CN104683649B - A kind of compression of compressed sensing vector geometrical model and restoration methods - Google Patents

Abstract

The invention discloses a method for compressing and recovering a geometric model of a compressed sensing vector. According to the present invention, the geometric information of the vector geometric model can be expressed as a sparse signal under the action of its Laplacian operator; a random matrix is used to sample the geometric information, and the compression is completed, and then the 0- Norm fitting function The sparse signal of the original signal is recovered, and then the original signal is recovered to complete the compression and recovery of the model. During the recovery process, the constrained optimization is converted to unconstrained optimization, and a new search direction is designed for solution. The method of the invention not only speeds up the compression speed of the model, but also realizes theoretical lossless compression.

Description

A Compression and Restoration Method of Compressed Sensing Vector Geometric Model

technical field

The invention belongs to the field of digital signal processing of computer graphics, in particular to a method for compressing vector geometric models based on the principle of rapid compression. The goal of model compression is achieved through efficient sampling technology, which has important application value in computer graphics application fields such as long-distance transmission, model maintenance, model retrieval and dimensionality reduction.

Background technique

With the in-depth application of computer graphics technology in computer animation, film and television games and other fields, the application of vector geometric models is more and more extensive. Data has become very easy. Users can reconstruct complex geometric models from the acquired high-precision data, and reuse existing geometric models through further processing to improve geometric design efficiency. The core of the remote transmission of vector geometric models lies in the model compression technology, and the compression technology also has high application value in the fields of model storage, retrieval and dimensionality reduction.

Two categories of compression research are currently being used: geometric compression technology (also known as space compression technology) and technology based on signal compression. The common geometric compression technology is the vertex simplified model compression technology, which merges some vertices according to the position of the model vertex coordinates, so as to reduce the number of vertices and complete the purpose of compression. The compression effect has a lot to do with it, but the geometric compression technology changes the topological structure of the model and has compression loss; the signal-based compression method only requires finding a set of suitable coordinate bases to decompose the model in the frequency domain. Compared with the geometric compression technology, it brings great convenience to users. The most classic signal compression method is the low-pass filter compression method. This method expresses the model in multiple resolutions and uses a low-pass filter to filter The high-frequency part of the model is dropped and the low-frequency part is retained, but the process is relatively complicated and lossy compression.

Contents of the invention

In view of the defects or deficiencies of the prior art, the purpose of the present invention is to provide a vector geometric model compression method with good compression speed and recovery effect, so as to increase the online transmission speed of the vector geometric model and reduce its storage space.

For realizing above-mentioned technical task, the present invention takes following technical solution:

In the method of the present invention, the geometric information of the two-dimensional vector geometric model is composed of geometric signal x ₂ and geometric signal y ₂ , and the geometric information of the three-dimensional vector geometric model is composed of geometric signal x ₃ , geometric signal y ₃ and geometric signal z ₃ , the method Specifically, it is realized through the following steps:

(1) For a two-dimensional vector geometric model: its Laplacian operator n ₁ is the total number of vertices of the two-dimensional vector geometric model, and n ₁ is a positive integer;

For 3D vector geometric models: its Laplacian operator Among them: A is the adjacency matrix of the three-dimensional vector geometric model, D is the vertex degree matrix of the three-dimensional vector geometric model, and

Wherein: d _i is the degree of the ith vertex of the three-dimensional vector geometric model, _n2 is the total number of vertices of the three-dimensional vector geometric model, and _n2 and i are all positive integers;

(2) For two-dimensional vector geometric models:

The Laplacian of the two-dimensional vector geometric model Act on the geometric signal x ₂ of the two-dimensional vector geometric model to obtain the vector λ' ₁ :

According to the set threshold ε ₁ , the elements in the vector λ' ₁ whose absolute value is smaller than ε ₁ are assigned 0, and the sparse geometric signal λ ₁ of the geometric signal x ₂ is obtained;

