Phase unwrapping algorithm based on differential manifold

Abstract

The invention discloses a phase unwrapping algorithm based on a differential manifold, and belongs to the technical field of phase unwrapping. Comprising the following steps: S1, converting a phase unwrapping problem; converting a phase unwrapping problem into a local coordinate transformation problem between M and N sub-manifolds; s2, calculating a local coordinate system of the sub-manifold through integration; s3, obtaining a relationship between two adjacent coordinate cards depending on a communication condition; and S4, realizing M-to-N integer mapping by using local coordinate transformation, and completing phase unwrapping. According to the invention, the differential manifold theory is introduced to solve the phase unwrapping problem; converting a phase unwrapping problem into a local coordinate transformation problem between M and N sub-manifolds; calculating a local coordinate system of the sub-manifolds through integration, obtaining a relationship between two adjacent coordinate cards by depending on a communication condition, and realizing integer mapping from M to N by utilizing local coordinate transformation to complete phase unwrapping; the efficiency bottleneck of a traditional phase unwrapping algorithm is broken through, and the robustness of phase unwrapping under the complex terrain is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of phase unwrapping, and in particular to a phase unwrapping algorithm based on differential manifolds. Background Art

[0002] Synthetic Aperture Radar (InSAR) technology uses the principal phase values obtained by conjugate multiplication of two SAR images of the same area to invert surface elevation information of the target area, outputting a geocoded, high-precision Digital Elevation Model (DEM) product. Surface deformation information can be derived from the phase differences of multiple SAR images after differential processing; hence, radar differential interferometry (DII) was proposed. Its wavelength-scale surface deformation monitoring capability provides important technical support for early warning of geological disasters such as earthquakes and volcanoes. To achieve continuous deformation monitoring over time, time-series InSAR techniques such as permanent scatterer radar interferometry (IPSI), small baseline set radar interferometry (SBSS), and distributed radar interferometry (DIRI) have been proposed. However, phase unwrapping, as an ill-posed problem, is a major bottleneck restricting the practical application of InSAR technology. Its quality directly affects the accuracy of DEMs and deformation monitoring.

[0003] Phase unwrapping methods can be categorized into local growth path integral methods and global optimization minimum norm methods, depending on their solution approach. The branch-cut method, a classic path integral method, first calculates the residual points of the interference pattern and divides them into positive and negative residual points based on the integral value. Then, the positive and negative residual points are connected to lay out branch cuts, and unwrapping is achieved using an integration method that does not pass through the branch cuts. However, in areas with dense residual points, different branch cut placement methods can produce different unwrapping results. Therefore, improving this method lies in finding a reasonable branch cut placement method to avoid erroneous unwrapping caused by the "islanding" phenomenon. One effective approach is to incorporate theoretical models from different disciplines to guide branch cut placement; for example, the flood diffusion model and ant colony algorithm are used to connect positive and negative residual points to minimize the total length of the branch cut placement. Branch-cut methods also incorporate additional continuity information to aid unwrapping. For example, the quality map method uses a priori quality map information to mask low-quality areas and place branch cuts in high-quality areas to eliminate noise. However, obtaining a reliable prior quality map poses a new challenge. To address this issue, intervention Figure 2The order partial derivative information is used as a quality map to guide the integral unwrapping path, which effectively solves the problem of unwrapping errors caused by incorrect branch tangent layout in areas with dense distribution of residual points. Subsequently, coherence coefficients, pseudo-coherence coefficients, improved coherence coefficients, and phase gradient maps were introduced into phase unwrapping as quality maps to correct the unwrapping results. Later, a special quality map auxiliary method, namely the Mask method, was proposed; the interference pattern is masked with 0 or 1, and low-quality areas are weighted with 0 or ignored during the unwrapping process. In addition, the Flynn minimum discontinuity algorithm realizes phase unwrapping by calculating the winding number matrix with the minimum phase discontinuity.

