Incremental Updating of 3D Indoor Data Considering Topological Linkages

Abstract

Indoor location-based services and applications are heavily dependent on the currentness of indoor data. Therefore, it is crucial to update indoor spatial information promptly and efficiently to ensure its relevance and reliability. Maintaining the topological consistency of geometric objects presents a significant challenge in updating indoor data. Consequently, this paper introduces an incremental updating method for 3D indoor data that considers topological linkages. The first step involves categorizing different types of building component changes and their corresponding indoor space alterations, followed by a detailed analysis of the topological linkage types for indoor features. On the basis of these identified changes, a set of updating operators is developed to handle various types of indoor space alterations. The experimental results demonstrate that the proposed updating operations effectively maintain the topological relationships of solids and the topological adjacency relationships of adjacent solids. This method facilitates efficient querying of indoor spatial information and topological adjacencies, thereby providing a robust data foundation for indoor location-based services and applications.

1. Introduction

With the rapid development of indoor positioning technology, indoor location-based services and applications, such as indoor navigation [1] and indoor emergency rescue [2], are gradually being realized and are expected to play an increasingly significant role in daily life. Indoor location-based services critically depend on timely and accurate indoor information regarding the layout and interrelationships of indoor spaces. Three-dimensional indoor models, which capture data on various functional spaces and building components, such as rooms, staircases, walls, ceilings, doors, and windows, organize indoor spaces into spatial units and their interrelationships [3,4,5]. Due to factors such as interior renovations, indoor spatial information is subject to frequent changes. When these changes are not incorporated into indoor spatial databases in a timely manner, discrepancies between the model and the actual indoor environment can lead to errors or incomplete results in applications [6,7]. Therefore, to maintain the accuracy and currency of indoor data, it is critical to continuously and effectively update the data reflecting spatial changes, ensuring the provision of reliable indoor applications [8,9].

Spatial data updating involves supplementing, correcting, and refreshing the existing database with current, high-accuracy data to reflect the real-world situation and maintain logical consistency in the data [10,11]. Research in this field has focused primarily on updates to cadastral databases, 3D cadastral objects, 3D city models, and 3D indoor scenes. Topological relationships among land parcels form the foundation for updating and maintaining cadastral data. Since various topological consistency constraints exist among parcels, boundary lines, and boundary points, such as the prohibition of gaps or overlap between adjacent parcels [12], the topological consistency of spatial data must be maintained during cadastral data updates [13]. Three-dimensional cadastre incorporates 3D property units within traditional 2D cadastral systems [14], providing a legal framework to define and describe rights in 3D space [15]. In 3D cadastre management, changes in property ownership necessitate updates to 3D land parcels, and the topological relationships among these parcels must be maintained during updates [16]. For example, Shi et al. [17] categorized update operations due to property changes as either subdivision or merging and proposed a 3D topological model-based algorithm that facilitates these operations while preserving the topological relationships. While 3D cadastral data models can represent indoor spaces enclosed by building components, they primarily address 3D property rights spaces defined by legal boundaries rather than physical boundaries [18,19]. Consequently, methods for updating 3D cadastral data are not entirely applicable for updating 3D indoor data.

In the process of updating 3D city models, it is essential to ensure that objects do not intersect, overlap, or penetrate one another and that updates do not disrupt the geometric and topological consistency of the dataset [20,21]. As a result, maintaining geometric and topological consistency during the update of 3D city objects presents a critical challenge. To address this, Gröger et al. [22] introduced a set of update rules, including face insertion, face deletion, solid splitting, and solid merging, to preserve the geometric and topological consistency. Gröger et al. [23] subsequently expanded these rules to incorporate face splitting and merging, as well as vertex movement of geometric objects, and provided comprehensive proofs of the validity and completeness of these rules. By mapping these rules to spatial databases, topological consistency is preserved when 3D city models are updated. However, since 3D city models do not represent detailed indoor building components, these update rules are only partially applicable for updating 3D indoor objects and are not suitable for handling updating operations of more complex indoor spatial objects [24].

Three-dimensional indoor scene updating involves processing changes within a real-world indoor environment (such as the movement of furniture) to maintain the current relevance of 3D indoor scenes, thereby facilitating 3D indoor location-based visualization services [25]. Liu et al. [26] proposed an automated method to achieve this by employing an RGB-D camera to capture scene changes, allowing for targeted updates through the matching of captured objects with those in the previously reconstructed indoor scene. Similarly, Ma et al. [27] introduced a real-time, image-based method for automatic matching and updating on the basis of indoor location and 3D visual perception. This approach uses an indoor localization algorithm combined with multi-sensor data to extract textures from real-time images, automatically integrating them into the 3D model to ensure automatic texture updates. Ma et al. [25] developed a method that leverages smartphone images to automatically update the textures of 3D indoor scenes. This method uses smartphone images as data sources and employs positional parameters to capture 3D layouts within the images, which are then mapped onto the indoor geometric model to facilitate automatic texture mapping and updating.

Current research on updating indoor spatial data has focused predominantly on updating 3D indoor scenes without addressing updates to the geometric representations of indoor spaces. This limitation restricts the ability to meet the demands of indoor location-based services and applications in increasingly dynamic indoor environments. Unlike updating 3D city models, cadastral ownership models, or 3D indoor scenes, 3D indoor data updating presents unique challenges and requirements. In this work, we identify and categorize the types of changes in building components and corresponding indoor spaces according to the characteristics of indoor data updating, and then analyze the topological linkages associated with these changes in building components. On the basis of these analyses, we propose updating methods tailored to various types of indoor space changes, achieving the effective updating of 3D indoor data and the maintenance of topological consistency.

The remainder of this paper is organized as follows. In Section 2, we discuss the issues of 3D indoor data updating. The different types of changes in building components are categorized in Section 3. In Section 4, we analyze the topological linkage types for indoor features on the basis of these changes. In Section 5, a set of updating operators is established to address indoor space alterations. In Section 6, experiments are conducted to update 3D indoor data. Some conclusions and future work are given in Section 7.

2. Issues with 3D Indoor Data Updating

Indoor data updating involves the targeted revision of spatial data within indoor environments to reflect modifications in layout, function, or use, thereby maintaining currency and accuracy. An incremental updating approach [28] is generally recommended for this purpose. Indoor space changes can be broadly categorized into two types: one where the attributes of indoor spaces change without altering the geometric characteristics, such as changes in the function, use, or occupants of a room, and the other where changes in the geometry accompany changes in attributes, such as room renovations. Updating the geometric changes in indoor spaces is the primary task in indoor data updating. These changes in geometric characteristics include alterations in the position, size, shape, or combination of geometric objects, and handling these changes may alter the topological combination of an indoor space and the topological adjacency relationship of adjacent spaces.

