A Morpho-Phase Feature-Based Method for Geometric Error Mitigation in InSAR Image Matching

Highlights

In InSAR-based scene matching navigation systems, parameter estimation errors cause significant interferometric fringe shifts, which severely degrade the accuracy of existing matching algorithms and the overall navigation positioning. To resolve this critical issue, this paper proposes the PRK-HMPD keypoint detection and matching algorithm specifically designed for interferograms, achieving robust matching under fringe shifts and significantly improving navigation accuracy.

What are the main findings?

• A Phase-Robust Keypoint (PRK) detection method is proposed to address textural shifts caused by parameter estimation errors. By constructing a 3D compensated phase space instead of the conventional scale space, and extracting stable phase extrema rather than gradient extrema, the proposed method effectively filters out invalid keypoints and ensures high repeatability under parameter uncertainties.
• To overcome the feature ambiguity of traditional descriptors on InSAR images with strong non-local similarity, a Hierarchical Morphological-Phase Descriptor (HMPD) is proposed, which fully characterizes the unique morphological features and phase statistical features of keypoints, thereby improving the discriminability of descriptors and reducing the false matching rate of keypoints.

What are the implications of the main findings?

• Academic Implication: It addresses the critical bottleneck of geometric parameter errors and interferogram non-local similarity that degrade conventional matching algorithms, providing a novel Morpho-Phase feature-based solution (integrating PRK detection and HMPD descriptor) to improve InSAR image matching robustness—offering a new research direction for feature design in interferometric image matching.
• Engineering Implication: By enabling stable and high-precision positioning under geometric parameter errors, the proposed method promotes the practical application of InSAR as a reliable payload for UAV scene matching navigation, which is particularly valuable for scenarios requiring high navigation accuracy (e.g., precision agriculture, disaster monitoring, and autonomous UAV operations).

Abstract

Interferometric Synthetic Aperture Radar (InSAR) is a promising payload for Unmanned Aerial Vehicle (UAV) scene matching navigation due to the rich textures in interferogram images compared to SAR intensity images. However, geometric parameter estimation errors during reference interferogram image generation cause significant textural discrepancies with real-time data. Compounded by inherent non-local similarity of InSAR images, these issues render conventional matching algorithms ineffective, degrading navigation accuracy. To address these challenges, this paper proposes a Morpho-Phase feature-based InSAR image matching method to mitigate the impact of parameter errors. Firstly, a Phase-Robust Keypoint (PRK) detection method is proposed, which overcomes the impact of parameter errors on keypoint detection by introducing a compensated phase and extracting phase extrema. Secondly, a Hierarchical Morphological-Phase Descriptor (HMPD) is constructed to resolve the feature ambiguity caused by the non-local similarity of interferograms by combining morphological features with phase statistics. Experimental results based on real-world InSAR data demonstrate that the proposed matching method effectively mitigates the impact of parameter errors on InSAR image matching, enhances navigation positioning accuracy, and provides stable, high-precision positioning capabilities in practical scene matching navigation tasks.

1. Introduction

Unmanned Aerial Vehicles (UAVs) serve as critical aerial platforms widely deployed in diverse flight missions; however, their operation relies heavily on high-precision positioning services provided by Global Navigation Satellite Systems (GNSS). In complex electromagnetic interference environments, GNSS are highly susceptible to suppression or spoofing, potentially leading to navigation failure [1,2]. While Inertial Navigation Systems (INS) offer the advantages of complete autonomy and strong immunity to interference, their positioning errors accumulate over time, making it difficult to meet the requirements of long-endurance and high-precision missions [3]. Consequently, the development of autonomous navigation technologies capable of correcting cumulative INS errors under GNSS-denied conditions has become a research hotspot. Scene Matching Navigation (SMN), which retrieves the absolute position of the aircraft by matching real-time ground images sensed by the UAV with pre-stored geo-referenced base maps, is characterized by high autonomy and drift-free errors [4,5]. It stands as a key technological approach for achieving all-day, high-precision autonomous navigation for aircraft.

Synthetic Aperture Radar (SAR) [6,7] is considered an ideal sensor for scene matching navigation [8,9]. Compared with optical and infrared imaging sensors, SAR possesses all-weather and all-day operational capabilities and can penetrate clouds, fog, and camouflage, offering significant advantages in adverse meteorological conditions [10]. With advancements in electronic technology, airborne SAR systems are evolving towards miniaturization, lightweight design, and high resolution, providing the hardware foundation for UAV platforms to carry SAR payloads [11]. Furthermore, research on scene matching using SAR images has matured significantly. Methodologies have progressed from early template matching algorithms based on grayscale correlation, to feature point matching algorithms utilizing local invariant features such as SAR-SIFT [12] and KAZE-SAR [13], and more recently to matching networks based on deep learning [14,15]. These algorithms have achieved favorable SAR image matching results in regions with salient features.

Despite the rapid development of SAR scene matching technology, severe challenges remain in practical applications. The imaging mechanism of SAR dictates that image quality is highly dependent on surface scattering characteristics [16]. For imaging regions containing strong scattering features such as roads, rivers, and buildings, SAR images can provide sufficient features for image matching [17]. However, in areas lacking distinct surface features, such as plains, hills, or uniformly vegetated regions, SAR images often exhibit weak texture and low contrast, and are susceptible to severe interference from speckle noise [18]. In such scenarios, geometric structural features like corners and edges within SAR images are extremely scarce, leading to a severe paucity of semantic information [19]. Consequently, traditional feature extraction algorithms struggle to detect a sufficient number of uniformly distributed stable feature points, which in turn leads to matching failure and the inability to provide effective positioning information for the aircraft.

