Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript presents a rigorous and well-structured framework for the quantitative characterization of complex 3D multiphase geometries. The work successfully integrates tools from topology, integral geometry, and tensorial descriptors, offering a multi-scale and multi-perspective mathematical analysis. The study is both conceptually sound and practically relevant, with strong potential for applications across porous media, materials science, and biological systems.

 

1. The paper is grounded in established mathematical theories, including Betti numbers, Minkowski functionals, etc. These descriptors form a complete morphological basis for additive geometric measures and are appropriately used as a baseline.
2. a particularly innovative contribution is the cut-response methodology, which Transforms scalar invariants into spatially dependent sensitivity functions and Evaluates descriptor variation under domain partitioning.
3. The manuscript clearly distinguishes between Topological count and Functional connectivity.
4. The introduction of orientation-weighted Betti-like descriptors is both Conceptually novel and Practically useful.

 

My Suggestions

A brief formalization of cut-response convergence properties could further strengthen the theory.

Including computational complexity estimates for large voxel grids would enhance reproducibility.

A small illustrative example with analytical geometry (e.g., torus or sphere) could help intuition.

Introduction section should be improved by adding recent work and remove some irrelevant references.

Conclusion section should reflect the limitation and future scope of the article.

 

 

 

 

 

 

Author Response

Reviewer 1

 

This manuscript presents a rigorous and well-structured framework for the quantitative characterization of complex 3D multiphase geometries. The work successfully integrates tools from topology, integral geometry, and tensorial descriptors, offering a multi-scale and multi-perspective mathematical analysis. The study is both conceptually sound and practically relevant, with strong potential for applications across porous media, materials science, and biological systems.

The paper is grounded in established mathematical theories, including Betti numbers, Minkowski functionals, etc. These descriptors form a complete morphological basis for additive geometric measures and are appropriately used as a baseline.

a particularly innovative contribution is the cut-response methodology, which Transforms scalar invariants into spatially dependent sensitivity functions and Evaluates descriptor variation under domain partitioning.

The manuscript clearly distinguishes between Topological count and Functional connectivity.

The introduction of orientation-weighted Betti-like descriptors is both Conceptually novel and Practically useful.

Answer.

Dear reviewer. We deeply appreciate youк expertize opinion in our case. All text that we want to delete according to your and other reviewer valuable comments, are highlighted in red and crossed-out letter. All added text is written in blue letters.

 

Reviewer 1

My Suggestions

A brief formalization of cut-response convergence properties could further strengthen the theory.

Answer.

Our sample is discrete, and the image may behave continuously. Therefore, the calculation result may actually vary depending on the cut frequency, and it may behave abruptly. We've made changes to Section 2.5 and the section on limitations.

Reviewer 1

Including computational complexity estimates for large voxel grids would enhance reproducibility.

Answer.

Dear reviewer, to address your comment, we've added a table to Section 2.8. Thank you very much; this has significantly improved the overall presentation of our article. Part of your response has also been incorporated into the limitations section.

 

Reviewer 1

A small illustrative example with analytical geometry (e.g., torus or sphere) could help intuition.

Answer.

We already have a section 3.7, but we presented it as an element of control for our research, now we will rename it and rewrite it a little to make it clearer what we mean.

Reviewer 1

Introduction section should be improved by adding recent work and remove some irrelevant references.

Answer.

Thank you very much, we have added it.

 

Reviewer 1

Conclusion section should reflect the limitation and future scope of the article.

Answer.

We already have a separate section, 4.8, called "Limitations." We insist that it should remain part of the discussion. Thanks to your comments, this section has now been significantly expanded. If we misunderstood your meaning, please let us know immediately, and we will make the appropriate changes to the manuscript.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the Report carefully

 

Comments for author File: Comments.pdf

Author Response

Reviewer 2 wrote:

Report of the article

Manuscript Number: Mathematics-4426566

Title: Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors The objective of this article is to investigate and proposes a uni_ed framework that extends scalar 3D topological and Minkowski-functional analysis in three directions.

(1) First, To introduce a cut-response analysis in which a phase is repeatedly split by coordinate planes normal to the x; y and z axes and normalized topological quantities are recomputed on the two resulting subdomains.

(2) Second, de_ne directional Betti-like descriptors based on spanning components, directional spanning indicators, local skeleton orientation and cavity-shape orientation.

(3) Third, compute the rank-two surface Minkowski tensor W0;2 1 , which measures the orientation distribution of surface normals and provides a rigorous tensorial descriptor of surface anisotropy.

