Comment on Ioannidou, S.; Pantazis, G. Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra. ISPRS Int. J. Geo-Inf. 2020, 9, 494

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the comments, the questioned equations should be given. The name of the questioned article should be given. The correct solutions to the questioned equations should be given, if possible. 

Comments on the Quality of English Language

average

Author Response

Two additional constraints for unity and orthogonality      

   ( r^Tr= 1,  r^Td= 0  )     

r₁² + r₂² + r₃² + r₄² = 1

r₁ d₁ + r₂ d₂ + r₃ d₃ + r₄ d₄ = 0                                                                                   

where 

r = [ r₁  r₂  r₃  r₄ ] ^T                               d = [d₁  d₂  d₃  d₄ ]^T        

Reviewer 2 Report

Comments and Suggestions for Authors

The reviewer's comments are correct in pointing out that the three parameterizations (Euler angles, quaternions, and dual-quaternions) uniquely determine the rotation matrix for the given parameter values. Note that the mappings are surjective and nonlinear. In the inverse problem, the objective is to compute the rotation matrix via the computations of each of the three parameterizations with a given point dataset. 

Assuming that the dataset is noise-free, that is, the relation y = f(p, x) is precisely satisfied for some value of parameter p, the exact solution of the nonlinear equation y-f(p,x)=0 to compute the parameters p, should lead to the precise calculation of the rotation matrix and translational parameters. 

However, the authors use the least-square method with linearized equations. Therefore, it is conceivable that the three parameterizations have different sensitivity to the linearization process and lead to different numerical results. Furthermore, noise in the measured data will likely be processed differently in the three parameterizations, thus leading to different numerical results and, therefore, different accuracies.  

Although I do agree with the reviewer's comment that several equations and dataset information are not provided in the 2020 paper, I am inclined to agree with the conclusions of the 2020 paper and I disagree with the overall point of the reviewer that the numerical results and their findings are wrong. 

Comments on the Quality of English Language

Good. 

Author Response

I am sorry. I cannot agree with Rewiever2's claims. The second referee's claim that noise in the data will lead to different results is not true. The same data is used in each method. The claim that possible noise in the data will give different results in different methods is invalid. Contrary to the referee's claim, many published article examples show that such a different result cannot be obtained.

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

No comment.

Comments on the Quality of English Language

No comment

Author Response

Extensive editing in English was done by a native speaker.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

There is nothing technical in the author's response to re-evaluate. They are just saying that they disagree with my claim. I am not sure which claim the authors disagree with. As such, there is nothing to review.

Reviewer 3 Report

Comments and Suggestions for Authors

Comments on author's sources of concerns:

1)
I agree with the author, that there are 2 constrains regarding the dual-quaternion method.
I agree, that the first constrain is the unity of the first quaternion (r) representing the rotation.
I am sure, that the orthogonality of 'r' and 'd' is not necessary, instead only d1 (the first
element of the second quaternion) should be zero.
The method is really well explained in (Zeng, H.; Frang, X.; Chang, G.; Yang, R. A dual quaternion algorithm of the Helmert transformation problem.
Earth Planets Space 2018, 70, 26.)
A further, very good reading is: https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/other/dualQuaternion/index.htm
I agree with the author, that The Ioannidou's Eq. 12. is not formulated properly, and I add, that
 its explanation is very confuse and inconsistent.

2)
I agree with the author, that data in the article are hidden, however the result are presented.

3)
I think the Ioannidou's result represent the problems in the least squares estimation,
namely wrong initial parameters, from where the parameters
did not converge, or - as the equations are not linear - not repeating the iteration enough times.

4)
I partially agree with the author, that the model was not properly presented in the papar, however
the model produced consistent results in many cases, so I do not agree with the statement, that
the model was set up incorrectly.

I recommend the author to withdraw this submission, and write a good, enjoyable paper, promoting the
usefullness of the dual-quaternion method with some easy to understand samples, and referring
Ioannidou's work as an example for using the method in practice, mentioning the shortcomings of
their results.