[0004] The key idea of the global optimization minimum norm method is to establish an optimization function between the interferometric phase derivative and the true absolute phase derivative, and to achieve phase unwrapping by minimizing this function. When the norm is 2, the method transforms into the classic least squares method, which reduces the influence of noise by setting different weights. Subsequently, Poisson estimation and the conjugate gradient method were introduced to the least squares method to improve the accuracy and convergence speed of phase unwrapping. When the norm is 1, the computational model of this method is consistent with the minimum cost flow (MCF). With the help of the coherence coefficient, the minimum solution of the model is calculated based on the residual point information, that is, the minimum cost of the network flow. Then, the interferogram is solved by setting branch tangents, achieving good unwrapping results. However, this problem is inherently nondeterministic and polynomially hard, and no algorithm with polynomial time complexity exists. The SNAPHU method uses statistical theory to calculate an approximate solution to the nondeterministic polynomially hard problem to achieve phase unwrapping. As the size of the interferogram increases, the computational complexity of this algorithm increases sharply, making it extremely difficult to obtain an exact solution in a short time. Summary of the Invention

[0005] The purpose of this invention is to propose a phase unwrapping algorithm based on differential manifolds. According to the homeomorphic topological structure of interference phase and true phase, phase unwrapping is converted into a coordinate card integer mapping problem between submanifolds. The coordinate transformation advantages of differential manifolds in nonlinear space are utilized to improve the robustness and computational efficiency of phase unwrapping under complex terrain.

[0006] In order to achieve the above object, the present invention adopts the following technical solutions:

[0007] A phase unwrapping algorithm based on differential manifolds includes the following steps:

[0008] S1, phase unwrapping problem transformation;

[0009] The phase unwrapping problem is transformed into a local coordinate transformation problem between M and N submanifolds;

[0010] S2, calculate the local coordinate system of the submanifold by integration;

[0011] S3, relying on the connectivity condition to obtain the relationship between two adjacent coordinate cards;

[0012] S4. Use local coordinate transformation to realize integer mapping from M to N and complete phase unwrapping.

[0013] In some embodiments, in step S1:

[0014] The corresponding relationship between the true phase unwrap(i,j) and the interference phase wrap(i,j) is:

[0015] unwrap(i,j)=wrap(i,j)+2k(i,j)π (1);

[0016] Where k(i,j) is the integer fuzzy number of point (i,j);

[0017] Considering the discrete echo signal as a set, the topological structure of the set and the discrete topological space are obtained;

[0018]

[0019] Where m and n are the interference pattern sizes;

[0020] The mapping f between M and N is defined as

[0021]

[0022] Where, i∈[1,n];

[0023] f:M→N is a homeomorphic mapping, M and N are homeomorphic;

[0024] Take any two coordinate cards of M and(U j ,θ); if Then U i ,U j The composite mapping between That is, M and N are diffeomorphisms on differential manifolds.

[0025] In some embodiments, in step S2:

[0026] Let the boundary point set of the edge region D be The applicable coordinate system of point p∈D is consistent with the orientation of M, and we get The local coordinate system

[0027] Then M The induced orientation on becomes a real analytic manifold with induced orientation; then, for any have

[0028]

[0029] Where, is the inclusion mapping, With orientation induced from M.

[0030] In some embodiments, in step S3:

[0031] Any two adjacent submanifolds satisfy

[0032] Any two adjacent points on the dependent manifold M |x i -x j |≤π, submanifold U i Make the following determination:

[0033]

[0034] If h(0)=a,h(1)=b, then the submanifold U i Connected, suitable for submanifold;

[0035] Otherwise, the submanifold U i Calculate formula (8)

[0036]

[0037] In this way, until all sub-manifolds U are connected, then determine whether they meet the orientation requirements. Find the relationship between two adjacent coordinate cards, f i -f i+1 =1or f i+1 -f i =1.

[0038] In some embodiments, in step S4:

[0039] When using formula (6) to calculate U i Homeomorphism f to N i , find the coordinate card of M (U i ,f i );

[0040] Use formula (9) to calculate the mapping f from M to N

[0041]

[0042] Thus, the mapping relationship between the interference phase and the true phase is obtained.

[0043] Compared with the prior art, the present invention provides a phase unwrapping algorithm based on differential manifolds, which has the following beneficial effects.

[0044] 1. The present invention introduces differential manifold theory for the first time to solve the phase unwrapping problem. A differential manifold can be constructed by the interference phase, the true phase set, and the mapping relationship between the two, which exactly conforms to the homeomorphic mapping. Therefore, the phase unwrapping problem is transformed into a local coordinate transformation problem between M and N submanifolds. The local coordinate system of the submanifold is calculated by integration, and the relationship between two adjacent coordinate cards is obtained based on the connectivity condition. Thus, the local coordinate transformation is used to realize the integer mapping from M to N, completing the phase unwrapping.

[0045] 2. The present invention, with the help of the efficient performance of differential manifolds in the field of coordinate transformation, breaks through the efficiency bottleneck of traditional phase unwrapping algorithms and improves the robustness of phase unwrapping under complex terrain.

