% | 0.48% |
TABLE V: Quantitative results for a droplet wave for two cases (1) interreflections on a diffuse surface and (2) interreflections on a surface with complex BRDF, with roughness parameter .
| BRDFs with interreflections | 0.09% | 0.10% | 0.57% |
TABLE VI: Quantitative results for a droplet wave object reconstructed from simulated BRDFs with interreflections. The lighting was set at .
| Weak | Moderate | Strong | |
| Spatially varying albedo | 0.03% | 0.06% | 1.31% |
TABLE VII: Quantitative results for a droplet wave object reconstructed from simulated spatially varying albedo modeled using a checkerboard pattern. The lighting was set at .
Fig. 11: Our imaging system consists of a thermal camera and a light source. The light source is placed close to the object so that heat conduction induced by light absorption can be detected by the thermal camera.
Fig. 12: Shape estimation results from a single thermal video of a 3D-printed object using shape-from-heat-conduction [ 9 ] and the proposed SFLS. From left to right, the reference 3D model, the captured image of the 3D-printed object, the shape estimated using shape-from-heat-conduction, and the shape estimated using SFLS. The depth profiles compare the estimated shapes with the ground truth along the vertical center line of the object. Detailed quantitative results are in Table VIII .
Fig. 13: Shape estimation results from a single thermal video of real-world objects. Figure (a) show the real-world objects. Figure (b) show the absorbed light estimated from the thermal video, which is used as the shading input. Figure (c) show the corresponding temperature gradients. Figures (d) and (e) present the shapes estimated from shape-from-heat-conduction [ 9 ] and the proposed SFLS. Figure (f) show the ground truth shapes obtained by SfM. Detailed quantitative results are summarized in Table IX .
| Droplet Wave | Bunny | |
| Shape from Heat Conduction [ 9 ] | 1.45% | 5.72% |
| SFLS (Ours) | 0.05% | 0.12% |
TABLE VIII: Average percentage distance error with the ground truth 3D-model for shapes estimated using shape-from-heat-conduction [ 9 ] with one thermal video, and the proposed shape from Laplacian and shading (Fig. 12 ), computed after scaling each shape to fit within a unit bounding box.
| Black-Bear | Painted-Bear | Sphere | Soda-can | Heart | |
| Shape from Heat Conduction [ 9 ] | 3.06% | 2.61% | 3.43% | 2.44% | 2.13% |
| SFLS (Ours) | 0.23% | 1.53% | 0.97% | 0.74% | 1.05% |
TABLE IX: Average percentage distance error with the ground truth SFM 3D-scans for shapes estimated using shape from heat conduction [ 9 ] with one thermal video, and the proposed shape from Laplacian and shading (Fig. 13 ), computed after scaling each shape to fit within a unit bounding box.
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