CN113791413B - Multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm - Google Patents

Abstract

The invention is disclosed inA multi-baseline InSAR branched delimitation pure integer programming phase unwrapping algorithm is developed, comprising the following steps: acquiring a multi-baseline InSAR interferogram; establishing an N-dimensional equation set of the winding phase differential fuzzy number according to the relation between the same relative elevation and each interference phase differential in the multi-baseline InSAR geometric model; converting a system of N-dimensional equations into an N-1-dimensional linear independent matrix equation, with k _i The intercept of the axis is an objective function, the directional rays intersected by the N-1 dimensional plane are used as constraint conditions, and a multi-baseline InSAR pure integer programming model is constructed; solving a multi-baseline InSAR pure integer programming model by adopting a branch delimitation method to determine a solution set of fuzzy numbers; multiplying the solved N groups of N-dimensional solution sets by 2 pi, and then superposing the original interference phases to obtain absolute phase values of N groups of baseline interferograms; the invention not only expands the non-fuzzy interval to [ -mpi, mpi), but also has better disentangling capability in the phase undersampling area and the terrain mutation area; and the baseline requirement on interference pairs is also weakened, and the unwrapping phase can be effectively calculated as long as the baseline lengths are unequal.

Description

Multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm

technical field

The invention relates to the technical field of multi-baseline InSAR data processing, in particular to a multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm.

Background technique

Interferometric Synthetic Aperture Radar (InSAR) technology can monitor ground targets all-weather, all-weather, and in a wide range, and is widely used in natural disasters, dam safety, engineering route selection, military target identification and positioning, etc. . As a key step in InSAR technology, phase unwrapping has always attracted the attention and exploration of experts and scholars. Generally speaking, by solving the 2kπ difference between the wrapped phase and the original phase, the process of restoring the obtained interferometric phase information from the interval [-π,π) to [-mπ,mπ) is called phase unwrapping. The traditional single-baseline phase unwrapping is based on the assumption of phase continuity, but the real terrain fluctuations often cannot meet the requirements of terrain continuity. The predecessors proposed a multi-baseline InSAR phase unwrapping technology that can add multiple interferometric phase images, which can weaken the phase The influence of factors such as undersampling, phase noise and spectral aliasing can improve the accuracy and reliability of phase unwrapping.

Pure Integer Programming (PIP) is a discrete optimization problem in which all decision variables are integers. From the point of view of seeking integer solutions, the two have the same purpose. Integer Programming (IP) formed an independent branch in 1958. It mainly solves the derivative problems of the original problem step by step, and determines the destination of the source problem through the solution of the relaxation problem until there are no more unsolved derivative problems. This theory is widely used in the field of transportation and computer communication, but the integer programming theory is less studied in the field of InSAR. Scholars at home and abroad have mentioned the term integer programming, and have also used the open source code SYMPHONY to solve the problem applicable to PS-InSAR. The formula for the expansion of the integer linear programming problem, but how to solve it is not presented to the readers. Solving the phase unwrapping problem using pure integer programming theory algorithm has not been seen in the existing literature.

Contents of the invention

The purpose of the present invention is to propose a new idea and a new method for solving the phase unwrapping problem, which is provided by using the pure integer programming branch and bound method to analyze and solve the multi-baseline phase unwrapping problem.

In order to achieve the above object, the technical solution of the present invention is: multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm, comprising the following steps:

1) Obtain multi-baseline InSAR interferogram;

1a) Prepare one SAR main image and multiple SAR auxiliary images of the same ROI;

1b) Perform interference processing on multiple auxiliary images and the main image respectively to obtain a multi-baseline InSAR interferogram;

2) According to the relationship between the same relative elevation and each interferometric phase differential in the multi-baseline InSAR geometric model, an N-dimensional equation system of winding phase differential fuzzy numbers is established;

3) Transform the N-dimensional equations into N-1-dimensional linear independent matrix equations, take the intercept of the ki axis as the objective function, and the directional rays intersecting the N-1-dimensional planes as constraints, and construct a multi-baseline InSAR pure integer programming model ;

4) Using the branch and bound method to solve the multi-baseline InSAR pure integer programming model to determine the solution set of fuzzy numbers;