 

The article is organized in _ve sections described as follows:

Section 1. Introduction.

Section 2. Materials and Methods.

2.1. Benchmark X-_CT Volume

Validating an X-ray micro-computed tomography (_CT) volume involves evaluating the quality, accuracy, and resolution of a 3D image. Researchers use standardized phantom objects to measure errors in volume, shape, and size.

2.2. Synthetic Control Geometries.

Is an advanced causal inference method used to evaluate the impact of an event (like a new law or marketing campaign). It selects comparison regions whose economies or features represent unique, independent trends. It ensures the data models distinct forces rather than overlapping trends.

2.3. Phase Domains.

Phase domains refer to distinct and separate regions within a material where the physical structure or chemical state is uniform.

2.4. Scalar Baseline Descriptors.

Reference scalar descriptors are the basic measurements of a system at its starting point. They establish a baseline value before any changes, treatments, or complex actions take place.

2.5. Cut-Response Analysis.

A cut response analysis assesses how a system, material, or participant alters its behavior when faced with a "cut" or threshold. Depending on the _eld of study, this typically involves analyzing test scores, diagnostic biomarkers, or physical stress.

2.6. Directional Betti-Like Descriptors.

Betti-type directional descriptors are mathematical tools used in topological data analysis (TDA) to study the shape and structure of data. They measure the number of connected parts, holes, and voids in a dataset, but they do so by observing it from speci_c angles or directions.

2.7. Surface Minkowski Tensor.

A Surface Minkowski Tensor is a mathematical tool used to measure the shape and direction of objects on curved or at surfaces. It helps to study anisotropy, that is, whether an object has dierent properties depending on which way it is facing.

2.8. Computational Implementation.

Computational implementation is the process of converting a mathematical algorithm (a list of step-by-step rules) into functional computer code. It acts as a bridge, connecting what a machine is theoretically expected to do with the specific programming language the computer understands.

Section 3. Results.

3.1. Scalar Descriptors De_ne Global Phase Roles

A scalar descriptor is a unique numerical value that denes a specific point in a system. In physics, these values act as controls that govern the overall state, or phase, of a system. Modifying these values determines how the system behaves, how it transitions, or how it changes its physical form.

3.2. Fractal Filling and Geometric Anisotropy.

Fractal _lling and geometric anisotropy describe how complex shapes pack into space in highly uneven or directional ways. These concepts are used in advanced materials science, physics, and computational modeling to create structures that outperform traditional, uniform designs.

3.3. Cut-Response Reveals Directional Heterogeneity.

The response to shear reveals directional heterogeneity, since a cut physically alters the material, causing its edges to react di_erently depending on the angle and direction of the force. In materials science, cutting a metal or polymer generates anisotropic (directional) damage at the edges. The properties change depending on whether the cut is made along the grain or perpendicular to it.

3.4. Directional Connectivity Separates Spanning from Fragmentation.

In network theory and physics, directional connectivity is the rule that determines whether a system can transmit data, water, or energy from one point to another. It ensures that the path is continuous (fully connected end-to-end) rather than fragmented (divided into small, isolated islands).

3.5. Orientation-Weighted Channel and Cavity Proxies.

Orientation-weighted channel and cavity models are mathematical tools used in 3D physics, antenna design, and structural biology to model how spaces (such as tubes, radio channels, or protein cavities) direct energy or particles based on their angles and physical shapes.

3.6. Surface Minkowski Tensor Quanti_es Surface Fabric.

Surface Minkowski tensors are advanced mathematical tools used to quantify surface structure. They go beyond simple averages to calculate the degree of alignment (anisotropic) or randomness (isotropic) of a material's surface. By measuring geometry and curvature, they help engineers predict material strength and performance.

3.7. Synthetic Control Results.

Synthetic controls create an arti_cial reference group by combining several similar, untreated units. This method estimates what would have happened to the treated unit if the intervention had not been applied. Outcomes are measured by comparing the results of the actual treated unit with those of the synthetic group after the intervention.

Section 4. Discussion

4.1. From Scalar Fractal and Topological Descriptors to Directional Characterization.

4.2. Why Scalar Descriptors Are Necessary but Insu_cient.

4.3. Cut-Response as a Structural Sensitivity Measure

4.4. Cut-Response as a Boundary-Sensitivity Diagnostic

4.5. Directional Connectivity Without Rede_ning Betti Numbers

The transition from scalar, fractal, and topological descriptors to directional characterization is a key step in understanding complex structures. This shift helps capture not only how much space an object occupies or its connectivity, but also where complexity is directed in space.

