Addressing incomplete tile phenomena in image tiling: Introducing the grid six-intersection model

Ling Zhang; Jinsongdi Yu; Ruiju Tong; Dandan Wei; Yu Fan

doi:10.1515/geo-2022-0628

Article Open Access

Addressing incomplete tile phenomena in image tiling: Introducing the grid six-intersection model

Ling Zhang , Jinsongdi Yu , Ruiju Tong , Dandan Wei and Yu Fan

Published/Copyright: May 15, 2024

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From the journal Open Geosciences Volume 16 Issue 1

Abstract

With the continuous development of Earth Observation technology, resolution of imagery and gridded data has significantly increased, leading to a rapid increase in data volume. To efficiently acquire and analyze these vast amounts of imagery and gridded data, image tiling technology has been developed to effectively access data of interested areas. Tiling technology divides large-scale image data into smaller tiles, providing fast, accurate, and efficient access support for imagery and gridded data. The spatial grid model, as the foundational framework of the new generation of geographic spatial information, plays a critical role in the retrieval, integration, services, and applications of imagery and gridded data resources. When tiling image data based on the spatial grid model, it always generates both complete and incomplete tiles. Particularly, when conducting image tile retrieval using the same rule-based grid in the retrieval area, incomplete tile phenomena along the boundary regions of the retrieved images often occur, resulting in gaps within the retrieval area. To tackle this issue, in this study, we present a new topological model called the Grid Six-Intersection Model (G-6IM), specifically designed for regular rectangular grids, to accurately represent boundary issues in image tiling. Through a practical case study, we demonstrate the effectiveness and practical application potential of the proposed G-6IM model, providing new insights and guidance for the improvement and optimization of imagery and gridded data tiling technology.

Keywords: imagery; grid; tiling; topology

1 Introduction

As Earth Observation technology continues to evolve and develop, the sources, resolution, and types of imagery and gridded data have significantly increased, resulting in a rapid growth in imagery and gridded data volume [1]. This development has facilitated widespread applications of imagery and gridded data in various fields, such as ecological environment monitoring [2] and climate change prediction [3]. However, the presence of multi-source, multi-resolution, multi-type, massive, and continuously expanding imagery and gridded datasets poses challenges in achieving faster, more accurate and efficient retrieval of imagery and gridded data [4]. To address these challenges, the technique of image tiling has emerged. It involves dividing large-scale image data into smaller blocks called tiles, enabling effective access to data of specific areas within expansive and continually growing repositories of imagery and gridded datasets [5]. The technology has been widely applied in the organization and management of imagery and gridded data across different domains [6], including online image services such as those provided by Google Maps and Tianditu.

Additionally, the ongoing development of Earth observation has propelled the advancement of spatial grid models, such as Quaternary Triangular Mesh (QTM) [7], Discrete Global Grid Systems(DGGS) [8,9], Spheroid Degenerated-Octree Grid (SDOG) [10], and GeoSOT grid [11]. These models serve as potential foundations for the next generation of geospatial reference frameworks, facilitating the retrieval, integration, and service of imagery and gridded data [12]. However, the application scope of QTM, DGGS, and SDOG is always restricted by high implementation difficulty, such as computational complexity [13]. By employing an appropriate spatial grid model, imagery and gridded data can be effectively tiled and managed, enabling multi-scale division and encoding, as well as the calculation of relationships and rapid access to image tile objects [14]. For example, GeoSOT-based approach is applied with metadata as the geolocation code to construct the tile’s rowkey and achieved efficient storage of remote sensing images [15], and to improve the retrieval and access efficiencies of large-scale and long time-series remote-sensing data in a cloud-computing environment [16].

When tiling imagery and gridded data based on the spatial grid model, it always generates both complete and incomplete tiles. Particularly, when conducting image tile retrieval using the same grid partitioning in the retrieval area, incomplete tile phenomenon along the boundary regions of the retrieved images often occur, such as resulting in gaps within the retrieval area, Figure 1. To illustrate this phenomenon, in this study, we present a new topological model called the Grid Six-Intersection Model (G-6IM), specifically designed for regular rectangular grids. The purpose of the model is to accurately represent topological relationships among tiles and ultimately enhance data access efficiency. Through a practical case study based on BeiDou grid [17] partitioning, developed based on GeoSOT, we demonstrate the effectiveness and practical application potential of the proposed G-6IM model, providing new insights and guidance for the improvement and optimization of imagery and gridded data tiling technology.