In Figure 1, the addition of a new wall within Room B divides it into two spaces. As a result, the original common wall, F₁, between Rooms A and B is subdivided into two walls, F₂ and F₃. This subdivision modifies the boundary faces of Room A and redefines its adjacent rooms to Rooms C and D. Failure to account for the topological relationships of indoor spaces during updating can lead to inconsistencies in the topology of geometric objects, thereby compromising the reliability of indoor topology queries, analyses, and reasoning processes. To address this issue, maintaining the topological relationships among indoor objects during geometric updates is crucial for ensuring the topological consistency of the geometric objects.

A building generally comprises six primary components: the foundation, floors, roof, walls, doors, and windows [29,30]. The floor and ground serve as horizontal dividing and load-bearing components, the roof functions as the uppermost enclosure and load-bearing structure, and the walls provide vertical enclosure and load-bearing components, while doors and windows connect and separate indoor and outdoor spaces. On the basis of these functions, ceilings, floors, walls, doors, and windows are adopted as the principal components defining various functional indoor spaces, such as rooms and corridors. When renovating indoor spaces, it is essential to ensure safety by preserving the original structural integrity and avoiding actions such as removing load-bearing components or creating openings in load-bearing walls. In line with the components in this work, ceilings and floors are horizontal load-bearing components; thus, the building components that may be modified in indoor spaces are limited to non-load-bearing walls, doors, and windows.

Changes in the geometric characteristics of indoor spaces arise from modifications in building components, with different types of component alterations producing various transformations in indoor spaces. Additionally, these changes often occur in conjunction with other changes rather than in isolation; for instance, doors and windows, which are key connectors between indoor spaces or between indoor and outdoor areas, are frequently modified alongside changes in walls. As a result, analyzing the types of modifications in indoor building components is essential for updating indoor spatial data. Alterations in building components lead to layout changes in indoor spaces, specifically reflected in the modifications of boundary faces, which subsequently alter the topological hierarchical relationships of a solid within indoor spaces. Consequently, updating indoor data requires reconstructing the topological relationships of a solid. Furthermore, given the space-filling characteristic of indoor spaces, adjacent spaces are invariably connected along common faces. Changes in boundary faces, therefore, inherently affect the topological adjacency relationships within indoor spaces, making it crucial to carefully handle the topological adjacency relationships of adjacent solids.

The analysis above indicates that changes in building components within indoor spaces are the primary reason for modifications to indoor environments. Updating indoor data requires processing indoor features on the basis of the specific types of changes in these building components. Categorizing the types of changes in indoor building components is thus the first step in updating indoor data. Furthermore, changes in building components lead to correlated changes in indoor features, which necessitates maintaining local topological relationships during the updating process. A detailed analysis of the topological linkages among the indoor features triggered by changes in building components is essential. In addition, updating the geometric characteristics of indoor spaces requires adjustments to the boundary faces defining each indoor solid. In this process, the topological adjacency relationships between solids serve as the basis for the updates and are also a critical task for maintaining the topological consistency of indoor data. On the basis of this framework, this paper follows the approach outlined in Figure 2 for incremental updates to indoor data.

3. Types of Changes in Building Components

Building components, considered geospatial entities, exhibit distinctive characteristics of spatial transformation. Changes in individual entities are typically categorized into three primary types: fundamental changes, deformation, and movement [31]. Fundamental changes refer to appearance, disappearance, and stability (changes in attributes); deformation involves changes in size or shape, such as expansion, contraction, and warping; and movement describes changes in position, including displacement and rotation. As previously noted, building components that are likely to undergo changes within indoor spaces include walls, doors, and windows. Considering practical scenarios of interior construction and renovation, walls generally do not undergo expansion, contraction, deformation, or rotation, whereas doors and windows are typically unaffected by deformation and rotation. Consequently, changes in individual building components within indoor spaces can be classified into five primary types: demolition (disappearance), addition (appearance), expansion, contraction, and movement, as outlined in

Table 1. Single types of changes in building components.

Changes in Building Components	Alterations in Indoor Spaces	Does the Layout of Indoor Spaces Change?	Do the Topological Relationships Change?
Demolition of a wall	merging	Y	Y
Addition of a wall	splitting	Y	Y
Movement of a wall	boundary modification	Y	Y
Demolition or addition of a door	boundary modification	N	Y
Expansion or contraction of a door		N	N
Movement of a door		N	N
Demolition or addition of a window	boundary modification	N	Y
Expansion or contraction of a window		N	N
Movement of a window		N	N

.

(1) Single types of changes to a wall

A wall serves as a building component that separates distinct indoor spaces within a building. Changes to walls can lead to alterations in the layout of indoor spaces. Single types of changes to a wall include demolition, addition, and movement. The demolition of a wall involves the removal of an interior wall shared by two adjacent spaces, resulting in the merging of the spaces. The addition of a wall refers to the construction of a new interior wall within an existing space, thereby splitting it into separate areas. Both the demolition and addition of a wall alter the topological relationships of solids and topological adjacency relationships of adjacent solids, requiring a local reconstruction of these topological relationships. Movement of a wall involves changing the position of a common wall between two adjacent spaces. Unlike demolition or addition, movement of an interior wall does not alter the number of spaces but instead modifies the boundary faces. This modification essentially involves the demolition of the existing wall and the addition of a new wall in a different location. Consequently, the movement of a wall, like the other modifications, requires local reconstruction of the topological relationships.

(2) Single types of changes to a door/a window

Changes to a door refer to alterations occurring on the wall that is semantically associated with it. These changes modify the boundaries of the indoor spaces without affecting the overall layout. The types of changes to a door include demolition, addition, expansion, contraction, and movement. The demolition of a door leads to changes in the shape and size of the wall to which it is attached, thereby altering the boundaries of the indoor spaces, whereas the number of indoor spaces remains unchanged. The demolition of a door causes changes in the topological relationships of solids, altering their topological adjacencies. Conversely, the addition of a door is the reverse process of demolition and similarly results in modifications of the boundaries of indoor spaces. Therefore, both the demolition and addition of doors necessitate local reconstruction of the topological relationships.

