A joint space-time decorrelation suppression method for tomosar
By combining spatiotemporal decorrelation suppression methods, high-precision complex covariance matrix estimation and phase inversion were achieved, solving the problem of spatiotemporal decorrelation suppression in TomoSAR and improving the quality of 3D imaging.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2026-05-19
- Publication Date
- 2026-06-30
AI Technical Summary
Existing TomoSAR technology cannot accurately reconstruct complex SLC images in spatiotemporal decorrelation suppression, resulting in insufficient phase fidelity and making it difficult to support the requirements of high-precision 3D imaging.
A joint spatiotemporal decorrelation suppression method is adopted to achieve high-precision estimation of the complex covariance matrix through spatiotemporal joint statistical homogeneous pixel identification. A joint observation model of interferometric observation and original phase prior is constructed, and stable phase inversion is achieved based on iterative weighted least squares (IRLS). Phase inversion is constructed through spatiotemporal joint statistical homogeneous pixel identification, realizing efficient technical application.
It significantly improves the 3D point cloud density and reconstruction accuracy of TomoSAR, making it suitable for high-precision 3D imaging of spaceborne TomoSAR.
Smart Images
Figure CN122307546A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of synthetic aperture radar technology, and particularly relates to a joint spatiotemporal decorrelation suppression method for TomoSAR. Background Technology
[0002] Synthetic Aperture Radar Tomography (TomoSAR) extends the synthetic aperture principle in the elevation direction, overcoming the inherent overlay effect of traditional 2D SAR imaging and achieving high-precision 3D spatial resolution of the observed scene. It has irreplaceable application value in fields such as urban 3D reconstruction, forest biomass inversion, and glacier internal structure detection. Spaceborne TomoSAR systems generally employ a multiple-passage, heavy-orbit data acquisition mode, with time baselines reaching months or even years. During this process, factors such as changes in surface scattering characteristics, orbital errors, and atmospheric disturbances introduce severe spatiotemporal decorrelation, directly leading to a significant increase in interferometric phase noise and a substantial reduction in elevation inversion accuracy and 3D reconstruction quality. Therefore, achieving efficient and robust spatiotemporal decorrelation suppression is a core technical aspect for improving TomoSAR imaging performance.
[0003] Traditional SAR denoising and decorrelation suppression methods primarily rely on spatial domain filtering. Early methods, such as Lee filtering, Frost filtering, and Kuan filtering, achieved adaptive noise suppression through local statistical properties, but they are prone to causing loss of edge and texture details in non-uniform regions. Goldstein filtering, a classic interferometric phase domain method, achieves noise smoothing through frequency domain weighting, but its fixed window and fixed parameters make it difficult to balance noise suppression and edge preservation. Non-local mean (NLM) filters utilize image patch similarity for denoising, performing well in uniform regions, but under strong decorrelation noise, patch similarity cannot be accurately calculated, resulting in a significant decrease in denoising performance.
[0004] In recent years, with the continuous enrichment of multi-temporal SAR data, multi-dimensional filtering algorithms have gradually extended spatial domain filtering to the spatiotemporal three-dimensional domain. Representative methods include three-dimensional block matching collaborative filtering (MSAR-BM3D), nonlocal filtering based on the complex covariance matrix (CCM), and context-aware filtering based on the three-dimensional quality index. These methods improve denoising performance by combining spatial and temporal dimensional information, and to some extent improve coherence and phase fidelity.
[0005] However, existing spatiotemporal decorrelation suppression methods still have obvious shortcomings: most methods only focus on noise smoothing and cannot explicitly reconstruct single-look complex (SLC) SAR images, making it difficult to directly support high-precision 3D reconstruction of TomoSAR based on compressed sensing; at the same time, most methods only perform spectral estimation and filtering on the complex covariance matrix (CCM), lacking an accurate phase recovery mechanism for inverting the original complex image from the CCM, resulting in insufficient phase fidelity and difficulty in meeting the requirements of high-precision tomographic imaging.