Figure 1

Retrieval of image tiles based on spatial grid model: complete and incomplete tiles in the retrieval area.

2 The state of art

Topological models based on point set topology theory [18] have been extensively studied in various domains [19]. These models include the Four-Intersection Model (4IM) [20], the Nine-Intersection Model (9IM) [21], as well as the Dimensionally Extended Four-Intersection Model, and the Dimensionally Extended Nine-Intersection Model (DE-9IM) [22].

The 4IM expresses the topological relationship between two geometric objects by considering the intersection or non-intersection of their interiors and boundaries. However, it does not take into account external conditions. In this case, overlap and meet are completely identical, leading to undetermined relationship descriptions [23]. The 9IM, building upon the 4IM, includes exterior of objects in its analysis, compensating this shortcoming with a finer granularity, as an expressive power improvement to distinguish more relations in a higher-dimensional space [23]. The dimensionally extended n-Intersection model (DE-nIM) extend these concepts by incorporating dimension extension, considering the intersection dimensions between various parts of the two spatial targets [19].

In 2D space, the DE-nIM matrix can have four possible values [22], indicating an empty intersection, intersection at a point, intersection along a line, or intersection within an area. The number of possible combinations theoretically reaches 4ⁿ. In 3D space, the DE-nIM matrix extends to include the possibility of intersection within a volume [19], resulting in five possible values and a theoretical total of 5ⁿ combinations. For example, the DE-9IM matrix has a total of 5⁹ = 1,953,125 potential combinations.

To reduce the complexity of topological models for spatiotemporal image queries, the EO-3IM [24], as a profile of DE-nIM, describes relationships between object internal and object internal, boundary, and external, to address image query of interested areas. However, the work does not consider the image partitioning cases. This study focuses on the context of relationships between tiled images and rectangular grids, without the need for dimensional extension. Therefore, G-6IM, as a special case of 9IM, is proposed to address the incomplete tile phenomena. In this model, the topological relationship between a rectangular grid retrieval space and image tile space can be described by binary decisions on spaces intersect or not, between the internal and external aspects of the grid element collection and the internal, boundary, and external aspects of the partitioned image.

Compared to 4IM, G-6IM takes into account the exterior of grid element collection, enhancing spatial retrieval accuracy and effectively addressing the issue of 4IM’s inability to distinguish between overlap and equal. This superiority is particularly notable in scenarios involving regularized tiled images, providing a more powerful tool for the precise description of topological relationships. Within the context of G-6IM, image tile space is composed of a rectangular grid and can be categorized into three distinct parts: the interior, the boundary, and the exterior. Similarly, the retrieval space, also constructed using the same grid partitioning, can be divided into two parts: interior and exterior, Figure 2.

Figure 2

Interior, boundary, and exterior of a tiled image.

3 Methodology

3.1 G-6IM

To illustrate spatial topological relationships of G-6IM, a 2 × 3 matrix is employed as below, which is described in formula (1). This matrix represents the intersections between the interior, boundary, and exterior of tiled image A, as well as the interior and exterior of retrieval area B in rectangular grids. The relationships are denoted as binary values, with 0 indicating non-intersecting regions and 1 representing intersecting regions. Theoretically, there exists 2⁶ = 64 possible topological relationships between two objects within G-6IM. However, not all of these relationships hold practical significance. Thus, it becomes necessary to identify and eliminate the non-existent topological relationships, while retaining those that are of practical relevance.

(1) G - 6 IM ( A , B ) = A ° ∩ B ° ∂ A ∩ B ° A − 1 ∩ B ° A ° ∩ B − 1 ∂ A ∩ B − 1 A − 1 ∩ B − 1 .