The expansion or contraction of a door refers to changes in its size on the same wall, which subsequently alter the dimensions of the wall to which it is attached, thereby modifying the boundaries of indoor spaces. However, the number of indoor spaces remains unchanged. The movement of a door involves a change in its position on the wall, which leads to modifications in the shape of the wall to which it is affixed, thereby altering the boundaries of indoor spaces without changing the number of indoor spaces. All of these changes, whether due to expansion, contraction, or movement, are classified as modifications to the boundaries of indoor spaces. Importantly, these modifications do not impact the topological relationships of solids or alter the topological adjacency relationships between adjacent solids. As a result, only the geometric information of the walls and doors requires being processed.

Changes to a window are defined as alterations made to the wall to which it is semantically associated. The types of changes to a window include demolition, addition, expansion, contraction, and movement. The indoor space alterations caused by the changes to windows are the same as those caused by the changes to doors, which can be handled according to the operations described above.

(3) Composite types of changes to walls, doors, and windows

In addition to single forms of changes, indoor spaces may also undergo composite alterations resulting from the combined changes in multiple building components. As previously discussed, walls and doors, as well as walls and windows, are semantically related by an inclusion relationship, where one feature encompasses the other. As a result, modifications to a wall typically entail corresponding changes to the associated doors or windows. Typically, windows serve as building components that connect indoor spaces with the exterior, whereas the walls facing the outdoors tend to remain unchanged. Thus, composite changes in walls are categorized as those involving the demolition, addition, or movement of both walls and doors.

The demolition of both walls and doors refers to the simultaneous removal of a wall and the door to which it is attached. This alteration, akin to the demolition of a wall, results in the merging of indoor spaces. The addition of walls and doors involves the simultaneous construction of a new wall and its corresponding door, which together split the original space into two distinct areas. The newly added wall and door form common boundaries between the two adjacent spaces. The movement of walls and doors entails a shift in the position of a common wall between two adjacent spaces, accompanied by a corresponding adjustment of the door to which it is attached. This change removes the original wall and door and replaces them with a new wall and door. The resulting alteration is classified as a boundary modification within the indoor spaces. As indicated by the previous analysis, the demolition, addition, or movement of both walls and doors leads to changes in the topological relationships of solids and topological adjacencies between indoor spaces. Therefore, local reconstruction of the topological relationships is needed.

4. Inference of Topological Linkage Types for Indoor Features on the Basis of Changes in Building Components

The above analysis of changes in building components reveals that changes to walls (both single and composite alterations), as well as the demolition or addition of doors and windows, alter the topological relationships of indoor spaces. Consequently, it is essential to analyze the resulting types of topological linkages for indoor features to ensure the effective maintenance of local topological relationships.

4.1. Topological Linkage Types of Changes in Walls

4.1.1. Topological Linkages for the Demolition of Walls

The demolition of a wall involves removing the common wall between two adjacent indoor spaces to combine them into one area. This transformation is accomplished by removing the common boundary faces between adjacent solids, thereby merging them into one.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. If a demolished interior wall face $F_k(k=\mathrm{1,2},\dots ,n)$ acts as a common boundary face between two adjacent solids $S_i,S_j(i,j=\mathrm{1,2},\dots ,m,i\ne j)$, then $S_i$ and $S_j$ should be merged.

Rule 1: If $disappearance\left(F_k\right)\mathrm{A}\mathrm{N}\mathrm{D}(S_i\cap S_j$=$F_k)$,

then $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}(S_i,S_j)$.

To merge the solids, the common face, as well as redundant nodes and edges within the face, are removed. Additionally, the edges and faces in the original solids related to the face should be merged, with a new solid constructed from the topologically reconstructed faces. The deletion of the edges and nodes in the face is determined by the topological adjacency relationships between the solids.

In Figure 3a, after removing the common face F₁ between the adjacent solids S₁ and S₂, every two faces in the solids S₁ and S₂ associated with each edge of the common face F₁ are reconstructed topologically, namely, the edge of face F₁ is deleted and the two faces associated with the edge are merged. However, if the two faces are shared with other solids, deleting the edge may lead to topological inconsistency. Specifically, in Figure 3a, faces F₂ and F₃ associated with edge E₁ belong to solids S₁ and S₂, respectively, which are also shared with solids S₃ and S₄. Therefore, edge E₁ should not be deleted, and faces F₂ and F₃ do not need to be merged.

When the two faces associated with the edge belong only to the adjacent solids, merging the faces may be necessary, but requires further assessment. In Figure 3a, faces F₅ and F₄ connected with edge E₂ belong to solids S₁ and S₂, respectively. However, since F₅ and F₄ are not coplanar, they cannot be merged, and edge E₂ should not be deleted. In contrast, if the faces are coplanar, merging is appropriate, as seen with faces F₆ and F₇, which are related to edge E₃ in Figure 3a.

When the two faces associated with the edge are shared not only by the adjacent solids but also by an additional solid, they may need to be merged. In Figure 3b, faces F₂ and F₃ connected with edge E₁ belong to solids S₁ and S₂, respectively, which are shared by solid S₃. Upon merging solids S₁ and S₂, the newly constructed solid is adjacent to solid S₃. In this scenario, merging faces F₂ and F₃ does not compromise the topological consistency between solid S₃ and the newly created solid. Thus, if faces F₂ and F₃ are coplanar, they should be merged.

The analysis above indicates that, in addition to the common face between adjacent solids, if an edge within the common face is associated with three or more faces (the degree of the edge is 4 or more), the two faces belonging to the adjacent solids should neither be merged nor the edge should be deleted. Conversely, if the degree of the edge is 3 and the two faces within the solids are coplanar, the edge should be removed, and the two faces should be merged to form a boundary face for a newly constructed solid.

Suppose that two adjacent solids in an indoor space are denoted $S_a$ and $S_b$, with a common face $F$. Let $E_i^F$ be an edge of the common face $F$ and $N_j^F$ be a node within $F$. Furthermore, let $F_i^a$ and $F_i^b$ be the faces associated with edge $E_i^F$ within solids $S_a$ and $S_b$, respectively. The degree of edge $E_i^F$ is given by ${deg(E}_i^F)$, and the degree of node $N_j^F$ is given by ${deg(N}_j^F)$.