[0046] Other advantages, objects and features of the present invention will be described in part in the following description; and in part will be apparent to those skilled in the art based on an examination of the following; or may be taught from the practice of the invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 Flowchart of the present invention.

[0048] Figure 2 This is the real DEM of Long's Peak National Park in the United States.

[0049] Figure 3 The interference pattern of the simulated baseline length B = 70m.

[0050] Figure 4 Estimated terrain height for a baseline length of B = 70 m.

[0051] Figure 5 This is the error distribution diagram for baseline length B = 70m. DETAILED DESCRIPTION

[0052] The technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0053] Reference Figure 1 , a phase unwrapping algorithm based on differential manifolds, including the following steps:

[0054] S1, phase unwrapping problem transformation;

[0055] The phase unwrapping problem is transformed into a local coordinate transformation problem between M and N submanifolds;

[0056] S2, calculate the local coordinate system of the submanifold by integration;

[0057] S3, relying on the connectivity condition to obtain the relationship between two adjacent coordinate cards;

[0058] S4. Use local coordinate transformation to realize integer mapping from M to N and complete phase unwrapping.

[0059] In the present invention, discrete topological spaces M and N can be obtained from the interference phase, the real phase set and their topological structure. The mapping relationship between the two conforms to the homeomorphic mapping and constitutes a differential manifold; therefore, the phase unwrapping problem can be converted into a local coordinate transformation problem between the M and N submanifolds; thus, with the help of the efficient performance of differential manifolds in the field of coordinate transformation, the efficiency bottleneck of traditional phase unwrapping algorithms is broken through.

[0060] In step S1:

[0061] The corresponding relationship between the real phase unwrap(i,j) and the interference phase wrap(i,j) is

[0062] unwrap(i,j)=wrap(i,j)+2k(i,j)π (1).

[0063] Where k(i,j) is the integer fuzzy number of point (i,j).

[0064] Each element of the interference phase matrix represents a discrete echo signal. These discrete echo signals are regarded as a set M, and the set and its subset family τ=2 are obtained. M .

[0065] At the same time, the topological structure τ of the set M and the discrete topological space {M,τ} are obtained.

[0066] Similarly, we obtain the topological structure ι of the set N and the discrete topological space {N,ι} it constitutes.

[0067]

[0068] Where m and n are the interference pattern sizes.

[0069] The mapping f between M and N can be defined as

[0070]

[0071] Where, i∈[1,n].

[0072] So, we can get f and f -1 It is continuous;

[0073] That is, f:M→N is a homeomorphic mapping, and M and N are homeomorphic.

[0074]

[0075] Take any two coordinate cards of M and(U j ,θ), according to the properties of discrete topological space, U i ,U j There is the following relationship between

[0076]

[0077] If the intersection is empty, then U i ,U j There is no composite mapping between them; therefore, we only discuss the case where the intersection is not empty.

[0078] like Then U i ,U j The composite mapping between

[0079]

[0080] That is, M and N are diffeomorphisms on differential manifolds.

[0081] Therefore, the mapping relationship between the interference phase and the true phase can be solved by local coordinate transformation between the M and N submanifolds.

[0082] In step S2:

[0083] Let the boundary point set of the edge region D be The applicable coordinate system of point p∈D is consistent with the orientation of M, and we get The local coordinate system Then M The induced orientation on becomes a real analytic manifold with induced orientation.

[0084] Then, for any have

[0085]

[0086] Where, is the inclusion mapping, With orientation induced from M.

[0087] In step S3:

[0088] Any two adjacent submanifolds satisfy And it is believed that there is only one submanifold with a local coordinate system that is a suitable submanifold;

[0089] Any two adjacent points on the dependent manifold M |x i -x j |≤π, submanifold U i Make the following determination:

[0090]

[0091] If h(0)=a,h(1)=b, then the submanifold U i Connected, suitable for submanifold.

[0092] Otherwise, the submanifold U i Calculate formula (8)

[0093]

[0094] In this way, until all sub-manifolds U are connected, then determine whether they meet the orientation requirements. Find the relationship between two adjacent coordinate cards, f i -f i+1 =1or f i+1 -f i =1.

[0095] In step S4:

[0096] Then use formula (6) to calculate U i Homeomorphism f to N i , find the coordinate card of M (U i ,f i ).

[0097] Use formula (9) to calculate the mapping f from M to N

[0098]

[0099] Thus, the mapping relationship between the interference phase and the true phase is obtained.

