An Efficient PSInSAR Method for High-Density Urban Areas Based on Regular Grid Partitioning and Connected Component Constraints

Abstract

Permanent Scatterer Interferometric Synthetic Aperture Radar (PSInSAR), with millimeter-level accuracy and full-resolution capabilities, is essential for monitoring urban deformation. With the advancement of SAR sensors in spatial and temporal resolution and the expansion of wide-swath observation capabilities, the number of permanent scatterers (PSs) in high-density urban areas has surged exponentially. To address these computational and memory challenges in high-density urban PSInSAR processing, this paper proposes an efficient method for integrating regular grid partitioning and connected component constraints. First, adaptive dynamic regular grid partitioning was employed to divide monitoring areas into sub-blocks, balancing memory usage and computational efficiency. Second, a weighted least squares adjustment model using common PS points in overlapping regions eliminated systematic inter-sub-block biases, ensuring global consistency. A graph-based connected component constraint mechanism was introduced to resolve multi-component segmentation issues within sub-blocks to preserve discontinuous PS information. Experiments on TerraSAR-X data covering Fuzhou, China (590 km²), demonstrated that the method processed 1.4 × 10⁷ PS points under 32 GB memory constraints, where it achieved a 25-fold efficiency improvement over traditional global PSInSAR. The deformation rates and elevation residuals exhibited high consistency with conventional methods (correlation coefficient ≥ 0.98). This method effectively addresses the issues of memory overflow, connectivity loss between sub-blocks, and cumulative merging errors in large-scale PS networks. It provides an efficient solution for wide-area millimeter-scale deformation monitoring in high-density urban areas, supporting applications such as geohazard early warning and urban infrastructure safety assessment.

1. Introduction

Time-series InSAR technology, which continuously captures phase information of surface deformation, has become an essential tool for urban subsidence monitoring and infrastructure health assessment. Its millimeter-level monitoring accuracy provides irreplaceable technical support for urban safety and maintenance [1,2,3,4,5]. Particularly for the health assessment of urban lifeline projects, high-efficiency, high-density, and high-precision time-series InSAR processing has become a prerequisite for accurate structural displacement inversion [6,7,8,9]. Among various time-series InSAR methods, Permanent Scatterer InSAR (PSInSAR) stands out due to its unique phase-unwrapping strategy [10,11,12]. By selecting stable ground scatterers (PS points), it constructs a joint inversion model based on arc segments for the deformation rate and elevation residual estimation, effectively suppressing spatiotemporal decorrelation and atmospheric phase disturbances. This enables the retrieval of high-precision deformation and position information, making it particularly suitable for urban areas with dense artificial structures [13,14].

In recent years, advancements in SAR satellite technology have led to three significant improvements: wide-swath observation capability (e.g., Sentinel-1 IW mode with a swath width of 250 km), high-frequency revisit cycles (e.g., LuTan-1 dual-satellite constellation achieving a 4-day revisit period [15]), and sub-meter spatial resolution (SAR spotlight mode resolution better than 1 m [16]). These improvements have drastically increased the density of PS points in urban areas, making it possible to resolve millimeter-scale deformations at individual building scales [17,18,19,20]. However, they also pose significant challenges to PSInSAR processing. For example, in this study’s case of Fuzhou’s main urban area, a single TerraSAR-X stripmap image can contain up to 90,000 PS points per km², with the entire area comprising approximately 1.4 × 10⁷ PS points, forming a massive interferometric network. The deformation inversion of such a large-scale PS network requires solving a coefficient matrix with dimensions exceeding 10⁷, far exceeding the capacity of a single computing unit. Traditional global PSInSAR methods, constrained by their O(N³) computational complexity and O(N²) memory requirements, struggle to efficiently process this enormous dataset.

To address the challenge of large-scale, high-density PS point processing, current research has primarily focused on two dimensionality reduction strategies. The first approach is the hierarchical network method [21,22,23,24,25,26], which draws inspiration from geodetic control networks with a “coarse-to-fine” framework. It first selects high-quality PS points using strict thresholds or sparse sampling to form a primary network. The remaining PS or Distributed Scatterer (DS) points are then used to densify the network into a secondary refined network. This method requires the primary network to maintain a high quality to ensure global network stability and connectivity [27]. To address potential ill-conditioning in the primary network, some studies [23,24,28] have employed ridge estimation techniques for stabilization. Meanwhile, other studies [29,30] have introduced adaptive network reconstruction to link disconnected subnetworks into the largest connected component, thereby ensuring global connectivity. However, despite reducing the computational complexity, this approach still requires global storage of all the PS points and arc segment information, leaving memory overflow issues unresolved. The second approach is the partition-based processing method [31,32,33,34], which divides the global processing task into smaller independent blocks, significantly reducing the memory usage and computational load per block. However, a major challenge is block merging. Previous studies have attempted different solutions: coherent target analysis has been used for block-based coherence estimation [31], and local control points have been employed to merge adjacent blocks [32,33]. Some studies [33,34] have adopted a regular grid partitioning strategy with overlapping regions, using common PS points in overlapping areas for block merging [35]. Nevertheless, PS point distributions are highly heterogeneous (e.g., dense urban areas vs. scattered villages), which can lead to the formation of multiple disconnected components within individual blocks. Existing methods lack mechanisms to identify these connected components, making it difficult to merge isolated blocks during the global least squares adjustment. This can result in a rank deficiency and reduce the mathematical completeness of the inversion model.