Rule 2: If ${deg(E}_i^F)=3 \mathrm{A}\mathrm{N}\mathrm{D}<F_i^a,\mathrm{c}\mathrm{o}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{a}\mathrm{r},F_i^b>$,

then $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}(F_i^a,F_i^b)$.

After each edge of the common face is judged, the next step is to determine whether the nodes associated with these edges can be deleted. The deletion of nodes is determined on the basis of the edges. If an edge associated with a node belongs to the common face and cannot be deleted, the node should be retained. As illustrated in Figure 3c, although edges E₃ and E₄ are eligible for deletion, all nodes are also connected to edges E₁ and E₂, which cannot be deleted. Therefore, nodes N₁, N₂, N_3,, and N₄ cannot be deleted.

If the two adjacent edges associated with a node in the common face can be removed, the node may be eligible for deletion. However, further evaluation is necessary.

(1) If the two edges associated with the node, which belong to the two original adjacent solids, are each connected to two faces of a different solid, the node cannot be deleted. As illustrated in Figure 3d, edges E₁ and E₂ associated with node N₁ belong to adjacent solids S₁ and S₂, respectively. However, these edges also connect to distinct faces, F₂ and F₃, of solid S₃. Consequently, edges E₁ and E₂ cannot be merged, and node N₁ cannot be deleted.

(2) If the two edges associated with the node, which belong to the two original adjacent solids, are connected to the same face of a different solid, the node can be deleted, and the two edges should be merged. As shown in Figure 3e, edges E₁ and E₂ associated with node N₁ belong to adjacent solids S₁ and S₂, respectively, but both edges are also connected to the same face, F₂, of solid S₃. Merging edges E₁ and E₂ does not introduce topological inconsistencies between solid S₃ and the newly constructed solid. Consequently, node N₁ can be removed, and edges E₁ and E₂ should be merged.

(3) If the two edges associated with the node belong exclusively to the two original adjacent solids, the node can be deleted, and the two edges should be merged, as shown in Figure 3f for nodes N₁ and N₂.

On the basis of the above analysis, in addition to the two edges of the common face, when the node is associated with three or more edges (the degree of the node is 5 or more), the node cannot be deleted, and the two edges associated with the node belonging to the original adjacent solids do not need to be merged. Therefore, if the two adjacent edges associated with a node in the common face can be removed and the degree of the node is 4, the node can be removed, and the two edges associated with the node belonging to the two original adjacent solids should be merged into a new edge.

$N_j^F$ is defined as above. $E_i^F$ and $E_{i+1}^F$ are the edges in the common face $F$ associated with node $N_j^F$. $E_j^a$ and $E_j^b$ are the edges of the two adjacent solids $S_a$ and $S_b$ that are associated with node $N_j^F$.

Rule 3: If ${deg(N}_j^F)=4 \mathrm{A}\mathrm{N}\mathrm{D}<E_i^F,\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}>\mathrm{A}\mathrm{N}\mathrm{D}<E_{i+1}^F,\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}>$,

then $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}(E_j^a,E_j^b)$.

Additionally, in the case of composite transformations of walls and doors, the removal of a common wall between two adjacent indoor spaces necessitates the simultaneous removal of the associated doors. Similarly, the common faces between two adjacent solids should be deleted, and the adjacent solids should be merged. Consequently, the topological linkages involved in the removal of walls and doors follow the same principles as those for the removal of walls, with the distinction being that the edges of the common faces may differ. When a wall is associated with doors, multiple edges may connect the same faces of adjacent solids. As illustrated in Figure 4, edges E₁, E₂, and E₃ of the common faces F₁ and F₂ are associated with faces F₃ and F₄ of solids S₁ and S₂, respectively. In such cases, it is necessary to assess each edge and its associated nodes individually to determine whether they can be deleted, which may result in redundant operations. Regardless of whether the faces of the two adjacent solids associated with these edges need to be merged, the edges will either be deleted (Figure 4a) or merged (Figure 4b). To avoid redundant operations, it is beneficial to merge the two faces of the wall and door beforehand and then assess the deletion conditions for the merged edges. Therefore, for the topological linkage operations caused by the demolition of walls and doors, it is advisable to first merge the common wall and door and then proceed with the topological linkage operations for the demolition of walls as previously outlined.

In conclusion, topological linkage operations for the demolition of walls need to consider the conditions for deleting the edges and nodes of the common face and merging the faces and edges of adjacent solids. On the basis of the inference rules outlined above, each edge and corresponding node of the common face should be analyzed sequentially, and the faces and edges of the adjacent solids should be processed accordingly.

4.1.2. Topological Linkages for the Addition of Walls

The addition of a wall refers to the insertion of a new wall within an indoor space, which divides the original space into two adjacent areas. This newly inserted wall serves as a dividing face within a solid, splitting it into two adjacent solids. Consequently, the wall acts as the common boundary between the adjacent solids.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. If $F_{n+1}$ is a newly added interior wall face within a solid $S_i(i=1,2,\dots ,m)$ and face $F_{n+1}$ intersects the boundary of solid $S_i$ at the exterior loop $Loop\left(F_{n+1}\right)$, then solid $S_i$ should be split.

Rule 4: If $appearance\left(F_{n+1}\right)$ AND ${(F}_{n+1}\cap boundary\left(S_i\right)$ = $Loop\left(F_{n+1}\right))$,

then $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\left(S_i\right)$.

The segmentation of a solid is characterized by the use of an inserted face to split its boundary faces and subsequently the construction of new solids on the basis of these segmented boundary faces. While the segmentation of faces involves subdividing the faces of the original solid using the edges of the inserted face, the nodes of the inserted face split the edges of the faces, and the segmented edges are used to reconstruct new faces. Consequently, the topological linkage operations for the addition of walls are to segment the edges and faces of a solid according to the inserted face.