To overcome the limitations of SAR images in weak-texture regions, Interferometric Synthetic Aperture Radar (InSAR) technology has entered the field of navigation. As a mature mapping technology, InSAR has traditionally been employed primarily for elevation measurement and terrain deformation monitoring [20,21]. It is precisely due to the extremely high sensitivity of InSAR to surface elevation changes that, even in weak-texture regions with uniform surface reflectivity, interferograms will exhibit distinct interferometric fringes as long as minor terrain undulations exist [22]. These fringes, which vary with terrain fluctuations, contain rich topographic semantic information, effectively filling the information gap of SAR images in feature-scarce areas. Figure 1a,b visually demonstrates the difference in textural information content between a SAR image and an InSAR interferogram of the same area. Currently, scholars have begun to explore scene matching navigation schemes based on InSAR. For instance, Jiang et al. [23] and Sun et al. [24] have successively proposed utilizing interferograms for scene matching, preliminarily validating the feasibility of this technological route.

Although scene matching schemes based on InSAR are theoretically feasible, existing studies mostly adopt classical image matching frameworks directly, extracting features with significant gradients by constructing various types of pyramid spaces. However, for InSAR-based scene matching navigation tasks, the reference interferogram is generated using DEMs and InSAR system parameters. The estimated values of these parameters inevitably contain errors due to factors such as the autonomous divergence of the INS and geometric calibration issues [25,26]. These errors lead to discrepancies in characteristics such as the position and density of interferometric fringes between the reference interferogram and the real-time interferogram acquired by the UAV, as illustrated in Figure 1b,c. Consequently, since the interferometric fringes serve as the primary source of gradient features, discrepancies in their spatial distribution directly result in a lack of spatial correspondence between the keypoints extracted from the reference and real-time interferograms, thereby causing the matching algorithm to fail.

On the other hand, existing studies have yet to fully account for the impact of the non-local similarity [27,28] of interferograms on scene matching. Due to the phase wrapping characteristic of interferometric phases, periodic jumps occur when the terrain elevation change exceeds the height of ambiguity [29]. This results in extremely high similarity in fringe structures and local neighborhood textures across different regions of the interferogram, as the red regions shown in Figure 1b,c. For keypoints distributed on different fringes, if traditional descriptors based on local neighborhood grayscale statistics or gradients such as SIFT descriptors are employed, the feature vectors of these keypoints tend to be highly convergent and lack discriminability. In the matching stage, this ambiguity of features readily leads to outliers, severely compromising the final navigation positioning accuracy.

In summary, leveraging the rich fringe features in InSAR interferograms for scene matching constitutes a superior solution for addressing the failure of SAR matching navigation in feature-poor regions. However, the matching frameworks in existing research have yet to effectively resolve the interferogram matching errors comprehensively caused by textural inconsistencies arising from system parameter estimation errors and feature confusion induced by phase wrapping. Consequently, this paper proposes an interferogram matching method based on Morpho-Phase features, which effectively mitigates the matching errors resulting from the aforementioned issues, thereby realizing higher-precision InSAR-based scene matching navigation. The primary contributions of this paper are twofold:

• A Phase-Robust Keypoint (PRK) detection method is proposed to address textural shifts caused by parameter estimation errors. By constructing a 3D compensated phase space instead of the conventional scale space, and extracting stable phase extrema rather than gradient extrema, the proposed method effectively filters out invalid keypoints and ensures high repeatability under parameter uncertainties.
• To overcome the feature ambiguity of traditional descriptors on interferograms with strong non-local similarity, a Hierarchical Morphological-Phase Descriptor (HMPD) is proposed, which fully characterizes the unique morphological features and phase statistical features of keypoints, thereby improving the discriminability of descriptors and reducing the false matching rate of keypoints.

Finally, the feasibility of the proposed method in real-world scene matching navigation tasks is demonstrated using a set of actual airborne InSAR data. The overall research motivation of this paper is illustrated in Figure 2.

It is worth noting that, compared with conventional InSAR mapping tasks, the proposed method relies only on the morphological structure of interferometric fringes rather than the absolute accuracy of the phase, and is therefore less sensitive to platform instability and phase noise. The airborne dual-antenna InSAR system used in our experiments ensures interferometric coherence through a rigid baseline structure. After standard motion compensation and phase filtering, the resulting interferograms adequately satisfy the requirements of the matching algorithm.

2. Materials and Methods

This paper proposes an autonomous navigation method for aircraft based on interferogram matching, which is illustrated in Figure 3. The aircraft acquires real-time interferograms using the airborne SAR system, while simultaneously generating reference interferograms based on the reference DEM. Subsequently, both interferograms undergo keypoint detection and matching. The aircraft positioning inversion is then performed based on the matched points, yielding the platform position corresponding to the azimuth time of each matched point. Finally, the integrated navigation result is obtained by fusing the data with the aircraft’s Inertial Navigation System (INS).

2.1. Data Acquisition & Generation

2.1.1. Real-Time Interferogram Acquisition

For a single-pass dual-antenna interferometric SAR system, a single flight pass is sufficient to acquire SAR images of the same area from the master and slave antennas. The real-time interferogram of the observed area is obtained by first co-registering these two SAR images to align the slave antenna image with the master antenna image [30], followed by interferometric phase extraction, flat-Earth phase removal, and phase filtering, as shown in Figure 4.

