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CN102801428A - Approximation optimization and signal acquisition reconstruction method for 0-1 sparse cyclic matrix - Google Patents

Approximation optimization and signal acquisition reconstruction method for 0-1 sparse cyclic matrix Download PDF

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CN102801428A
CN102801428A CN2012102857212A CN201210285721A CN102801428A CN 102801428 A CN102801428 A CN 102801428A CN 2012102857212 A CN2012102857212 A CN 2012102857212A CN 201210285721 A CN201210285721 A CN 201210285721A CN 102801428 A CN102801428 A CN 102801428A
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CN102801428B (en
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朱国宾
程涛
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Guangxi University of Science and Technology
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Abstract

The invention discloses an approximation optimization and signal acquisition reconstruction method for a 0-1 sparse cyclic matrix, belongs to the technical field for the design and optimization of measurement matrix in compressive sensing, and provides a posteriori optimizing method which is easy to implement by hardware and can ensure signal reconstruction effect, wherein the 0-1 sparse cyclic matrix is adopted in the measurement stage, and a Gaussian matrix is adopted in the reconstruction stage. The method comprises the following steps: orthonormalizing the row vector and unitizing the column vector of the measurement matrix obtained by the i-1th iteration by the ith iteration; and optimizing the 0-1 sparse cyclic matrix by taking the maximum value of the absolute value of the correlated coefficient between each row and column vector, the convergence stability of each row vector module and the row and column number of each row and column subjected to Gaussian distribution as the criteria. The posteriori optimization of the measured data and measured matrix is completed by solving a transition matrix and an approximate matrix. The method establishes the foundation for the compressive sensing to be practical from the theoretical study.

