Curvature-based pebble segmentation for reconstructed surface meshes

Rheinwalt, Aljoscha; Purinton, Benjamin; Bookhagen, Bodo

doi:https://doi.org/10.5194/egusphere-2025-1110

Preprints

https://doi.org/10.5194/egusphere-2025-1110

Preprints

31 Mar 2025

| 31 Mar 2025

Curvature-based pebble segmentation for reconstructed surface meshes

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

Abstract. Accurate segmentation of pebbles in complex 3D scenes is essential to understand sediment transport and river dynamics. In this study, we present a curvature-based instance segmentation approach for detecting and characterizing pebbles from 3D surface reconstructions. Our method is validated using high-resolution ground truth models, allowing for a quantitative assessment of segmentation accuracy. The workflow involves reconstructing a sandbox scene using available open-source or commercial software packages, segmenting individual pebbles based on curvature features, and evaluating segmentation performance using detection metrics, primary axes estimation, 3D orientation retrieval, and surface area comparisons. Results show a high detection precision (0.980), with segmentation errors primarily attributed to under-segmentation caused by overly smooth surface reconstructions. Primary axis estimation via bounding box fitting proves more reliable than ellipsoid fitting, particularly for the A and B axes, while the C-axis remains the most challenging due to partial occlusion. 3D orientation estimation reveals variability, with cumulative errors ranging from less than 5° to more than 45°, highlighting the difficulty in retrieving full orientations from incomplete segments. Surface area metrics indicate that our approach prioritizes precision over recall, with nine out of ten ground truth pebbles achieving IoU values above 0.8. In addition, we introduce a Python-based segmentation tool that provides detailed morphological and color-based metrics for each detected pebble. Our findings emphasize the advantages of true 3D analysis over traditional 2D photo-sieving approaches and suggest future improvements through refined segmentation algorithms and enhanced surface reconstructions.

Received: 07 Mar 2025 – Discussion started: 31 Mar 2025

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.

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Journal article(s) based on this preprint

24 Sep 2025

Curvature-based pebble segmentation for reconstructed surface meshes

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

Earth Surf. Dynam., 13, 923–940, https://doi.org/10.5194/esurf-13-923-2025,https://doi.org/10.5194/esurf-13-923-2025, 2025

Short summary

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1110', Anonymous Referee #1, 28 Apr 2025

The authors present a novel method and proof-of-concept for pebble segmentation in 3D meshes of surfaces from topographic point clouds. This is a persistently tricky (especially in 3D) but essential task. Therefore, a method that achieves precise and accurate pebble segmentation in 3D is a timely and most welcome contribution to the field. The authors present convincing-looking results for their method, for which they also release a Python-based software for segmentation. However, the presentation of the work is currently hard to follow, due to:
1) A convoluted and occasionally confusing method section that presents some results (the table tennis ball experiment) and some discussion (e.g., of table tennis ball results, camera recommendations). For more details and specific suggestions, please refer to the in-line comments in the attached pdf.
2) A too-short results section with more bullet points than text (see in-line comments in attached pdf).
These points make it hard for readers to comprehend all the detailed information in the method section and lead to confusion in parts of the validation metrics (i.e., see in-line comments related to some IoU calculations). To address these, I’d propose to describe the main experiments of the study more clearly (maybe with a graphical overview of the central workflow); to re-order parts of the method section, and to move some parts of the section not crucial to the central workflow to an Appendix. This would allow for some needed clarifications (see attached comments), and include the table tennis ball results in the results section. Furthermore, I’d suggest converting the results section to a more continuous text. In addition to these two major points, there are some minor comments on some phrasings (e.g., regarding “true 3D” and “outperforming 2D methods") in the attached in-line comments.
I suspect addressing all points would amount to moderate to major revisions. Nevertheless, I want to emphasize that the results seem impressive, and the approach seems well-conceived and meticulously tested. I suspect that the over-packed method section results from this careful testing. Therefore, I am confident that the authors will be able to address all points raised without complications, resulting in a manuscript that will appeal to an even wider audience.

