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

I have read the article by Ioannidou and Pantazis [1]. However, I respectfully disagree with their article. I would like to point out several methodological points and conclusions that need to be examined more closely. I have serious doubts about Ionnadis and Pantazis’ article.

My sources of concern are as follows:

1.

The authors say in that study (page 4, line 31–33), “the unknown parameters are nine: four for each quaternion that represents the rotation and translation and one for scale factor”. Their model had nine parameters. The authors did not provide detailed information about the adjustment model. Only one equation, Equation (12), is given for the dual quaternion transformation.

$${\mathit{P}}_{\mathit{i}}^{\prime }=2{\mathit{W}}_{\left(\mathit{r}\right)}^{\mathit{T}}\mathit{d}+\mathbf{\lambda }{\mathit{W}}_{\left(\mathit{r}\right)}^{\mathit{T}}{\mathit{Q}}_{\left(\mathit{r}\right)}{\mathit{P}}_{\mathit{i}}\mathbf{,}\mathbf{\cdots }\mathbf{i}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{\dots }\mathbf{ }\mathbf{m}$$

(12)

It is assumed that they use unit quaternions in their study, but additional constraint equations would have to be found for unit quaternions to exist. However, no additional constraints are mentioned in the study.

Attempting to solve it using the model based on Equation (12) in that article, it is seen that the normal equations will be singular, so the solution cannot be obtained.

For this model to be valid, two additional conditions (unity and orthogonality) are required.

These are as follows:

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

2.

The data and results in the article are hidden. Therefore, it is not possible to check the results of the article.

3.

The main evidence that the results of the article are wrong:

In this article (pages 7–13), the authors compared the transformation parameters they found with Euler angles, the quaternion method, and the dual quaternion method. Presumably, they concluded from this comparison that some parameters are sensitive to the Euler angle method, some to the quaternion method, and some to the dual quaternion method. However, it is indisputable that Euler angles, the quaternion method, and the dual quaternion method give the same results as [2,3,4,5]. Since all three methods are derived from the Helmert similarity transformation, this shows that the model and results of the article are incorrect.

4.

The errors in the article cannot be explained by some minor processing errors. In the article, the model was set up incorrectly and extremely erroneous judgments were made. As I have explained above, the authors claim in the article that the Euler angle method, the quaternion method, and the dual quaternion method give different results. This judgement is completely wrong. It is undeniable that Euler angles, the quaternion method, and the dual quaternion method give the same results because all three methods are derived from the Helmert similarity transformation. This judgement of the authors contradicts the findings in the existing literature.

We feel that it is appropriate to publish this comment so that readers are properly informed.