[0006] In summary, existing technologies struggle to achieve high-precision SLC complex image reconstruction while suppressing spatiotemporal decorrelation noise, failing to fully meet the 3D imaging quality requirements of high-resolution spaceborne TomoSAR. Therefore, there is an urgent need to propose a decorrelation suppression method that can jointly utilize spatiotemporal information, accurately estimate the complex covariance matrix, and achieve phase inversion through iterative optimization, fundamentally improving the point cloud density, integrity, and elevation accuracy of TomoSAR reconstruction. Summary of the Invention
[0007] To address the shortcomings of existing TomoSAR spatiotemporal decorrelation suppression techniques, such as inaccurate reconstruction of complex SLC images, insufficient phase fidelity, and difficulty in supporting compressed sensing tomography, this invention provides a joint spatiotemporal decorrelation suppression method for TomoSAR. This method achieves high-precision complex covariance matrix (CCM) estimation through spatiotemporal joint statistical homogeneous pixel identification, constructs a joint observation model combining interferometric observation and original phase prior, and achieves stable phase inversion based on iterative weighted least squares (IRLS). This approach suppresses decorrelation noise while accurately recovering complex SAR images, significantly improving the 3D point cloud density and reconstruction accuracy of TomoSAR.
[0008] The technical solution of this invention is as follows: A joint spatiotemporal decorrelation suppression method for TomoSAR includes: Step 1: Input multi-temporal reorbit SLC SAR images, complete registration and deskewing processing, construct multi-dimensional complex scattering vectors pixel by pixel, and generate the initial complex covariance matrix (CCM) using multiple view windows. Step 2: Construct a search window centered on the target pixel, identify statistically homogeneous pixels (SHP) based on the complex Wishart distribution and likelihood ratio test, calculate similarity weights, and perform a weighted average of the reference pixel CCM to obtain a high-precision denoised CCM. Step 3: Extract interferometric phase, coherence coefficient, and intensity information from high-precision CCM to construct an interferometric observation model; simultaneously, construct a phase prior observation model based on the phase of the original SAR image; fuse the two models to establish a nonlinear joint observation equation. Step 4: Linearize the joint observation model by performing a first-order Taylor expansion, construct a joint weighting matrix based on the coherence coefficients, and use IRLS iterative solution to solve the phase increment; the iterative process only retains the real part for updating to ensure the physical meaning of the phase, until convergence to obtain the final inverted phase; Step 5: Traverse all pixels in the entire image and output the SLC complex image after spatiotemporal decorrelation suppression; use the compressed sensing method based on orthogonal matching pursuit (OMP) to realize TomoSAR 3D tomographic reconstruction and output the 3D point cloud results.
[0009] Furthermore, step 1 includes: Step 1.1, Input M Multi-temporal single-look complex SAR images after scene registration, at the pixel level x Construction M Complex scattering vector: (1) Step 1.2: Using an L×L multiview window, generate the initial sample complex covariance matrix: (2) in For the first i The complex scattering vector of each independent sample. Let represent the Hermitian transpose. Then the complex covariance matrix can be written as: .
[0010] Furthermore, step 2 includes: Step 2.1, with target pixels x Set a search window centered on the reference pixels and iterate through the reference pixels within the window. y Generate in the same way ; Step 2.2, calculate the likelihood ratio statistic. Characterizes the similarity between two pixel CCMs: (3) Step 2.3, calculate similarity weights The pixel is obtained by taking a weighted average of the CCM of all reference pixels within the window. x High-precision CCM at the location: (4) in The normalization coefficient is... This is the search window.
[0011] Furthermore, step 3 includes: Step 3.1: Extract the interference phase and coherence coefficient from the high-precision CCM to construct the interferometric observation model; Step 3.2: Using the phase of the original SAR image as the initial value, construct an absolute phase prior observation model; Step 3.3: Integrate interferometric observations and prior observations to establish a nonlinear joint observation model: (5) Where X is the phase vector to be inverted, Y is the joint observation vector, and ε is the observation noise vector.
[0012] Furthermore, step 4 includes: Step 4.1, for the joint observation model in the current ( t The first-order Taylor expansion at the iterative phase yields a linearized model: (6) Where J is the Jacobian matrix and ΔX is the phase increment; Step 4.2: Construct a joint weighting matrix W based on the coherence coefficient and CRLB; Step 4.3, solve for the phase increment under the weighted least squares criterion: (7) Where r is the observation residual vector; Step 4.4: Update the phase only by retaining the real part to ensure physical realizability. (8) Step 4.5: Repeat the iteration until |ΔX| is less than the threshold or the maximum number of iterations is reached, and output the final phase.