4a) If a _n =max(a ₁ , a ₂ ,..., a _n ), set the basic equations In the case of m _n =max(m ₁ ,m ₂ ,...,m _n ), then use k _n as the basis to solve the N-1-dimensional fuzzy number solution set, and when m _i =max(m ₁ ,...,m _{i -1} , m _i ,..., m _n ), then use _ki as the base to solve the N-1-dimensional fuzzy number solution set, and finally combine the base to construct the N-dimensional fuzzy number solution set;

4b) If a _i =max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), set the rebasing equation system The conversion to the basic equations is realized through the axis symmetry theory, and the solution set of N-dimensional fuzzy numbers is solved by the method of solving the basic equations;

4c) If a ₁ =a ₂ =...=a _n , directly determine the N-dimensional fuzzy number solution set without considering the integer programming;

5) The absolute phase values of N sets of baseline interferograms can be obtained by multiplying the solved N sets of N-dimensional solution sets by 2π and superimposing the original interferometric phase.

Preferably, in the above 2), it is assumed that the phases of interferometric interferometry corresponding to the multi-baseline InSAR are respectively The corresponding relationship between the same relative elevation and the interference phase differential can be obtained as

Among them, B ₀ =[B ₁ ,B ₂ ,…,B _n ] is the least common multiple of the baselines B ₁ , B ₂ ,…,B _n , dZ is the relative elevation, λ is the wavelength of the radar wave, /> is the interference phase differential, _ki is /> The ambiguity of m _i =B ₀ /B _i (i=1,2,…,n) is the modulus;

make The equations of fuzzy numbers k ₁ , k ₂ ,...,k _n that can be established for winding phase differential are

In , all parameters are integers, X is the smallest integer solution of the equation system; a ₁ , a ₂ ,…, a _n are interferogram winding phase differential functions; m ₁ , m ₂ ,…, m _n are moduli.

Preferably, the parameters in the equations in 3) are all integers. If the equations are superimposed, X can be written as an expression containing fuzzy numbers k ₁ , k ₂ ,...,k _n as

nX＝a ₁ +a ₂ +...+a _n +k ₁ m ₁ +k ₂ m ₂ +...+k _n m _n

in phase difference After the sum modulus _mi is determined, a ₁ , a ₂ ,..., a _n can be found, so find the minimum value of X, that is, find the minimum value of Y=k ₁ m ₁ +k ₂ m ₂ +...+k _n m _n ;

general style Y＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n is transformed into an integer programming problem, which is recorded as the IP model

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

So far, the multi-baseline InSAR pure integer programming model has been constructed.

Preferably, to solve the integer programming problem in the above 4), it is necessary to first remove the integer constraint, convert it to an LP model, and solve the relaxation problem

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

Preferably, it is assumed in 4a) that a _n = max(a ₁ , a ₂ , ..., a _n ), if the optimal solution of the equation system is set to be positive, n-1 linearly independent equations are obtained , among them, k ₁ , k ₂ ,..., k _n-1 are independent variables, k _n is dependent variable, and set this system of equations as the base equation system

According to the branch and bound method, solve the optimal solution of the basic equation system

Branching for k _i component, denoted as LP ₁ , LP ₂ model, [] is the rounding function

For the LP ₁ model, since a _n = max(a ₁ , a ₂ ,..., a _n ), there is the following relationship

In , the condition of the equal sign on the left is that a _n -a _i can be divisible by m _i , and at this time k _i takes the maximum value, that is

According to the equation The relational expression corresponding to k _n and k _i is

k _i m _i -k _n m _n = a _n -a _i

Simplify to the solution formula of k _n

For the LP ₁ model, substitute the maximum value of _ki into it, then k _n = 0, if _ki decreases by one unit, then the value of k _n is

At this time, k _n must be less than 0, so consider the LP ₂ model;

Set the initial value of k ₁ , k ₂ ,..., k _n

Determine the path of traversal according to the size of the modulus. If the _modulus m _n is the largest, then traverse k _{n ;} =1, 2, ..., n-1); because m ₁ , m ₂ , ..., m _n are not equal, there is no situation that any two modules are equal.