In Figure 5, S₁ is the solid being segmented, and F is an inserted face within it. Edges E₁, E₂, and E₃ of face F lie in faces F₁, F₂, and F₃ of solid S₁, respectively. As a result, edges E₁, E₂, and E₃ divide faces F₁, F₂, and F₃, respectively, such as face F₂ being subdivided into F₂₁ and F₂₂. While edge E₄ of face F coincides with edge E₅ of face F₄ in solid S₁, edge E₅ is also associated with face F₅ of solid S₁. After segmentation of solid S₁, faces F₄ and F₅ become the boundary faces of the two newly constructed solids, meaning that faces F₄ and F₅ do not require segmentation. Therefore, when an edge of the inserted face coincides or partially coincides with the edge of a face in the segmented solid (the intersection of the two edges has a dimensionality of 1), the faces associated with the edge in the segmented solid do not need to be subdivided.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_l\}$ be the set of faces within solid $S$ in a 3D indoor space, $F_m$ be the inserted face in solid $S$, and $E_k^{F_m}$ be an edge of face $F_m$. If the intersection of edge $E_k^{F_m}$ with face $F_i(i=\mathrm{1,2},\dots ,l)$ of solid $S$ is edge $E_k^{F_m}$, and if the dimensionality of the intersection between edge $E_k^{F_m}$ and edge $E_j^{F_i}$ of face $F_i$ is unequal to 1, then edge $E_k^{F_m}$ divides face $F_i$.

Rule 5: If ${(E}_k^{F_m}\cap F_i=E_k^{F_m}) \mathrm{A}\mathrm{N}\mathrm{D} {\left(\mathrm{d}\mathrm{i}\mathrm{m}\right(E}_k^{F_m}\cap E_j^{F_i})\ne 1)$,

then $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\left(F_i\right)$.

A face is segmented by splitting the edges of the face using the nodes of the inserted face. After the faces of the original solid to be divided are determined, the edges of these faces are examined sequentially. In Figure 5, F₂ is a face to be divided, and nodes N₁ and N₂ of edge E₂ of the inserted face split edges E₆ and E₇ of face F₂, respectively. However, for face F₁, edges E₈ and E₉ are associated with faces F₄ and F₅, respectively; hence, edges E₈ and E₉ are not split. Therefore, when the nodes of the inserted face coincide with the nodes of the edges within the face to be divided, the corresponding edges do not need to be split.

Let $F_i(i=\mathrm{1,2},\dots ,l)$ be a face to be divided in the set of faces $\mathcal{F}$ within the solid $S$; $F_m$, $E_k^{F_m}$ and $E_j^{F_i}$ are defined as above. ${N(E}_k^{F_m})$ is a node of edge $E_k^{F_m}$, and ${N(E}_j^{F_i})$ is a node of edge $E_j^{F_i}$. If node ${N(E}_k^{F_m})$ lies in edge $E_j^{F_i}$ and ${N(E}_k^{F_m})$ is not equal to ${N(E}_j^{F_i})$, then node ${N(E}_k^{F_m})$ divides edge $E_j^{F_i}$ associated with node ${N(E}_j^{F_i})$.

Rule 6: If ${\left(N\right(E}_k^{F_m})\cap E_j^{F_i}={N(E}_k^{F_m})) \mathrm{A}\mathrm{N}\mathrm{D} {\left(N\right(E}_k^{F_m})\cap {N(E}_j^{F_i})=\mathrm{\varnothing })$,

then $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\left(E_j^{F_i}\right)$.

Additionally, for the composite changes in walls and doors, the added wall and door jointly divide the original indoor space into two adjacent spaces. Both the wall and the door serve as common faces between two adjacent spaces. Therefore, the topological linkage operations for the addition of walls and doors follow the same rules as those for the addition of walls, with the difference being the edges of the inserted faces. When a wall and door are added, the edges that divide the boundary faces of the original solid include not only the edges of the wall but also those of the door, such as edge E₂ in Figure 6. As a result, the edges of the added door should be incorporated into the set of edges of the added wall, and the same edges of the wall and door should be removed. The topological linkage operations for the addition of walls are then applied as described above.

To summarize, the topological linkage operations for the addition of walls should consider the division conditions for faces and edges in the solid to be divided according to the inserted face. Following the inference rules above, the faces and edges of the solid are analyzed sequentially, and the edges and faces that need to be divided are processed accordingly.

4.1.3. Topological Linkages for the Movement of Walls

The movement of a wall involves a change in the position of the common wall between adjacent indoor spaces, resulting in modifications to the boundary faces of the indoor spaces. This can be viewed as the removal of the common wall between adjacent spaces and then the addition of a new wall. Essentially, this involves deleting the common face between two adjacent solids, merging them, and then inserting a new face inside the solid to split it, as illustrated in Figure 7.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. If the position of an interior wall face $F_k(k=\mathrm{1,2},\dots ,n)$ is changed and $F_k$ is the common face between adjacent solids $S_i$ and $S_j$, where $(i,j=\mathrm{1,2},\dots ,m,i\ne j)$, then solids $S_i$ and $S_j$ should be merged and subsequently divided.

Rule 7: If $movement\left(F_k\right) \mathrm{A}\mathrm{N}\mathrm{D} {(S}_i\cap S_j$=$F_k)$,

then $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}(S_i,S_j)\to S_{m+1}$ AND $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\left(S_{m+1}\right)$.

On the basis of the above statement, the merging of solids requires consideration of the conditions for deleting the edges and nodes of the common face, whereas dividing a solid involves the conditions for splitting the faces and edges of the solid to be divided. Therefore, the topological linkage operations for the movement of walls should follow the inference rules for the sequential demolition and addition of walls, with the faces and edges of the two adjacent solids being merged and the faces and edges of the newly constructed solid being divided accordingly.

4.2. Topological Linkage Types of Changes in Doors and Windows

4.2.1. Topological Linkages for the Demolition of Doors and Windows

The demolition of a door changes the shape and size of the associated wall, resulting in changes to the boundary faces between two adjacent indoor spaces. Consequently, the common faces between the two spaces shift from a wall with a door to only a wall. Therefore, the removal of a door involves merging the two faces of the wall and the door while preserving the attributes of the wall, as illustrated in Figure 8.

Let ${\mathcal{F}}^W=\{F_1^W,F_2^W,\dots ,F_n^W\}$ be the set of all interior wall faces in a 3D indoor space, and let ${\mathcal{F}}^D=\{F_1^D,F_2^D,\dots ,F_r^D\}$ be the set of all door faces. If $F_i^D$($i=\mathrm{1,2},\dots ,r)$ is a demolished door face with an associated wall face $F_j^W(j=\mathrm{1,2},\dots ,n)$, then $F_i^D$ and $F_j^W$ should be merged.

Rule 8: If $disappearance\left(F_i^D\right)$ AND $<F_i^D,relate,F_j^W>$,

then $\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}(F_i^D,F_j^W)\to F_j^W$.