[0013] Furthermore, step 5 includes: Step 5.1: Traverse all pixels in the entire image and repeat steps 1–4 to obtain the SLC composite image after correlation suppression of the entire image; Step 5.2: TomoSAR 3D tomographic reconstruction is performed using an OMP-based compressed sensing method, and the 3D point cloud and reconstruction accuracy indicators are output.
[0014] This completes all the steps.
[0015] Beneficial effects: 1. By using the spatiotemporal joint likelihood ratio test to identify statistically homogeneous pixels, high-precision CCM estimation is achieved, strong noise suppression is achieved in homogeneous regions, and structural details are preserved in edge regions; 2. Construct a joint observation model of interferometric relative phase and original absolute phase to solve the problem that traditional methods only filter without reconstruction and cannot support SLC domain TomoSAR processing; 3. A phase inversion method based on IRLS is proposed. The iterative process only retains the real part for updating, which ensures the physical meaning of the phase and improves the stability and convergence accuracy of the algorithm. 4. Directly outputs denoised SLC composite images, which can be directly used for compressed sensing TomoSAR reconstruction, significantly improving point cloud density and ground and elevation reconstruction accuracy, and is suitable for high-precision 3D imaging of spaceborne TomoSAR. Attached Figure Description
[0016] Figure 1. TomoSAR observation geometry configuration; Figure 2. Overall processing flowchart of the present invention; Figure 3. Simulation scene diagram; Figure 4. Target point cloud distribution map; Figure 5. Simulated spatiotemporal baseline distribution; Figure 6. Comparison of noise-free and decorrelation-free SAR images; Figure 7. Comparison of interference phase results using different methods; Figure 8. Comparison of point cloud reconstruction results using different methods; Figure 9. Comparison of point cloud reconstruction accuracy in ground areas; Figure 10. Comparison of point cloud reconstruction accuracy in facade areas. Detailed Implementation
[0017] Figure 1 shows the TomoSAR observation geometry. The satellite acquires SAR images from multiple locations, selects one scene as a reference, and completes registration and deslope processing for the remaining images. Within the same range-azimuth resolution cell, the echo signal can be represented as the superposition of multiple scattering points along the elevation direction.
[0018] Figure 2 shows the overall process of this invention, which is completed in sequence as follows: multi-temporal data input, initial CCM generation, spatiotemporal joint SHP identification and high-precision CCM estimation, joint observation modeling, IRLS phase inversion, full-map complex image reconstruction, and TomoSAR three-dimensional tomography reconstruction.
[0019] The present invention will now be described in detail with reference to the accompanying drawings, and the specific steps are as follows: Step 1: Input multi-temporal SLC SAR images, complete registration and deskewing processing, construct multi-dimensional complex scattering vectors pixel by pixel, and generate the initial complex covariance matrix (CCM) using multiple view windows.
[0020] Step 1.1, Input MMulti-temporal single-look complex SAR images after scene registration, at the pixel level x Construction M Complex scattering vector: (9) Step 1.2: Using an L×L multiview window, generate the initial sample complex covariance matrix: (10) in For the first i The complex scattering vector of each independent sample. Let represent the Hermitian transpose. Then the complex covariance matrix can be written as: .
[0021] Step 2: Construct a search window centered on the target pixel, identify statistically homogeneous pixels (SHP) based on the complex Wishart distribution and likelihood ratio test, calculate similarity weights, and perform a weighted average of the reference pixel CCM to obtain a high-precision denoised CCM.
[0022] Step 2.1, with target pixels x Set a search window centered on the reference pixels and iterate through the reference pixels within the window. y Generate in the same way ,Right now: (11) Step 2.2, calculate the likelihood ratio statistic. Characterizes the similarity between two pixel CCMs: (12) Step 2.3, calculate similarity weights The pixel is obtained by taking a weighted average of the CCM of all reference pixels within the window. x High-precision CCM at the location: (13) in The normalization coefficient is... This is the search window.
[0023] From the precise CCM, we can extract the intensity, phase, and coherence of pixel x after spatiotemporal decorrelation suppression, represented as: (14) in, It is the first u The intensity of each image. It is the first v The intensity of each image. It is the first u and thev The phase of the interferogram generated from the image. It is the first u and the v Coherence of the interferograms generated by the images.
