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The manuscript is much improved. I recommend that the paper should be accepted for publishing.
The presentation is largely improved. Most typos are fixed. I suggest publishing it.
The paper presents a method to obtain high order approximations material distribution. Authors should revise the paper according to the reviewers and the below suggestions.
1. The summary needs to be more informative. Particularly, the authors should clearly highlight their contributions since the paper is based on the previous work and the existing open source software. Correct the grammatical errors. Change is shown to are shown.
2. Extend the introduction and related work. In those two sections, the contexts and detailed descriptions of the topics and the motivation of this work should be clearly presented.
3. In the fourth section, the presentation of the algorithm needs to be much improved. Authors should use other approaches (pseudo code, flow chart, or equations) to make it more understandable.
4. Writing and presentation. Revise the entire paper to avoid the grammatical errors and typos. In addition, when using equations, authors need to explain the used variables and cite the equations using the numbers in the text.
This paper use their open source mesh generator named Seeder to implement a robust first order method in obtain polynomial representations of geometrical objects to describe non-smooth material distributions. The seeder is introduced very well in the paper, the different experiment figure shown the good experiment results as well. Most important part is “Numerical properties”, the figure 7 and figure 8 shown the value and contribution of this paper. However, there are some place I want to point out:
1. The equation (1) - (4) should explain as well.
2. In the three steps: Voxelization, Probe and Conversion, what is probe is not very clear.
3. The summary part is very weak.
I think the article meets all the basic standards here.
I think the article meets the experimental design standards.
I think the findings are valid.
This article proposes a numerical method to generate high order approximations of 3D surfaces. I think the proposed method and experimental results are valid.
The presentation of this paper may be further improved:
1. For the proposed algorithm, other than the English description, I suggest the authors to also describe it in formal mathematical language, that is, clearly formulate the inputs, outputs and each step of the algorithm.
2. Figure 1 is not clear. What I can see is only a white sphere, voxelization in red, and others in blue. I do not see “the thick black lines outline the elements of the actual mesh”, or “the voxelization within elements follows the octree refinement towards the sphere and is indicated by the dark blue lines.” I suggest the authors to either improve the figure or change the description.
3. Minor typos, e.g., in the description of Figure 6 “Blue is the sphere and yellow the isosurface of the color value at 0.5.” I suggest the authors to perform another round of grammar check.
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