2.1.2. Reference Interferogram Generation

The reference interferogram can be generated using a DEM and the imaging geometry of the InSAR system. Assuming a dual-antenna InSAR system, where B is the baseline length, $\alpha$ is the baseline inclination angle, $\theta$ is the look angle of the master antenna, and $R_1$ and $R_2$ are the distances from the master and slave antennas to a ground target point, respectively. The relationship between $R_1$ and $R_2$ can be expressed as [23]:

$$R_2=\sqrt{R_1^2+B^2-2B{R}_{1}sin(\theta -\alpha )}$$

(1)

Subsequently, the absolute phases corresponding to $R_1$ and $R_2$ can be computed based on the relationships between wavelength, phase, and slant range, as shown below:

$${\Psi }_1=-{\displaystyle \frac{4\pi R_1}{\lambda }},{\Psi }_2=-{\displaystyle \frac{4\pi R_2}{\lambda }}$$

(2)

Therefore, the interferometric phase for any target point on the DEM relative to the airborne InSAR system is given by:

$$\Delta \Psi ={\Psi }_2-{\Psi }_1=-{\displaystyle \frac{4\pi }{\lambda }}(R_1-R_2)$$

(3)

The interferometric phase shown in Equation (3) still contains the reference flat-Earth phase. After removing the flat-Earth phase and phase wrapping, the reference interferogram can be obtained. Figure 5a shows the reference DEM, and Figure 5b is the reference interferogram generated based on this DEM and the estimated imaging geometry parameters corresponding to the data shown in Figure 4.

2.2. Interferogram Matching

As shown in Figure 4 and Figure 5, significant texture differences, especially interferometric fringe shifts, exist between the real-time acquired interferogram and the reference interferogram generated from DEM for the same imaging area. The core reason for this issue lies in the inevitable errors introduced by the InSAR system observation parameters used to generate the reference interferogram. To be more precise, estimation errors may exist for the baseline length, baseline inclination angle, and even the slant range of the master antenna in Equation (1) when computing the interferometric phase with Equation (1) through Equation (3). Since the interference phase is extremely sensitive to parameter errors, the textures of the two images are inherently inconsistent. Direct matching using these two images would introduce significant errors, thereby affecting navigation and positioning accuracy.

To achieve robust image matching under parameter errors, this paper constructs a three-dimensional space by introducing a compensation phase dimension based on the range and azimuth directions. Stable feature points are extracted through morphological processing, and a multi-level grayscale-morphology joint feature is designed with corresponding similarity measurement criteria, thus achieving robust feature point matching.

2.2.1. 3D Space Construction

The texture representation capability of a single interferogram is limited by its unique phase distribution. To enhance the robustness of feature representation, this paper generates a set of compensation phase sequences within the range of $[0,2\pi )$:

$${\varphi }_k={\displaystyle \frac{2\pi k}{K}},k=0,1,\dots ,K-1$$

(4)

Each compensation phase is superimposed onto the original interferogram $\Phi (x,y)$ and undergoes phase rewrapping:

$${\Phi }_{{\varphi }_k}(x,y)=\mathrm{mod}\left(\Phi (x,y)+{\varphi }_k,2\pi \right)-\pi$$

(5)

where $\mathrm{mod}(·)$ denotes the modulo operation. This process constructs a three-dimensional space $(\varphi ,x,y)$ composed of compensation phase, range direction, and azimuth direction, as shown in Figure 6. The fundamental significance of this space is that for any point $(x,y)$ in the range-azimuth plane, it corresponds to multiple phase values ${\Phi }_{{\varphi }_k}(x,y)$ under different compensation phases ${\varphi }_k$, thereby enabling the generation of multiple feature descriptors. This multi-view representation provides rich discriminative information for subsequent feature extraction and matching.

2.2.2. Keypoint Detection

The primary features of interferograms manifest as fringe patterns formed by phase jumps, which can be categorized into closed fringes and open fringes. When local extreme points exist in the terrain, the surrounding fringes typically exhibit closed forms. As the compensation phase varies, these closed fringes undergo radial movement around the terrain extreme points, analogous to the evolution of equiphase surfaces around extreme points.

This observation indicates that the interference phase extreme points within each region enclosed by closed fringes can serve as stable feature points. Feature point detection mainly involves three steps: phase jump point extraction, morphological-based closed fringe extraction, and extreme point localization.

First, the phase gradient field is constructed for ${\Phi }_{{\varphi }_k}(x,y)$:

$$\begin{array}{c}{G}_x^{{\varphi }_k}\left(x,y\right)={\Phi }_{{\varphi }_k}(x+1,y)-{\Phi }_{{\varphi }_k}(x,y)\hfill \\ G_y^{{\varphi }_k}\left(x,y\right)={\Phi }_{{\varphi }_k}(x,y+1)-{\Phi }_{{\varphi }_k}(x,y)\hfill \end{array}$$

(6)

Considering the phase wrapping characteristics, cyclic differences are used to compute the gradients:

$$\begin{array}{c}{\nabla }_x{\Phi }_{{\varphi }_k}(x,y)=min\left(|G_x^{{\varphi }_k}\left(x,y\right)|,2\pi -|G_x^{{\varphi }_k}\left(x,y\right)|\right)\hfill \\ \nabla y{\Phi }_{{\varphi }_k}(x,y)=min\left(|G_y^{{\varphi }_k}\left(x,y\right)|,2\pi -|G_y^{{\varphi }_k}\left(x,y\right)|\right)\hfill \end{array}$$

(7)

Based on the gradient components, the combined gradient magnitude is defined as:

$$E_{{\varphi }_k}(x,y)=max\left(|{\nabla }_x{\Phi }_{{\varphi }_k}(x,y)|,|{\nabla }_y{\Phi }_{{\varphi }_k}(x,y)|\right)$$

(8)

A binary edge map is obtained through adaptive threshold segmentation:

$$B_{{\varphi }_k}(x,y)=\left\{\begin{array}{c}1,\mathrm{if}{E}_{{\varphi }_k}(x,y)>\alpha ·2\pi \hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.$$

(9)

where $\alpha \in (0,1)$ is the threshold ratio coefficient, typically set to 0.6–0.8.

Since the initial edge map $B_{{\varphi }_k}(x,y)$ may contain discontinuities, morphological closing operation [31] is applied to connect discrete edge segments $B_{{\varphi }_k}^{\mathrm{closed}}$:

$$B_{{\varphi }_k}^{\mathrm{closed}}=\left(B_{{\varphi }_k}\oplus S\right)\ominus S$$

(10)

where S is a circular structuring element, and ⊕ and ⊖ represent morphological dilation and erosion operations, respectively.