Regarding the definition of topological relationships between spatial objects, five predicates [22] were used to test spatial relations: disjoint, touch, within, cross, and overlap. Since the research objects are two-dimensional image tiles in a rectangular grid, the cross relationship does not exist in this case. Referring to additional test practice on tile retrieval, this study provides additional predicates based on OpenGIS spatial relation test methods [25], including contain, equal, and intersect, to improve the model expressive power. A detailed sematic of each topological relationship predicate in G-6IM, let A be the tiled image and B be the retrieval area in rectangular grids, is shown in Table 1.

Table 1

Description of the topological relation predicates

Predicates	Description	Semantic
Disjoint	The interior and boundary of A do not intersect with the interior of B	A ° ∩ B ° = 0 ∧ ∂ A ∩ B ° = 0
Touch	The boundary of A intersects with the interior of B	A ° ∩ B ° = 0 ∧ ∂ A ∩ B ° = 1
Within	A is completely contained within the interior of B, and the interior and boundary of A do not intersect with the exterior of B	A ∩ B = B ∧ A ≠ B
Contain	B is completely contained within the interior of A, and the interior of B does not intersect with the exterior of A	A ∩ B = A ∧ A ≠ B
Equal	A equals B	A = B
Overlap	The interior, exterior, or boundary of A intersect with the interior of B	A ° ∩ B ° = 1 ∧ A ‒ 1 ∩ B ° = 1
Intersect	The interior or boundary of A intersect with the interior of B	A ° ∩ B ° = 1 ∨ ∂ A ∩ B ° = 1

Note: A = A ° + ∂ A , B = B°.

3.2 Model verification

Shuttle Radar Topography Mission ocean floor depth data [26] are used to validate the G-6IM model. The data are partitioned based on the BeiDou first-level grid code coding standards [17], Figure 3. The retrieval area is designated using the BeiDou grid location code, with N22D serving as the starting grid and longitude and latitude step codes of 2 and 3, respectively, Figure 4.

Figure 3

Encoding structure and symbolic representation of 1–8 level BeiDou grid code [17].

Figure 4

Validation area under the BeiDou first-level reference grid.

Utilizing the G-6IM model, we validated the topological relationships described in Table 1. To present the findings, verification results for each topological condition is shown, distinguishing between those with boundary tiles and those without boundary tiles, Figure 5.

Figure 5

Topological predicate verification designed for G-6IM.

In Figure 5, formula (1) result is expressed as a binary string, such as “011101.” This topological code is arranged based on intersections between the interior, boundary, and exterior of the image as well as the interior and exterior of the retrieval area. Each digit (1 or 0) in this combination indicates whether there is an intersection between the corresponding positions of the two objects. Specifically, these are the results of interior–interior, boundary–interior, exterior–interior, interior–exterior, boundary–exterior, and exterior–exterior relationships. For instance, the second digit indicates whether the boundary of image A intersects with the interior of retrieval area B, and the fifth digit indicates whether the boundary of image A intersects with the exterior of retrieval area B.

In this way, G-6IM predicates are verified. Furthermore, the second and fifth digit in this matrix can be used to determine if there are boundary tiles, concretely incomplete tiles, in the results.

4 Study case

Marine species distribution modeling aims to predict the potential geographic spatial distribution of marine species by utilizing marine environmental data and species observation data [27]. Typically, the Biomod2 model, combined with the R programming language, can be employed for distribution forecasting. The research contributes to the understanding of ecological niche characteristics, potential spatial distribution, and the response of marine species to climate change [28–30]. It is of significant importance for marine environmental protection, fisheries management, and ecological disaster response [31–33]. However, compared to terrestrial environmental data, the collection of marine environmental data presents challenges due to its relative scarcity and multiple dispersed sources. The searching and identification of available environmental variables within the research area pose significant difficulties in this field.

4.1 Data and methods

In this study, the topological relationships defined by G-6IM were employed as constraints to integrate data of research areas in previous studies, facilitating the efficient integration of research data in the specified area of interest. There are a number of studies on marine species distribution prediction. Early work by Stirling et al. [34], Palialexis et al. [35], Tyberghein et al. [36], Guillaumot et al. [37], Robinson et al. [38]; and recent efforts, such as those by Waldock et al. [39], Maureaud et al. [40], Chaudhary et al. [41], Gamliel et al. [42]. Seventeen study areas for species distribution modeling are collected in Table 2. The geographic data are partitioned according to BeiDou grid code rules, Figure 3, including complete and incomplete tiles at different division levels. Starting grid code and step codes are used to construct retrieval areas based on regions of research interest. By using G-6IM, users can integrate complete tiles from different images, thus achieving accurate retrieval of interested area. The corresponding technical route is shown in Figure 6. BeiDou grid is calculated in Javascript, BeiDou grid code is calculated in JAVA, and result is represented in ArcGIS.