Citation: https://doi.org/10.5194/egusphere-2025-1110-RC1
- AC1: 'Reply on RC1', Aljoscha Rheinwalt, 08 Jun 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1110/egusphere-2025-1110-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1110-AC1
RC2:
'Comment on egusphere-2025-1110', Anonymous Referee #2, 01 May 2025

The manuscript presents a new technique for the in-field measurement of the shape properties of pebbles without extracting them from their original location. The surface of a layer of pebbles is reconstructed from photos using the structure from motion technique. Recognition and separation of individual pebble segments are achieved by taking the divergence of the normal field of the mesh using finite differences. Although a large part of the pebbles remain hidden, the recovered parts represent the original pebbles well in terms of the principal axes. The authors validated their technique for different settings and error measurements. This work is of great value and interesting for a broad audience by offering an automated evaluation of the geometry of large pebble populations. The figures are clear, nicely formatted, and communicative. The Python script with the Jupiter notebook and sample dataset is delivered in a user-friendly way.

However, the structure of the paper needs improvement to emphasize the results better and provide a clearer line of thought. Now, the reader needs to jump forward and backward in the text. The appearance of the references of the figures does not follow the figure order (e.g. Fig. 2 is referenced sooner than Fig. 1). Terms and methods are used before they are explained, and some conclusions are drawn before presenting the supporting data. Secondly, the main findings of the manuscript are not well separated from existing techniques. Section 2 contains forward and backward references, and the subsections do not follow the order of the workflow (taking photos, reconstructing the surface, evaluating the mesh), while Section 4 is redundant and contains repeated information from Section 3.

The main result of the manuscript is the segmentation algorithm. The techniques to obtain the axes and orientation from the segments are possibly new, but their novelty is emphasized neither in the abstract nor the text. The abstract and the introduction should be revised accordingly.

I suggest a new structure of the paper that delivers the main results sooner and removes the forward and backward jumps of the current structure.
1. Introduction: Section 1
2. The segmentation algorithm: Section 2.3
3. Materials and methods:
3.1. setup: Section 2.1 presenting the three test cases with numbers
Case 1: tennis balls
Case 2: 10 numbered pebbles
Case 3: 318 pebbles
3.2. photo taking: Section 2.5.
3.3. surface reconstruction: first paragraph of Section 2.2, Section 2.7 can be here or in the appendix
4. Validation of the technique
4.1. Surface reconstruction accuracy (case 1): second and third paragraphs of Section 2.2.
4.2. Pebble metrics accuracy (case 2):
Description of pebble metrics: Section 2.4.
4.2.1. primary axes: Sections 3.2 and 4.2
4.2.2. pebble orientation: Sections 3.3 and 4.3
4.2.3 surface area metrics: Sections 3.4 and 4.4
4.3. Pebble detection accuracy (case 3): Sections 3.1 and 4.1
5. Conclusion: Section 4.6, Section 5.
Appendix:
(1) Section 2.6 (and possibly also Section 2.7)
(2) Section 4.5, table and old Fig. 16

I also suggest a new order of the figures that follows the logic of the proposed new structure and provides a clearer understanding of the text.
Figure 1: old Fig. 2
Figure 2: I suggest creating a new figure explaining the method in general for the new Section 2. The figure should show a zoomed part of a mesh with the normals. The triangles should be colored according to their convexity.
Figures 3-4: old Fig. 1 and Fig. 7., they explain well the suggested workflow for the new Section 3
Figures 5-7: old Figs. 4-6. for the new section 4.1
Figure 8: old Fig. 3 showing the ground truth pebbles for the new section 4.2
Figures 9-11: old Figs. 8-10. Illustrating pebble metrics for the new section 4.2
Figures 12-15: old Figs. 12-15. showing the results for the new section 4.2
Figure 16: old Fig. 11. Showing the pebble detection results for the new section 4.3

