A Fourier Fitting Method for Floating Vehicle Trajectory Lines
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThis paper proposes an approach based on Fourier series for more efficient modeling of discrete position-time data from floating vehicles. In this paper, the trajectories of moving vehicles are treated as a time-varying signal, which is transferred to the frequency domain via Fourier transform. The trajectory data is modeled in the spectral domain in terms of both position and velocity components and transformed into a continuous function. Furthermore, error thresholds are defined based on visual resolution and velocity accuracy criteria based on human perception, and the number of terms to represent the Fourier series is calculated according to these accuracy levels. The experiments show that the proposed method provides high accuracy and adaptability, and that the Fourier approach offers significant advantages when compared with traditional Bézier curves. In this respect, the study contributes to the effective representation of trajectory data in urban mobility analysis and geographic information systems.
Abstract: The abstract section of the paper provides information about the general purpose of the study and the methodology used, but does not include numerical data on the findings and results obtained. However, the abstract is expected to present not only the methodology but also the scope of the study and the main conclusions reached, briefly and, if possible, with quantitative outputs. Therefore, it is suggested that the conclusions section of the abstract be strengthened to be more specific and data-driven.
Introduction: The Introduction outlines the main challenges (high dimensionality, irregularity, semantic complexity, etc.) encountered in the analysis of moving objects with increasing spatiotemporal data generation and justifies the need for data simplification/functional modeling. In addition, it presents a brief literature review of existing data fitting approaches (linear regression, polynomial and spline interpolation, machine learning based methods, etc.) and demonstrates the importance of the topic.
However, this introduction does not clearly enough emphasize the shortcomings of previous works in the literature and the original contribution of this study. The existing literature generally focuses on fitting in the spatial dimension only; models that are integrated with velocity information, continuously changing over time, and transformable into the frequency domain are rarely considered. In this context, the simultaneous modeling of position and velocity components with Fourier series emerges as an important innovation, which should be emphasized more clearly in the introduction.
The ways in which studies in the literature are inadequate (e.g., not including velocity information, not performing Fourier-based holistic modeling, ignoring visual accuracy criteria) should be discussed more clearly.
How the presented method fills these gaps and how it differs from previous approaches (e.g. robustness of the Fourier transform, parametric flexibility, possibility of joint velocity-location modeling) should be clearly stated.
At the end of the introduction, the structure of the paper (a brief summary of the chapters) should be included, but the objective and contribution of the paper should be concluded with a clearer paragraph emphasizing its originality.
Fourier Series Model of Trajectory Data:
Although subheadings 2.1 and 2.2 are functional in their current form, the section “2.2 Fourier Transform of Trajectory Data” is quite dense and contains several separate conceptual stages. This sub-heading could be made more readable by dividing it into sub-headings such as “Fourier Series Construction” and “Periodic Extension and Gibbs Phenomenon Avoidance”.
If the process of applying the Fourier model is summarized in a step-by-step diagram or algorithmic flow, the chapter becomes more accessible, especially for practitioner readers.
It should be more clearly explained why the trajectory data should be periodicized and how this avoids Gibbs phenomena. The advantages and limitations of this transformation can be briefly discussed in a theoretical context.
A brief methodological limitation could be emphasized that the Fourier series is only suitable for smooth functions or functions satisfying certain continuity conditions.
Fitting Method of Trajectory Data:
How control points are determined from the road network is well explained in the logical steps of Algorithm 1. However, this process is quite technical and lacks description, except for visual support.
A brief discussion of the situations in which the algorithm may fail, e.g. sensor failure, GPS drift, etc., could be added.
The three sub-topics seem to be separate from each other. A short synthesis paragraph at the end of the chapter should bring the whole fitting process together and briefly answer the question “why does this method work better?”.
Test and Analysis:
It is stated that the data used in the chapter is taxi trajectory data from Wuhan city, dated September 1, 2023, and the data set details are presented in a table. This transparency is positive.
However, more explanation about the diversity and representativeness of the dataset (e.g. different time periods, different traffic conditions, different road types) is important for the generalizability of the results.
The four-stage implementation process of the proposed method is systematically described. These steps complement the previous methodology section well.
However, there is some repetition between these steps; for example, instead of summarizing the previously described formulas here, the results of their direct application could be emphasized.
The graphical distribution of the error values (e.g. box plot or histogram) could be given to show the stability of the method more clearly.
A direct comparison of the proposed method with other methods (e.g. Bézier fitting) could also be summarized in a short table under this heading (the current comparison is in chapter 5).
The experimental results are generally successful, but it is not discussed when the method may perform poorly (e.g. in cases of extremely sparse sampling, sudden speed changes, sharp turns, etc.).
Identifying such potential limitations would increase the scientific credibility of the study.