To overcome these limitations, we present a novel PSInSAR method based on regular grid partitioning and connected component constraints. At the partitioning strategy level, a dynamic regular grid partitioning algorithm was designed to adaptively adjust the grid size and extend block boundaries, achieving an optimal balance between memory usage and computational efficiency. At the merging model level, a weighted least squares adjustment model was constructed using common PS points in overlapping regions, leveraging the spatial continuity of phase observations (deformation rate and elevation residuals) to eliminate systematic biases between blocks. At the mathematical completeness level, a graph-theoretic connectivity analysis algorithm was introduced to identify independent connected components within each block by constructing an adjacency matrix. This enables parallel inversion across connected subdomains, resolving the issue of isolated blocks and rank deficiency caused by uneven PS point distributions. The key breakthrough of this method lies in the first-ever integration of connected component theory into PSInSAR block processing, providing a new paradigm for the automated, high-efficiency processing of large-scale PS networks at the urban scale.

The structure of this paper is as follows. Section 2 describes the proposed method in detail. Section 3 introduces the study area and datasets. Section 4 presents the experimental results and analysis. Section 5 provides a discussion. Finally, Section 6 concludes this paper.

2. Methods

To address the challenges of high computational complexity and large memory consumption in PSInSAR processing for large-scale, high-density urban areas, we propose a fully automated processing framework based on data block partitioning, independent sub-block PSInSAR processing, and sub-block result merging. First, the method takes as input the following SAR-derived datasets: co-registered SAR images, temporal and spatial baseline files, differential interferograms after terrain phase removal, intensity images, coherence maps, slant range, and incidence angle parameters. Based on these inputs, the proposed method consists of three key stages: regular grid-based data block partitioning, independent PSInSAR processing for each sub-block, and global spatially consistent merging of results. In this workflow, regular grid partitioning employs an adaptive grid size adjustment and boundary expansion mechanism to ensure controllable memory usage and sufficient overlap for point matching in subsequent merging. Independent PSInSAR processing for each sub-block transforms the high-dimensional global matrix operations into parallelized low-dimensional sub-problems. The global merging stage integrates common PS points from overlapping regions and topological relationships of connected components to construct a weighted least squares adjustment model to eliminate systematic biases from the sub-block processing. This enables the complete retention of PS point information and seamless integration of the PS dataset. The workflow of the proposed method is illustrated in Figure 1.

2.1. Adaptive Data Block Partitioning Strategy

To reduce the data volume processed per PSInSAR computation and enable adaptive partitioning, we propose a dynamic partitioning strategy based on a regular grid, utilizing a sliding window mechanism and edge sub-block merging to ensure seamless coverage of the entire area. The method is implemented as follows: first, an initial grid size $B_h\times B_w$ and overlap size $O$ are predefined. The sliding step sizes are then computed as $S_h=B_h-O$ and $S_w=B_w-O$. Based on the original data matrix size $H\times W$, initial sub-blocks are generated with row and column sizes defined as

$$B_{h0}^{\prime }=\min \left(B_h,H-i\cdot S_h\right),B_{w0}^{\prime }=\min \left(B_w,W-j\cdot S_w\right)$$

(1)

where $i$ and $j$ are the sub-block indices, with $i\in \left[0,⌈H/S_h⌉\right)$ and $j\in \left[0,⌈W/S_w⌉\right)$.

For edge sub-blocks (with row index $i_{\max }$ or column index $j_{\max }$) that may be smaller than the predefined size, a dynamic edge adjustment mechanism is introduced. If $B_{h0}^{\prime }<B_h$ (i.e., the remaining height in the original data is less than the initial grid size), the last row sub-block $(i_{\max },j)$ is merged with its adjacent upper sub-block $(i_{\max }-1,j)$, adjusting its height using

$$B_h^{\prime }=H-\left(i_{\max }-1\right)\cdot S_h$$

(2)

The adjusted sub-block size satisfies $B_h^{\prime }\in \left[B_h,\hspace{0.17em} 2B_h\right)$, with a similar adjustment applied in the column direction. This adaptive boundary extension strategy ensures complete coverage of the monitoring area, avoiding fragmented sub-blocks caused by non-integer multiples of the grid size in traditional partitioning methods. It also maintains strict consistency in overlapping regions between adjacent sub-blocks. For example, given an original data size of 8300 × 6700 pixels, with grid size $B_h=B_w=2000$ and overlap size $O=500$, the proposed partitioning method generates 20 sub-blocks. The last row sub-block has a height of 2300 pixels, and the last-column sub-block has a width of 2200 pixels, while all other sub-blocks remain at the standard size (Figure 2). This strategy effectively balances the data completeness and computational resource allocation efficiency while significantly simplifying the sub-block adjacency relationship retrieval in subsequent merging stages.