General comments:
- The phrase “ground truth pebble” is unclear. Some parts of the text suggest that ground truth is a 3D shape of the pebble, while others indicate that they are high-quality scanning of the pebbles sitting in the sandbox. If ground truth means the separated 3D shape of the pebble, the manuscript lacks an explanation of the method of matching the ground truth to the scene.
- The validation of the orientations is unclear. How were the reference orientations of the “ground truth pebbles” determined?
- The tennis balls were evaluated by fitting spheres to their segments. What is the reason of not using the mean curvature for validation determined during the segmentation?
- Many referenced papers aim to determine the roundness of pebbles. Approximating the roundness of the total pebble from its segments would be worth mentioning in the part listing future directions. I also suggest referencing the paper by Ludmány et al. 2023, which also handled the problem of analyzing the geometry of a partly reconstructed body.
- The text does not clarify how the pebbles were placed in the sandbox (distributing them by human hand one by one or by some other technique). There are multiple possible ways of putting them into the box, which could lead to different results. It would be very interesting to repeat the sandbox experiments multiple times in the future.

Specific comments:
- Page 2, line 53: I suggest including Fehér et al. 2023 next to the two referenced papers, which aim to retrieve shape properties from 3D models.
- Page 3, line 61: The last sentence should be followed by a statement that this manuscript aims to measure the accuracy of the SfM for small pebbles.
- Page 3, lines 64-69: These sentences aim for clarity but cause confusion here. It is clear from the previous sentence what “mesh” means and it is also clear what the “volume of an individual pebble” means without further explanation.
- Page 4, line 84: There should be some additional sentences stating the paper's goals to motivate the reader.
- Page 5, Fig. 3: The word “ground truth” is unclear in the caption. The small images show partly reconstructed pebbles sitting on other pebbles, while the table contains properties (e.g. sphericity) suggesting that these pebbles were fully reconstructed. The matching technique also needs further explanation.
- Page 6, line 116: How was the sphere fitting achieved?
- Page 7, Fig. 5, caption line 3: It is unclear how fitting a sphere to a single triangle is possible.
- Page 8, first paragraph of Section 2.3: Instead of referencing Fig. 7B, I suggest referencing the new Figure 2.
- Page 9, Fig.7: The colors of subfigures C and D are not explained in the caption.
- Page 10, line 169: How are the concave, convex, and flat regions distinguished from the mean curvature?
- Page 10, lines 170-186: The text mentions concave and convex triangles that are quite confusing without a clear definition. I suggest either avoiding these terms or defining them at the end of line 169.
- Page 11, lines 202-203: It is unclear how the alignment of a 3D model can be compared to a pebble segment of a scene.
- Page 11, lines 210-231: The description of calculating the axes and orientation of the 3D models and the segments should be separated. Now, the descriptions of manual measurements of the physical object, calculations of the 3D model and pebble segments are mixed, which is confusing.
- Page 12, Fig. 8: Instead of referencing another figure, I suggest using the number of the pebble (2). The last sentence should be removed because it is not visible from this figure and corresponds to a later part of the manuscript.
- Page 12, Fig. 9: The caption should reference pebble 2.
- Page 13, lines 234-236: The last two sentences jump forward in the text and should be removed from here.
- Page 13, lines 241-244: I did not understand how to determine the retrievable part. The clarification of the phrase “ground truth model” might also clarify this part. Is the 3D model aligned somehow to the reconstructed surface?
- Page 17, Fig. 11: What do the red circles correspond to?
- Page 19, Fig. 13: The last two sentences of the caption should be moved into the body of the text.
- Page 19: I would also mention that Fig. 12 and Fig. 14. show that the errors in the axis and the orientation are related: the bottom five and the top five pebbles are the same in both cases.
- Pages 22-23: Many sentences repeat sentences from Section 3. These repeated sentences can be removed after combining old Sections 3 and 4 in the new Section 4.
- Page 25-26: The conclusion should emphasize the main result, the segmentation technique, and how to calculate the axes and orientations.

