A Novel Robust Approach for Computing DE-9IM Matrices Based on Space Partition and Integer Coordinates

Round 1

Reviewer 1 Report

Paper “A novel robust approach for computing DE-9IM matrices based on space partition and integer coordinates” describes a method to build intersection matrices (DE-9IM) using integer coordinates and overcome the rounding effect of coordinates that might compromise the correct extraction of topological relations.

The article is of interest to the journal, as topological queries between vector features are a part of regular GIS operation, and it follows previous published research on the topic.

Basically, the proposed method starts by calculating the extent of the envelope where the features lie and uses this to represent endpoints using “virtual” integer coordinates guaranteeing the topological equivalence between the entry data and the represented model.

I suggest some changes in the text to improve the presentation.

• Figure 1. It is not clear if features “a” (point) and “b” (line) belong to the set with areas 1 to 5 or to the set with areas A and B.
• Line 42. Please refer here (and not only later in the text) which “conventional geo tools” [sic] are used in the comparison. Are they really conventional, and why are they conventional?
• Lines 48-49. It is not clear how integer coordinates can be used to check geometrical properties of features.
• Line 91. Please correct “from of” to “from”
• Line 146. Please correct “In Figure 3” to “Figure 3”
• Line 155. The “floor” operator is well known, so there is no need to say that “operator returns the integer parts”
• Lines 174-180 and Figure 4 discuss and explain how the “translation” to integer coordinates might generate topological inconsistencies. The text refers to the need of checking out these cases, but no further details are provided. How could that test be made? Is it needed only when the 64-bit integers are not used?
• Line 196. Section 3,4 title should be “Space Partition” instead of “Space Partion”. Please confirm
• Section 3.4 and Figures 6, 7 and 8 are very confusing. For instance, in Figure 6, what is the green arrow symbolizing? What is the rotative arrow in image b)?
• In Figure 9, please change “Interieur” and “Exterieur” to “Interior” and “Exterior”. Or just delete Figure 9, as it is trivial and the reader should understand the I/B/E model by now.
• Lines 374-377: “In this way (…) should be covered”. This is repeated.
• Table 8. Please refrain from using colors in the table. It is possible to distinguish the data loss cases using symbols or something in regular writing
• Change all occurrences of “you” to a more neutral English discourse. For instance, in line 26, replace “If you compare features from the same source, you may be able to avoid this problem” with “If features from the same source are compared, it is possible to avoid this problem”. Look for other calls to the readed in this style along the text.
• If possible, please discuss, with presented examples, some of the apparent fails that are due to the truncated coordinates after scaling. If the scaling factor is itself “converted” to an integer, it would not be an issue, or would it? I think this deserves more input from the authors as it appears that the conclusion is that the method to build up the intersection matrices is only reliable for “all cases” when 64-bit integers are used.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

In this paper, a new algorithm is proposed to guarantee consistency between geometrical and topological relations of 2D features. Overall, the topic is interesting. However, there are some problem for improvement by authors. Some specific suggestions and comments are as follows.

(1) Figure 1: What is the relations between Figure 1(c) and Figure 1(d)? Is Figure 1(d) generated based on Figure 1(c)? If yes, how to generate Figure 1(d) from Figure 1(c)?

(2) Section 3.5: The OGC type of contains not only Point, Line and Region, but also MultiPoint, MultiLine and MultiPolygon and GeometryCollection. Why Multipoint, MultiPolygon and Geometrycollection are not considered in this article.

(3) Table 3: The bottom edge of Table 3 is missing. The meaning of "X", "Interior0" and other elements need to be explained further.

(4) Equation 12: Equation 12 has some ambiguities. From the perspective of Equation 12, the exterior and the union of interior and boundary are not equal. In fact, the exterior is the complement of the union of interior and boundary. Equation 12 cannot express the relationship between them very well.

(5) Table 4 and Figure 10: The content of the "Calculation" field description in Table 4 is inconsistent with Figure 10. For instance, does "Interior2A" in Table 4 refer to the A.2 in Figure 10?

(6) Based on the 9IM, 33 topological relations can be realized between simple lines, 19 topological relations can be realized between a region and a line (Egenhofer and Herring, 1991). The DE-9IM can distinguish more topological relations than 9IM (Shen et al., 2018). Why only part of topological relations are selected in Figure 12? Is there anything special and representative of these topological relations in Figure 12?

References:

Egenhofer, M. & Herring, J., 1991. Categorizing binary topological relations between regions, lines, and points in geographic databases. 9, 1-28.

Shen, J., Chen, M., & Liu, X. . (2018). Classification of topological relations between spatial objects in two-dimensional space within the dimensionally extended 9-intersection model. Transactions in Gis, 22(2), 514-541.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

The approach of this article is very interesting in that it uses an integer representation of coordinates in order to try to provide an exact calculation of intersections in 2D Euclidean geometry. However, there are some basic unclear issues which need to be addressed in order to make this article publishable. 