The Laplacian of the two-dimensional vector geometric model Act on the geometric signal y ₂ of the two-dimensional vector geometric model to obtain the vector λ' ₂ :

According to the set threshold ε ₂ , the elements in the vector λ' ₂ whose absolute value is smaller than ε ₂ are assigned a value of 0, and the sparse geometric signal λ ₂ of the geometric signal y ₂ is obtained;

Where: the dimensions of λ ₁ and λ ₂ are both n ₁ , ε ₁ and ε ₂ satisfy: the number of non-zero elements in λ ₁ and λ ₂ is equal;

For 3D vector geometry models:

The Laplacian of the three-dimensional vector geometric model Act on the geometric signal x ₃ of the three-dimensional vector geometric model to obtain the vector λ' ₃ :

According to the set threshold ε ₃ , the elements in the vector λ' ₃ whose absolute value is smaller than ε ₃ are assigned a value of 0, and the sparse geometric signal λ ₃ of the geometric signal x ₃ is obtained;

The Laplacian of the three-dimensional vector geometric model Act on the geometric signal y ₃ of the three-dimensional vector geometric model to obtain the vector λ' ₄ :

According to the set threshold ε ₄ , the elements in the vector λ' ₄ whose absolute value is smaller than ε ₄ are assigned 0 to obtain the sparse geometric signal λ ₄ of the geometric signal y ₃ ;

The Laplacian of the three-dimensional vector geometric model Act on the geometric signal z ₃ of the three-dimensional vector geometric model to obtain the vector λ' ₅ :

According to the set threshold ε ₅ , the elements in the vector λ' ₅ whose absolute value is smaller than ε ₅ are assigned a value of 0, and the sparse geometric signal λ ₅ of the geometric signal z ₃ is obtained;

Among them: the dimensions of λ ₃ , λ ₄ and λ ₅ are all n ₂ , and ε ₃ , ε ₄ and ε ₅ satisfy: the number of non-zero elements in λ ₃ , λ ₄ and λ ₅ is equal;

(3) For a two-dimensional vector geometry model: record the number r ₁ of the non-zero elements in the sparse geometry signal λ ₁ or the sparse geometry signal λ ₂ , r ₁ << n ₁ ;

For a three-dimensional vector geometry model: record the number r ₂ of non-zero elements in the sparse geometry signal λ ₃ , the sparse geometry signal λ ₄ or the sparse geometry signal λ ₅ , r ₂ << n ₂ ;

Step 2, generate a random matrix to sample the geometric information of the vector geometric model:

For 2D vector geometry models:

Generate Random Sampling Matrix

use Sampling the geometric signal x ₂ of the two-dimensional vector geometric model to obtain the sampled signal θ ₁ :

use Sampling the geometric signal y ₂ of the two-dimensional vector geometric model to obtain the sampled signal θ ₂ :

Where: the lengths of θ ₁ and θ ₂ are both 4r ₁ , 4r ₁ << n ₁ , thus completing the compression;

For 3D vector geometry models:

Generate Random Sampling Matrix

use Sampling the geometric signal x ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₃ ,

use Sampling the geometric signal y ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₄ ,

use Sampling the geometric signal z ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₅ , Wherein: θ ₃ , θ ₄ and θ ₅ are all 4r ₂ in length, 4r ₂ << n ₂ ;

And then complete the compression;

Step 3, restore the model:

For 2D vector geometry models:

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₁ of the geometric signal x ₂ is recovered from the sampled signal θ ₁ , h _j represents the jth component of λ ₁ , j=1,2,3,...,n ₁ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₂ of the geometric signal y ₂ is recovered from the sampled signal θ ₂ , h _i represents the i-th component of λ ₂ , i=1,2,3,...,n ₁ :

Then, using the inverse Laplacian of the two-dimensional vector geometric model and λ ₁ to restore the original geometric signal x ₂ :

Inverse Laplacian using two-dimensional vector geometry and λ ₂ to recover the original geometry signal y ₂ :