From the edge map, boundaries that are both continuous and innermost contours are extracted, thereby screening out closed solid regions $R_i$ enclosed by these contours, where i denotes the region index. Local maximum points are detected within each solid closed region $R_{{\varphi }_k,i}$, and denoted as $P_{{\varphi }_k,i}$, which can serve as detected keypoint for subsequent matching. Figure 7 shows the processing results of the above steps.

2.2.3. Descriptor Extraction

As mentioned earlier, keypoints are extreme points within closed fringes, and thus descriptors can be established based on the morphological characteristics of the closed fringes and the interferometric phase within their enclosed regions.

For each keypoint $P_{{\varphi }_k,i}$ corresponding to the solid closed region $R_{{\varphi }_k,i}$, a 5-dimensional morphological feature vector is defined:

$${\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_k,i}=\left[L_{{\varphi }_k,i},A_{{\varphi }_k,i},L_{\mathrm{bbox}}^{{\varphi }_k,i},W_{\mathrm{bbox}}^{{\varphi }_k,i},D_c^{{\varphi }_k,i}\right]\in {\mathbb{R}}^5$$

(11)

where $L_{{\varphi }_k,i}$ is the perimeter of the closed region, $A_{{\varphi }_k,i}$ is the area of the region, $L_{\mathrm{bbox}}^{{\varphi }_k,i}$ and $W_{\mathrm{bbox}}^{{\varphi }_k,i}$ are the length and width of the minimum enclosing rectangle, and $D_c^{{\varphi }_k,i}$ is the distance between the centroid of the closed region and the geometric center of the enclosing rectangle, as illustrated in Figure 8.

In addition to morphological features, the phase distribution within solid closed regions also has descriptive capabilities. The interferogram ${\Phi }_{{\varphi }_k}(x,y)$ is mapped to the interval $[0,255]$ to obtain the grayscale image $I_{{\varphi }_k}(x,y)$, and the histogram of grayscale values within each solid closed region $R_{{\varphi }_k,i}$ is calculated to obtain the normalized histogram as the phase feature vector:

$${\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_k,i}=[h_0^{{\varphi }_k,i},h_1^{{\varphi }_k,i},...,h_{255}^{{\varphi }_k,i}]\in {\mathbb{R}}^{256}$$

(12)

The above two feature vectors will serve as the descriptors for subsequent keypoint matching.

2.2.4. Keypoint Aggregation and Matching

It is evident that after constructing a three-dimensional space by introducing compensation phases, closed regions and keypoints can be detected under different compensation phases. More importantly, due to the wrapping characteristics of the interferometric phase, although the shapes of certain closed regions may vary under different compensation phases, the keypoints corresponding to these closed regions remain at the same position. This means that within the three-dimensional space, keypoints at identical positions can be aggregated along the compensation phase dimension, as shown in Figure 9. The aggregated keypoints will then have more descriptors, with each descriptor corresponding to a specific compensation phase.

The aggregated keypoints can serve as robust features of the interferogram under different compensation phases, thus we call them Phase-Robust Keypoints (PRK); while the descriptors corresponding to these keypoints contain both morphological features and phase statistical features of the point under multiple compensation phases, therefore we term this description as Hierarchical Morphological-Phase Descriptor (HMPD).

Specifically, the steps for keypoint aggregation are as follows. Assume that point $P_{{\varphi }_1,i}=(r_i^{{\varphi }_1},a_i^{{\varphi }_1})$ and $P_{{\varphi }_2,j}=(r_j^{{\varphi }_2},a_j^{{\varphi }_2})$ correspond to keypoints in solid closed regions $R_{{\varphi }_1,i}$ and $R_{{\varphi }_2,j}$ at compensation phases ${\varphi }_1$ and ${\varphi }_2$, respectively. If the following condition is met:

$$\sqrt{{\left(r_i^{{\varphi }_1}-r_j^{{\varphi }_2}\right)}^2+{\left(a_i^{{\varphi }_1}-a_j^{{\varphi }_2}\right)}^2}<{\tau }_{\mathrm{merge}}$$

(13)

where ${\tau }_{\mathrm{merge}}$ is the aggregation threshold, it is then assumed that these two keypoints correspond to the same physical location, and they are merged into a new keypoint $P=(r,a)$, with its descriptor set $\mathbf{D}$ constructed:

$$\left\{\begin{array}{cc}\hfill P& =(r,a)\hfill \\ \hfill & =\left(\mathrm{round}\left({\displaystyle \frac{r_i^{{\theta }_1}+r_j^{{\theta }_2}}{2}}\right),\mathrm{round}\left({\displaystyle \frac{a_i^{{\theta }_1}+a_j^{{\theta }_2}}{2}}\right)\right)\hfill \\ \hfill \mathbf{D}& =\left\{{\mathbf{d}}^{{\theta }_1,i},{\mathbf{d}}^{{\theta }_2,j}\right\}\hfill \\ \hfill & =\left\{\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\theta }_1,i},{\mathbf{d}}_{\mathrm{phase}}^{{\theta }_1,i}\right],\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\theta }_2,j},{\mathbf{d}}_{\mathrm{phase}}^{{\theta }_2,j}\right]\right\}\hfill \end{array}\right.$$

(14)

The pairwise merging procedure described above for two layers can be rigorously extended to all K layers. Let ${\mathcal{P}}_{{\varphi }_k}$ denote the set of keypoints detected under compensation phase ${\varphi }_k$. Define the union of all raw keypoints across all layers as:

$${\mathcal{P}}_{\mathrm{all}}={\displaystyle \bigcup _{k=0}^{K-1}}{\mathcal{P}}_{{\varphi }_k}$$

(15)