Similar to the demolition of doors, the removal of windows leads to changes in the shape and size of the associated wall, thereby altering the boundary faces of the indoor spaces. Consequently, the common faces between the indoor and outdoor spaces transition from a wall with a window to only a wall. Therefore, the topological linkage operations for the demolition of windows should follow the inference rules for the demolition of a door.

4.2.2. Topological Linkages for the Addition of Doors and Windows

The addition of doors is the reverse of its demolition, leading to changes in the boundary faces between two adjacent indoor spaces. Specifically, the common faces between these spaces transition from a wall to a wall with a door. Consequently, the addition of a door requires that the face of the associated wall be divided using the face of the door, as shown in Figure 9.

Let ${\mathcal{F}}^W=\{F_1^W,F_2^W,\dots ,F_n^W\}$ be the set of all interior wall faces in a 3D indoor space, and let ${\mathcal{F}}^D=\{F_1^D,F_2^D,\dots ,F_r^D\}$ be the set of all door faces. If $F_{r+1}^D$ is a newly added door face, with an associated wall face $F_j^W(j=\mathrm{1,2},\dots ,n)$, then $F_j^W$ should be split.

Rule 9: If $appearance\left(F_{r+1}^D\right) \mathrm{A}\mathrm{N}\mathrm{D}<F_{r+1}^D,relate,F_j^W>$,

then $\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}\left(F_j^W\right)\to F_j^W$.

Similar to the addition of doors, the addition of windows changes the boundary faces of the indoor space, which alters the common faces between the indoor and outdoor spaces from a wall to a wall with a window. Therefore, adding a window requires that the face of the wall be divided using that of the window, and the topological linkage operations for the addition of windows can follow the inference rules for the addition of a door.

5. Updating Indoor Data Considering Topological Consistency

5.1. Updating Operators

Indoor data updating involves updating records in a database on the basis of incremental information in building components and their corresponding topological linkages and maintaining the topological consistency of indoor data. During the updating process, a series of operations, such as addition, modification, and deletion, are performed on the database, which are executed through the appropriate updating operators [32].

On the basis of the types of change to the building components above, the set of these types of changes is denoted as

$$X_1=\{appearance,disappearance,movement,expansion,contraction\}$$

The set of alterations in indoor spaces caused by changes in building components is denoted as

$$X_2=\{merging,splitting,boundary\_modification\}$$

Indoor space merging is characterized by the disappearance of two original adjacent spaces and the appearance of a new merged indoor space. Conversely, indoor space splitting involves the disappearance of the original space and the appearance of two new adjacent spaces after the split. Adjustment to indoor space boundaries is defined by the modification of the boundary faces of the original space on the basis of the new boundary configurations.

Additionally, when the geometric characteristics of indoor spaces and building components remain unchanged but attributes, such as functions or occupants, are altered, they are classified as property changes (stability). Therefore, the set of the types of changes to the indoor features is denoted as

$$X=\{appearance,disappearance,movement,expansion,contraction,boundary\_modification,stability\}$$

Considering the characteristics of changes in building components, the following update operations are proposed. The “appearance” of building components can be handled using the update operation “create”; the “disappearance” of components can be addressed using the update operation “delete”; and the “movement” of walls (or walls and doors), as well as the “movement”, “expansion”, and “contraction” of doors and windows, can be managed using the update operation “geometrically modify.”

For indoor spaces, “merging” requires the “deletion” of two adjacent spaces and the “creation” of a new merged space; “splitting” requires the “deletion” of the original space and the “creation” of new subdivided spaces; “boundary_modification” involves modifying the boundary faces of the original indoor spaces using the operation “modify” according to the topologically reconstructed faces; and changes in the attributes of indoor spaces can be addressed using the update operation “semantic_modify.”

To implement these updates, this study designs five update operators to map changes in the state of indoor features to the indoor database, as shown in Figure 10. The definitions of these update operators are as follows.

create: create a new database record;

delete: delete a database record;

geometrical_modify: modify the geometric information of the current object without altering its existing record in the database;

modify: modify the boundary faces of indoor spaces using the topology reconstructed faces;

semantic_modify: modify the semantic information of the current object without altering its existing record in the database.

The updating operators described above can be formally expressed as follows.

$\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}(\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}):->\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}//$ create a new spatial object, with the operand consisting of the complete information of the spatial object;

$\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}(\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\_\mathrm{i}\mathrm{d}):->\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}//$ delete a spatial object, with the operand being the identifier of the spatial object;

$\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\_\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\_\mathrm{i}\mathrm{d},\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}):->\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}//$ modify the geometric information of a spatial object, with the operands consisting of the spatial object identifier and the new geometric information;

$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\_\mathrm{i}\mathrm{d}):->\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}//$ modify a spatial object, with the operand being the identifier of the spatial object;

$\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}\_\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(\mathrm{o}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\_\mathrm{i}\mathrm{d},\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{s}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{c}):->\mathrm{b}\mathrm{o}\mathrm{o}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{n}//$ modify the semantic information of a spatial object, with the operands consisting of the spatial object identifier and the new semantic information.

The operation result is a Boolean value, $\{\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{e},\mathrm{f}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\}$, which indicates whether the update operation was successful.

5.2. Update Processing Based on Change Mapping

Alterations to indoor spaces involve diverse changes to indoor features, necessitating the coordinated utilization of multiple updating operators to execute the update process effectively. According to active database research, an updating operator is defined as an atomic event, whereas a complete indoor data update process is characterized as a composite event. Composite events are formed by applying event operators to atomic events or other composite events, which can be expressed via the event pattern language (EPL) [33]. This section examines the handling of indoor space updates, specifically merging, splitting, and boundary modification, and provides a formalized representation of the indoor data update process using the EPL.

5.2.1. Merging of Indoor Spaces

The process of merging indoor spaces can be outlined as follows: (a) the common face between two adjacent solids is removed; (b) redundant edges within the common face are removed, and the associated faces to be merged are determined; (c) the redundant nodes within the common face are eliminated, and the associated edges to be merged are identified; (d) the corresponding edges and faces are merged sequentially; (e) the merged solid is built according to the topologically reconstructed faces, and semantic attributes are assigned; and (f) the topological adjacency relationships between the newly formed solid and other adjacent solids are established according to the topologically reconstructed faces.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. $F_k(k=\mathrm{1,2},\dots ,n)$ is a demolished interior wall face associated with two adjacent solids $S_i,S_j(i,j=\mathrm{1,2},\dots ,m,i\ne j)$. The process of merging indoor spaces can be defined as a composite event $E$, which can be described using the EPL as

$$E=\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\left(F_k\right)∆\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\left(S_{m+1}\right)∆\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\left(S_i\right)∆\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\left(S_j\right)$$

In the expression above, “∆” denotes the event operator “AND,” which signifies that the composite event $E=E_1∆E_2$ occurs only when both events $E_1$ and $E_2$ are triggered. $S_{m+1}$ represents the newly constructed solid after merging.