[0024] Step 3: Extract interferometric phase, coherence coefficient and intensity information from high-precision CCM to construct an interferometric observation model; at the same time, construct a phase prior observation model based on the phase of the original SAR image; fuse the two models to establish a nonlinear joint observation equation.
[0025] Step 3.1: Extract the interference phase and coherence coefficient from the high-precision CCM to construct the interferometric observation model.
[0026] For any given pixel, let the vector of true phase parameters to be estimated be defined as: (15) in, Let be the true phase value of the m-th image at a given pixel. For M Images can form a total The interference pairs. From the CCM, the interferometric observation term can be extracted, and the interferometric observation vector can be constructed as follows: (16) in, To enclose the interference phase, the theoretical model for interferometric observation is as follows: (17) in, k and q They represent the first k and the q 1 image. Therefore, the interferometric observation equation is expressed as: (18) in, This refers to the error in interferometric observation. Obtained from the original SAR image through interferometric processing, and represented as: (19) Step 3.2: Using the phase of the original SAR image as the initial value, construct an absolute phase prior observation model; Since the phase of the initial SAR image is known, the prior observation vector is constructed as follows: (20) in, Indicates the first mThe original observation phase. It is worth noting that the above equation is constructed through amplitude normalization to eliminate the influence of amplitude. Considering the presence of noise in reality, the prior observation equation is expressed as: (twenty one) in, This represents the phase noise of the original SAR image. Derived from the original SAR image, the nonlinear vector function representing the prior observation components is expressed as: (twenty two) Step 3.3: Integrate interferometric observations and prior observations to establish a nonlinear joint observation model: (twenty three) Where Y is the joint observation vector. It is a joint nonlinear vector function, where ε is the observation noise vector, and is expressed as follows: (twenty four) Step 4: Linearize the joint observation model by performing a first-order Taylor expansion, construct a joint weighting matrix based on the coherence coefficients, and use IRLS iterative solution to solve the phase increment; the iterative process only retains the real part update to ensure the physical meaning of the phase, until convergence to obtain the final inverted phase.
[0027] Step 4.1: Since the joint observation model is nonlinear, the Gauss-Newton method is used for iterative solution. Let... Indicates the first t The estimated value of the next iteration. The relationship between the true value and the current estimated value is determined by... The phase increment vector is given. (25) For the joint observation model in the current ( t The first-order Taylor expansion at the iterative phase yields a linearized model: (26) Where J is the Jacobian matrix, expressed as: (27) Step 4.2: Construct a joint weighting matrix W based on the coherence coefficient and CRLB; The phase measurement accuracy varies for different interferometric pairs. Therefore, a weighted scheme is required during the estimation process. The weights can be determined based on the accuracy of the observed signals. According to parameter estimation theory, the Cramer-Rao lower bound (CRLB) provides a measure of the phase measurement accuracy of the interferometric phase, expressed as: (28) Therefore, the interferometric observation weight matrix can be constructed as follows: (29) The amplitude of the original SAR image follows a Rayleigh distribution, while the phase follows a uniform distribution. Therefore, the accuracy of the prior information can be determined by the phase variance of the SAR image. According to CRLB, the lower bound of the phase estimation variance is: (30) Therefore, the weight matrix of the prior observation equation is expressed as: (31) The corresponding joint weight matrix is: (32) Step 4.3: Solve for the phase increment under the iterative weighted least squares criterion, and obtain: (33) Where r is the observation residual vector, expressed as: (34) Step 4.4: Update the phase only by retaining the real part to ensure physical realizability. (35) Step 4.5: Repeat the iteration until |ΔX| is less than the threshold or the maximum number of iterations is reached, and output the final phase.
[0028] Step 5: Traverse all pixels in the entire image and output the SLC complex image after spatiotemporal decorrelation suppression; use the compressed sensing method based on orthogonal matching pursuit (OMP) to realize TomoSAR 3D tomographic reconstruction and output the 3D point cloud results.