For any two points $P_{{\varphi }_i,u}$ and $P_{{\varphi }_j,v}$ in ${\mathcal{P}}_{\mathrm{all}}$, they are considered neighbors if their spatial distance satisfies the condition in Equation (13). Based on this adjacency relation, global clustering is performed using the principle of transitive closure: if there exists a path along which every pair of consecutive points satisfies the adjacency condition, then all points on that path belong to the same cluster $C_m$. Consequently, ${\mathcal{P}}_{\mathrm{all}}$ is partitioned into M mutually disjoint clusters $\{C_0,C_1,\dots ,C_{M-1}\}$. For each cluster $C_m$, the coordinates of the corresponding PRK are defined as the mean of the coordinates of all member points in the cluster:

$$P_m=(r_m,a_m)=\left({\displaystyle \frac{1}{|C_m|}}\sum _{P\in C_m}{r}_P,{\displaystyle \frac{1}{|C_m|}}\sum _{P\in C_m}{a}_P\right)$$

(16)

The descriptor set ${\mathbf{D}}_m$ of this PRK is then the union of the descriptors corresponding to all members in $C_m$.

According to the above method, all keypoints under the K compensation phases $\left\{{\varphi }_1,{\varphi }_2,\cdots ,{\varphi }_{K-1}\right\}$ are aggregated, resulting in the keypoints and descriptor sets corresponding to an interferogram. For any keypoint $P_m$, $0\le m\le M-1$, where M represents the total number of merged keypoints, its descriptor set ${\mathbf{D}}_m$ contains at least 1 descriptor vector and at most K descriptor vectors. The relationship between any keypoint $P_m$ and its corresponding descriptors in an interferogram is as follows:

$$P_m(r_m,a_m)\leftrightarrow {\mathbf{D}}_m\left\{\begin{array}{c}\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_1,m},{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_1,m}\right]\\ ⋮\\ \left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_k,m},{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_k,m}\right]\\ ⋮\\ \left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_{l_m},m},{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_{l_m},m}\right]\end{array}\right.,1\le l_m\le K$$

(17)

For any two points $P_m$ and $P_n$, assuming their descriptor sets are ${\mathbf{D}}_m$ and ${\mathbf{D}}_n$, respectively, the similarity between descriptors can be represented as:

$$\mathrm{sim}\left(P_m,P_n\right)=\underset{\begin{array}{c}{\mathbf{d}}^{{\varphi }_i,m}\in {\mathbf{D}}_m\\ {\mathbf{d}}^{{\varphi }_j,n}\in {\mathbf{D}}_n\end{array}}{max}\mathrm{sim}\left({\mathbf{d}}^{{\varphi }_i,m},{\mathbf{d}}^{{\varphi }_j,n}\right)$$

(18)

and

$$\begin{array}{c}\hfill \mathrm{sim}\left({\mathbf{d}}^{{\varphi }_i,m},{\mathbf{d}}^{{\varphi }_j,n}\right)=\omega ·{\mathrm{sim}}_{\mathrm{morpho}}\left({\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_i,m},{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_j,n}\right)\\ \hfill +\left(1-\omega \right)·{\mathrm{sim}}_{\mathrm{phase}}\left({\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_i,m},{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_j,n}\right)\end{array}$$

(19)

where ${\mathrm{sim}}_{\mathrm{morpho}}\left(·\right)$ represents the morphological similarity, ${\mathrm{sim}}_{\mathrm{phase}}\left(·\right)$ represents the phase similarity, and $\omega$ is the weight coefficient, with a value range of [0, 1].

The adoption of the max operator in Equation (18) has a clear theoretical basis. The fundamental purpose of constructing the three-dimensional compensated phase space is to simulate all possible phase translation states between the real-time and reference interferograms arising from estimation errors in the interferometric system parameters. For a pair of truly matching keypoints, among the K compensation phase layers there exists at least one layer under which the local phase patterns of the two interferograms achieve a high degree of consistency—i.e., the parameter errors are exactly compensated at that layer. The max operator essentially serves to automatically identify this optimal compensation state from among all possible layer combinations. In contrast, using the mean would mix in information from uncompensated erroneous layers, diluting the discriminative power of the optimal layer; using the min lacks any meaningful physical interpretation. The max strategy thus constitutes the key theoretical link connecting the three-dimensional space construction and the final keypoint matching.

Based on the forms of the morphological feature vector and the phase feature vector as shown in Equations (11) and (12), the morphological similarity calculation formula and the phase similarity calculation formula can be established, respectively:

$$\begin{array}{c}\hfill {\mathrm{sim}}_{\mathrm{morpho}}\left({\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_i,m},{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_j,n}\right)={\displaystyle \frac{1}{5}}{\displaystyle \sum _{k_{\mathrm{m}}=1}^5}{\displaystyle \frac{min({\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_i,m}\right]}_{k_{\mathrm{m}}},{\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_j,n}\right]}_{k_{\mathrm{m}}})}{max({\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_i,m}\right]}_{k_{\mathrm{m}}},{\left[{\mathbf{d}}_{\mathrm{morpho}}^{{\varphi }_j,n}\right]}_{k_{\mathrm{m}}})}}\end{array}$$

(20)

$${\mathrm{sim}}_{\mathrm{phase}}({\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_i,m},{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_j,n})=exp\left(-{∥{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_i,m}-{\mathbf{d}}_{\mathrm{phase}}^{{\varphi }_j,n}∥}_2\right)$$

(21)

The similarity calculation is performed to match all keypoints, and the matching results are further verified by the RANSAC algorithm to obtain the coordinates of the matched points.

2.3. Positioning Inversion

After matching the real-time interferogram and the reference interferogram, the keypoints in the real-time interferogram are assigned real geographic coordinates. The purpose of the positioning inversion is to calculate the aircraft’s position through the real geographic coordinates of these keypoints.