Preferably, in said 4a), it is assumed that m _n is the largest, and k _n is traversed, and the basic equation system is changed into a matrix form

Simplified to MK-k _n M _n = A, the corresponding parameter expression can refer to the following formula

Since the modulus is not 0, M is reversible, and the solution set K of the equation system can be solved

K=M ^-1 (A+k _n M _n )

Since m _n is the largest, at this time k _n can be traversed sequentially from 0. k _n increases by one unit length in turn, solves the solution set K of the base equation system, and judges whether the elements in K satisfy integers. If this condition is met, the solved solution set K is combined with the base k _n to complete the solution of the N-dimensional fuzzy number.

Preferably, in 4a) it is assumed that m _{i is} the largest, and traversing k _i

base equation can be rewritten as

Simplified to M _x K _x -k _i M _i ＝A _x , the corresponding parameter expression can refer to the following formula

Since the modulus is not 0, M _x is reversible, and the solution set K _x of the equation system can be solved

K _x ＝M _x ^-1 (A _x +k _i M _i )

Since m _i is the largest, the initial value has the following restrictions, and k _i traverses sequentially from 1

k _i increases by one unit length in turn, solve the solution set K _x of the equation system, judge whether the elements of K _x satisfy integers, find out if this condition is met, and if this condition is met, the solution set K _x combined with the base _ki , that is Complete the solution of N-dimensional fuzzy numbers.

Preferably, in 4b) it is assumed that a _i = max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), if the optimal solution of the system of equations is set to be positive, At this time, k ₁ ,..., _ki-1 , ki ₊₁ ,..., k _n are the independent variables, k _i is the equation of the dependent variable, and the equations are set as rebasing equations

In the optimal solution parameters to be solved, m _i and m _n , a _i and a _n are interchanged, and using the theory of axis symmetry, the variable base equations can also be rewritten into basic equations

At this time, the model obtained is still a _n = max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), and the corresponding N-dimensional Fuzzy number k _i value.

Preferably, if a ₁ =a ₂ =...=a _n in the above 4c), the pure integer programming model is not considered at this time, and the remainder value is directly assigned to X, that is, X=a _i (i=1,2,..., n), then the corresponding fuzzy number k _i is

k ₁ =k ₂ = . . . =k _n =0.

Compared with the prior art, the invention has the following beneficial effects: the pure integer programming theory is introduced into the multi-baseline InSAR phase unwrapping. Based on the relationship between the same relative elevation and each interference phase differential, a pure integer programming model is constructed, and a multi-baseline InSAR branch-and-bound phase unwrapping algorithm based on pure integer programming is proposed. The algorithm limits the number of conditional equations based on the number of baselines, constructs an N-dimensional space with the fuzzy number as the axis, takes the intercept of the _ki axis as the objective function, and the directed ray intersecting the N-1-dimensional plane is Constraint conditions, the optimal solution of k _i is the starting point of the solution, and the unit length is increased successively to verify whether the solution set composed of the remaining fuzzy numbers satisfies the integer condition until the solution set meets the integer condition, then the optimal integer of N fuzzy numbers After the solution is determined, the phase unwrapping is completed. It not only extends the unambiguous interval to [-mπ, mπ), but also has better unwrapping ability in the phase undersampling area and the terrain mutation area; and the baseline requirement for the interference pair is also weakened. As long as the baseline length is not equal, the It can effectively solve the unwrapped phase.

Description of drawings

Fig. 1 is a flow chart of the steps of the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm of the present invention.

Fig. 2 is a DEM diagram of the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm of the present invention.

Fig. 3 is a simulation multi-baseline interferogram of the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm of the present invention.

Fig. 4 is the pure integer programming phase unwrapping result of the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm of the present invention.

Fig. 5 is a pure integer programming phase error diagram of the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm of the present invention.

Detailed ways

The present invention is described in further detail now in conjunction with accompanying drawing. The accompanying drawings are simplified schematic diagrams, which only schematically illustrate the basic structure of the present invention, so that they only show the components relevant to the present invention.