After restructuring the text and the figures and clarifying the points above, I suggest the manuscript for publication.

Citation: https://doi.org/10.5194/egusphere-2025-1110-RC2
- AC2: 'Reply on RC2', Aljoscha Rheinwalt, 08 Jun 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1110/egusphere-2025-1110-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1110-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1110', Anonymous Referee #1, 28 Apr 2025

The authors present a novel method and proof-of-concept for pebble segmentation in 3D meshes of surfaces from topographic point clouds. This is a persistently tricky (especially in 3D) but essential task. Therefore, a method that achieves precise and accurate pebble segmentation in 3D is a timely and most welcome contribution to the field. The authors present convincing-looking results for their method, for which they also release a Python-based software for segmentation. However, the presentation of the work is currently hard to follow, due to:
1) A convoluted and occasionally confusing method section that presents some results (the table tennis ball experiment) and some discussion (e.g., of table tennis ball results, camera recommendations). For more details and specific suggestions, please refer to the in-line comments in the attached pdf.
2) A too-short results section with more bullet points than text (see in-line comments in attached pdf).
These points make it hard for readers to comprehend all the detailed information in the method section and lead to confusion in parts of the validation metrics (i.e., see in-line comments related to some IoU calculations). To address these, I’d propose to describe the main experiments of the study more clearly (maybe with a graphical overview of the central workflow); to re-order parts of the method section, and to move some parts of the section not crucial to the central workflow to an Appendix. This would allow for some needed clarifications (see attached comments), and include the table tennis ball results in the results section. Furthermore, I’d suggest converting the results section to a more continuous text. In addition to these two major points, there are some minor comments on some phrasings (e.g., regarding “true 3D” and “outperforming 2D methods") in the attached in-line comments.
I suspect addressing all points would amount to moderate to major revisions. Nevertheless, I want to emphasize that the results seem impressive, and the approach seems well-conceived and meticulously tested. I suspect that the over-packed method section results from this careful testing. Therefore, I am confident that the authors will be able to address all points raised without complications, resulting in a manuscript that will appeal to an even wider audience.

Citation: https://doi.org/10.5194/egusphere-2025-1110-RC1
- AC1: 'Reply on RC1', Aljoscha Rheinwalt, 08 Jun 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1110/egusphere-2025-1110-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1110-AC1
RC2:
'Comment on egusphere-2025-1110', Anonymous Referee #2, 01 May 2025

The manuscript presents a new technique for the in-field measurement of the shape properties of pebbles without extracting them from their original location. The surface of a layer of pebbles is reconstructed from photos using the structure from motion technique. Recognition and separation of individual pebble segments are achieved by taking the divergence of the normal field of the mesh using finite differences. Although a large part of the pebbles remain hidden, the recovered parts represent the original pebbles well in terms of the principal axes. The authors validated their technique for different settings and error measurements. This work is of great value and interesting for a broad audience by offering an automated evaluation of the geometry of large pebble populations. The figures are clear, nicely formatted, and communicative. The Python script with the Jupiter notebook and sample dataset is delivered in a user-friendly way.

However, the structure of the paper needs improvement to emphasize the results better and provide a clearer line of thought. Now, the reader needs to jump forward and backward in the text. The appearance of the references of the figures does not follow the figure order (e.g. Fig. 2 is referenced sooner than Fig. 1). Terms and methods are used before they are explained, and some conclusions are drawn before presenting the supporting data. Secondly, the main findings of the manuscript are not well separated from existing techniques. Section 2 contains forward and backward references, and the subsections do not follow the order of the workflow (taking photos, reconstructing the surface, evaluating the mesh), while Section 4 is redundant and contains repeated information from Section 3.