5.2.2. Splitting of Indoor Spaces

The process of splitting an indoor space involves the following steps: (1) the faces of the original solid are divided using the inserted face; (2) two new adjacent solids are constructed from the divided faces, and semantic information is assigned; (3) the topological adjacency relationships between the two newly constructed solids are established; and (4) the topological adjacency relationships among the new solids and other adjacent solids are rebuilt.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. $F_{n+1}$ is an inserted interior wall face within solid $S_i(i=\mathrm{1,2},\dots ,m)$. The process of splitting an indoor space can be defined as a composite event $E$, which can be described using the EPL as

$$E=\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\left(F_{n+1}\right)∆\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\left(S_{m+1}\right)∆\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\left(S_{m+2}\right)∆\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\left(S_i\right)$$

In the expression above, $S_{m+1}$ and $S_{m+2}$ are the two newly constructed solids.

5.2.3. Boundary Modification of Indoor Spaces

(1) Movement of Walls

Modification of the boundaries of indoor spaces resulting from the movement of walls can be addressed by first following the process of solid merging and then reconstructing the topological adjacency relationships between the merged solid and neighboring solids. The merged solid is subsequently divided according to the process of solid splitting, which is followed by the reconstruction of topological adjacency relationships among the solids. Additionally, if a common wall is associated with a door, the same procedure can be applied for updating.

Let $\mathcal{F}=\{F_1,F_2,\dots ,F_n\}$ be the set of all interior wall faces in a 3D indoor space, and let $S=\{S_1,S_2,\dots ,S_m\}$ be the set of all solids. $F_k(k=\mathrm{1,2},\dots ,n)$ is a common interior wall face that is moved between two adjacent solids $S_i$, and $S_j$ $(i,j=\mathrm{1,2},\dots ,m,i\ne j)$. The modification of indoor space boundaries caused by the movement of walls is defined as a composite event $E$, which can be described using the EPL as

$$E=\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\_\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(F_k,\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{d}\_1,\dots )∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_i\right)∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_j\right)$$

In the expression above, $\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{d}$ represents the positional information of the nodes on the common wall face after movement.

(2) Demolition of Doors

The demolition of a door involves merging the faces of the wall and the door. Following this, the boundary faces of the two adjacent solids are adjusted, and the topological adjacency relationships are reconstructed.

Let ${\mathcal{F}}^W=\{F_1^W,F_2^W,\dots ,F_n^W\}$ be the set of all interior wall faces in a 3D indoor space, and let ${\mathcal{F}}^D=\{F_1^D,F_2^D,\dots ,F_r^D\}$ be the set of all door faces. $F_i^D$($i=\mathrm{1,2},\dots ,r)$ is a demolished door face, with an associated wall face $F_j^W(j=\mathrm{1,2},\dots ,n)$, and $S_a$ and $S_b$ denote the two adjacent solids associated with faces $F_i^D$ and $F_j^W$, respectively. The modification of indoor space boundaries caused by the demolition of doors is defined as a composite event $E$, which can be described using the EPL as

$$E=\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{e}\mathrm{t}\mathrm{e}\left(F_i^D\right)∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_a\right)∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_b\right)$$

(3) Addition of Doors

The addition of a door requires splitting the associated wall face using the door face. The boundary faces of the two adjacent solids are subsequently modified, and the topological adjacencies are rebuilt to maintain consistency.

Let ${\mathcal{F}}^W=\{F_1^W,F_2^W,\dots ,F_n^W\}$ be the set of all interior wall faces in a 3D indoor space, and let ${\mathcal{F}}^D=\{F_1^D,F_2^D,\dots ,F_r^D\}$ be the set of all door faces. $F_{r+1}^D$ is a newly added door face, with an associated wall face $F_j^W(j=\mathrm{1,2},\dots ,n)$, and $S_a$ and $S_b$ denote the two adjacent solids associated with face $F_j^W$. The modification of indoor space boundaries caused by the addition of doors is defined as a composite event $E$, which can be described using the EPL as

$$E=\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{t}\mathrm{e}\left(F_{r+1}^D\right)∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_a\right)∆\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}\left(S_b\right)$$

(4) Movement of Doors

The movement of a door requires the modification of only the geometric information of the door face and its associated wall face. Let ${\mathcal{F}}^W=\{F_1^W,F_2^W,\dots ,F_n^W\}$ be the set of all interior wall faces in a 3D indoor space, and let ${\mathcal{F}}^D=\{F_1^D,F_2^D,\dots ,F_r^D\}$ be the set of all door faces. $F_i^D$($i=\mathrm{1,2},\dots ,r)$ is a moved door face, with an associated wall face $F_j^W(j=\mathrm{1,2},\dots ,n)$. The modification of indoor space boundaries caused by the movement of doors is defined as a composite event $E$, which can be described using the EPL as

$$\begin{gathered} E=\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\_\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(F_i^D,\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{d}\_1,\dots )∆ \\ \mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}\_\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{y}(F_j^W,\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{d}\_1,\dots ) \end{gathered}$$

In the expression above, $\mathrm{n}\mathrm{e}\mathrm{w}\_\mathrm{n}\mathrm{o}\mathrm{d}\mathrm{e}\_\mathrm{i}\mathrm{d}$ represents the positional information of the nodes on the door face after movement.

Changes in the size of the attached walls caused by the expansion or contraction of doors can be updated following the procedure for the modification of indoor space boundaries caused by the movement of doors. Additionally, the demolition or addition of windows, which alter the shape and size of the corresponding walls, can be addressed by following the process for the demolition or addition of doors, respectively. The movement, expansion, and contraction of windows can likewise be handled by applying the procedure for the modification of indoor space boundaries triggered by the movement of doors.