Description

The near-optimal of the sparse circular matrix of a kind of 0-1 and signals collecting reconstructing method
Technical field
The invention belongs to the compressed sensing technical field, the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 specifically is provided.
Background technology
Design, optimization and the character of measuring matrix in the compressed sensing are the key factors that concerns signal reconstruction.Random matrix (Gauss, Bernoulli Jacob's equal matrix) though signal reconstruction ability and universality are preferably arranged, but owing to be difficult to that hardware is realized people then research character is relatively poor, be easy to hard-wired certainty matrix (Teoplitz, circulation, polynomial matrix etc.).The circular matrix of measuring in the matrix is easy to the hardware realization, and can adopt the DFT rapid solving; The 0-1 sparse matrix not only is easy to hardware and realizes but also the little fast operation of required memory space.Therefore, the sparse circular matrix of 0-1 that both is mixed (be called for short: sparse circular matrix) realize greatly by the design of simplified measurement matrix and hardware.But the ranks irrelevance of sparse matrix is relatively poor, and adopts sparse matrix can cause each element in the measured value can only comprise a part of information of signal, and each element no longer is in par, the anti-packet loss ability variation.
In any case current measurement matrix optimal design, constant is all to adopt same measurement matrix in measurement and two stages of reconstruct.If be easy to the hard-wired certainty matrix of not optimizing, just can't guarantee the reconstruct effect of signal in reconstruction stage in the measuring phases employing; If adopt the optimization matrix, just can't guarantee the easy realization of measuring phases measurement matrix in reconstruction stage.Adopt the certainty matrix that hardware is realized easily, character is relatively poor in measuring phases, adopt in reconstruction stage that hardware is difficult for realizing, character preferably Gauss's matrix be measurement matrix design and the data processing general layout that people expect.
Summary of the invention
The present invention is low and measure the problem of matrix design in order to solve sparse circular matrix signal reconstruction ability, and the spy provides the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1.
The present invention is achieved through following proposal: the near-optimal of the sparse circular matrix of a kind of 0-1 and signals collecting reconstructing method, and the process of said method is:
Step 1: generate the sparse circular matrix of 0-1
Figure 513714DEST_PATH_IMAGE001
; With optimizing matrix season; Wherein
Figure 148275DEST_PATH_IMAGE003
,
Figure 481167DEST_PATH_IMAGE004
.The initial row vector of
Figure 295539DEST_PATH_IMAGE001
is 1 the 0-1 loose line vector of ( ) the individual random distribution that comprises
Figure 55685DEST_PATH_IMAGE005
, and each row vector all is the result of previous row vector each element moves to right
Figure 459301DEST_PATH_IMAGE007
successively ( and satisfy
Figure 59227DEST_PATH_IMAGE009
) position.The complementation of
Figure 748310DEST_PATH_IMAGE010
expression
Figure 485322DEST_PATH_IMAGE011
,
Figure 946390DEST_PATH_IMAGE012
,
Figure 313918DEST_PATH_IMAGE013
,
Figure 555543DEST_PATH_IMAGE005
and they are natural numbers;
Step 2: check
Figure 349504DEST_PATH_IMAGE001
In whether identical row or row are arranged, if return execution in step one, otherwise set iterations iInitial value be 0, set iteration error
Figure 684670DEST_PATH_IMAGE014
Step three: The Halk - Bella (Jarque-Bera) test calculation
Figure 351275DEST_PATH_IMAGE015
each column and each row Gaussian distribution of the number of rows
Figure 430089DEST_PATH_IMAGE016
and columns
Figure 865750DEST_PATH_IMAGE017
; calculation
Figure 637397DEST_PATH_IMAGE015
The correlation coefficient between the column vectors, remove its maximum absolute value
Figure 525718DEST_PATH_IMAGE018
; calculated for each row The correlation coefficient between vectors, remove its maximum absolute value
Figure 408224DEST_PATH_IMAGE019
; calculation
Figure 633144DEST_PATH_IMAGE015
each row vector mode, remove its maximum
Figure 575692DEST_PATH_IMAGE020
and the minimum value
Figure 951310DEST_PATH_IMAGE021
;
Step 4: quadrature standardization
Figure 371927DEST_PATH_IMAGE015
Each row vector, each column vector of unitization makes then i= i + 1, matrix just can be optimized
Figure 782180DEST_PATH_IMAGE022
Calculate transition matrix
Figure 895629DEST_PATH_IMAGE023
simultaneously; Approximate matrix
Figure 758543DEST_PATH_IMAGE024
, and make
Figure 920534DEST_PATH_IMAGE025
;
Step 5: and
Figure 204065DEST_PATH_IMAGE027
and
Figure 616592DEST_PATH_IMAGE028
and and
Figure 763856DEST_PATH_IMAGE030
and
Figure 911720DEST_PATH_IMAGE031
that judge the optimization matrix; If execution in step six, otherwise return execution in step three;
Step 6: obtain and optimize matrix
Figure 811543DEST_PATH_IMAGE032
, transition matrix
Figure 315337DEST_PATH_IMAGE033
and approximate matrix
Figure 617005DEST_PATH_IMAGE034
;
Step 7: through sparse circular matrix with the conventional method image data shown in the constraint equation in the following formula:
Figure 915262DEST_PATH_IMAGE035
Step 8: the measurement data to conventional method collects is passed through transition matrix optimization:
Figure 302381DEST_PATH_IMAGE036
Step 9: through traditional signal reconstruction algorithm reconstruction signal:
Figure 609866DEST_PATH_IMAGE037
。
The present invention has realized the optimization of the sparse circular matrix of 0-1 with the iterative cycles computing of the unitization of quadrature standardization of each row vector to
Figure 500462DEST_PATH_IMAGE001
and each column vector.Then through the traditional method of optimizing the transition matrix approximation collected measurement data
Figure 235199DEST_PATH_IMAGE038
and the measurement matrix
Figure 844035DEST_PATH_IMAGE001
.Optimize matrix and approximate matrix and both had the universality of Gauss's matrix, improved the signal reconstruction ability again.Method of the present invention has not only been simplified the hardware designs and the realization of measuring matrix; And improved the signal reconstruction effect, have a wide range of applications in fields such as the image processing of compressed sensing, video analysis, radar remote sensing, communication code, DABs.
Description of drawings