For 3D vector geometry models:

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₃ of the geometric signal x ₃ is restored by the sampled signal θ ₃ , h _m represents the mth component of λ ₃ , m=1,2,3,...,n ₂ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₄ of the geometric signal y ₃ is restored by the sampled signal θ ₄ , h _p represents the pth component of λ ₄ , p=1,2,3,...,n ₂ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₅ of the geometric signal z ₃ is restored by the sampled signal θ ₅ , h _q represents the qth component of λ ₅ , q=1,2,3,...,n ₂ :

Then, using the inverse Laplacian of the 3D vector geometry model and the sparse geometric signal λ ₃ to restore the original geometric signal x ₃ :

Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ ₄ to restore the original geometric signal y ₃ :

Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ ₅ to restore the original geometric signal z ₃ :

Optionally, the solution process of the formula (1) is to convert it into unconstrained optimization:

In formula (6):

is the 0-space orthonormal basis of Ψ, is a free vector of n ₁ -4r ₁ dimension;

A is n ₁ -dimensional vector, a ₁ a ₂ ... are real numbers;

is a matrix of size n ₁ ×(n ₁ -4r ₁ ), v ₁ v ₂ … Both are n ₁ -4r ₁ -dimensional row vectors;

The solution process of the formula (6) is as follows:

initial value

Step1, calculate k is 0 and a natural number, δ= ^10-3 ;

Step2, when r>0, take As the search direction, I is the identity matrix,

When r≤0, with as the search direction,

Step3, if hour, Otherwise, go to Step1.

According to the present invention, the geometric information of the vector geometric model can be expressed as a sparse signal under the action of its Laplacian operator; a random matrix is used to sample the geometric information, and the compression is completed, and then the 0- Norm fitting function The sparse signal of the original signal is recovered, and then the original signal is recovered to complete the compression and recovery of the model. During the recovery process, the constrained optimization is converted to unconstrained optimization, and a new search direction is designed for solution. The method of the invention not only speeds up the compression speed of the model, but also realizes theoretical lossless compression. In actual operation, the recovery effect can be controlled according to the user's precision requirements.

Description of drawings

Fig. 1 (a) is the original model figure of embodiment 1 two-dimensional vector geometric model; Fig. 1 (b) is the recovery effect diagram after embodiment 1 two-dimensional vector geometric model compression;

Fig. 2 (a) is the original model diagram of the two-dimensional vector geometric model of embodiment 2; Fig. 2 (b) is the recovery effect diagram of the compressed two-dimensional vector geometric model of embodiment 2.

The present invention will be further described below in conjunction with embodiment and accompanying drawing.

detailed description

The basic idea of the present invention is to improve the compression speed by only using one-step compression of sparse matrix sampling, and restore the sparse expression of the original geometric signal by optimizing the 1-norm method, and finally use the inverse Laplacian operator to complete the recovery of the geometric signal . This speeds up compression and is theoretically lossless.

The following are specific examples provided by the inventors. It should be noted that the provided examples are further explanations of the present invention, and the protection scope of the present invention is not limited to the given examples.

Example 1:

This embodiment is to compress the two-dimensional vector geometric model (as shown in Figure 1 (a)), the geometric information of this two-dimensional vector geometric model is made up of geometric signal x ₂ and geometric signal y ₂ , and specific method is as follows:

(1) For the two-dimensional vector geometric model: its Laplacian operator n ₁ is the total number of vertices of the two-dimensional vector geometric model, n ₁ =896;

(2) The Laplacian operator of the two-dimensional vector geometric model The geometric signal x ₂ applied to the two-dimensional vector geometric model results in the vector λ' ₁ :

According to the set threshold ε ₁ , the elements in the vector λ ₁ ' whose absolute value is smaller than ε ₁ are assigned 0, and the sparse geometric signal λ ₁ of the geometric signal x ₂ is obtained;