Please refer to Figure 1-5, the multi-baseline InSAR branch and bound pure integer programming phase unwrapping algorithm, including the following steps:

1) Obtain the multi-baseline InSAR interferogram.

1a) Prepare one SAR main image and multiple SAR auxiliary images of the same ROI.

1b) Perform interference processing on multiple auxiliary images and the main image respectively to obtain multi-baseline InSAR interferograms.

2) According to the relationship between the same relative elevation and each interferometric phase differential in the multi-baseline InSAR geometric model, an N-dimensional equation system of winding phase differential fuzzy numbers is established. In 2), it is assumed that the interferometric phases corresponding to multi-baseline InSAR are The corresponding relationship between the same relative elevation and the interference phase differential can be obtained as

Among them, B ₀ =[B ₁ ,B ₂ ,…,B _n ] is the least common multiple of the baselines B ₁ , B ₂ ,…,B _n , dZ is the relative elevation, λ is the wavelength of the radar wave, /> is the interference phase differential, _ki is /> The ambiguity of m _i =B ₀ /B _i (i=1,2,…,n) is the modulus;

make The equations of fuzzy numbers k ₁ , k ₂ ,...,k _n that can be established for winding phase differential are

In , all parameters are integers, X is the smallest integer solution of the equation system; a ₁ , a ₂ ,…, a _n are interferogram winding phase differential functions; m ₁ , m ₂ ,…, m _n are moduli.

3) Transform the N-dimensional equations into N-1-dimensional linear independent matrix equations, take the intercept of the ki _- axis as the objective function, and the directional rays intersecting the N-1-dimensional planes as constraints, and construct a multi-baseline InSAR pure integer programming Model; 3) The parameters in the equation system are all integers. If the equations are superimposed, X can be written as an expression containing fuzzy numbers k ₁ , k ₂ ,...,k _n as

nX＝a ₁ +a ₂ +...+a _n +k ₁ m ₁ +k ₂ m ₂ +...+k _n m _n

in phase difference After the sum modulus _mi is determined, a ₁ , a ₂ ,..., a _n can be found, so find the minimum value of X, that is, find the minimum value of Y=k ₁ m ₁ +k ₂ m ₂ +...+k _n m _n ;

general style Y＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n is transformed into an integer programming problem, which is recorded as an integer programming (Integer Programming, IP) model

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

So far, the multi-baseline InSAR pure integer programming model has been constructed.

4) Using the branch and bound method to solve the multi-baseline InSAR pure integer programming model to determine the solution set of fuzzy numbers. 4) To solve the integer programming problem, you need to remove the integer constraint first, convert it to a linear programming (Linear Programming, LP) model, and solve the relaxation problem

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

4a) If a _n =max(a ₁ , a ₂ ,..., a _n ), set the basic equations In the case of m _n =max(m ₁ ,m ₂ ,...,m _n ), then use k _n as the basis to solve the N-1-dimensional fuzzy number solution set, when m _i =max(m ₁ ,.. ., m _i-1 , m _i ,..., m _n ), use _ki as the base to solve the N-1-dimensional fuzzy number solution set, and finally combine the base to construct the N-dimensional fuzzy number solution set . In 4a), it is assumed that a _n = max(a ₁ , a ₂ , ..., a _n ), if the optimal solution of the equation system is set to be positive, n-1 linearly independent equations are obtained, where k ₁ , k ₂ ,..., k _n-1 is the independent variable, k _n is the dependent variable, and set this system of equations as the base equation system

According to the branch and bound method, solve the optimal solution of the basic equation system

Branching for k _i component, denoted as LP ₁ , LP ₂ model, [] is the rounding function

For the LP ₁ model, since a _n = max(a ₁ , a ₂ ,..., a _n ), there is the following relationship

In , the condition of the equal sign on the left is that a _n -a _i can be divisible by m _i , and at this time k _i takes the maximum value, that is

According to the equation The relational expression corresponding to k _n and k _i is

k _i m _i -k _n m _n = a _n -a _i

Simplify to the solution formula of k _n

For the LP ₁ model, substitute the maximum value of _ki into it, then k _n = 0, if _ki decreases by one unit, then the value of k _n is

At this time, k _n must be less than 0, so consider the LP ₂ model;