2.2. Block-Based PSInSAR Processing

2.2.1. Sub-Block PS Candidate Point Selection and Optimization

In block-based PSInSAR processing, traditional methods perform relative radiometric calibration independently for each sub-block, which may lead to a misjudgment of the same physical pixel in overlapping regions between adjacent sub-blocks due to differences in the calibration coefficients. Specifically, the radiometric calibration mean difference between sub-blocks causes discrepancies in the calibrated intensity values at the same location, which, in turn, affects the calculation of the amplitude deviation value $D_{A,k}\left(i\right)$, leading to a contradiction in the “selection-rejection” of PS candidate points in the overlapping region. To solve this issue, we designed a global consistency optimization framework based on the candidate point selection in the overlapping regions. First, each sub-block $k$ undergoes independent radiometric calibration, and an initial set of candidate points $P_k^{\mathrm{init}}$ is selected based on the amplitude deviation threshold ${\tau }_D$ and the average coherence threshold ${\tau }_{\gamma }$. The mathematical definition is as follows:

$$P_k^{\mathit{init}}=\left\{i | D_{A,k}\left(i\right)<{\tau }_D, \overline{{\gamma }_k}\left(i\right)>{\tau }_{\gamma }\right\}$$

(3)

where $D_{A,k}\left(i\right)$ represents the amplitude deviation value of pixel $i$ in sub-block $k$; reflecting the temporal stability of the intensity; and $\overline{{\gamma }_k}\left(i\right)$ represents the temporal average coherence, which measures the level of phase noise.

Then, for each pair of adjacent sub-blocks $\left(k,l\right)$, the union of PS candidate points in the overlapping region $R_{kl}$ is extracted, denoted as $O_{kl}$, which is defined as

$$O_{kl}=\left(P_k^{\mathit{init}}\cup P_l^{\mathit{init}}\right)\cap R_{kl}$$

(4)

By forcing the merging of candidate points in the overlapping region, consistent selection results for the same physical location are ensured across different sub-blocks. Finally, the candidate point set for sub-block $k$, $P_k^{\mathrm{final}}$, is updated using

$$P_k^{final}=\left(\bigcup _{l\in \mathcal{N}\left(k\right)}{O}_{kl}\right)\cup \left(P_k^{init}\setminus \bigcup _{l\in \mathcal{N}\left(k\right)}{R}_{kl}\right)$$

(5)

where $\mathcal{N}\left(k\right)$ represents the set of sub-blocks adjacent to sub-block $k$.

The candidate point selection and optimization strategy in this section effectively eliminates the local bias introduced by block-wise calibration through the global fusion of redundant candidate points, ensuring the continuity of the spatial distribution of candidate points and providing a consistent input for subsequent parameter calculation.

2.2.2. Block-Based PSInSAR Parameter Calculation

Once the candidate point set has been optimized, each sub-block independently performs PSInSAR parameter calculations [11,36]. The core process includes constructing the adjacency network, selecting reference points, performing a weighted least squares adjustment, and iterative optimization. First, an initial adjacency network is constructed based on the spatial distribution of candidate points, connecting candidate point pairs whose spatial distance is smaller than a preset threshold (default is 1 km) to suppress the phase noise introduced by atmospheric decorrelation in long baseline arc segments. Furthermore, arc segments are screened using overall coherence, retaining highly reliable connections to form the largest connected subnetwork. To ensure automated processing, the reference point selection strategy uses the candidate point with the minimum amplitude deviation within the sub-block. Based on this reference point [37], the deformation rate and elevation residuals are jointly solved using a weighted least squares adjustment. The weight design assigns a greater contribution to highly coherent arc segments, thus improving the robustness of the calculation results. Finally, the deformation rate field and elevation residual field of the sub-block are output as independent solving units for global stitching. To further improve the accuracy, the residual phase can undergo spatiotemporal filtering to separate nonlinear deformation components from the atmospheric phase, and the deformation model parameters can be iteratively updated. This process reduces the global $O\left(N^3\right)$ complexity of traditional PSInSAR by converting it into multiple $O\left(n^3\right)$ (where $n\ll N$) subproblems, significantly reducing the computational burden.

2.3. Global Stitching and Error Adjustment

To achieve the seamless integration of block-based solution results, we propose a global stitching framework based on overlapping region constraints and graph theory-connected component analysis. The core process includes an initial adjustment estimation, connected component identification, and least squares adjustment. First, the validity criterion for the overlapping region between adjacent sub-blocks $\left(k,l\right)$ is defined: if the number of common PS points $N_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}}$ exceeds a preset threshold ${\tau }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}}$, the overlap is considered valid; otherwise, the number of common points is set to 0 to exclude low-confidence connections. For valid overlapping regions, outliers in the deformation rate differences and elevation residual differences are removed using the three-sigma criterion, and the initial adjustment quantities are calculated based on the remaining valid common points’ deformation rate difference $\mathsf{\Delta }{v}_{kl}^{\left(m\right)}$ and elevation residual difference $\mathsf{\Delta }{ϵ}_{kl}^{\left(m\right)}$ [35]:

$$\mathsf{\Delta }{v}_{kl}={\displaystyle \frac{1}{N_{\mathit{valid}}}}\sum _{m\in V}\mathsf{\Delta }{v}_{kl}^{\left(m\right)},\mathsf{\Delta }{ϵ}_{kl}={\displaystyle \frac{1}{N_{\mathit{valid}}}}\sum _{m\in V}\mathsf{\Delta }{ϵ}_{kl}^{\left(m\right)}$$

