Helmert Transformation Problem. From Euler Angles Method to Quaternion Algebra

Round 1

Reviewer 1 Report

The manuscript investigates three transformation methods for the reverse problem, and via experiments, some properties of each method are found. The paper is well written (the theory of each method is described in detail, and the experiment is well designed).

 

But the following comments should be considered in the revised version:

1. The Helmert transformation in the title should have the same name throughout the manuscript. For example, in the Abstract, it is called Helmert’s transformation.
2. I cannot clearly know what is the novelty of the manuscript in this version. Maybe the comparison between different transformation methods, and the authors find some characteristics of each method. The authors should show the contribution of your paper apparently.
3. Do you have some ground dataset rather than the synthetic data? In general, the transformation is utilized to perform a large dataset such as LiDAR point clouds, or other large-range geodesy coordinates with millions of points. 

Author Response

1. Notes for Reviewer 1:

   1. In all the manuscript, the transformation is written as “Helmert transformation”. [Lines: 12, 22, 198]
   2. The novelty of the manuscript is to test and compare the three different methods, for each parameter. The main result is that none of them have the optimal solution not only at angles but also at translations and scale. This result is presented in the discussion [Lines: 344-357] and some details are added so as to be clearer. [Lines: 358-362]
   3. A real dataset was tested, which contains points for monitoring a bridge. The dataset was big but only ten points are used and presented in this thesis, because after a point there are no differences when calculating the unknown parameters. [Lines 273-315]

Reviewer 2 Report

The authors address the very common problem of setting up 3D Helmert transformations between 3d point clouds. Thus the findings are of interest for a large audience.

However, the manuscript and the experiments caried out have to be improved significatly before publication.

My major concerns with respect to the manuscript:

1) Please use commonly used terms and phrases like e.g. translation/shift (better than transition), rotation and scale in a consistent manner throughout the paper. Other examples "coordinate (system) transformation" instead of "coordinating system transformation" or "origin" instead of "beginning" or "rotate/rotation" instead of "turns".

2) I´m not a native speaker at all but many sentences are hardly to ready because of the used English language. I would strongly recommend to make use of a professional translator to improve the used language.

3) It is a well known fact that Euler angles or any other angle representation and sequence have their limits beause of the ambiguities of the sine and cosine functions. Thus literature especially in photogrammetry is full of recommendations when to use which representation. This knowlegde should also be used in this study and also the design of the experiments should be "fair" with respect to this limitations. I also disagree that the 27 (?) rotations combinations (line 69) yield 27 different rotation matrices! To my understanding the rotation matrix represent the axis of one coordinates system in the second one. Consequently there is only ONE valid rotation matrix which can be calculated using many different angle representations and sequences.

4) I learned from the manuscript that "dual quaternion" representation might overcome the limitations described before. If thisis true, this should be one major finding of the manuscript!

5) Unfortunately, there is little information about the process of finding the transformation parameters. I guess it is based on an iterative least squares approach using linearisations of the transformation equations. This should be explained in more detail.

6) Moreover, there is no information how the initial values for the transformation parameters are found. However, in the interpretation of the results it is stated several times that gross errors are because of this initialisation. Please explain in more detail.

7) The set-up of the experiments is not quite clear for me. First, one should consider the most common use cases for the Helmert transformation (e.g. ITRF - ETRF date conversions, ...). This common use cases should also already be described in the introduction. The experiments then should be based on the identified use cases (e.g. ITRF-ETRF conversion very small shifts, rotation angles and scale differences but at least mm-accuracy required).

8) There is also little to no information and discussion about how the distribution of the 3d points clouds were designed. The distribution is shown in figure 1, however it is hard to interpret whether e.g. all points lie on a plane or not.

9) To highlight the "dual quaternion" method of course one experiment could also include high variations of the rotation angles. However, here I would suggest to vary only one angle and investigate only a few problematic and non-problematic areas.

10) Please rethink the presentation of the results. Figure 9 is to my view very informative whereas figures 3-8 are not. This is mainly due to the large number of experiments which make it difficult to identify the used set-up.

 

As I would suggest a major revision of the manuscript and an extensive editing of the text, I do not add a more detailed review.