[0029] Step 5.1: Traverse all pixels in the entire image and repeat steps 1–4 to obtain the SLC composite image after correlation suppression of the entire image; Step 5.2: TomoSAR 3D tomographic reconstruction is performed using an OMP-based compressed sensing method, and the 3D point cloud and reconstruction accuracy indicators are output.
[0030] This completes all the steps.
[0031] The following provides an implementation example with specific parameters.
[0032] This embodiment uses simulated TomoSAR data to verify the effectiveness of the method. The system parameter settings are shown in the table below.
[0033] Table 1 System Parameter Information
[0034] Perform the steps described in this invention sequentially: Step 1: Input 7 registered simulated SLC complex images, construct a 7-dimensional complex scattering vector pixel by pixel, and generate the initial complex covariance matrix using a 3×3 multi-view window.
[0035] Step 2: Set a 5×5 search window centered on the target pixel, traverse the reference pixels and calculate the corresponding CCM; obtain the similarity weight based on the likelihood ratio statistic, and calculate the weighted average of the reference pixel CCM to obtain the high-precision denoised CCM.
[0036] Step 3: Extract the interferometric phase and coherence coefficient from the high-precision CCM to construct the interferometric observation term; use the phase of the original SAR image as the initial value to construct the absolute phase prior term; and fuse them to obtain the nonlinear joint observation model.
[0037] Step 4: Perform a first-order Taylor expansion linearization on the joint observation model; construct a joint weighting matrix based on the coherence coefficient and CRLB; iteratively solve the phase increment based on IRLS, and update the phase by retaining only the real part until convergence.
[0038] Step 5: Traverse all pixels in the entire image to obtain the SLC complex image after spatiotemporal decorrelation suppression; use the OMP compressed sensing method to perform TomoSAR 3D tomography reconstruction and output 3D point cloud.
[0039] Step 6, select Goldstein filtering as the comparison method, in Figure 8 (c) Calculate ENL, SSI, and MSRE in the homogeneous region within the red box, and calculate them in the edge region. , And count the number of point clouds, Figure 8 (c) The RMS accuracy of the black frame (facade area) and red frame (ground area) reconstruction is shown in Tables 2 and 3.
[0040] Table 2 Indicator Evaluation Results
[0041] Table 3 Point Cloud Evaluation Results
[0042] Figure 3 shows the simulation scene; Figure 4 shows the target point cloud distribution; Figure 5 shows the simulation spatiotemporal baseline distribution. Figure 6(a) Noise-free image, (b) SAR image with decorrelation noise; Figure 7 shows the comparison results of interferometric phase after processing by the method, (a) original interferometric phase, (b) interferometric phase after decorrelation suppression; Figure 8 shows the comparison of tomographic point cloud reconstruction results by different methods, (a) original reconstruction result, (b) reconstruction result after GoldStein filtering, (c) reconstruction result after decorrelation suppression; Figure 9 shows the comparison of point cloud reconstruction accuracy in the ground area, (a) point cloud profile, (b) evaluation area; Figure 10 shows the comparison of point cloud reconstruction accuracy in the elevation area, (a) point cloud profile, (b) evaluation area.
[0043] Quantitative results show that: In homogeneous regions, this invention achieves the highest ENL (128.7821), the lowest SSI (0.1774), and the lowest MSRE (0.1430), demonstrating significantly better noise suppression performance than Goldstein filtering. In the edge regions, the invention maintains a high level of [efficacy / reliability]. With near-optimal It effectively preserves edge structure while reducing noise; The tomographic reconstruction results show that the present invention increases the total number of point clouds by 13.05%, the ground reconstruction accuracy by 70.5%, and the facade reconstruction accuracy by 48.2%.
[0044] The above results demonstrate that the present invention can effectively suppress spatiotemporal decorrelation noise, accurately recover SLC complex images, and significantly improve the point cloud density, integrity, and elevation accuracy of TomoSAR 3D reconstruction.
[0045] Of course, the present invention may have other various embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and modifications according to the present invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.