(6)

where $V$ is the set of valid common points; $N_{\mathit{valid}}$ is the number of such points; and $\mathsf{\Delta }{v}_{kl}^{\left(m\right)}=v_k^{\left(m_1\right)}-v_l^{\left(m_2\right)}$ and $\mathsf{\Delta }{ϵ}_{kl}^{\left(m\right)}={ϵ}_k^{\left(m_1\right)}-{ϵ}_l^{\left(m_2\right)}$ represent the rate difference and residual difference for common points $\left(m_1,m_2\right)$, respectively.

Next, the connected components between sub-blocks are calculated [20], i.e., the set of sub-blocks is mapped to a graph $G$, and a depth-first search (DFS) is performed to identify all the connected components $\mathcal{C}$:

$$G=\left(V,E\right)$$

(7)

$$\mathcal{C}=\{{\mathcal{C}}_1,{\mathcal{C}}_2,\dots ,{\mathcal{C}}_{\mathcal{M}}\}$$

(8)

where the node set $V=\{1,2,\dots ,K\}$ represents sub-block indices, with $K$ being the total number of sub-blocks; the edge set $E=\left\{\left(p,q\right)\right\}$ represents adjacent sub-block pairs $p$ and $q$ having a valid overlapping relationship. Each connected component ${\mathcal{C}}_{\mathcal{m}}=\{k_1,k_2,\dots ,k_{K_m}\}$ represents a set of sub-blocks that can be directly or indirectly connected via an overlap, with $K_m$ being the number of sub-blocks in this connected component.

To suppress the error propagation, the sub-block with the most connecting edges in each connected component ${\mathcal{C}}_{\mathcal{m}}$ is selected as the reference sub-block $k_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}$ (its adjustment quantity is set to $X_{k_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}}=0$), and the coefficient matrix $B$ and observation vector $L$ are constructed. The matrix elements are defined as

$$B_{(\mathit{a},\mathit{p})}=\left\{\begin{array}{cc}1& ifedgeacorrespondstosub-blockpair(k,l)andp=k,\\ -1& ifedgeacorrespondstosub-blockpair(k,l)andp=l,\\ 0& otherwise\end{array}\right.$$

(9)

The weight matrix $P$ can be set as a diagonal matrix with diagonal elements representing the ratio of the number of valid common points for each sub-block pair, thus enhancing the weight of highly redundant connections. The adjustment quantities $X={\left[X_1,X_2,\dots ,X_{K_m}\right]}^{\top }$ are then solved using a least squares adjustment:

$$X=(B^{\top }PB{)}^{-1}{B}^{\top }PL$$

(10)

where $B\in {\mathbb{R}}^{L_m\times K_m}$, $P\in {\mathbb{R}}^{L_m\times L_m}$, $L_m$ is the number of valid edges in connected component ${\mathcal{C}}_{\mathcal{m}}$, and $K_m$ is the number of nodes (sub-blocks). $L\in {\mathbb{R}}^{L_m}$ represents the initial deformation rate adjustment $\mathsf{\Delta }{v}_{kl}$ or elevation residual adjustment $\mathsf{\Delta }{\mathsf{ϵ}}_{kl}$ for each edge $a$ corresponding to the sub-block pair $(k,l)$.

Finally, the adjusted quantities $X_k$ are added to the original solution results of each sub-block, generating a globally consistent deformation rate field and elevation residual field within each connected component. For multiple connected components, independent adjustments are performed. Absolute corrections align each component to a unified reference frame using manually identified stable ground control points (GCPs, e.g., non-deforming pavement areas or building corners), with the deformation rates and elevation residuals set to zero. This method, by using graph theory connectivity constraints and a weighted least squares adjustment, effectively solves the rank deficiency problem caused by isolated sub-blocks in traditional block stitching and achieves the high-precision seamless integration of large-scale PS networks.

3. Experimental Area and Data Preparation

We used the typical high-density urban area of Fuzhou, the capital city of Fujian Province in China, located on the southeastern coast of China (covering the core areas of Gulou District, Taijiang District, Cangshan District, and Jin’an District), as the experimental region. The geographical range was from 119°15′E to 119°25′E and from 26°00′N to 26°12′N (Figure 3a). The area was situated on the lower alluvial plain of the Min River, characterized primarily by plains (with an average elevation of approximately 10 m) interspersed with hills, such as Gaogai Hill and Qingliang Hill, which correspond to the two small red triangular regions marked in Figure 3b. The area had a dense water network, where water bodies accounted for 4.8% of the region [38], and the main stream of the Min River runs through the city (Figure 3c), forming a natural geographical boundary. The region contained diverse building types, including high-rise residential clusters, commercial complexes, transportation hubs, and urban villages, which are dense human-made features. The geological disaster risk induced by surface deformation was high, making it an ideal test area for validating the PSInSAR monitoring method in high-density urban areas.