However, the coordinate systems of the reference interferogram and the real-time interferogram are usually inconsistent. The reference interferogram is generated from DEM, with the coordinate system being a projection coordinate system; while the real-time interferogram is in the SAR’s range-azimuth coordinate system. There exists a rotational relationship between these two coordinate systems, with the rotation angle being the SAR’s heading angle. Therefore, the positioning inversion process includes coordinate system transformation and platform positioning inversion, as shown in Figure 10.

For airborne SAR systems, the geometric relationship between the airborne platform and the target is shown in Figure 11. $P(x_p,y_p,h_p)$ represents the position of the airborne platform in the projection coordinate system, $T(x_t,y_t,h_t)$ represents the position of a matched point in the interferogram, ${\theta }_{heading}$ represents the heading angle, and ${\theta }_{sq}$ represents the squint angle [32].

Based on the process in Figure 10 and the geometric relationship in Figure 11, the process of aircraft position inversion can be obtained, as shown in Equations (22)–(24):

(a)

Coordinate System Transformation

$$\left\{\begin{array}{c}P{x}_t=x_{t}cos{\theta }_{\mathrm{heading}}-y_{t}sin{\theta }_{\mathrm{heading}}\hfill \\ P{y}_t=x_{t}sin{\theta }_{\mathrm{heading}}+y_{t}cos{\theta }_{\mathrm{heading}}\hfill \end{array}\right.$$

(22)

(b)

Positioning Inversion

$$\left\{\begin{array}{c}P{x}_p=P{x}_t-Rsin\left({\theta }_{sq}\right)\hfill \\ P{y}_p=P{y}_t-\sqrt{R^2{cos}^2\left({\theta }_{sq}\right)-{\left(h_p-h_t\right)}^2}\hfill \end{array}\right.$$

(23)

(c)

Coordinate System Back Transformation

$$\left\{\begin{array}{c}{x}_p=P{x}_{p}cos{\theta }_{\mathrm{heading}}+P{y}_{p}sin{\theta }_{\mathrm{heading}}\hfill \\ y_p=-P{x}_{p}sin{\theta }_{\mathrm{heading}}+P{y}_{p}cos{\theta }_{\mathrm{heading}}\hfill \end{array}\right.$$

(24)

3. Results

This paper conducts validity experiments and robustness experiments. The validity experiments aim to conduct scene matching navigation experiments using the interferogram matching algorithm and evaluate the performance of different methods using positioning error as an evaluation metric. The robustness experiments aim to verify the suppression ability of the proposed interferogram matching algorithm against parameter estimation errors by introducing different degrees of baseline length errors.

The control group matching algorithms used in this paper include KAZE, AKAZE, SIFT, ORB, and RIFT2. These methods are all high-performance and widely used image matching algorithms that can well represent the mainstream level of current image matching algorithms.

3.1. Data Preparation

The experimental data used in this paper is a set of C-band airborne single-pass dual-antenna InSAR data, with a flight altitude of approximately 3600 m and an observation area located in Shandong Province, China. The DEM plane precision used to generate the reference interferogram is 2 m. The aircraft is equipped with a high-precision differential GPS system, which can provide high-precision trajectory data as a reference trajectory.

The complete test data include 10 scenes, and the imaging area, reference DEM, and reference trajectory for each scene are shown in the Figure 12. Each SAR image, real-time interferogram, reference DEM, and reference interferogram are shown in Figure 13.

3.2. Metric and Parameters Setting

The positioning error metric used in this paper is the 2D DRMS, as shown below:

$${\mathrm{DRMS}}_{2\mathrm{D}}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n\left[{\left(x_i^{\mathrm{ref}}-x_i^{\mathrm{inv}}\right)}^2+{\left(y_i^{\mathrm{ref}}-y_i^{\mathrm{inv}}\right)}^2\right]}$$

(25)

where $x_i^{\mathrm{inv}}$ and $y_i^{\mathrm{inv}}$ are the inverted coordinates in azimuth time $t_i$, and $x_i^{\mathrm{ref}}$ and $y_i^{\mathrm{ref}}$ are the reference coordinates in the same azimuth time obtained from reference trajectory.

The values of the main parameters involved in the experiments are as follows. The number of phase compensation layers is set to $K=20$, which was determined by progressively increasing K and observing the saturation of the matching inlier ratio. The edge detection threshold coefficient in Equation (9) is set to $\alpha =0.7$, which provides a balanced trade-off between noise suppression and phase-jump edge sensitivity. The weight coefficient in Equation (19) is set to $\omega =0.5$, and the keypoint aggregation threshold in Equation (13) is set to ${\tau }_{\mathrm{merge}}=2\sqrt{2}$ pixels.

3.3. Validity Experiment of Matching Algorithms

First, this paper verifies the positioning error performance of the proposed PRK-HMPD interferogram matching algorithm and other comparison methods on the complete set of 10 scenes.

Table 1. Summary of positioning errors ${\mathrm{DRMS}}_{2\mathrm{D}}$ of various matching algorithms on the 10 scenes.

Scene Index	Matching Methods
	PRK-HMPD	KAZE	AKAZE	SIFT	ORB	RIFT2
Scene 1	10.29	87.78	93.05	1336.26	932.96	1521.21
Scene 2	12.08	80.06	238.44	27.72	1279.35	1281.29
Scene 3	8.35	28.85	183.50	63.77	988.23	9.91
Scene 4	15.84	21.40	176.57	302.20	456.55	772.26
Scene 5	19.28	36.39	45.79	166.75	551.99	984.45
Scene 6	17.54	32.28	38.20	10.07	511.23	1136.34
Scene 7	12.26	17.86	46.10	35.37	199.65	671.01
Scene 8	7.76	71.54	156.75	12.00	334.50	315.25
Scene 9	9.80	20.63	411.27	114.82	935.91	931.40
Scene 10	7.08	22.69	195.32	540.18	366.58	940.52
Average	12.03	41.95	158.50	260.91	655.70	856.36

shows the summary results of the positioning errors of various matching algorithms on the 10 scenes.