Claims
1. A joint space-time decorrelation suppression method for TomoSAR, characterized in that, include: Step 1: Input multi-temporal reorbit SLC SAR images, complete registration and deskewing processing, construct multi-dimensional complex scattering vectors pixel by pixel, and generate the initial complex covariance matrix (CCM) using multiple view windows. Step 2: Construct a search window centered on the target pixel, identify statistically homogeneous pixels (SHP) based on the complex Wishart distribution and likelihood ratio test, calculate similarity weights, and perform a weighted average of the reference pixel CCM to obtain a high-precision denoised CCM. Step 3: Extract interferometric phase, coherence coefficient, and intensity information from high-precision CCM to construct an interferometric observation model; simultaneously, construct a phase prior observation model based on the phase of the original SAR image; fuse the two models to establish a nonlinear joint observation equation. Step 4: Linearize the joint observation model by performing a first-order Taylor expansion, construct a joint weighting matrix based on the coherence coefficients, and use IRLS iterative solution to solve the phase increment; the iterative process only retains the real part for updating to ensure the physical meaning of the phase, until convergence to obtain the final inverted phase; Step 5: Traverse all pixels in the entire image and output the SLC complex image after spatiotemporal decorrelation suppression; use the compressed sensing method based on orthogonal matching pursuit (OMP) to realize TomoSAR 3D tomographic reconstruction and output the 3D point cloud results.
2. A joint space-time decorrelation suppression method for TomoSAR according to claim 1, characterized in that, Step 3 includes: Step 3.1: Extract the interference phase and coherence coefficient from the high-precision CCM to construct the interferometric observation model; For any given pixel, let the vector of true phase parameters to be estimated be defined as: ； in, Let m be the true phase value of the m-th image at a given pixel; for M Images can form a total The interference pairs; by extracting the interferometric observation terms from the CCM, the interferometric observation vector can be constructed as follows: ； in, To enclose the interference phase, the theoretical model for interferometric observation is as follows: ； in, k and q They represent the first k and the q 1 image; therefore, the interferometric observation equation is expressed as: ； in, This refers to interferometric observation error; Obtained from the original SAR image through interferometric processing, and represented as: ； Step 3.2: Using the phase of the original SAR image as the initial value, construct an absolute phase prior observation model; Since the phase of the initial SAR image is known, the prior observation vector is constructed as follows: ； in, Indicates the first m The original observation phase; it is worth noting that, in order to eliminate the influence of amplitude, the above equation is constructed through amplitude normalization; considering the existence of noise in reality, the prior observation equation is expressed as: ； in, Phase noise of the original SAR image; Derived from the original SAR image, the nonlinear vector function representing the prior observation components is expressed as: ； Step 3.3: Integrate interferometric observations and prior observations to establish a nonlinear joint observation model: ； Where Y is the joint observation vector. It is a joint nonlinear vector function, where ε is the observation noise vector, and is expressed as follows: 。 3. The joint spatiotemporal decorrelation suppression method for TomoSAR as described in claim 1, characterized in that, Step 4 includes: Step 4.1: Since the joint observation model is nonlinear, the Gauss-Newton method is used for iterative solution; let... Indicates the first t The estimated value of the next iteration; the relationship between the true value and the current estimated value is determined by... The phase increment vector is defined as follows: ； For the joint observation model in the current ( t The first-order Taylor expansion at the iterative phase yields a linearized model: ； Where J is the Jacobian matrix, expressed as: ； Step 4.2: Construct a joint weighting matrix W based on the coherence coefficient and CRLB; The phase measurement accuracy varies for different interferometric pairs; therefore, a weighted scheme is required during estimation. The weights can be determined based on the accuracy of the observed signals. According to parameter estimation theory, the Cramer-Rao lower bound (CRLB) provides a measure of the phase measurement accuracy of the interferometric phase, expressed as: ； Therefore, the interferometric observation weight matrix can be constructed as follows: ； The amplitude of the original SAR image follows a Rayleigh distribution, while the phase follows a uniform distribution; therefore, the accuracy of the prior information can be determined by the phase variance of the SAR image; according to CRLB, the lower bound of the phase estimation variance is: ； Therefore, the weight matrix of the prior observation equation is expressed as: ； The corresponding joint weight matrix is: ； Step 4.3: Solve for the phase increment under the iterative weighted least squares criterion, and obtain: ； Where r is the observation residual vector, expressed as: ； Step 4.4: Update the phase only by retaining the real part to ensure physical realizability. ； Step 4.5: Repeat the iteration until |ΔX| is less than the threshold or the maximum number of iterations is reached, and output the final phase.