Comparison between Fourier Fitting and Bézier Fitting of Trajectory Data:
The comparison was made only with 3rd order Bézier curves. While this limitation is justified, it is not methodologically justified why this order was chosen.
If different degrees of Bézier curves have been tried, the results should be shared; if not, this should be mentioned as a limitation.
It is stated that the Fourier approach can also model velocity information, whereas Bézier only models position. This important difference could be emphasized more strongly with a table or example.
Images such as Figure 10 make the comparison intuitively understandable. However, the explanations underneath these figures could be more technical and analytical. For example, why the Bézier curves deviate more and under what conditions this difference grows.
It is emphasized that the Fourier model is more easily questionable due to its continuous and singular function structure, whereas the Bézier method is complex because it consists of multiple parts. This comparison is very appropriate and original contribution. Perhaps sample query scenarios should be presented in a short text box for a clearer understanding of this section.
Discussion and Summary:
The “Discussion and Summary” section summarizes the overall contributions of the study and collectively evaluates the advantages of the method. However, the discussion aspect of this section is weak; it is mostly repetitive and praise-based, with limited critical evaluation and future directions.
It is reiterated that the method is successful in terms of accuracy, flexibility and visual resolution, but aspects such as under which conditions these successes are more pronounced and for which data types they may be limited are not discussed.
The aspects in which the proposed method is more successful compared to previous studies should be discussed in comparison not only with Bézier but also with other methods mentioned in the literature (e.g. spline, polynomial, machine learning based).
Potential uses of the method (e.g. traffic density forecasting, event analysis, road network optimization) are only mentioned in general terms. This section would be more meaningful if application scenarios are briefly mentioned with concrete examples.
In the current version, the limitations section is very short and superficial. The following should be explicitly mentioned:
- How scalable is the Fourier approach for large data sets in terms of processing time and computational cost?
- Can it be used for real-time analysis?
- How is its performance affected by data with dense intersections or GPS errors?
“Future work” section is only a few sentences. Future work should be presented more clearly.
Author Response
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Author Response File: 
Author Response.pdf
Reviewer 2 Report
Comments and Suggestions for AuthorsPresent contribution represent interesting and relevant work with significant outcomes for territorial planning and policies. Nevertheless, some minor changes are required:
a. english language must be improved, further proof-reading is suggested;
b. State-of-the-art considering applied methodologies and algorithms in Section 1 could be more valued. I would suggest to separate introductory part from this subsection. In this direction, introduction could be enriched describing past case-studies (from academic literature) and/or potential urban applications;
c. I would consider summing up investigated methodolologies, through a dedicated table showing respective pros and cons;
d. I suggest further expansion of conclusive remarks, thus highlighting planning and policies operational implications.
Comments on the Quality of English LanguageI would suggest further proof-reading since general syntax prove to be quite complicated and hard-to-be read. Anyway, no typos are present.
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe paper proposes a method that aims at improving the integrity and accuracy of trajectory data by combining the network topology information with the computation of Fourier series data. The proposed approach is interesting, sound and relevant. Some comments for the authors are listed as follows.
1) Section 1 does not introduce the work done since only the last paragraph of this section is about the content of the current manuscript.
2) To improve the readability of the text, it is important to align the line numbering of Algorithm 1.
3) The text is dense, full of mathematic definitions and contains many processing specifications in algorithmic language. Thus, it would be interesting to have a running example to be used throughout the paper or some examples in the text to illustrate the content presented by the authors.
4) The main disadvantage of the article is that it does not present the state of the art. For example, the area of transportation systems is cited but the state of the art in this area is left out of the current version of the article.
5) Section 6 is a mere summary of what has been presented in the paper. What is missing though is a discussion, carefully comparing strengths and weaknesses of the proposed approach in comparison with related work. Many basic definitions are given but the state of the art is not discussed in the current article.
Author Response
Please see the attachment.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Comments and Suggestions for AuthorsDear Editor,
The authors have revised the manuscript in line with the detailed feedback provided in the previous review. Notable improvements have been made in several key sections. The abstract has been expanded with more specific and quantitative findings. The introduction now more clearly emphasizes the limitations of existing studies and the original contribution of this work, particularly in integrating velocity information into Fourier-based trajectory modeling.
Section 2.2 has been restructured with clearer subheadings and a process diagram, improving readability. The rationale for periodic extension and its relation to the Gibbs phenomenon is now more thoroughly explained. The limitations and assumptions of the Fourier approach are appropriately discussed.
In the methodology and experimental sections, additional clarifications have been added regarding data diversity and potential limitations under specific conditions. The comparison with Bézier curves has been deepened, both conceptually and visually, with improved technical commentary. The discussion and conclusion sections have also been strengthened with more critical reflection and forward-looking insights.
In summary, the authors have addressed the prior concerns comprehensively. I find the revised version to be significantly improved and suitable for publication.
Kind regards,
