CN113791413B - Multi-baseline InSAR branched delimitation pure integer programming phase unwrapping algorithm - Google Patents
Multi-baseline InSAR branched delimitation pure integer programming phase unwrapping algorithm Download PDFInfo
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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
Technical Field
The invention relates to the technical field of multi-baseline InSAR data processing, in particular to a multi-baseline InSAR branch delimitation pure integer programming phase unwrapping algorithm.
Background
The synthetic aperture radar interferometry (Interferometric Synthetic Aperture Radar, inSAR) technology can monitor the ground targets all the day, all the weather and in a large scale, and is widely applied to the fields of natural disasters, dyke safety, engineering line selection, military target identification and positioning and the like. Phase unwrapping has been a key step in InSAR technology and has been of interest and investigation by expert students. In general, the process of recovering the acquired interference phase information from the interval of [ -pi, pi) to the interval of [ -m pi, m pi) by solving the phase difference of 2k pi between the winding phase and the original phase is called phase unwrapping. The traditional single-base-line phase unwrapping is based on a phase continuity assumption, but the actual topography fluctuation change often cannot meet the requirement of topography continuity, so that the ancestors propose a multi-base-line InSAR phase unwrapping technology capable of increasing a plurality of interference phase maps, the influences of factors such as phase undersampling, phase noise, spectrum aliasing and the like can be weakened, and the accuracy and reliability of phase unwrapping are improved.
Pure integer programming (Pure Integer Programming, PIP) is a discrete optimization problem where all decision variables are integers. From the perspective of looking for integer solutions, the two have different similarities. Integer programming (Integer Programming, IP) was independently branched in 1958, mainly by gradually solving derivative problems of the original problem, and by determining the destination of the source problem by relaxing the solution of the problem until there are no more outstanding derivative problems. The theory is widely applied to the fields of transportation and computer communication, but the integer programming theory is less studied in the InSAR field, domestic and foreign scholars mention the word of integer programming, and open source code SYMPHONY is also utilized to solve a formula suitable for developing the PS-InSAR integer linear programming problem, but how to solve the problem is not presented in the eyes of readers. Solving the phase unwrapping problem using pure integer programming theory algorithms is not seen in the existing literature.
Disclosure of Invention
The invention aims to provide a new thought and a new method for solving the phase unwrapping problem, which are provided by utilizing a pure integer programming branch delimitation method to analyze and solve the multi-baseline phase unwrapping problem.
In order to achieve the above purpose, the technical scheme of the invention is as follows: a multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm comprising the steps of:
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 the N-dimensional equation set into an N-1-dimensional linear independent matrix equation, and constructing a multi-baseline InSAR pure integer programming model by taking the intercept of the ki axis as an objective function and the directional rays intersected by the N-1-dimensional plane as constraint conditions;
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) And 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.
Preferably, the winding interference phases corresponding to the multi-baseline InSAR in the 2) are respectively assumed to beThe 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 1 ，m 2 ，…，m n is a mould.
Preferably, the parameters in the equation set in 3) are integers, and if the equations are superimposed, 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.
Preferably, the integer programming problem is solved in 4), the integer constraint is removed first, the LP model is converted, and the relaxation problem is solved
minY＝k 1 m 1 +k 2 m 2 +…+k n m n
Preferably, in said 4 a), a is assumed 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 walk through k 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.
Preferably, m is assumed in said 4 a) n Maximum, traverse k n Changing the basic equation set into a momentArray 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 0. k (k) n Sequentially increasing a unit length, solving a solution set K of a basic equation set, judging whether elements in the K meet integers, and if the elements meet the conditions, combining the solution set K with the base K n And (5) completing the solution of the N-dimensional fuzzy number.
Preferably, m is assumed in said 4 a) 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.
Preferably, in said 4 b) it is assumed that a 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.
Preferably, in the 4 c), if a 1 ＝a 2 ＝…＝a n At this time, the pure integer programming is not consideredModel, the remainder value is given to X directly, i.e. x=a i (i=1, 2, …, n), the corresponding ambiguity k i The value 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 multi-baseline InSAR phase unwrapping. A pure integer programming model is built based on the relation between the same relative elevation and each interference phase differential, and a multi-baseline InSAR branching delimitation phase unwrapping algorithm based on pure integer programming is provided. The algorithm takes the number of the base lines as a reference to limit the number of the conditional equation sets, constructs an N-dimensional space taking the fuzzy number as an axis, and takes k as i The intercept of the axes is an objective function, the directional rays intersected by the N-1 dimensional plane are constraint conditions, and k i The optimal solution of the number N of fuzzy numbers is used as a solution starting point, a unit length is gradually increased, whether a solution set formed by the residual fuzzy numbers meets an integer condition is verified, and if the solution set meets the integer condition, the determination of the optimal integer solution of the number N of fuzzy numbers is completed, so that phase unwrapping is completed. The non-fuzzy interval is expanded to [ -mpi, mpi), and the method has better disentangling capability in a phase undersampling area and a 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.