Set the initial value of k ₁ , k ₂ ,..., k _n

Determine the path of traversal according to the size of the modulus. If the _modulus m _n is the largest, then traverse k _{n ;} =1, 2, ..., n-1); because m ₁ , m ₂ , ..., m _n are not equal, there is no situation that any two modules are equal. Assuming that m _n is the largest, traverse k _n and change the basic equation system into matrix form

Simplified to MK-k _n M _n = A, the corresponding parameter expression can refer to the following formula

Since the modulus is not 0, M is reversible, and the solution set K of the equation system can be solved

K=M ^-1 (A+k _n M _n )

Since m _n is the largest, at this time k _n can be traversed sequentially from 0. k _n increases by one unit length in turn, solves the solution set K of the base equation system, and judges whether the elements in K satisfy integers. If this condition is met, the solved solution set K is combined with the base k _n to complete the solution of the N-dimensional fuzzy number. Assuming that m _i is the largest, traverse k _i

base equation can be rewritten as

Simplified to M _x K _x -k _i M _i ＝A _x , the corresponding parameter expression can refer to the following formula

Since the modulus is not 0, M _x is reversible, and the solution set K _x of the equation system can be solved

K _x ＝M _x ^-1 (A _x +k _i M _i )

Since m _i is the largest, the initial value has the following restrictions, and k _i traverses sequentially from 1

k _i increases by one unit length in turn, solve the solution set K _x of the equation system, judge whether the elements of K _x satisfy integers, find out if this condition is met, and if this condition is met, the solution set K _x combined with the base _ki , that is Complete the solution of N-dimensional fuzzy numbers.

4b) If a _i =max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), set the rebasing equation system The conversion to the basic equations is realized through the axial symmetry theory, and the solution set of N-dimensional fuzzy numbers is solved by the method of solving the basic equations. In 4b), it is assumed that a _i = max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), if the optimal solution of this equation system is set to be positive, then k ₁ can be obtained , ..., ki _-1 , ki ₊₁ , ..., k _n is the independent variable, _ki is the equation of the dependent variable, set this equation system as a rebasing equation system

In the optimal solution parameters to be solved, m _i and m _n , a _i and a _n are interchanged, and using the theory of axis symmetry, the variable base equations can also be rewritten into basic equations

At this time, the model obtained is still a _n = max(a ₁ ,..., a _i-1 , a _i ,..., a _n ), and the corresponding N-dimensional Fuzzy number k _i value.

4c) If a ₁ =a ₂ =...=a _n , the N-dimensional fuzzy number solution set can be directly determined without considering integer programming. In 4c), it is assumed that a ₁ =a ₂ =...=a _n . At this time, the pure integer programming model is not considered, and the remainder value is directly assigned to X, that is, X=a _i (i=1,2,...,n), then the corresponding The value of fuzzy number k _i is

k ₁ =k ₂ = . . . =k _n =0.

5) The absolute phase values of N sets of baseline interferograms can be obtained by multiplying the solved N sets of N-dimensional solution sets by 2π and superimposing the original interferometric phase.

In order to verify the effectiveness of the multi-baseline InSAR pure integer programming phase unwrapping algorithm, the multi-baseline InSAR interferogram simulated by the real DEM of the ISOLA National Park in the United States (as shown in Figure 2) is selected to verify the main parameters. Table 1. Fig. 3 is the interferogram of the simulated baselines of 120m, 192m, 320m and 420m respectively.

Table 1 Main parameters of interferogram

As shown in Figure 3, there is a phase undersampling region in the interferogram. After solving by the branch-and-bound pure integer programming method, the phase unwrapping results of different baseline interferograms are shown in Figure 4, which can be obtained in good agreement with Figure 2 The unwrapping phase of . According to Figure 5-(a), 5-(b), it can be seen that for short baselines, the phase error is concentrated in the area with a small phase change on the left side of the interferogram; according to 5-(c), 5-(d), it can be seen , for the long baseline, the phase error is concentrated in the area with a large phase mutation on the right side of the interferogram, the magnitude of the error is stable at 10^-15 to 10^-16, and the unwrapping accuracy is high.