Table 2

Geographical scopes of the study areas in previous works on marine species distribution modeling

Citation	X _min	X _max	Y _min	Y _max
Braga-Henriques et al. [43]	−35.75°	−21.08°	33.77°	48°
White et al. [44]	−20°	−8.7°	46.3°	58.9°
Hu et al. [45]	−97°	−82°	22°	30.5°
Modica et al. [46]	−6.67°	−5.41°	43.58°	44.08°
Abad-Uribarren et al. [47]	−3.5°	−2.5°	43.5°	44.5°
Kutti et al. [48]	4.7°	6.5°	59.7°	60°
Leverette et al. [49]	−76°	−60°	38°	48°
Brooke et al. [50]	−76°	−60°	36.5°	38.5°
Stevenson et al. [51]	−15°	−5°	45°	55°
Garcia-Guillen et al. [52]	−6.6°	−5.47°	43.74°	44.03°
Gomez-Ballesteros et al. [53]	−6.78°	−5.31°	43.63°	44.31°
Somoza et al. [54]	−12.25°	−11.5°	42.25°	43°
Hebbeln et al. [55]	−7.25°	−6.67°	35°	35.5°
Guinan et al. [56]	−15.2°	−12.2°	52.2°	53.9°
Guinan et al. [57]	−17.7°	−7.8°	49.7°	56.6°
De Froe et al. [58]	−16.1°	−15.8°	55.4°	55.7°
De Mol et al. [59]	−5.15°	−2.86°	46.75°	47.08°

Figure 6

Research technical route.

The North Atlantic was used as a case study, Figure 7. The interested area and each of the spatial areas of the 17 species distribution modeling works were mapped onto the first and second-level grids of the BeiDou grid location code [17].

Figure 7

Interested area and research areas in BeiDou grid partitioning.

To explore the interested area 1, which utilizes the BeiDou grid location code N26I in the first level grid, with a longitude step code of 3 and a latitude step code of 7, the G-6IM is employed to reveal the incomplete tile phenomena as indicated by the topological codes, Table 3.

Table 3

G-6IM values for the relationship between interest area 1 and research area

G-6IM	Interest area 1 (BeiDou grid location code = N26I, longitude step = 3, latitude step = 7)
Research area	Touch	Within	Contain	Equal	Overlap
Braga-Henriques et al. [43]	×	111001	×	×	×
White et al. [44]	011011	×	×	×	×

The bold “1” represents that incomplete tiles are included in the results.

As illustrated in Figure 8, in the retrieval results [43,44], the blue blocks represent the complete tile area, and the yellow blocks represent the incomplete tile area in first level grid. Consequently, by refining the partitioning levels, which utilizes the BeiDou grid location code N26IB0 in the second level grid, with a longitude step code of 36 and a latitude step code of 56. As illustrated in Figure 8, the blue and yellow blocks represent the complete tile area, and the green blocks represent the incomplete tile area. In this way, more complete tiles can be included and further referenced for the subsequent research. As more and more species distribution modeling works are incorporated, the interested area will ultimately be completely covered by complete tiles of different sources, as shown in the interested area 2, Figure 9.

Figure 8

Retrieval results constrained using G-6IM intersection under the first and second-level BeiDou grid.

Figure 9

Retrieval results constrained using G-6IM intersection under the first-level BeiDou grid.

To explore the interested area 2, which utilizes the BeiDou grid location code N29M in the first level grid, with a longitude step code of 2 and a latitude step code of 1, G-6IM is employed to eliminate the incomplete tiles as indicated by the topological codes, Table 4. The interested area is ultimately completely covered by complete tiles from White et al. [44], Stevenson et al [51], and Guinan et al. [57].

