Total Least-Squares Iterative Closest Point Algorithm Based on Lie Algebra
Round 1
Reviewer 1 Report
The work is at a very good scientific level. I recommend publishing the full text of the thesis in Journal Applied Sciences.
The paper presents two least squares TLS-ICP ( total least-squares - iterative closest point) algorithms based on Lie algebra using the GHM structure. These methods include random noise from the source coordinate. To solve the GHM model, an iterative optimization algorithm was used in which the Euler angle was replaced by Lie algebra. Definitions and properties of Lie algebra and Lie groups as well as a variant of the Gauss-Helmert model used in the ICP model are presented. The Jacobian matrix used in the Gauss-Helmert model was derived. The final part of the work contains a numerical example verifying the developed TLS model.The work is at a very good scientific level. I recommend publishing the full text of the thesis in Journal Applied Sciences.
Author Response
Thank you very much for your comments about our work. We modify several grammatical mistakes in the new submission.
Reviewer 2 Report
The authors provide a satisfactory review of research done in the field. They present clearly their methodology and their approach to solve the problem. Their experiments though are only limited in using 4 point for the estimation of the rotation and translation parameters. Usually ICP algorithms use thousands of points to estimate the translation and rotation parameters between 2-point clouds. So it would be useful if the performed some experiments using a lot of points and also provide some information about the running time of the algorithm.
Author Response
Thank you for pointing this out. We add cost time of our algorithm in Section 3 as follows: “TLS-ICP algorithm often uses 4-8 points to calculate transformation between two coordinates in geological mapping. So in previous experiments, datasets from [8] are used to evaluate accuracy. But in the field of three-dimensional reconstruction, ICP algorithm also can be used to calculate robot pose. So the cost time of ICP is very important. In our experiment, all of codes run on a laptop with 2.7 GHz quad cores in Ubuntu. We use 100 points, 500 points and 1000 points, which are copied from [8], to evaluate the cost time of every step in our algorithm. The cost time of every iterative step is 0.2ms using 4 points. The cost time of every iterative step is 380ms using 100 points. The cost time of every iterative step is 46 seconds. The cost time of every iterative step is 369 seconds.”
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
1. The present manuscript reported an interesting algorithm registers the source model to the target model. As the authors mentioned, it is critical in coordinate conversion. The academic language should be further improved, in order to improve the readabilities of the paper.
2. Please try to improve the writing organization of the abstract. It would be better if you can present a brief background, main objectives, the employed methodologies, and the summary of the conclusion in the abstract.
3. For the introduction/ literature review. It would be better to present the applications of your presented algorithm in engineering since there are so many computations or calculations in engineering used you mentioned algorithm. If the proposed algorithm is good, it will be critical to improving the calculation accuracy and efficiency in engineering. For example, in the Eq. (15), during I review your paper, I found an interesting paper is related to your methodology, i.e., Applied Mathematical Modelling, 61, pp.726-743.
4. Starts from section 2.1, please try to add some theoretical background in the derivations. For example, the interactive closet points have their basic theoretical background. Please clarify it.
5. Eqs. (6-7), please change the symbol * to x, in the derivations. Also, please try to check all the equations.
6. The authors should take care of the format of the symbols in the equations, normal or italic???
7. Where are the figures 1 and 2??? BTW, please try to use figures to present your numerical examples or experiments, in order to ensure your results understood by the technical reviewers or potential readers.
Overall, it is an interesting paper.
Reviewer 2 Report
The authors attempt to solve the iterative closest point problem with a total least-squares cost using the special orthogonal group and the special Euclidean group. This seems in principle worth pursuing.
Unfortunately the presentation of is of low quality: English language editing is necessary, figures are missing, equations need to be formatted. It seems like this manuscript has not been carefully reviewed by the authors before submission for peer-review. This makes it difficult to judge the scientific merit of the contribution.
Before a complete review can be conducted, these issues need to be fixed. Here are some additional comments from a preliminary review. I hope these will help the authors to submit a decent manuscript for peer-review.
1) The introduction should include a more thorough review of the literature on total least squares solutions for 3D similarity transformation problems. Some references that come to mind are as follows.
Chang, G. On least-squares solution to 3D similarity transformation problem under Gauss–Helmert model. J Geod. (2015) 89: 573. https://doi.org/10.1007/s00190-015-0799-z
Fang, X. Weighted total least-squares with constraints: a universal formula for geodetic symmetrical transformations. J Geod (2015) 89: 459. https://doi.org/10.1007/s00190-015-0790-8
B. Wang, J. Li, C. Liu, J. Yu. Generalized total least squares prediction algorithm for universal 3D similarity transformation. Adv. Space Res., 59 (2017), pp. 815-823
2) There is a lot of notation, which is hard to keep track of and not always clearly introduced.
A table of notation might be useful. More detail is needed, for example "P is the information matrix" should be replaced by "P is the Fisher information matrix with $P_{ij} = ...$. In equation (4), is x=X? The x's in equation (6) are not the same as in equation (4). What is B_0 in equation (10)? Isn't there a factor 2 missing in equation (10)? What is A_0 in equation (11)?
3) The definition of a group is lacking both identity element and inverses.
4) More details on how the ^ operator is defined is necessary to make the manuscript self-contained.
4) Methods 1 and 2 appear without a motivation. Later it is stated that M1 is more accurate in estimating rotation and M2 in estimating translation and some explanation is given. Adding the motivation for these two methods already when the methods first appears would make the manuscript easier to follow.
5) The choice of error measure (equation (38)) should be explained.
6) In the experiments, the methods are only compared to a least squares solution. To understand the contribution of the new methods, they should be compared with other total least squares methods, such as the ones in references 8, 9, 11, 14 and others.