[0100] Reference Figure 2-5 , untangling the simulated interference pattern of Long's Peak National Park in the United States; even for Figure 3 In the dense fringe area of the interference pattern shown in the figure, the algorithm can still achieve good disentanglement; Figure 4 、 Figure 5 As shown in the unwrapping results and errors, it can be seen that the unwrapping efficiency is high, the unwrapping effect is good, and it has high robustness.

[0101] The present invention introduces the differential manifold theory for the first time to solve the phase unwrapping problem. The differential manifold can be formed by the interference phase, the real phase set and the mapping relationship between the two, which just conforms to the homeomorphic mapping. Therefore, the phase unwrapping problem is transformed into a local coordinate transformation problem between M and N submanifolds. The local coordinate system of the submanifold is calculated by integration, and the relationship between the two adjacent coordinate cards is obtained based on the connectivity condition, so that the integer mapping from M to N is realized by using the local coordinate transformation to complete the phase unwrapping. The present invention, with the help of the efficient performance of differential manifolds in the field of coordinate transformation, breaks through the efficiency bottleneck of traditional phase unwrapping algorithms and improves the robustness of phase unwrapping under complex terrain.

[0102] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any technician familiar with the technical field, within the technical scope disclosed by the present invention, who makes equivalent replacements or changes based on the technical solution and inventive concept of the present invention, should be covered by the scope of protection of the present invention.

[0103] In the description of this specification, the reference terms "one embodiment", "some embodiments", "example", "specific example", or "some examples" mean that the specific features, structures, materials or characteristics described in conjunction with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner. In addition, those skilled in the art can combine and combine different embodiments or examples described in this specification and features of different embodiments or examples without contradiction.

[0104] Although the embodiments of the present invention have been shown and described above, it will be understood that the above embodiments are illustrative and are not to be construed as limitations on the present invention. A person skilled in the art may change, modify, replace and modify the above embodiments within the scope of the present invention.

Claims

1. A phase unwrapping algorithm based on differential manifolds, characterized in that: The following steps are involved: S1, phase unwrapping problem transformation; The phase unwrapping problem is transformed into a local coordinate transformation problem between M and N submanifolds; S2, calculate the local coordinate system of the submanifold by integration; S3, relying on the connectivity condition to obtain the relationship between two adjacent coordinate cards; S4. Use local coordinate transformation to realize integer mapping from M to N and complete phase unwrapping.

2. The phase unwrapping algorithm based on differential manifold according to claim 1, characterized in that: In step S1: The corresponding relationship between the true phase unwrap(i,j) and the interference phase wrap(i,j) is: unwrap(i,j)=wrap(i,j)+2k(i,j)π (1); Where k(i,j) is the integer fuzzy number of point (i,j); Considering the discrete echo signal as a set, the topological structure of the set and the discrete topological space are obtained; Where m and n are the interference pattern sizes; The mapping f between M and N is defined as Where, i∈[1,n]; f:M→N is a homeomorphic mapping, M and N are homeomorphic; Take any two coordinate cards of M and(U j ,θ); if Then U i ,U j The composite mapping between That is, M and N are diffeomorphisms on differential manifolds.

3. The phase unwrapping algorithm based on differential manifold according to claim 1, characterized in that: In step S2: Let the boundary point set of the edge region D be The applicable coordinate system of point p∈D is consistent with the orientation of M, and we get The local coordinate system Then M The induced orientation on becomes a real analytic manifold with induced orientation; then, for any have Where, is the inclusion mapping, With orientation induced from M.

4. The phase unwrapping algorithm based on differential manifold according to claim 3, characterized in that: In step S3: Any two adjacent submanifolds satisfy Any two adjacent points on the dependent manifold M |x i -x j |≤π, submanifold U i Make the following determination: If h(0)=a,h(1)=b, then the submanifold U i Connected, suitable for submanifold; Otherwise, the submanifold U i Calculate formula (8) In this way, until all sub-manifolds U are connected, then determine whether they meet the orientation requirements. Find the relationship between two adjacent coordinate cards, f i -f i+1 =1 or f i+1 -f i =1.

5. The phase unwrapping algorithm based on differential manifold according to claim 4, characterized in that: In step S4: When using formula (6) to calculate U i Homeomorphism f to N i , find the coordinate card of M (U i ,f i ); Use formula (9) to calculate the mapping f from M to N Thus, the mapping relationship between the interference phase and the true phase is obtained.

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