Apparently, the above-mentioned embodiments are only examples for clear description, rather than limiting the implementation. For those of ordinary skill in the art, other changes or changes in different forms can be made on the basis of the above description. It is not necessary and impossible to exhaustively list all the implementation manners here. And the obvious changes or changes derived therefrom are still within the scope of protection of the present invention.

Claims (8)

1. The multi-baseline InSAR branched delimitation pure integer programming phase unwrapping algorithm is characterized by comprising the following steps:

1) Acquiring a multi-baseline InSAR interferogram;

1a) Preparing a SAR main image and a plurality of SAR auxiliary images of the same region of interest;

1b) Respectively carrying out interference processing on a plurality of auxiliary images and the main image to obtain a multi-baseline InSAR interferogram;

2) Establishing an N-dimensional equation set of the winding phase differential fuzzy number according to the relation between the same relative elevation and each interference phase differential in the multi-baseline InSAR geometric model;

3) Converting a system of N-dimensional equations into an N-1-dimensional linear independent matrix equation, with k _i The intercept of the axis is an objective function, the directional rays intersected by the N-1 dimensional plane are used as constraint conditions, and a multi-baseline InSAR pure integer programming model is constructed;

4) Solving a multi-baseline InSAR pure integer programming model by adopting a branch delimitation method to determine a solution set of fuzzy numbers;

4a) If a is _n ＝max(a ₁ ，a ₂ ，...，a _n ) Setting a basic equation setAt m _n ＝max(m ₁ ，m ₂ ，...，m _n ) In the case of (1), then at k _n Solving the N-1-dimensional fuzzy number solution set for the substrate, at m _i ＝max(m ₁ ，...，m _i-1 ，m _i ，...，m _n ) In the case of (1), then at k _i Solving an N-1-dimensional fuzzy number solution set for the substrate, and finally combining the substrate to construct the solution set of the N-dimensional fuzzy number;

4b) If a is _i ＝max(a ₁ ，...，a _i-1 ，a _i ，...，a _n ) Setting a system of variable basis equationsThe conversion to a basic equation set is realized through the shafting symmetry theory, and then a solution set of N-dimensional fuzzy numbers is solved by utilizing a method for solving the basic equation set;

4c) If a is ₁ ＝a ₂ ＝…＝a _n The N-dimensional fuzzy number solution set can be directly determined without considering integer programming;

5) Multiplying the solved N groups of N-dimensional solution sets by 2 pi, and then superposing the original interference phases to obtain absolute phase values of N groups of baseline interferograms;

the assumption in 2) that the corresponding winding interference phases of the multi-baseline InSAR are respectively

The corresponding relation between the same relative elevation and the interference phase differential is obtained

In (B) ₀ ＝[B ₁ ,B ₂ ,…,B _n ]Is baseline B ₁ ，B ₂ ，…，B _n dZ is the relative elevation, lambda is the radar wavelength,/-the least common multiple of (2)>To interfere with phase differentiation, k _i Is->Ambiguity of m _i ＝B ₀ /B _i (i=1, 2, …, n) is a modulus;

order theFuzzy number k capable of establishing winding phase differentiation ₁ ，k ₂ ，…，k _n Is as follows

Wherein, all parameters are integers, X is the minimum integer solution of the equation set; a, a ₁ ，a ₂ ，…，a _n Winding a phase derivative function for the interferogram; m is m ₁ ，m ₂ ，…，m _n Is a mould.

2. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the parameters in the equation set in the 3) are integers, if the equations are overlapped, X can be written to contain the fuzzy number k ₁ ，k ₂ ，…，k _n The expression of (2) is

nX＝a ₁ +a ₂ +…+a _n +k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

In phase differenceSum of modes m _i After determination, a ₁ ，a ₂ ，…，a _n Can be found, so find the minimum value of X, i.e. find y=k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n Is the minimum of (2);

will be described inY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n Conversion to integer programming problem, noted as IP model

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

Thus, the multi-baseline InSAR pure integer programming model is built.

3. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the integer programming problem is solved in the 4), integer constraint is removed firstly, the integer programming problem is converted into an LP model, and the relaxation problem is solved

minY＝k ₁ m ₁ +k ₂ m ₂ +…+k _n m _n

4. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the assumption a in 4 a) is that _n ＝max(a ₁ ，a ₂ ，...，a _n ) If the optimal solution of the equation set is set to be positive, n-1 equations with independent linearity are obtained, wherein k is ₁ ，k ₂ ，…，k _n-1 As an independent variable, k _n Is a dependent variable and sets the equation set as a basic equation set

According to the branch-and-bound method, solving the optimal solution of the basic equation set

For k _i The components are branched and denoted as LP ₁ ，LP ₂ Model []As a function of rounding

For LP ₁ Model due to a _n ＝max(a ₁ ，a ₂ ，...，a _n ) Then the following relation exists

The condition that the left equal sign is satisfied is a _n -a _i Can be m _i Integer division of k at this time _i Take the maximum value, i.e

According to the system of equationsCorresponds to k _n And k is equal to _i The relational expression of (2) is

k _i m _i -k _n m _n ＝a _n -a _i

Reduce to k _n Solution formula of (2)

For LP ₁ Model, let k _i Taking the maximum value substitution, then k _n =0, if k _i Decreasing by one unit, then k _n The value of (2) is that

At this time, k _n Necessarily less than 0, thus consider LP ₂ Modeling;

setting k ₁ ，k ₂ ，…，k _n Initial value of

Determining the traversing path according to the size of the module, if the module m _n Maximum, then traverse k _n The method comprises the steps of carrying out a first treatment on the surface of the If there is a certain mode m _i (i=1, 2, …, n-1) is maximum, then k is traversed _i (i=1, 2,) n-1; because m is ₁ ，m ₂ ，...，m _n Not equal, there is no case where any two modes are equal.

5. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the assumption m in 4 a) is that _n Maximum, traverse k _n Changing the basic equation set into matrix form

Simplified to MK-k _n M _n Corresponding parameter expression can be referred to as follows =a

Since none of the modes is 0, M is reversible, the solution set K of the equation set can be solved

K＝M ^-1 (A+k _n M _n )

Due to m _n Max, at time k _n Traversing from 0 to k _n Sequentially increasing by one unit lengthDegree, solving a solution set K of the basic equation set, judging whether elements in the K meet integers, and if so, combining the solution set K with the base K _n And (5) completing the solution of the N-dimensional fuzzy number.

6. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the assumption m in 4 a) is that _i Maximum, traverse k _i

The basis equation setRewritable as

Simplified to M _x K _x -k _i M _i ＝A _x The corresponding parameter expression can be referred to as the following formula

Since the modes are not 0, M _x Reversible, solution set K capable of solving equation set _x

K _x ＝M _x ^-1 (A _x +k _i M _i )

Due to m _i Maximum, the initial value is limited as follows, k _i Traversing sequentially from 1

k _i Sequentially increasing a unit length, and solving solution set K of solution equation set _x Judging K _x If the elements of (2) satisfy the integer, find out the solution set K solved if the condition is satisfied _x Bond substrate k _i And (5) completing the solution of the N-dimensional fuzzy number.

7. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: the assumption a in 4 b) is that _i ＝max(a ₁ ，...，a _i-1 ，a _i ，...，a _n ) If the optimal solution of the equation set is set to be positive, k is obtained ₁ ，…，k _i-1 ，k _i+1 ，…，k _n As an independent variable, k _i The equation set is set as a variable basis equation set

M in optimal solution parameters to be solved _i And m is equal to _n ，a _i And a _n The system of variable basis equations can be rewritten into the system of basis equations by using the shafting symmetry theory

At this time, a is still obtained _n ＝max(a ₁ ，...，a _i-1 ，a _i ，...，a _n ) The fuzzy number k corresponding to the N dimension can be solved by repeating the solving process of the basic equation set _i Values.

8. The multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of claim 1, characterized by: if a in 4 c) ₁ ＝a ₂ ＝…＝a _n The remainder value is given to X at this point without consideration of the pure integer programming model, i.e., x=a _i (i=1, 2, …, n), the corresponding ambiguity k _i The value is

k ₁ ＝k ₂ ＝…＝k _n ＝0。