The Laplacian of the two-dimensional vector geometric model The geometric signal y ₂ applied to the two-dimensional vector geometric model results in the vector λ' ₂ :

According to the set threshold ε ₂ , the elements in the vector λ' ₂ whose absolute value is smaller than ε ₂ are assigned a value of 0, and the sparse geometric signal λ ₂ of the geometric signal y ₂ is obtained;

Wherein: the dimensions of λ ₁ and λ ₂ are both n ₁ , ε ₁ and ε ₂ satisfy: the number of non-zero elements in λ ₁ and λ ₂ is equal, ε ₁ =0.17, ε ₂ =0.15;

(3) Record the number r ₁ of the non-zero elements in the sparse geometric signal λ ₁ or the sparse geometric signal λ ₂ , r ₁ << n ₁ ;

Step 2, generate a random matrix to sample the geometric information of the vector geometric model:

Generate Random Sampling Matrix

use Sampling the geometric signal x ₂ of the two-dimensional vector geometric model to obtain the sampled signal θ ₁ ,

use Sampling the geometric signal y ₂ of the two-dimensional vector geometric model to obtain the sampled signal θ ₂ , Where: the lengths of θ ₁ and θ ₂ are both 4r ₁ , 4r ₁ << n ₁ ; and then the compression is completed;

Step 3, restore the model:

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₁ of the geometric signal x ₂ is recovered from the sampled signal θ ₁ , h _j represents the jth component of λ ₁ , j=1,2,3,...,n ₁ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₂ of the geometric signal y ₂ is recovered from the sampled signal θ ₂ , h _i represents the i-th component of λ ₂ , i=1,2,3,...,n ₁ :

Then, using the inverse Laplacian of the two-dimensional vector geometric model and λ ₁ to restore the original geometric signal x ₂ :

Inverse Laplacian using two-dimensional vector geometry and λ ₂ to recover the original geometry signal y ₂ :

The solution process of the formula (1) is to convert it into unconstrained optimization:

In formula (6):

Using SVD, is the 0-space orthonormal basis of Ψ, is a free vector of n ₁ -4r ₁ dimension;

A is n ₁ -dimensional vector, a ₁ a ₂ ... are real numbers;

is a matrix of size n ₁ ×(n ₁ -4r ₁ ), v ₁ v ₂ … Both are n ₁ -4r ₁ -dimensional row vectors;

Described (formula 6) solving process is as follows:

initial value

Step1, calculate k is 0 and a natural number, δ= ^10-3 ;

Step2, when r>0, take As the search direction, I is the identity matrix,

When r≤0, with as the search direction,

Step3, if hour, Otherwise, go to Step1.

The solution process of (Formula 2) is the same as that of (Formula 1); the recovered model is shown in Figure 1(b).

In the above steps, using or In the process of sampling the geometric signal of the vector geometric model, the random sampling matrix is adjusted according to the degree of sparsification or matrix the number of rows.

Example 2:

This embodiment is to compress the three-dimensional vector geometric model (as shown in Fig. 2(a)), the geometric information of the _three -dimensional vector geometric model is composed of geometric signal _x3 , geometric signal y3 and geometric signal _z3 , the method specifically passes The following steps are implemented:

(1) For the three-dimensional vector geometric model: its Laplacian operator Among them: A is the adjacency matrix of the three-dimensional vector geometric model, D is the vertex degree matrix of the three-dimensional vector geometric model, and

Wherein: d _i is the degree of the ith vertex of the three-dimensional vector geometric model, _n2 is the total number of vertices of the three-dimensional vector geometric model, _n2 and i are all taken as positive integers, _n2 =7609, can be according to the topological structure of the read model , A and D can be determined according to graph connection theory,

(2) The Laplacian operator of the three-dimensional vector geometric model Act on the geometric signal x ₃ of the three-dimensional vector geometric model to obtain the vector λ' ₃ :