Fig. 1 is the near-optimal of the sparse circular matrix of embodiment one described a kind of 0-1 and the flow chart of signals collecting reconstructing method; Fig. 2 uses embodiment 128 * 256 sparse circular matrix is handled optimization matrix and the capable module maximum of approximate matrix and the graph of a relation of iterations that obtains; Fig. 3 uses embodiment 128 * 256 sparse circular matrix is handled optimization matrix and the ranks coefficient correlation of approximate matrix and the graph of a relation of iterations that obtains; Fig. 4 uses embodiment optimization 128 * 256 sparse circular matrix to be handled optimization matrix and the ranks number that meets Ha Erke-Bei La (Jarque-Bera) check of approximate matrix and the graph of a relation of iterations that obtains; Fig. 5 uses embodiment 128 * 256 sparse circular matrix is handled the reconstruct probability of the optimization matrix, approximate matrix and the sparse circular matrix that obtain and the graph of a relation of degree of rarefication.
Embodiment
Embodiment one: specify this execution mode according to Figure of description 1.The near-optimal of the sparse circular matrix of a kind of 0-1 and signals collecting reconstructing method, the process of said method is:
Step 1: generate the sparse circular matrix of 0-1
Figure 955211DEST_PATH_IMAGE001
; With optimizing matrix season; Wherein
Figure 871531DEST_PATH_IMAGE003
, .The initial row vector of
Figure 616950DEST_PATH_IMAGE001
is 1 the 0-1 loose line vector of (
Figure 555749DEST_PATH_IMAGE006
) the individual random distribution that comprises
Figure 482138DEST_PATH_IMAGE005
, and each row vector all is the result of previous row vector each element moves to right
Figure 139177DEST_PATH_IMAGE007
successively (
Figure 326576DEST_PATH_IMAGE008
and satisfy ) position.The complementation of
Figure 559291DEST_PATH_IMAGE010
expression
Figure 630015DEST_PATH_IMAGE011
,
Figure 621105DEST_PATH_IMAGE012
,
Figure 398568DEST_PATH_IMAGE013
,
Figure 879228DEST_PATH_IMAGE005
and
Figure 109352DEST_PATH_IMAGE007
they are natural numbers;
Step 2: check
Figure 966450DEST_PATH_IMAGE001
In whether identical row or row are arranged, if return execution in step one, otherwise set iterations iInitial value be 0, set iteration error
Figure 598419DEST_PATH_IMAGE014
Step three: The Halk - Bella (Jarque-Bera) test calculation
Figure 249981DEST_PATH_IMAGE015
each column and each row Gaussian distribution of the number of rows
Figure 232980DEST_PATH_IMAGE016
and columns
Figure 628189DEST_PATH_IMAGE017
; calculation
Figure 111736DEST_PATH_IMAGE015
The correlation coefficient between the column vectors, remove its maximum absolute value
Figure 199777DEST_PATH_IMAGE018
; calculated for each row The correlation coefficient between vectors, remove its maximum absolute value
Figure 404494DEST_PATH_IMAGE019
; calculation
Figure 603394DEST_PATH_IMAGE015
each row vector mode, remove its maximum
Figure 944377DEST_PATH_IMAGE020
and the minimum value
Figure 203320DEST_PATH_IMAGE021
;
Step 4: quadrature standardization
Figure 895332DEST_PATH_IMAGE015
Each row vector, each column vector of unitization makes then i= i + 1, matrix just can be optimized
Figure 570027DEST_PATH_IMAGE022
Calculate transition matrix simultaneously; Approximate matrix
Figure 460940DEST_PATH_IMAGE024
, and make
Figure 436986DEST_PATH_IMAGE025
;
Step 5: and and
Figure 831692DEST_PATH_IMAGE028
and
Figure 560614DEST_PATH_IMAGE029
and
Figure 574182DEST_PATH_IMAGE030
and
Figure 9842DEST_PATH_IMAGE031
that judge the optimization matrix; If execution in step six, otherwise return execution in step three;
Step 6: obtain and optimize matrix
Figure 515910DEST_PATH_IMAGE032
, transition matrix and approximate matrix
Figure 489999DEST_PATH_IMAGE034
;
Step 7: through sparse circular matrix with the conventional method image data shown in the constraint equation in the following formula:
Figure 842483DEST_PATH_IMAGE035
Step 8: the measurement data to conventional method collects is passed through transition matrix optimization:
Figure 722715DEST_PATH_IMAGE036
Step 9: through traditional signal reconstruction algorithm reconstruction signal:
。
Embodiment two: this embodiment is to the further specifying of the described a kind of Gauss's matrix optimizing method based on compressed sensing of embodiment one, and sets iteration error in the step 2 Err1 does
Figure 518949DEST_PATH_IMAGE039
, Err2 do
Figure 866885DEST_PATH_IMAGE039
, Err3 do
Figure 649509DEST_PATH_IMAGE039
Embodiment three: this embodiment is further specifying the near-optimal of the sparse circular matrix of embodiment one described a kind of 0-1 and signals collecting reconstructing method; Each row vector of the described quadrature standardization of step 4
Figure 574740DEST_PATH_IMAGE015
; The detailed process of each column vector of unitization is then: at first to
Figure 736731DEST_PATH_IMAGE015
each the row vectorial orthogonalization; Each row of unitization is vectorial then, last each column vector of unitization.
Embodiment four: specify this execution mode below in conjunction with Fig. 2-Fig. 5.This execution mode is to adopt the gaussian signal of different degree of rarefications to be applied to optimize matrix, approximate matrix and sparse circular matrix respectively, relatively the reconstruct probability after its each 500 times experiments.Band " " mark is the maximum curve among Fig. 2; What be with "
Figure 223524DEST_PATH_IMAGE041
" mark is the minimum value curve; What be with "
Figure 636051DEST_PATH_IMAGE042
" mark is reference line.Band "
Figure 336153DEST_PATH_IMAGE042
" mark is the row curves among Fig. 3-Fig. 4; What be with "
Figure 517736DEST_PATH_IMAGE040
" mark is the row curve.(a) representing optimized matrix among Fig. 2-Fig. 4; (b) represent approximate matrix.The curve of band "
Figure 910671DEST_PATH_IMAGE043
;
Figure 748177DEST_PATH_IMAGE044
;
Figure 251971DEST_PATH_IMAGE045
" mark is respectively to adopt to optimize matrix among Fig. 5; The reconstruct probability curve of approximate matrix and sparse circular matrix.
Experimental result such as Fig. 2-shown in Figure 5.Visible by Fig. 2, the extreme difference that sparse circular matrix is optimized optimization matrix and the corresponding with it vectorial mould of each row of approximate matrix in the iterative process convergence that constantly diminishes; Visible by Fig. 3, the convergence that constantly diminishes of the maximum of optimizing matrix and each ranks coefficient correlation absolute value of approximate matrix; Visible by Fig. 4, sparse circular matrix optimize optimize in the iterative process matrix and with it the ranks number of corresponding each ranks Gaussian distributed of approximate matrix become many rapidly in the iteration later stage, Gaussian distributed nearly all; Visible by Fig. 5, the reconstruct probability curve of optimizing matrix and approximate matrix is positioned at the right side of the curve of sparse circular matrix fully very near similar.