By comparing the data in Table 1, in the complete set of 10 test scenes, the proposed PRK-HMPD interferogram matching algorithm significantly outperforms other comparison methods in terms of positioning error, with an average positioning error of 12.03 m, while the average positioning errors of other methods exceed 40 m, with the highest reaching 856.36 m. Among them, the PRK-HMPD method did not achieve ideal results only in scene 6, mainly because the interferogram of this scene has a high density of fringes, resulting in relatively small differences between the real-time interferogram and the reference interferogram, allowing SIFT to extract matchable keypoints. Nevertheless, the positioning error of PRK-HMPD in this scene is only 17.54 m, still better than other comparison methods. Therefore, overall, the proposed PRK-HMPD interferogram matching algorithm performs excellently in terms of positioning error and can effectively support navigation and positioning tasks based on interferogram matching.

Taking the scene 1 data as an example, the interferogram matching results corresponding to different methods are shown in Figure 14. From the figure, it can be clearly seen that for the control group matching algorithms, their keypoints are mainly concentrated in regions with significant gradients, specifically at the edges of the fringes. However, it is evident that the textures of the real-time interferogram and the reference interferogram cannot be completely consistent, which leads to significant errors in the matching results of the control group methods, such as KAZE and AKAZE, despite appearing to match. On the other hand, the significant non-local similarity of interferograms also leads to a decrease in the discriminability of traditional matching algorithm descriptors, resulting in significant matching errors, such as in the SIFT matching results.

The matching results of other scene data using the proposed PRK-HMPD method are shown in Figure 15. Obviously, PRK-HMPD can effectively extract keypoints with significant morphological features and perform accurate matching.

Compared the inverted aircraft position with the reference trajectory, the effectiveness of the proposed method can also be visually compared. Figure 16 shows the comparison results of the aircraft position inverted through interferogram matching using PRK-HMPD and KAZE methods with the reference trajectory. It can be clearly seen from the figure that the aircraft position inverted by the PRK-HMPD method is highly consistent with the reference trajectory, while the KAZE method shows significant deviations.

3.4. Robustness Experiment of Matching Algorithms

The core purpose of the proposed PRK-HMPD is to suppress the incorrect matching caused by the interference fringe shift due to the estimation error of the InSAR system parameters. Therefore, the purpose of the robustness experiment is to verify that the algorithm can maintain stable matching ability under different noise levels.

For example, scene 1 is taken as an example. When generating the reference interferogram, we artificially introduced baseline length estimation errors of different proportions, namely 10%, 20%, 30%, and 40%. Then, KAZE and PRK-HMPD were used to match the real-time interferogram and the reference interferogram, respectively, and the results are shown in Figure 17.

It can be clearly seen that under different baseline length error conditions, the PRK-HMPD algorithm can detect stable keypoints during matching, meaning that most of the keypoint positions are basically consistent and are not affected by fringe shifts. In contrast, the KAZE algorithm exhibits significant instability; as the baseline length error increases, the position of the fringes shifts, and the corresponding matched keypoints keep changing. Coupled with the influence of the non-local similarity of interferograms, this ultimately leads to obvious mismatches in the matched points.

Other matching methods also exhibit similar situations, with errors even higher than those of the KAZE algorithm. Figure 18 shows the positioning error of the SAR platform after inverting the positioning error of the SAR platform through interferogram matching using various matching algorithms under four baseline length error conditions for scene 1.

Table 2. Positioning error results of interferogram matching navigation experiments using various matching algorithms under different baseline length error conditions.

Scene Index	Baseline Length Error (%)	Matching Methods
		PRK-HMPD	KAZE	AKAZE	SIFT	ORB	RIFT2
Scene 1	10	10.55	15.27	221.27	1313.19	1146.28	1629.11
	20	14.72	221.77	346.61	1488.47	1238.56	1655.13
	30	8.45	200.80	321.73	812.90	1012.16	1708.11
	40	10.70	62.00	1207.73	644.93	1100.75	1561.43
Scene 2	10	13.96	10.12	133.71	51.21	1048.89	1400.61
	20	16.60	53.84	63.73	1081.39	979.32	1708.88
	30	16.51	1604.71	1685.17	1038.55	1164.52	1696.74
	40	10.27	1393.02	517.64	1628.85	1009.00	491.49
Scene 3	10	12.17	7.62	31.58	15.04	1506.20	9.66
	20	8.88	33.79	16.60	20.28	289.79	1369.85
	30	10.97	22.03	63.34	26.55	1480.98	1339.60
	40	11.32	26.98	53.16	20.50	1406.05	1166.20
Average	/	12.09	304.33	388.52	678.49	1115.21	1311.40

shows the complete results of SAR platform positioning inversion errors of various matching algorithms under different baseline length error conditions for three scenes, as well as the statistical results of average errors.

From the data in Table 2, it can be seen that under different degrees of baseline length error conditions, the proposed PRK-HMPD interferogram matching algorithm significantly outperforms other comparison methods in terms of positioning error, with an average positioning error of 12.09 m, while the average positioning errors of other methods exceed 300 m, with the highest reaching 1311.40 m. On the other hand, PRK-HMPD can maintain stable matching ability, while other algorithms exhibit significant fluctuations. For example, KAZE has positioning errors as high as 1604.71 m and 1393.02 m for scene 2 data with baseline length errors of 30% and 40%, respectively, while performing well under other conditions. Therefore, overall, PRK-HMPD has strong robustness and can adapt to interferogram changes caused by parameter estimation errors, thereby achieving stable interferogram matching navigation.