According to the set threshold ε ₃ , the elements in the vector λ' ₃ whose absolute value is smaller than ε ₃ are assigned a value of 0, and the sparse geometric signal λ ₃ of the geometric signal x ₃ is obtained;

The Laplacian of the three-dimensional vector geometric model The geometric signal y ₃ acting on the three-dimensional vector geometric model results in the vector λ' ₄ :

According to the set threshold ε ₄ , the elements in the vector λ' ₄ whose absolute value is smaller than ε ₄ are assigned 0 to obtain the sparse geometric signal λ ₄ of the geometric signal y ₃ ;

The Laplacian of the three-dimensional vector geometric model The geometric signal z ₃ acting on the three-dimensional vector geometric model results in the vector λ' ₅ :

According to the set threshold ε ₅ , the elements in the vector λ ₅ ' whose absolute value is smaller than ε ₅ are assigned 0 to obtain the sparse geometric signal λ ₅ of the geometric signal z ₃ ;

Among them: the dimensions of λ ₃ , λ ₄ and λ ₅ are all n ₂ , in order to satisfy that the number of non-zero elements in λ ₃ , λ ₄ and λ ₅ is equal, ε ₃ =0.0073465,ε ₄ =0.04653,ε ₅ = 0.0045784;

(3) For the three-dimensional vector geometric model: record the number r ₂ of non-zero elements in the sparse geometric signal λ ₃ , sparse geometric signal λ ₄ or sparse geometric signal λ ₅ , r ₂ << n ₂ ;

Step 2, generate a random matrix to sample the geometric information of the vector geometric model:

Generate Random Sampling Matrix

use Sampling the geometric signal x ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₃ ,

use Sampling the geometric signal y ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₄ ,

use Sampling the geometric signal z ₃ of the three-dimensional vector geometric model to obtain the sampled signal θ ₅ , Among them: θ ₃ , θ ₄ and θ ₅ are all 4r ₂ in length, 4r ₂ << n ₂ , and then the compression is completed;

Step 3, restore the model:

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₃ of the geometric signal x ₃ is restored by the sampled signal θ ₃ , h _m represents the mth component of λ ₃ , m=1,2,3,...,n ₂ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₄ of the geometric signal y ₃ is restored by the sampled signal θ ₄ , h _p represents the pth component of λ ₄ , p=1,2,3,...,n ₂ :

Using the Fit Function to Minimize Sparse Signals The sparse geometric signal λ ₅ of the geometric signal z ₃ is restored by the sampled signal θ ₅ , h _q represents the qth component of λ ₅ , q=1,2,3,...,n ₂ :

Then, using the inverse Laplacian of the 3D vector geometry model and the sparse geometric signal λ ₃ to restore the original geometric signal x ₃ :

Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ ₄ to restore the original geometric signal y ₃ :

Inverse Laplacian Using 3D Vector Geometric Models and the sparse geometric signal λ ₅ to restore the original geometric signal z ₃ :

The solution process of formula (3), formula (4) and formula (5) is the same as that of formula (1); the restored model is shown in Fig. 2(b).

Claims (2)

1. the geometry of two-dimensional vector geometrical model in compression and the restoration methods of a kind of compressed sensing vector geometrical model, this method Information is by geometry signals x₂With geometry signals y₂Constitute, the geological information of trivector geometrical model is by geometry signals x₃, geometry letter Number y₃With geometry signals z₃Constitute, method is realized especially by the following steps：

(1) for two-dimensional vector geometrical model：Its Laplace operatorn₁For two The summit sum of n dimensional vector n geometrical model, n₁Take positive integer；

For trivector geometrical model：Its Laplace operatorWherein：A is trivector geometry The adjacency matrix of model, D is the Vertex Degree matrix of trivector geometrical model, and

Wherein：d_iFor i-th summit of trivector geometrical model Degree, n₂For the summit sum of trivector geometrical model, n₂Positive integer is taken with i；