4.6. Minkowski Tensor as a Rigorous Tensorial Extension

4.7. Generality Beyond the Geological Demonstration

The Minkowski tensor is a rigorous tensor extension of classical Minkowski functionals. While standard functionals measure scalar properties of geometric shapes (such as volume, surface area, and Euler characteristic), Minkowski tensors capture directional and anisotropic (orientation-dependent) information.

4.8. Limitations

A limitation is a limit, restriction, or defect that controls, reduces, or stops what a person, thing, or system can achieve. It is a barrier that prevents something from being done beyond a certain point.

Section 5. Conclusions

The present work introduced a uni_ed framework for directional and tensorial extensions of scalar 3D microstructure analysis. The central result is methodological: scalar fractal, topological and integral-geometric descriptors remain necessary, but they should be complemented by response, connectivity and tensorial quantities when the aim is to characterize anisotropic or spatially heterogeneous 3D geometries. The benchmark application shows that these descriptor families separate structural properties that would otherwise be merged into a small set of global numbers.

The results show that the proposed extensions reveal spatial sensitivity, boundaryto-boundary connectivity and surface fabric that are not captured by scalar phaselevel invariants alone. The proposed framework can be used to analyze segmented 3D images of multiphase geological, porous, composite and engineered samples, thereby expanding quantitative knowledge about their internal structure beyond scalar phase-level descriptors. In general, the proposed framework should be understood as a bridge between compact scalar morphology and interpretable directional structure. It retains the mathematical role of classical topological invariants while incorporating more suitable descriptors for anisotropy, localization, and tensor surface structure in segmented multiphase 3D images. In relation with the article the Methodology is the adequate, In relation with the References all of them are mentioned in the article. The authors used many tables that are interpreted carefully. At this point, I want to make it clear that this article is framed within the _eld of Materials Science and only uses mathematical concepts that are merely stated without any supporting evidence. It also uses some topological concepts that are neither described nor proven. I believe it is necessary for the authors to answer the following questions, and I am still unsure whether, given the above, this article is appropriate for the journal.

Answer:

Dear reviewer. We are fascinated by the attention you have paid to our paper, especially it detalization of our sections. We have tried to answer all of your highly valuable comments. All text that we want to delete according to your and the other reviewer valuable comments, are highlighted in red and crossed-out letter. All added text is written in blue letters.

 

 

 

 

Reviewer 2 wrote:

I suggest that the following questions be answered in a clear and precise manner by the authors.

Question 1. The authors wrote this in lines 346 to 348, Could you explain these statements more clearly?. How to understand this statement that its structural function is dominated by isolated components rather than by the topology of tunnels or cavities.

“The magnetite-density phase, although highly fragmented, has very low betta1 and betta2, indicating that its structural role is dominated by 347 isolated components rather than by tunnel or cavity topology.”

Answer:

Dear reviewer! Thank you very much for your incredible attention to detail. The relevant text of Section 3.1 has been rewritten, as indicated in the text.

 

Reviewer 2 wrote:

Question 2. The authors wrote this in lines 364 to 368, Could you explain these statements more clearly. How to clearly understand all the mathematical things that are said but not are proven.

“These results indicate that the strongest topological signals are not explained simply by a single coherent grain orientation fabric. The olivine-density phase has the strongest tunnel and cavity topology (Table 2), but not the strongest phase anisotropy or grain orientation strength (Table 3). Its high fractal dimension instead suggests that the tunnel- rich topology occurs in a comparatively volume-lling phase.”

Answer:

Dear reviewer! Thank you again for your attention. The relevant part of Section 3.2 has been rewritten, as indicated in the text.

 

 

Reviewer 2 wrote:

Question 3. The authors wrote this in lines 392 to 393, Could you explain the process of converting scalar topology into a spatial sensitivity diagnostic in a clearer way?

“The main methodological point is that cut-response profiles convert scalar topology into a spatial sensitivity diagnostic”.

Answer:

Dear reviewer, We have attempted to provide more detail in this section of the manuscript (section 3.3).

 

Reviewer 2 wrote:

Question 4. The authors wrote this in lines 440 to 443, Could you explain these statements more clearly.

“These tensorial results are complementary to topology: the olivine-density phase is topologically rich, whereas the magnetite-density phase is surface-anisotropic and fragmented without being spanning.”

Answer:

Dear reviewer! Thanks to your attention to our article, it is becoming clearer and more precise. Section 3.6 has been rewritten.

 

Reviewer 2 wrote:

Question 5. The authors wrote this in lines 469 to 471, Could you explain these statements more clearly.