4. Discussion

4.1. Interpretation of Results and Comparison with Previous Studies

Scene matching navigation in GNSS-denied environments has traditionally employed SAR images. However, conventional SAR images lack sufficient geometric features in regions with uniform scattering characteristics such as mountains and hills, which frequently leads to matching failures. Although utilizing InSAR interferograms provides rich topographic semantic information to fill this gap, research based on classical image matching frameworks represented by SIFT, KAZE, or ORB is difficult to apply to interferometric phase maps. The experimental results of this study clearly demonstrate that traditional gradient-based descriptors suffer from feature ambiguity caused by the non-local similarity arising from phase wrapping. In contrast, the proposed PRK-HMPD method effectively resolves this issue by substituting gradient extrema with stable phase extrema and utilizing a three-dimensional compensated phase space. Tested on ten real-world airborne InSAR scenes, PRK-HMPD significantly outperforms classical matching algorithms including KAZE, AKAZE, and ORB in terms of interferometric phase map matching performance.

4.2. Robustness Against Parameter Estimation Errors

A major challenge faced by InSAR-based scene matching is the texture and fringe shift caused by parameter estimation errors such as baseline length divergence during the generation of reference interferograms. This problem has not been adequately addressed in previous studies. The robustness experiments in this study highlight a critical breakthrough. When baseline length errors ranging from 10% to 40% are artificially introduced, traditional methods including KAZE produce significant mismatched points that directly lead to increased positioning errors. Conversely, the PRK-HMPD algorithm maintains consistent keypoint detection and low positioning errors. This proves that the Phase-Robust Keypoints aggregated along the compensation phase dimension are inherently invariant to fringe position shifts, providing a highly reliable solution for practical UAV applications where parameter uncertainties are inevitable.

4.3. Strengths and Broader Implications

The primary strength of this study lies in achieving an innovative paradigm shift from traditional gradient-based feature extraction to a Morpho-Phase representation. While traditional matching algorithms view the phase wrapping phenomenon as an obstacle causing non-local similarity, this study maximizes the inherent advantages of interferometric phases by transforming it into highly distinctive morphological features represented by closed fringes. The implications of these findings extend beyond the field of UAV scene matching navigation. The proposed three-dimensional space construction and hierarchical descriptor extraction framework can be generalized to other complex remote sensing tasks including multi-temporal InSAR image registration, deformation monitoring, and cross-modal image matching exhibiting strong structural discrepancies.

4.4. Limitations and Sources of Uncertainty

Despite the promising results, this study still possesses certain limitations that warrant attention. Firstly, the performance of the algorithm is influenced by fringe density. As observed in the validity experiment involving Scene 6, extremely dense interferometric fringes caused by dramatic terrain variations or specific baseline configurations can slightly degrade the matching accuracy. High fringe density may cause adjacent closed regions to overlap or merge during morphological processing, which subsequently reduces the distinctiveness of local phase features. Secondly, the generation of reference interferograms heavily relies on the quality of prior Digital Elevation Models. This study utilized a Digital Elevation Model with a planar precision of 2 m. If the models available in actual scenarios are outdated or of low resolution, the simulated reference interferograms might exhibit fundamental differences from the real-time data, thereby introducing unpredictable matching uncertainties. Finally, constructing a three-dimensional compensated phase space and calculating descriptors under multiple compensation phases inevitably increases computational complexity. Performing real-time onboard processing under constrained computational resources remains a challenge for highly dynamic UAV platforms.

4.5. Future Research Directions

Future work will focus on resolving the aforementioned limitations. To improve the timeliness and computational efficiency of the algorithm, we plan to optimize the three-dimensional space generation process and explore lightweight parallel computing strategies suitable for airborne embedded systems. Furthermore, integrating deep learning techniques to adaptively extract phase and morphological features holds the promise of further reducing processing time while maintaining robustness. We will also expand the validation of the PRK-HMPD algorithm to more diverse and extreme terrain scenarios including highly urbanized areas and steep mountainous regions, and investigate the algorithm’s performance under varying qualities of prior Digital Elevation Models to further mature the autonomous navigation technology based on InSAR.

5. Conclusions

To address the image matching error problem caused by significant interferometric fringe shifts due to parameter estimation errors in the reference interferogram and the inherent significant non-local similarity of interferograms in InSAR-based scene matching navigation, this paper proposes an interferogram matching algorithm based on Phase-Robust Keypoints and Hierarchical Morphological-Phase Descriptors. By constructing a compensation phase-range-azimuth three-dimensional space, Phase-Robust Keypoints that do not change with fringe position shifts caused by parameter errors are detected, and Hierarchical Morphological-Phase Descriptors are constructed based on these keypoints to achieve robust matching between real-time and reference interferograms. Finally, the proposed PRK-HMPD interferogram matching algorithm is validated using measured airborne InSAR data, demonstrating significant superiority over other comparison methods in terms of positioning error and robustness. The positioning error for scene matching navigation using the PRK-HMPD method can be controlled to around 10m on the test data, while the positioning errors of comparison methods reach hundreds of meters and exhibit significant fluctuations with parameter errors. In summary, the proposed PRK-HMPD interferogram matching algorithm can effectively support high-precision scene matching navigation tasks based on InSAR interferograms. Moreover, although validated using UAV-borne InSAR data, the PRK-HMPD framework constitutes a general-purpose interferogram matching method that is not limited to a specific platform or flight configuration. Its underlying principle—extracting phase-robust keypoints via compensated phase space construction and describing them through morphological-phase features—is broadly applicable to any InSAR interferogram, regardless of whether the platform is spaceborne, airborne, or UAV-based.

Future work will continue to optimize the proposed methods in terms of timeliness, robustness, and accuracy. Building upon the general-purpose nature of the PRK-HMPD framework, we plan to extend its application beyond UAV navigation to broader InSAR-related tasks, including spaceborne InSAR image registration and multi-temporal interferogram change detection for deformation monitoring.