Table 4

G-6IM values for the relationship between interest area 2 and research area

G-6IM	Interest area 2 (BeiDou grid location code = N29M, longitude step = 2, latitude step = 1)
Research area	Touch	Within	Contain	Equal	Overlap
(2) White et al. [44]	×	×	110111	×	×
(3) Stevenson et al. [51]	×	×	×	×	010011
(4) Guinan et al. [57]	011011	×	×	×	×

The bold “1” represents that incomplete tiles are included in the results.

4.2 Results discussion

In this study, a marine species distribution modeling case based on the BeiDou grid partitioning is used to demonstrate the practical application potential of the G-6IM model. The main objective of the model is to accurately represent topological relationships among tiles, thus enhancing data access efficiency.

The BeiDou grid partitioning serves as a representative example of the challenges encountered in image tiling. When tiling imagery and gridded data based on a spatial grid model, both complete and incomplete tiles are generated. The incomplete tile phenomenon, particularly along the boundary regions of the retrieved images, often results in gaps within the retrieval area. These gaps can significantly impact the data access and analysis, leading to incomplete or inaccurate results.

The G-6IM model addresses this issue by providing a topological representation that accounts for the relationships among tiles. Through a detailed analysis of the BeiDou grid partitioning, how the G-6IM model effectively characterizes the topological structure of the grid, identifies incomplete tiles, and closes gaps within the retrieval area is demonstrated. By accurately representing topological relationships, the model provides a potential to facilitate more effective spatial analysis and querying in a range of applications, such as remote sensing, environmental monitoring, and urban planning.

5 Conclusion and future work

In order to accurately and effectively access data of interested areas, this study proposes a G-6IM model for grid-based spatial relationships for imagery and gridded data tiling. It differentiates between the internal and external regions of rectangular grid retrieval areas, as well as the internal, boundary, and external regions of tiled images. This model enables the description of spatial topological relationships between rectangular grid retrieval areas and tiled image regions under the same grid subdivision. The model’s application is demonstrated through the prediction of marine species distribution under BeiDou grid, aligning with the needs of marine species distribution modeling. It provides support for the acquisition of imagery and gridded data. Future research holds potential for further improvement and optimization of the grid-based topological model. This includes exploring the topological relationships between grid retrieval areas and tiled image data under different subdivision rules, levels, and dimensions. For instance, this may encompass, but is not limited to, QTM, DGGS, or SDOG.

Regarding application cases, future work can further explore several domain implementation avenues, such as research in the field of species distribution modeling. A comprehensive and up-to-date database can be established to archive various marine species distribution modeling studies and include relevant grid-based metadata information. The development and maintenance of such a database would have profound implications for the advancement of marine species distribution modeling, marine conservation, and resource management. It could serve as a valuable resource for researchers, practitioners, and policymakers seeking to deepen their understanding of species distribution patterns and inform effective conservation strategies. Additionally, continuously updating the metadata associated with these works would ensure that the scientific community has access to the most recent and pertinent information in this field.

Acknowledgements

This work is supported by the National Key R&D Program of China, No. 2019YFE0127100.

Funding information: This work was funded by the National Key R&D Program of China, No. 2019YFE0127100.
Author contributions: L.Z.: conceptualization, data curation, formal analysis, investigation, methodology, study case, visualization, validation, writing – original draft preparation, writing, review, and editing; J.Y.: conceptualization, data curation, formal analysis, investigation, methodology, funding acquisition, project administration, resources, supervision, validation, writing – original draft preparation, writing, review, and editing; R.T.: data curation, study case, visualization, validation, writing, review, and editing; D.W.: data curation, formal analysis, validation, writing, review, and editing; Y.F.: data curation, study case, visualization, writing, review, and editing.
Conflict of interest: The authors have no conflicts of or competing interests to declare that are relevant to the content of this article.
Data availability statement: The data associated with this study are listed in Table 2. The data are processed by python and visualized in ArcMap.

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Received: 2023-09-16

Revised: 2024-01-10

Accepted: 2024-03-08

Published Online: 2024-05-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/geo-2022-0628

Keywords for this article

imagery; grid; tiling; topology

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