For these reasons I recommend a major revision.

My specific concerns are the following:

1. Introduction
The authors state: 

"If, for example, features are given in a topological valid half-edge data structure, this does not ensure that the explicitly specified topology equals the geometrically calculated implicit topology."

This is precisely the issue that led the authors of 

A. Giovanella et al. Evaluation of Topological Consistency in CityGML.
ISPRS Int. J. Geo-Inf. 2019, 8, 278

to introduce their version of "topological consistency". Their definition was seen to be equivalent with the ISO 19107 Standard. 

Further, the authors promise to provide

"2. An algorithm for generating a complete, gapless and non-overlapping spatial decomposition."

In the article 

Jahn, M. W. et al. (2017). Topologically Consistent Models for Efficient Big Geo-Spatio-Temporal Data Distribution. ISPRS Annals of the Photogrammetry, Remote Sensing and Spatial Information Sciences, 12th 3D Geoinfo Conference 2017, Melbourne, Australia, 26th - 27th October 2017, 65–72

an overlay method was defined for precisely this purpose.

For these reasons, the authors should compare their approach with these existing approaches.

3.2 Integer coordinates

In fact, integer coordinates are interesting because of their potential for exact calculation. However, this presupposes that all geometric entities have a similar range of coordinates and the uncertainty is not too large.
The authors assume an accuracy in the millimeter range for a region of up to 2000 km. It is highly questionable that one actually obtains such an accuracy range for real-world data, if their sizes are in the kilometer range. Even for actual buildings, this might be quite a challenge. This limits the applicability of their approach, I think. The authors should discuss this issue a bit more in depth.

Figure 4.
I did not understand, why the degeneration error or orientation error depicted does not occur. I see that their transformation to integer coordinates is a rounding procedure. So, please clarify this point better than just stating that conversion is lossless.

3.3 Intersections as Rational Positions
The authors claim that their intersection points are not represented by coordinates. But then as what? I did not understand this. In any case, it is not true that intersection points are always rational numbers, even if the endpoints of line segments have integer coordinates: e.g. take the unit square and intersect the two diagonals. The coordinates will not be rational numbers!
So, please clarify the representation of intersection points more explicitly, because this is most crucial for the remainder of your article.

Equations 9 and 10.

I do not understand what these determinant expressions have to do with the intersection points. Please clarify. Also, the intersection point of two line segements, if it exists, is unique. So, what is the meaning of the two expressions pos_ab and pos_cd ? Here, I am puzzled.

Table 8. Does N yellow mean that N have passed? And N red means that N failed?

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors have solved my concerns. I would like to see the paper published.

Author Response

No changes required.

Reviewer 3 Report

The authors have duly addressed all my concerns except for one: 

"Point 6: Equations 9 and 10. I do not understand what these determinant expressions have to do with the intersection points. Please clarify. Also, the intersection point of two line segements, if it exists, is unique. So, what is the meaning of the two expressions pos_ab and pos_cd ? Here, I am puzzled"

I still do not understand two things:

1. What does pos_{ab} mean? How is the value between 0 and 1 calculated "relative to edge ab"? (same question for pos_{cd}).
2. Why can this be expressed by a determinant?

This still puzzles me, I am afraid.

Once this is clarified, I can go on with reviewing the remainder of the article.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 3

Reviewer 3 Report

The authors have provided an example [eq. (11)] in an attempt to help me understand their approach in eqs. (9) and (10). This example reveals that the determinantal expressions in eqs. (9) and (10) do not yield the coordinates of the intersection point: the asserted coordinates in this example are wrong, because the intersection lies inside the quadrangle defined by the points a,b,c,d, but the asserted intersection point does not. The authors need to correct the determinantal expressions in eqs. (9) and (10) before the article to be publishable.   

Author Response

Reviewer:

The authors have provided an example [eq. (11)] in an attempt to help me understand their approach in eqs. (9) and (10). This example reveals that the determinantal expressions in eqs. (9) and (10) do not yield the coordinates of the intersection point: the asserted coordinates in this example are wrong, because the intersection lies inside the quadrangle defined by the points a,b,c,d, but the asserted intersection point does not. The authors need to correct the determinantal expressions in eqs. (9) and (10) before the article to be publishable.  

Response:

To make it clearer that formulas 9, 10 and 11 are not about coordinate calculations, but about the calculation of relative positions, the following changes have been made:

1. Caption of figure 5 changed.
2. Formula 11 with new values.
3. Figure 6 to show the values in formula 11.
4. Changes in lines 210-214 and 219-221.

Round 4

Reviewer 3 Report

The corrections are now made, and the article can be accepted.