The main result of the manuscript is the segmentation algorithm. The techniques to obtain the axes and orientation from the segments are possibly new, but their novelty is emphasized neither in the abstract nor the text. The abstract and the introduction should be revised accordingly.

I suggest a new structure of the paper that delivers the main results sooner and removes the forward and backward jumps of the current structure.
1. Introduction: Section 1
2. The segmentation algorithm: Section 2.3
3. Materials and methods:
3.1. setup: Section 2.1 presenting the three test cases with numbers
Case 1: tennis balls
Case 2: 10 numbered pebbles
Case 3: 318 pebbles
3.2. photo taking: Section 2.5.
3.3. surface reconstruction: first paragraph of Section 2.2, Section 2.7 can be here or in the appendix
4. Validation of the technique
4.1. Surface reconstruction accuracy (case 1): second and third paragraphs of Section 2.2.
4.2. Pebble metrics accuracy (case 2):
Description of pebble metrics: Section 2.4.
4.2.1. primary axes: Sections 3.2 and 4.2
4.2.2. pebble orientation: Sections 3.3 and 4.3
4.2.3 surface area metrics: Sections 3.4 and 4.4
4.3. Pebble detection accuracy (case 3): Sections 3.1 and 4.1
5. Conclusion: Section 4.6, Section 5.
Appendix:
(1) Section 2.6 (and possibly also Section 2.7)
(2) Section 4.5, table and old Fig. 16

I also suggest a new order of the figures that follows the logic of the proposed new structure and provides a clearer understanding of the text.
Figure 1: old Fig. 2
Figure 2: I suggest creating a new figure explaining the method in general for the new Section 2. The figure should show a zoomed part of a mesh with the normals. The triangles should be colored according to their convexity.
Figures 3-4: old Fig. 1 and Fig. 7., they explain well the suggested workflow for the new Section 3
Figures 5-7: old Figs. 4-6. for the new section 4.1
Figure 8: old Fig. 3 showing the ground truth pebbles for the new section 4.2
Figures 9-11: old Figs. 8-10. Illustrating pebble metrics for the new section 4.2
Figures 12-15: old Figs. 12-15. showing the results for the new section 4.2
Figure 16: old Fig. 11. Showing the pebble detection results for the new section 4.3

General comments:
- The phrase “ground truth pebble” is unclear. Some parts of the text suggest that ground truth is a 3D shape of the pebble, while others indicate that they are high-quality scanning of the pebbles sitting in the sandbox. If ground truth means the separated 3D shape of the pebble, the manuscript lacks an explanation of the method of matching the ground truth to the scene.
- The validation of the orientations is unclear. How were the reference orientations of the “ground truth pebbles” determined?
- The tennis balls were evaluated by fitting spheres to their segments. What is the reason of not using the mean curvature for validation determined during the segmentation?
- Many referenced papers aim to determine the roundness of pebbles. Approximating the roundness of the total pebble from its segments would be worth mentioning in the part listing future directions. I also suggest referencing the paper by Ludmány et al. 2023, which also handled the problem of analyzing the geometry of a partly reconstructed body.
- The text does not clarify how the pebbles were placed in the sandbox (distributing them by human hand one by one or by some other technique). There are multiple possible ways of putting them into the box, which could lead to different results. It would be very interesting to repeat the sandbox experiments multiple times in the future.