“The olivine-density phase has the strongest scalar tunnel and cavity topology, but the scalar values alone do not specify the direction in which this topology is most spatially sensitive.”

Answer:

Dear reviewer! Thanks to your attention to our article, it is becoming clearer and more precise. Section 4.1 has been rewritten. We have also added a definition to Section 2.4 regarding homology groups.

 

Reviewer 2 wrote:

Question 6. The authors wrote this in lines 469 to 471, Could you explain how tunnel topology is de_ned and how to determine that this topology is more spatially sensitive in a clearer way?.

“The olivine density phase exhibits the most pronounced scalar tunneling and cavity topology, but scalar values alone do not indicate the direction in which this topology is most spatially sensitive.

Answer:

Dear Reviewer, This question is a complete replication of the previous one.

 

 

 

Reviewer 2 wrote:

Question 7. The authors wrote this in lines 518 to 519, Could you explain these statements more clearly, as well as how to understand the difference between global and classical topology?

“The method can be understood as a diagnostic counterpart to global topology. Classical topology asks what the object is, up to topological equivalence.”

Answer:

Dear reviewer, we have replaced a truly incorrect term Global topology to topology of the whole sample. Thank you for your attentiveness.

 

Reviewer 2 wrote:

Question 8. The authors wrote this in lines 650 to 651, Could you explain these statements more clearly and how to do so without redefining the mathematical term Homology?. How explain this

“This distinction is important for mathematical clarity. It allows directional connectivity to be discussed without redefining homology.”

Answer:

Dear reviewer! You're absolutely right, these sentences are confusing for the reader. We've decided to remove them entirely; they're essentially unnecessary.

 

Reviewer 2 wrote:

Question 9. The authors wrote this in lines 656 to 661, The authors can clarify that the main result is methodological: scalar, fractal, topological and geometric integral descriptors are still necessary, but must be complemented with response, connectivity and tensor quantities when the goal is to characterize anisotropic or spatially heterogeneous 3D geometries.?

“The present work introduced a unified framework for directional and tensorial extensions of scalar 3D microstructure analysis. The central result is methodological: scalar fractal, topological and integral-geometric descriptors remain necessary, but they should be complemented by response, connectivity and tensorial quantities when the aim is to char acterize anisotropic or spatially heterogeneous 3D geometries. The benchmark application shows that these descriptor families separate structural properties that would otherwise be merged into a small set of global numbers.”

Answer:

Dear reviewer! Indeed, we claim that our main result is methodological. Apparently, this was poorly described in Section 5. Conclusions; we've rewritten that section slightly.

 

Reviewer 2 wrote:

Question 10. The authors wrote this in lines 699 to 702, Could you explain these statements more clearly?

“Overall, the proposed framework should be understood as a bridge between compact scalar morphology and interpretable directional structure. It preserves the mathematical role of classical topological invariants while adding descriptors that are better suited to anisotropy, localization and tensorial surface fabric in segmented multiphase 3D images.”

Answer:

Dear reviewer, our conclusion was indeed written in somewhat free language; we have rewritten it in a strictly scientific register.

 

 

Reviewer 2 wrote:

Dear Editor: I recommend a major revision of the article "Mathematical Method for the Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors." This is because there are several questions related with the topological aspect in the submitted version. I will review the new version and provide my decision.

Sincerely yours

The Referee

Answer:

We are deeply grateful for such a careful reading of our manuscript. We have endeavored to address all your insightful and meaningful comments.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The revised version of this paper is written according to the reviewers comment. So, I recommend acceptance of this paper.

Reviewer 2 Report

Comments and Suggestions for Authors

This new version is more complete.

Comments for author File: Comments.pdf

Author Response

The Reviewer 2 wrote:

New Report of the final version of the article Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors ID: 4426566

Dear Editor, I have carefully and thoroughly read the article ”Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors,” paying particular attention to all the sections highlighted in blue in this new version. I have also focused on the authors’ answers to the questions posed in the previous report, All of them were answered satisfactorily, clearly and precisely. I wish to express my satisfaction and congratulate the authors on their meticulous work in significantly improving the article’s content. Therefore, I recommend this article, ”New revision of the article Mathematical Pipeline for Quantitative Analysis of Multiphase 3D Material Structures Using Fractal, Topological, and Minkowski Descriptors,” in this new version, for publication in your prestigious journal.

Sincerely yours
The Referee

Anwer of the authors:

Dear reviewer!

Thank you for such a deep read of our work! We are glad that it was you who made the comments, work with them helped us improve our Manuscript a lot.