6. Implementation

A CityGML 3.0 dataset of an office building is utilized as raw data, and the algorithm from [34] is employed to construct a 3D indoor topological data model. Incremental updating of 3D indoor data was conducted based on this model. The building comprises four floors, with a total of 91 indoor spaces (e.g., rooms and corridors), as illustrated in Figure 11. As previously discussed, the alterations in indoor spaces are categorized into merging, splitting, and boundary_modification. Accordingly, the experiment includes three parts: (1) the demolition of a common wall face triggers the merging of two adjacent indoor spaces; (2) the addition of a new wall face results in the division of the indoor space; and (3) boundary faces modification of indoor spaces, exemplified through demolition and addition of a door.

6.1. Merging of Indoor Spaces

The merging of indoor spaces involves updating the indoor model on the basis of a common wall face being demolished and the associated adjacent indoor spaces, according to the merging process described previously. The steps for indoor space merging are outlined as follows.

1. The common wall face F to be removed, along with its associated adjacent solids S_i and S_j, is identified.

2. The edges to be removed on the common face F are determined by applying the topological linkages rules for the demolition of walls described previously, and the faces to be merged in S_i and S_j, are identified.

3. The nodes to be removed on the common face F are determined by applying the topological linkage rules for the demolition of walls described previously, and the edges to be merged in S_i and S_j, are identified.

4. The identified edges and faces are merged sequentially within S_i and S_j.

5. A new solid S_{n + 1} is constructed on the basis of the merged faces.

The adjacent rooms to be merged, as shown in Figure 12, are identified as “bld_ac_Institute_S_10” and “bld_ac_Institute_S_14”, with the common wall face to be removed identified as “face_209…”. Following the outlined procedure, the two adjacent rooms are merged, with the merged room retaining the identifier “bld_ac_Institute_S_10”. To evaluate the topological adjacency relationships of the merged room, queries are performed using its wall “face_229…” and floor “face_10…,”, as shown in Figure 13. The query results indicate that the original adjacent rooms are successfully updated, with their topological adjacency relationships properly maintained.

6.2. Splitting of an Indoor Space

The splitting of an indoor space involves inserting a new wall face into a specific indoor space, followed by updating the indoor model using the splitting process described previously. The procedure for indoor space splitting consists of the following steps.

1. The inserted wall F and the solid S_i to be segmented are identified.

2. The faces to be segmented within solid S_i are determined by applying the topological linkages rules for the addition of walls described previously.

3. The edges to be segmented within solid S_i are determined by applying the topological linkages rules for the addition of walls described previously.

4. The identified edges and faces within S_i are split sequentially.

5. New solids S_{n + 1} and S_{n + 2} are constructed on the basis of the segmented faces.

The room to be divided is identified as “bld_ac_Institute_S_0”, as illustrated in Figure 14. By following the outlined procedure, the room is segmented into two adjacent rooms, “bld_ac_Institute_S_0_split_0” and “bld_ac_Institute_S_0_split_1”. The topological adjacency relationships between two rooms can be identified through the newly added wall, “face_new…” (Figure 15a). Additionally, when a floor face of one of the split rooms, “splitface_1…”, is used, a query for adjacent rooms on neighboring floors identifies “bld_ac_Institute_S_0_split_0” and “bld_ac_Institute_S_32” (Figure 15b). The results demonstrate that the original indoor space is updated after the division operation, with their topological adjacency relationships properly maintained.

6.3. Boundary Modification of Indoor Spaces

As discussed above, apart from demolition and addition of a wall, the other types of changes in building components result in boundary modification of indoor spaces. This section demonstrates the updating operations associated with such modifications, using demolition and addition of a door as examples. Demolition of a door involves merging the two faces of the door and the wall, followed by modifications of the boundary faces of the solids and reconstruction of the topological adjacency between the two adjacent solids. The door to be demolished is identified as “face_74_door_2…”, as shown in Figure 16a. After the merging of the indoor spaces above, the door and the associated wall are combined into a new face, “newface_182_wall_2…”, which preserves the original attribute of the wall, as shown in Figure 16b. To evaluate the topological adjacency relationships, queries are performed using the new wall “newface_182_wall_2…”, as shown in Figure 17. The query results indicate that the original door and wall are successfully updated, with the topological adjacency relationships properly maintained.

The addition of a door involves splitting the associated wall face using the door face, followed by modifications of the boundary faces of the solids and reconstructions of the topological adjacency between the two adjacent solids. The wall face to be split is identified as “face_191_wall_2…”, as shown in Figure 18a. After the splitting of the indoor spaces above, the wall is divided accordingly. The resulting split wall face is identified as “splitFace1_191_wall_2_…”, and the added door is represented as “face_0_door_2_new…”, as shown in Figure 18b. To evaluate the topological adjacency relationships, queries are performed using the split wall face “splitFace1_191_wall_2_…”, as shown in Figure 19. The query results indicate that the original wall is successfully updated, with the topological adjacency relationships properly maintained.

7. Conclusions and Future Work

This paper proposes a method for incrementally updating 3D indoor data that incorporates topological linkages and addresses the critical challenge of maintaining topological relationships among geometric objects during updates. The approach systematically identifies and classifies changes in building components, such as walls, doors, and windows, along with their corresponding alterations in indoor spaces. On this basis, detailed topological linkage types are analyzed, and inference rules are developed for indoor features. To implement the updates, a set of updating operators is designed to map changes in the states of indoor features to an indoor spatial database. Furthermore, tailored update methods are proposed to handle alterations in indoor spaces. This approach ensures efficient incremental data updates and preserves the consistency of topological relationships among indoor spaces, enabling the effective querying of essential indoor spatial information and topological adjacency relationships.

The proposed method for indoor data updating considering topological linkages is based on topological relationships and imposes high demands on the indoor data quality. Specifically, basic topological relationships, which include both hierarchical combinations of a solid for an indoor space and the topological adjacencies between solids, are established for indoor data before updating. These relationships are crucial for constructing 3D indoor spaces, such as rooms and corridors. Moreover, when building components undergo changes, the updated data should meet the specified quality standards outlined by the updating rules; failure to do so may disrupt the execution of incremental updates and hinder the maintenance of topological relationships. Additionally, the updating rules and methods presented in this work are designed primarily for structurally regular buildings, such as office buildings. Indoor spaces are modeled using typical building components such as walls, ceilings, and floors. However, they are currently unable to address highly complex indoor environments and architectural components, such as those found in large-scale shopping malls. Therefore, future work will focus on extending the proposed methods to accommodate more complex building structures and further refining the indoor data updating rules.