Author Response

Notes for Reviewer 2:

1. The main terminology has been replaced in all the manuscript

• Transitionà Translation [Lines: 20, 109, 122, 131, 136, 139, 142, 149, 150, 150, 181, 202, 210, 214, 217, 218, 229, 234, 236, 237, 252, 254, 261, 262, 319, 323, 331, 337, 341, 354]
• Coordinating system transformation à Coordinate (system) transformation [Line 26]
• TurnsàRotate/rotation [Lines: 20, 43]
• BeginningàOrigin [Line 104, 319]

 

2. The English language was checked both in grammar and in syntax by an English teacher who is a native English speaker.
3. The combinations are 27 because we have 3 rotations at 3 different axes. Sometimes the rotation matrix is the same independently of the sequence. The main problem at Euler angle method is the gimbal lock, that a degree of freedom is missing and different angles will lead at the same rotation matrix. This is explained in summary, by adding new text in the manuscript. [Lines 73-76, 79-81]
4. It is true that dual quaternion method might overcome the limitations of the rotation matrix, but this method leads to bigger deviations on the translation vector. This is explained in detail at the discussion, and some text has been also added. [Lines: 358-362]
5. A new paragraph was added, that explains the least square method and how the initial values of the parameters have been calculated. [Lines 151-162]
6. The initial values are found by solving the minimum equation at each method. In quaternion method for some scenarios, the minimum equations are solved in an incorrect way so the final results, after the least square method, are not correct, and so the transformation parameters.
7. Some examples of Helmert transformation problem, were added in the introduction. [Lines 27-30] The scenarios are plenty so as to cover the most cases. We don’t want to cover only the small shifts, or the mm-accuracy, because in geodesy there is a huge variety of data transformations.
8. The 3D points are chosen randomly, and there are not on a plane. We try to cover all the space, so as to have good distribution in the requested space. [Lines 173, 183]
9. In the example of the real data, only a rotation through z-axis is calculated. So even if the scenarios have 3 rotations, the example tests a different rotation problem, and in that way, we have a full “access” at the problems of each method.
10. In figures 3-8, it is important to see the exact deviations of each method, at each parameter. This is why they present plenty of information. In discussion, the figure 9 (after the corrections figure 12) shows the average of the deviations of all independent axis parameters. It is derived from the average of the deviations for each data set. In essence, an average value of the deviations per axis was calculated for each dataset as it would be wrong to compare dissimilar data. And finally, the average for each method and for the 27 scenarios was obtained.

Reviewer 3 Report

The paper is very interesting, especially for the geodetic community since this is one of the most important transformations, just as the authors stated in the paper.

There are several thighs that need to be resolved before publishing.

First, the English language needs moderate changes. The term "transition" is used throughout the paper, but the term "translation" should be used instead.

I would suggest to the authors to read the papers from R.E.Deakin which can be found at http://www.mygeodesy.id.au/documents/Similarity%20Transforms.pdf  and http://www.mygeodesy.id.au/documents/COTRAN_1.pdf, I found them very interesting and useful.

It would really be interesting to see how this methodology work on a realistic set of data (geodetic, TRF) and not artificially created.

 

 

Author Response

Notes for Reviewer 3:

1. Any changes for the terminology have been done at all text [Lines: 20, 109, 122, 131, 136, 139, 142, 149, 150, 150, 181, 202, 210, 214, 217, 218, 229, 234, 236, 237, 252, 254, 261, 262, 319, 323, 331, 337, 341, 354]
2. The papers that proposed to us are so helpful, and one of them is also used as a reference when describing the least square method. [Line in the text: 154] [Lines in the references: 406-407]
3. A real data example is presented, and also some results in discussion. [Lines 273-315, 344-345, 352-353]

Reviewer 4 Report

Dear Authors,

I found this paper very interesting - especially to geodetic scientific population.

I have several recommendations:

• English proof reading
• line 45 - do you have original paper by Hamilton to quote?
• please use geodetic terminology - e.g. translation vector instead of transition vector
• line 61 - 72 - would recommend to read the article of Deakin ==> http://www.mygeodesy.id.au/documents/Similarity%20Transforms.pdf
• line 108 - do you have original paper by Awange and Grafarend to quote?
• line 126 - do you have original paper by Clifford to quote?
• dot goes after quoting - e. g. lines 128 & 150
• would like to see your methodology applied on real terrestrial coordinate reference frames

Author Response

Notes for Reviewer 4:

• The English language is proven to be acceptable both in grammar and in syntax, cause the manuscript is checked by native English speaker.
• The original paper of Hamilton was found, checked and added as a reference. [Line in the text: 49] [Lines in the references: 375-379]
• The main terminology have been replaced in all the manuscript [Lines: 20, 109, 122, 131, 136, 139, 142, 149, 150, 150, 181, 202, 210, 214, 217, 218, 229, 234, 236, 237, 252, 254, 261, 262, 319, 323, 331, 337, 341, 354]
• This article was so useful and was used as a reference when describing the least square method. [Line in the text: 154] [Lines in the references: 406-407]
• The original paper of Awange and Grafarend was found, checked and added as a reference. [Line in the text: 117] [Lines in the references: 391-393]
• Also the original paper of Clifford was added as a reference. [Line in the text: 135] [Lines in the references: 396-397]
• The dot correction have been done. [Lines: 137, 171]
• A real data example is presented, and also some results in discussion. . [Lines 273-315, 344-345, 352-353]