This experiment used 25 scenes of TerraSAR-X ascending strip-mode data (range resolution: 0.9 m, azimuth resolution: 1.8 m, incidence angle: 32.6°) obtained from 20 May 2023, to 29 October 2024, as the primary dataset. The image from 17 January 2024, was selected as the reference image for registration, with the spatiotemporal baseline parameters shown in Figure 4a. To verify the reliability of the results, 45 scenes of Sentinel-1 ascending IW-mode data (range resolution: 2.3 m, azimuth resolution: 13.9 m, incidence angle: 44.0°) were also obtained, with the image from 20 January 2024, used as the reference image (spatiotemporal baseline shown in Figure 4b). The external DEM used was the Copernicus DEM GLO-30m to remove terrain effects. Data preprocessing was carried out using the GAMMA 2023 software platform, and the main steps included (1) spatial clipping of the study area (Figure 3); (2) multi-temporal data precise registration based on the reference image; and (3) differential interferometry processing and geocoding to generate differential interferograms after terrain removal, coherence maps, and range geometry parameter files, which provided standardized input for the PSInSAR processing.

4. Experimental Results and Analysis

4.1. Experimental and Results Analysis

In the monitoring area of 590 km² in the main urban area of Fuzhou, based on TerraSAR-X data (image matrix of 14,500 × 13,000 pixels), a dynamic rule grid partitioning strategy (grid size of 1200 × 1200 pixels, overlap of 300 pixels, overlap rate of 25%) was applied to divide the entire region into 210 sub-blocks (Figure 4a). After an adaptive adjustment, the maximum height of the edge sub-blocks was 1900 pixels, and the maximum width was 1300 pixels, while the other sub-blocks maintained the standard size. Using a dual threshold selection method with amplitude deviation ($D_A\le 0.65$) and average coherence ($\overline{\gamma }\ge 0.5$), the maximum number of candidate points in a single sub-block was 396,800 (Figure 5b), with the corresponding spatial density reaching 88,742 points/km². The total number of processed points reached 23,837,354, and after de-duplication in the overlapping regions, 13,867,836 PS candidate points were retained (

Table 1. Statistics of sub-blocks and PS points for different stitching methods.

	Initial Sub-Block Results	Traditional Stitching Method	Proposed Stitching Method
Number of sub-blocks (with PS points)	202	157	202
Number of PS points	12,921,707	12,204,302	12,921,707

), with an average density of 25,000 points/km² across the entire region. Under the 32 GB memory limitation of the machine, this method significantly surpassed the data-bearing limits of the traditional global methods. Based on the candidate point selection strategy described in Section 2.2.1, the distribution differences of the initial PS candidate points in the overlapping regions of each sub-block were significantly improved by global merging: after updating the overlapping regions through additional selection (Figure 5c), the number of candidate points in each sub-block increased by an average of 1600 points (a growth rate of 4.5%), providing sufficient redundant observations for subsequent stitching.

By setting the overlap region validity threshold to ${\tau }_{\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{a}\mathrm{p}}=500$, 494 valid overlap regions were selected (the valid overlap relationships of the sub-blocks are shown in Figure 6). For example, sub-block 75 (located in the sixth row and fifth column) had valid overlaps with seven adjacent sub-blocks (the overlapping sub-block indexes were 60, 61, 62, 74, 76, 89, and 90), indicating that it was in a core hub position within the connected components. Based on graph theory and connected component analysis, nine independent connected components were identified across the entire region. Among them, connected component ① contained 19 isolated sub-blocks (which were not connected with other sub-blocks due to PS point absence or invalid overlap regions), while connected component ③, the core connected domain, covered 157 sub-blocks (which accounted for 75.1% of valid overlapping sub-blocks) and formed the main deformation monitoring network for the entire region.

The independent PSInSAR processing of each sub-block yielded a total of 21,694,364 cumulative PS points. After merging and de-duplication, 12,921,707 PS points were retained, which accounted for approximately 93.2% of the original PS candidate points. However, the initial merging results exhibited significant systematic deviations in both the deformation rate and elevation residuals (Figure 7a). After correction using a weighted least squares adjustment, the stitched results (Figure 7b,c) demonstrate that the adjustment model effectively eliminated the systematic bias between sub-blocks.

The traditional stitching method (Figure 7b) refers to a block-merging approach that directly combines adjacent sub-blocks without analyzing the connectivity relationships or preserving isolated PS clusters. This method enforced a full-rank coefficient matrix by retaining only the largest connected domain (157 sub-blocks), which inevitably excluded fragmented components beyond this domain. As a result, PS point information from 45 sub-blocks (22.3% of the total) was discarded, and thus, only 94.4% of the PS points were retained compared with the proposed method. These excluded points predominantly corresponded to non-urban areas (e.g., scattered villages and suburban zones), which led to critical gaps in the local deformation monitoring.