(2) for two-dimensional vector geometrical model：

By the Laplace operator of two-dimensional vector geometrical modelIt is applied to the geometry signals x of two-dimensional vector geometrical model₂, obtain To vectorial λ '₁：

<mrow> <msubsup> <mi>&amp;lambda;</mi> <mn>1</mn> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>;</mo> </mrow>

According to the threshold epsilon of setting₁, by vectorial λ '₁In absolute value be less than ε₁Element be entered as 0, obtain geometry signals x₂It is dilute Dredge geometry signals λ₁；

By the Laplace operator of two-dimensional vector geometrical modelIt is applied to the geometry signals y of two-dimensional vector geometrical model₂, Obtain vectorial λ '₂：

<mrow> <msubsup> <mi>&amp;lambda;</mi> <mn>2</mn> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>;</mo> </mrow>

According to the threshold epsilon of setting₂, by vectorial λ '₂In absolute value be less than ε₂Element be entered as 0, obtain geometry signals y₂It is dilute Dredge geometry signals λ₂；

Wherein：λ₁And λ₂Dimension be n₁, ε₁And ε₂Meet：λ₁And λ₂In non-zero element number it is equal；

For trivector geometrical model：

By the Laplace operator of trivector geometrical modelIt is applied to the geometry signals x of trivector geometrical model₃, Obtain vectorial λ '₃：

<mrow> <msubsup> <mi>&amp;lambda;</mi> <mn>3</mn> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>;</mo> </mrow>

According to the threshold epsilon of setting₃, by vectorial λ '₃In absolute value be less than ε₃Element be entered as 0, obtain geometry signals x₃It is dilute Dredge geometry signals λ₃；

By the Laplace operator of trivector geometrical modelIt is applied to the geometry signals y of trivector geometrical model₃, Obtain vectorial λ '₄：

<mrow> <msubsup> <mi>&amp;lambda;</mi> <mn>4</mn> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>;</mo> </mrow>

According to the threshold epsilon of setting₄, by vectorial λ '₄In absolute value be less than ε₄Element be entered as 0, obtain geometry signals y₃It is dilute Dredge geometry signals λ₄；

By the Laplace operator of trivector geometrical modelIt is applied to the geometry signals z of trivector geometrical model₃, Obtain vectorial λ '₅：

<mrow> <msubsup> <mi>&amp;lambda;</mi> <mn>5</mn> <mo>&amp;prime;</mo> </msubsup> <mo>=</mo> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <msub> <mi>z</mi> <mn>3</mn> </msub> <mo>;</mo> </mrow>

According to the threshold epsilon of setting₅, by vectorial λ '₅In absolute value be less than ε₅Element be entered as 0, obtain geometry signals z₃It is dilute Dredge geometry signals λ₅；

Wherein：λ₃、λ₄And λ₅Dimension be n₂, ε₃、ε₄And ε₅Meet：λ₃、λ₄And λ₅In non-zero element number it is equal；

(3) for two-dimensional vector geometrical model：Record sparse geometry signals λ₁Or sparse geometry signals λ₂In non-zero element Number r₁, r₁＜ ＜ n₁；

For trivector geometrical model：Record sparse geometry signals λ₃, sparse geometry signals λ₄Or sparse geometry signals λ₅In The number r of non-zero element₂, r₂＜ ＜ n₂；

(4) generation random matrix is sampled to the geological information of vector geometrical model：

For two-dimensional vector geometrical model：

Generate random sampling matrix

UtilizeTo the geometry signals x of two-dimensional vector geometrical model₂It is sampled, the signal θ after being sampled₁：

UtilizeTo the geometry signals y of two-dimensional vector geometrical model₂It is sampled, the signal θ after being sampled₂：

Wherein：θ₁And θ₂Length is 4r₁, 4r₁＜ ＜ n₁, thus complete compression；

For trivector geometrical model：

Generate random sampling matrix

UtilizeTo the geometry signals x of trivector geometrical model₃It is sampled, the signal θ after being sampled₃,