Specific comments:
- Page 2, line 53: I suggest including Fehér et al. 2023 next to the two referenced papers, which aim to retrieve shape properties from 3D models.
- Page 3, line 61: The last sentence should be followed by a statement that this manuscript aims to measure the accuracy of the SfM for small pebbles.
- Page 3, lines 64-69: These sentences aim for clarity but cause confusion here. It is clear from the previous sentence what “mesh” means and it is also clear what the “volume of an individual pebble” means without further explanation.
- Page 4, line 84: There should be some additional sentences stating the paper's goals to motivate the reader.
- Page 5, Fig. 3: The word “ground truth” is unclear in the caption. The small images show partly reconstructed pebbles sitting on other pebbles, while the table contains properties (e.g. sphericity) suggesting that these pebbles were fully reconstructed. The matching technique also needs further explanation.
- Page 6, line 116: How was the sphere fitting achieved?
- Page 7, Fig. 5, caption line 3: It is unclear how fitting a sphere to a single triangle is possible.
- Page 8, first paragraph of Section 2.3: Instead of referencing Fig. 7B, I suggest referencing the new Figure 2.
- Page 9, Fig.7: The colors of subfigures C and D are not explained in the caption.
- Page 10, line 169: How are the concave, convex, and flat regions distinguished from the mean curvature?
- Page 10, lines 170-186: The text mentions concave and convex triangles that are quite confusing without a clear definition. I suggest either avoiding these terms or defining them at the end of line 169.
- Page 11, lines 202-203: It is unclear how the alignment of a 3D model can be compared to a pebble segment of a scene.
- Page 11, lines 210-231: The description of calculating the axes and orientation of the 3D models and the segments should be separated. Now, the descriptions of manual measurements of the physical object, calculations of the 3D model and pebble segments are mixed, which is confusing.
- Page 12, Fig. 8: Instead of referencing another figure, I suggest using the number of the pebble (2). The last sentence should be removed because it is not visible from this figure and corresponds to a later part of the manuscript.
- Page 12, Fig. 9: The caption should reference pebble 2.
- Page 13, lines 234-236: The last two sentences jump forward in the text and should be removed from here.
- Page 13, lines 241-244: I did not understand how to determine the retrievable part. The clarification of the phrase “ground truth model” might also clarify this part. Is the 3D model aligned somehow to the reconstructed surface?
- Page 17, Fig. 11: What do the red circles correspond to?
- Page 19, Fig. 13: The last two sentences of the caption should be moved into the body of the text.
- Page 19: I would also mention that Fig. 12 and Fig. 14. show that the errors in the axis and the orientation are related: the bottom five and the top five pebbles are the same in both cases.
- Pages 22-23: Many sentences repeat sentences from Section 3. These repeated sentences can be removed after combining old Sections 3 and 4 in the new Section 4.
- Page 25-26: The conclusion should emphasize the main result, the segmentation technique, and how to calculate the axes and orientations.

After restructuring the text and the figures and clarifying the points above, I suggest the manuscript for publication.

Citation: https://doi.org/10.5194/egusphere-2025-1110-RC2
- AC2: 'Reply on RC2', Aljoscha Rheinwalt, 08 Jun 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1110/egusphere-2025-1110-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1110-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

AR by Aljoscha Rheinwalt on behalf of the Authors (08 Jun 2025) Author's response Author's tracked changes Manuscript

ED: Publish as is (25 Jun 2025) by Anne Baar

ED: Publish as is (17 Jul 2025) by Wolfgang Schwanghart (Editor)

AR by Aljoscha Rheinwalt on behalf of the Authors (17 Jul 2025)

Journal article(s) based on this preprint

24 Sep 2025

Curvature-based pebble segmentation for reconstructed surface meshes

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

Earth Surf. Dynam., 13, 923–940, https://doi.org/10.5194/esurf-13-923-2025,https://doi.org/10.5194/esurf-13-923-2025, 2025

Short summary

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

Data sets

Curvature-based pebble segmentation for reconstructed surface meshes Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen https://doi.org/10.5281/zenodo.14987824

Aljoscha Rheinwalt, Benjamin Purinton, and Bodo Bookhagen

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Our study presents a computer-based method to detect and measure pebbles in 3D models reconstructed from camera photos. We tested it in a controlled setup and achieved 98 % accuracy in detecting pebbles. Unlike traditional 2D methods, our approach provides full 3D size and orientation data. This improves sediment analysis and riverbed studies by offering more precise measurements. Our work highlights the potential of 3D modeling for studying natural surfaces.


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