In contrast, the proposed stitching method (Figure 7c) utilizes global connected component identification and regionally independent adjustment strategies. By analyzing graph-based topological relationships and performing weighted least squares adjustments for each connected component, this approach ensured complete retention of the PS points from all 202 sub-blocks (Table 1). A comparison shows that the traditional stitching method discarded 5.6% of the PS points outside the largest connected domain (the area outside the red rectangle in Figure 7c), corresponding to scattered villages and suburban areas at the city’s edge, which led to the loss of crucial local deformation monitoring information. By incorporating connected component identification and regional adjustment, the proposed method effectively mitigated this issue, and thus, preserved the deformation signals in peripheral regions and enhanced the completeness and reliability of the global monitoring results.

The final results are shown in Figure 8 and Figure 9, where the PS points in the areas with man-made features are fully covered, and the overall point count reached the ten million level, which supported refined deformation analysis for sparse village areas or individual buildings. The global elevation residuals were concentrated in the range of [−30 m, 110 m], which was consistent with the actual building height distribution, with the InSAR monitoring points densely distributed along the contours of high-rise residential areas and large single buildings (Figure 8). The global deformation rates were concentrated in the range of [−10 mm/y, 4 mm/y]. The severely subsiding areas (deformation rate ≤ −10 mm/y) were mostly concentrated in self-built houses in urban villages, construction areas, riverbank protections along the Min River, and major roads. Most areas had deformation rates around 0 mm/y (Figure 9).

4.2. Accuracy Verification

To verify the reliability of the proposed method, internal consistency validation was performed by comparing the results of the proposed method with those of the traditional global PSInSAR method using TerraSAR-X data. Cross-validation with multi-source data was also conducted using Sentinel-1 data. The traditional global PSInSAR method operates as a non-partitioned, full-scene-processing technique that constructs a global adjacency network across the monitoring area and solves deformation parameters through a least squares adjustment. This approach requires full network connectivity to maintain the coefficient matrix’s full-rank condition—a critical prerequisite for stable parameter estimation. To accommodate memory constraints in monolithic processing, we applied 5 × 5 spatial down-sampling (5:5 multi-looking) to the TerraSAR-X dataset, reducing the PS density from 25,000 points/km² to 930 points/km² and original candidate PS points from 13,867,836 to 547,133. This density reduction fragmented the network topology into isolated subnetworks. Consequently, only the largest connected subnetwork (416,089 PS points) was retained to prevent coefficient matrix singularity, while the discontinuous regions (e.g., rural villages, mountainous areas) where subnetworks failed to satisfy the rank condition for stable least squares adjustment were systematically excluded. For the Sentinel-1 dataset, we used data from the same period, with cropped images of size 9000 × 1900 (5:1 multi-looking), which yielded 349,674 PS points (

Table 2. Data specifications for different methods.

Method	Image Size	Multi-Looking Ratio	Number of PS Points	Peak Memory Usage
Traditional Method—TerraSAR-X	14,500 × 13,000	5:5	416,089	~20.7 GB
Traditional Method—Sentinel-1	9000 × 1900	5:1	349,674	~22.0 GB
Proposed Method—TerraSAR-X	14,500 × 13,000	1:1	12,921,707	~15.1 GB

). The elevation residuals and deformation rates of matched PS points obtained from different methods were quantitatively evaluated using three metrics: mean bias, standard deviation (SD), and correlation coefficient (Cor). The results are shown in

Table 3. Accuracy comparison of different methods.

Method	PS Point Ratio (vs. Proposed Method)	Elevation Residual (vs. Proposed Method, m)			Deformation Rate (vs. Proposed Method, mm/y)
		Mean Bias	SD	Cor	Mean Bias	SD	Cor
Traditional Method—TerraSAR-X	3.2%	−0.85	3.38	0.99	0.08	0.48	0.98
Traditional Method—Sentinel-1	2.7%	0.02	2.77	0.99	−0.63	2.43	0.75
Proposed Method—TerraSAR-X	100%	0	0	1	0	0	1

. Notably, during cross-validation with Sentinel-1 data, geometric consistency between TerraSAR-X and Sentinel-1 datasets was ensured by projecting line-of-sight (LOS) deformation rates to the vertical direction through

$$v_{\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{l}}={\displaystyle \frac{v_{\mathrm{L}\mathrm{O}\mathrm{S}}}{\cos \theta }}$$

(11)

where $v_{\mathrm{L}\mathrm{O}\mathrm{S}}$ is the LOS deformation rate, and $\theta$ is the radar incidence angle (32.6° for TerraSAR-X, 44.0° for Sentinel-1).