UtilizeTo the geometry signals y of trivector geometrical model₃It is sampled, the signal θ after being sampled₄,

UtilizeTo the geometry signals z of trivector geometrical model₃It is sampled, the signal θ after being sampled₅,Wherein：θ₃、θ₄And θ₅Length is 4r₂, 4r₂＜ ＜ n₂；

And then complete compression；

(5) recovery of model is carried out：

For two-dimensional vector geometrical model：

Utilize the fitting function for minimizing sparse signalPass through the signal θ after sampling₁Recover Go out geometry signals x₂Sparse geometry signals λ₁, h_jRepresent λ₁J-th of component, j=1,2,3 ..., n₁：

Utilize the fitting function for minimizing sparse signalPass through the signal θ after sampling₂Recover Geometry signals y₂Sparse geometry signals λ₂, h_iRepresent λ₂I-th of component, i=1,2,3 ..., n₁：

Then, the inverse Laplace operator of two-dimensional vector geometrical model is utilizedAnd λ₁Recover original geometry signals x₂：

<mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;lambda;</mi> <mn>1</mn> </msub> <mo>;</mo> </mrow>

Utilize the inverse Laplace operator of two-dimensional vector geometrical modelAnd λ₂Recover original geometry signals y₂：

<mrow> <msub> <mi>y</mi> <mn>2</mn> </msub> <mo>=</mo> <msup> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;lambda;</mi> <mn>2</mn> </msub> <mo>;</mo> </mrow>

For trivector geometrical model：

Utilize the fitting function for minimizing sparse signalPass through the signal θ after sampling₃Recover Geometry signals x₃Sparse geometry signals λ₃, h_mRepresent λ₃M-th of component, m=1,2,3 ..., n₂：

Utilize the fitting function for minimizing sparse signalPass through the signal θ after sampling₄Recover Go out geometry signals y₃Sparse geometry signals λ₄, h_pRepresent λ₄P-th of component, p=1,2,3 ..., n₂：

Utilize the fitting function for minimizing sparse signalPass through the signal θ after sampling₅Recover Go out geometry signals z₃Sparse geometry signals λ₅, h_qRepresent λ₅Q-th of component, q=1,2,3 ..., n₂：

Then, the inverse Laplace operator of trivector geometrical model is utilizedWith sparse geometry signals λ₃Recover original Geometry signals x₃：

<mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;lambda;</mi> <mn>3</mn> </msub> <mo>;</mo> </mrow>

Utilize the inverse Laplace operator of trivector geometrical modelWith sparse geometry signals λ₄Recover original geometry Signal y₃：

<mrow> <msub> <mi>y</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;lambda;</mi> <mn>4</mn> </msub> <mo>;</mo> </mrow>

Utilize the inverse Laplace operator of trivector geometrical modelWith sparse geometry signals λ₅Recover original geometry Signal z₃：

<mrow> <msub> <mi>z</mi> <mn>3</mn> </msub> <mo>=</mo> <msup> <msub> <mi>L</mi> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> <mo>&amp;times;</mo> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>&amp;lambda;</mi> <mn>5</mn> </msub> <mo>.</mo> </mrow>

2. compression and the restoration methods of compressed sensing vector geometrical model as claimed in claim 1, it is characterised in that

(formula 1) solution procedure is to convert thereof into unconstrained optimization：

In formula (6)：

It is Ψ 0 orthogonal space base,It is n₁-4r₁The free vector of dimension；

A is n₁Dimensional vector,It is real number；

For n₁×(n₁-4r₁) size matrix,It is n₁-4r₁The row of dimension Vector；

Formula (6) solution procedure is as follows：

Initial value

Step1, is calculatedK is 0 and natural number, δ=10^-3；

Step2, as r ＞ 0, withAs the direction of search, I is identity matrix,

When r≤0, withAs the direction of search,

Step3, ifWhen,Otherwise it is transferred to Step1。