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 plot of the multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of the present invention.
FIG. 3 is a simulated multi-baseline interferogram of the multi-baseline InSAR branched delimiter pure integer programming phase unwrapping algorithm of the present invention.
Fig. 4 is a pure integer programming phase unwrapping result of the multi-baseline InSAR branching delimiter pure integer programming phase unwrapping algorithm of the present invention.
Fig. 5 is a graph of pure integer programming phase error for the multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm of the present invention.
Detailed Description
The invention will now be described in further detail with reference to the accompanying drawings. The drawings are simplified schematic views illustrating the basic structure of the present invention by way of illustration only, and thus show only the constitution related to the present invention.
Referring to fig. 1-5, a multi-baseline InSAR branch-and-bound pure integer programming phase unwrapping algorithm includes the steps of:
1) A multi-baseline InSAR interferogram is acquired.
1a) One SAR main image and a plurality of SAR auxiliary images of the same region of interest are prepared.
1b) And respectively carrying out interference processing on the plurality of auxiliary images and the main image to obtain a multi-baseline InSAR interference pattern.
2) And establishing an N-dimensional equation set of the winding phase differential ambiguity number according to the relation between the same relative elevation and each interference phase differential in the multi-baseline InSAR geometric model. 2) In the assumption that the corresponding winding interference phases of the multi-baseline InSAR are respectivelyThe 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.
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; 3) The parameters in the equation set 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 integer programming (Integer Programming, 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.
4) And solving the multi-baseline InSAR pure integer programming model by adopting a branch delimitation method to determine a solution set of fuzzy numbers. 4) The integer programming problem is solved by firstly removing integer constraint, converting the integer constraint into a linear programming (Linear Programming, LP) model and solving the relaxation problem
minY＝k 1 m 1 +k 2 m 2 +…+k n m n
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 And solving an N-1-dimensional fuzzy number solution set for the substrate, and finally combining the substrate to construct the N-dimensional fuzzy number solution set. 4a) Let a be 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 As dependent variable and setting the equation set as the baseProgram 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 walk through k 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. Let m be 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 0. k (k) n Sequentially increasing a unit length, solving a solution set K of a basic equation set, judging whether elements in the K meet integers, and if the elements meet the conditions, combining the solution set K with the base K n And (5) completing the solution of the N-dimensional fuzzy number. Let m be 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 the equationSolution set K of groups 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.
4b) If a is i ＝max(a 1 ，...，a i-1 ，a i ，...，a n ) Setting a system of variable basis equationsAnd (3) converting into a basic equation set through a shafting symmetry theory, and solving a solution set of the N-dimensional fuzzy number by utilizing a method for solving the basic equation set. 4b) Let a be 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.
4c) If a is 1 ＝a 2 ＝…＝a n The N-dimensional fuzzy number solution set can be directly determined without considering integer programming. 4c) Let a be 1 ＝a 2 ＝…＝a n At this time, no consideration is given to pure integersPlanning model, giving the remainder value to X, i.e. x=a i (i=1, 2, …, n), the corresponding ambiguity k i The value is
k 1 ＝k 2 ＝…＝k n ＝0。
5) And 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.
In order to verify the effectiveness of the multi-baseline InSAR pure integer programming phase unwrapping algorithm, a multi-baseline InSAR interferogram simulated by the real DEM (shown in figure 2) of the national park in ISOLA was selected for verification, and the main parameters are shown in Table 1. FIG. 3 is an interferogram with simulated baselines 120m,192m,320m, and 420m, respectively.
TABLE 1 interference pattern principal parameters
As shown in FIG. 3, the interferogram has a phase undersampling region, and after the interferogram is solved by a branch-and-bound pure integer programming method, the phase unwrapping results of different baseline interferograms are shown in FIG. 4, and unwrapped phases which are well matched with those of FIG. 2 can be obtained. From fig. 5- (a), 5- (b), it can be seen that for a short baseline, the phase error is concentrated in the left side phase abrupt small region of the interferogram; according to the results of 5- (c), 5- (d), for a long base line, the phase error is concentrated in a region with larger phase abrupt change on the right side of the interference pattern, the order of magnitude of the error is stable from 10-15 to 10-16, and the unwrapping precision is higher.
It is apparent that the above examples are given by way of illustration only and are not limiting of the embodiments. Other variations or modifications of the above teachings will be apparent to those of ordinary skill in the art. It is not necessary here nor is it exhaustive of all embodiments. While still being apparent from variations or modifications that may be made by those skilled in the art are within the scope of the 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。
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