The results of the elevation residuals and deformation rates from different methods are shown in Figure 10 and Figure 11. The traditional global PSInSAR method using the TerraSAR-X and Sentinel-1 datasets produced fewer PS points (3.2% and 2.7% of the total points from the proposed method, respectively), which significantly reduced its ability to monitor fine details, such as individual buildings. Additionally, the PS points in the left lower corner of the image, which corresponded to urban villages and sparsely distributed villages and roads on the mountains (highlighted in red in Figure 10), were not effectively captured. The SD of the elevation residuals for the traditional and proposed methods were 3.38 m and 2.77 m, respectively, with correlation coefficients of 0.99 for both, indicating a high consistency between the two methods. In areas with concentrated deformation rates (shown in the blue rectangles in Figure 11), both methods provided similar results. However, for densely populated low-rise buildings (highlighted in yellow in Figure 11), significant deformation was observed in the proposed method, while the traditional method did not detect any PS deformation points in these areas. The SD of the deformation rates for the traditional and proposed methods were 0.48 mm/y and 2.43 mm/y, respectively, with correlation coefficients of 0.98 and 0.75. Overall, the deformation rates and elevation residuals from the proposed method demonstrated high reliabilities.

5. Discussion

5.1. Advantages of Grid-Based Block Partitioning

In the grid-based block-partitioning strategy, the selection of the sub-block size and overlap degree is essentially a dynamic balancing process between memory efficiency, computational accuracy, and processing time. Smaller sub-block sizes significantly reduce the memory usage by decreasing the number of PS points processed at a time. Meanwhile, the increased local network connection density (more redundant arcs) enhances the accuracy and robustness of the adjustment calculation. A higher overlap degree enlarges the interaction area between adjacent sub-blocks, which not only improves the connectivity (ensuring sufficient common points in the overlap area) but also increases the observation redundancy for the sub-block adjustment calculations, thereby mitigating error propagation. As demonstrated in Section 4.1, the selected partitioning parameters (grid size of 1200 × 1200 pixels, overlap of 300 pixels) produced 210 sub-blocks with a total of 23,837,354 PS points. After de-duplication in the overlapping regions, 13,867,836 points were retained, indicating that ~9.97 million points (42% of the total) resided in overlapped areas. This highlights the dual effect of a higher overlap: redundant observations for a robust adjustment (elevating solution precision) versus an increased computational load from duplicate processing. Critically, these parameters (sub-block size, overlap degree, sliding step) primarily influence the memory management and connectivity during stitching but do not inherently alter the deformation or elevation solutions at individual PS points. The core PSInSAR inversion (e.g., phase unwrapping, weighted least squares) operates identically within each sub-block, and systematic biases introduced by partitioning are corrected during the global adjustment (Section 2.3). As demonstrated in Section 4.2, the deformation rates and elevation residuals from the proposed method exhibited near-perfect consistency with traditional global PSInSAR results (correlation coefficient ≥ 0.98, standard deviation ≤ 0.48 mm/y for deformation, ≤3.38 m for elevation residuals; Table 3). This confirms that the parameter choices affected the computational workflow rather than solution fidelity. However, this optimization came at the cost of increased repeated processing—smaller sub-blocks led to a linear increase in the number of blocks, and a higher overlap degree led to exponentially increased duplicate computation. Therefore, partitioning parameters need to be carefully balanced based on hardware resources and task objectives.

The grid-based partitioning was the key preliminary step for the block-based PSInSAR processing in this study. Its core advantage lies in the strict geometric structure and deterministic spatial logic. The precise correspondence between sub-block row/column indices and geographic coordinates ensures that the block positions remain consistent in the global coordinate system, avoiding the need for dynamic spatial index reconstruction, as required by irregular block partitioning. This significantly simplifies the retrieval of overlapping regions and the matching of common points, reducing the complexity of the process. The parameterized nature of the grid allows for critical parameters, such as the sub-block size, overlap degree, and sliding step size, to be pre-defined through mathematical formulas. Combined with the boundary adjustment mechanism, this approach achieves fully automated seamless coverage of the data without relying on prior knowledge of PS point distribution, ensuring both efficiency and robustness in the partitioning process. The spatial symmetry between blocks further optimizes the subsequent adjustment process: the uniformly distributed overlap areas not only guarantee redundancy in common point observations but also give the coefficient matrix a structured sparse feature, greatly improving the efficiency of large-scale matrix computations. Meanwhile, the adjacency relationship based on row/column indices allows for rapid identification of the independent connected domains, helping to avoid global network rank deficiency by segmenting the adjustment. Compared with adaptive block partitioning methods, grid partitioning replaces the uncertainty of dynamic shape adjustments with fixed geometric constraints, achieving a better balance between algorithmic complexity, computational efficiency, and result consistency. Its spatial logic transparency and controllability provide a reusable technical framework for wide-area deformation monitoring in high-density urban environments.

5.2. Feasibility Evaluation of PSInSAR Result Stitching Based on Overlap Constraints

Traditional PSInSAR methods rely on a single control point and a global connected network for phase unwrapping and parameter transfer. The global network construction faces computational bottlenecks when dealing with a vast number of PS points. The block strategy proposed in this paper adjusts for systematic biases between sub-blocks by analyzing the statistical characteristics of deformation rate differences and elevation residual differences at common points in the overlap region. The theoretical assumption is that block differences arise from variations in local control points and network structures, manifesting as overall offsets (systematic biases) rather than random noise. By analyzing the statistical properties of the differences at common points in the overlap regions, the mathematical rationality and practical applicability of the method can be verified.

The statistical analysis results of the effective overlap regions (Figure 12) show that the stitching error between sub-blocks was within a controllable range: (1) The mean standard deviations of elevation residual differences and deformation rate differences were 0.326 m and 0.126 mm/y, respectively. Among these, 88% of the effective overlap regions had an elevation residual standard deviation ≤0.6 m and a deformation rate standard deviation ≤0.2 mm/y, indicating that the differences between the sub-blocks were mainly driven by systematic biases. (2) The mean absolute values of the adjustment for elevation residuals and deformation rate were 18.201 m and 2.532 mm/y, respectively. The corresponding standard deviations (0.326 m and 0.126 mm/y) led to relative errors of 1.8% and 5.0%, respectively. This demonstrates that the adjustment values had far less dispersion than the mean, confirming the effectiveness of adjusting the entire block based on the mean value of differences at common points in the overlap region.

5.3. Advantages and Limitations of the PSInSAR Method Based on Grid Partitioning and Connectivity Constraints

The PSInSAR method based on grid partitioning and connectivity constraints has demonstrated significant technical advantages when dealing with high-resolution or ultra-high-resolution SAR imagery (strip-map or spotlight mode) with large data volumes and heterogeneous PS point distributions. This method breaks down massive amounts of data into independently processed sub-blocks via grid partitioning, reducing memory requirements to levels that are manageable by standard computational devices, thus overcoming the memory limitations that prevent traditional methods from processing ultra-large images. It significantly lowers the technical barriers for processing large-scale high-resolution data and ensures efficient full-area coverage. The independent processing characteristic of sub-blocks naturally adapts to distributed computing frameworks, allowing for parallel acceleration based on multi-core CPUs or computing clusters, drastically reducing the processing time and providing the technical foundation for near-real-time large-area deformation monitoring. In addition, the global constraint mechanism for the overlap regions between sub-blocks effectively suppresses discontinuities in the deformation field or elevation residual field at the stitching boundaries, ensuring spatial consistency across the entire region. Notably, the connectivity analysis module introduced in this method automatically identifies independent connected components within sub-blocks (such as isolated building groups or scattered villages) and applies adaptive adjustment, thus avoiding the loss of isolated PS point signals and achieving the dual goals of “global coverage without omissions” and “high-fidelity local deformation monitoring”.

However, this method is currently primarily suitable for large-scale partitioning and the stitching of single-scene SAR imagery or cropped single-scene images. Its applicability to multi-scene SAR imagery for wide-area coverage still has limitations. Multi-scene images may suffer from mismatches at the stitching boundaries due to geometric registration errors, radiometric differences, and deformation model parameter transfer across scenes. Further research into cross-scene overlap region constraint mechanisms or adaptive registration algorithms is needed to improve the overall consistency of the deformation field. Additionally, fixed-size grid partitioning may interfere with the identification of connected components in complex terrain areas (such as mountainous–plain transition zones). Future work will explore dynamic grid-partitioning strategies or machine learning-assisted connected domain optimization methods. Potential local parameter estimation biases due to independent sub-block adjustment also need to be addressed by global iterative optimization or by introducing external observation data constraints. In this study, the manual selection of stable ground reference points within connected components ensured absolute calibration but introduced subjectivity. For instance, in Section 4.1, 19 ground points were manually selected across nine components to unify the reference frames. Future work will integrate GNSS data or automated algorithms to mitigate systematic error accumulation and improve the robustness of large-area deformation field inversion.

6. Conclusions

This paper proposes an efficient automated method based on grid partitioning and connectivity constraints to address the challenges of memory overflow and block stitching due to the heterogeneous distribution of PS points in large-scale PSInSAR processing for high-density urban areas. Through grid partitioning and overlap region adjustment mechanisms, the method reduces memory usage while ensuring spatial consistency across the global deformation field. By combining graph theory-based connectivity analysis, it solves the problem of retaining information from discrete PS point clusters within sub-blocks, achieving the collaborative optimization of “global continuity” and “local precision”. Experiments based on the TerraSAR-X data from Fuzhou’s central urban area show that the method can process up to 14 million PS points on standard computational devices (32 GB memory), improving the processing efficiency by 25 times compared with traditional global methods. The deformation rate and elevation residual results are highly consistent with traditional methods (correlation coefficient ≥ 0.98), and cross-validation with the Sentinel-1 data further confirmed its reliability. The innovation of this method lies in the construction of an integrated “block-adjustment-fusion” technical framework, significantly lowering the technical barriers for wide-area PSInSAR applications and providing a feasible path for operational deformation monitoring using high-resolution SAR data. Future research will focus on optimizing multi-scene image stitching and dynamic grid partitioning to extend its applicability to complex terrain and wide-area monitoring.