Claims (7)

1. the near-optimal of the sparse circular matrix of 0-1 and signals collecting reconstructing method, it is characterized in that: the process of said method is:
Step 1: generate the sparse circular matrix of 0-1
Figure 562352DEST_PATH_IMAGE001
; With optimizing matrix
Figure 188505DEST_PATH_IMAGE002
season; Wherein
Figure 513307DEST_PATH_IMAGE003
;
Figure 555213DEST_PATH_IMAGE004
; The initial row vector of
Figure 649071DEST_PATH_IMAGE001
is 1 the 0-1 loose line vector of (
Figure 992644DEST_PATH_IMAGE006
) the individual random distribution that comprises ; Each row vector all is the result of previous row vector each element moves to right
Figure 166137DEST_PATH_IMAGE007
successively ( and satisfy ) position; The complementation of expression
Figure 562297DEST_PATH_IMAGE011
,
Figure 627818DEST_PATH_IMAGE012
,
Figure 766675DEST_PATH_IMAGE013
,
Figure 287786DEST_PATH_IMAGE005
and
Figure 475185DEST_PATH_IMAGE007
they are natural numbers;
Step 2: check
Figure 460458DEST_PATH_IMAGE001
In whether identical row or row are arranged, if return execution in step one, otherwise set iterations iInitial value be 0, set iteration error
Figure 707900DEST_PATH_IMAGE014
Step three: The Halk - Bella (Jarque-Bera) test calculation
Figure 778624DEST_PATH_IMAGE015
each column and each row Gaussian distribution of the number of rows
Figure 769714DEST_PATH_IMAGE016
and columns
Figure 609494DEST_PATH_IMAGE017
; calculation
Figure 27837DEST_PATH_IMAGE015
The correlation coefficient between the column vectors, remove its maximum absolute value
Figure 320278DEST_PATH_IMAGE018
; calculating the correlation coefficient between the row vector, remove the absolute value of the maximum
Figure 115059DEST_PATH_IMAGE019
; calculation
Figure 809345DEST_PATH_IMAGE015
each row vector, we remove its maximum and the minimum value ;
Step 4: quadrature standardization
Figure 773869DEST_PATH_IMAGE015
Each row vector, each column vector of unitization makes then i= i+ 1, matrix just can be optimized
Figure 322662DEST_PATH_IMAGE022
Calculate transition matrix simultaneously
Figure 348387DEST_PATH_IMAGE023
, approximate matrix
Figure 553103DEST_PATH_IMAGE024
, and order
Figure 486424DEST_PATH_IMAGE025
Step 5:
Figure 92986DEST_PATH_IMAGE026
and
Figure 351929DEST_PATH_IMAGE027
and
Figure 778362DEST_PATH_IMAGE028
and
Figure 780953DEST_PATH_IMAGE029
and
Figure 242022DEST_PATH_IMAGE030
and
Figure 406287DEST_PATH_IMAGE031
that judge the optimization matrix; If execution in step six, otherwise return execution in step three;
Step 6: obtain and optimize matrix , transition matrix
Figure 126298DEST_PATH_IMAGE033
and approximate matrix
Figure 441873DEST_PATH_IMAGE034
;
Step 7: through sparse circular matrix with the conventional method image data shown in the constraint equation in the following formula:
Figure 42618DEST_PATH_IMAGE035
Step 8: the measurement data to conventional method collects is passed through transition matrix optimization:
Step 9: through traditional signal reconstruction algorithm reconstruction signal:
Figure 785108DEST_PATH_IMAGE037
。
2. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1; It is characterized in that step 1 described
Figure 220768DEST_PATH_IMAGE006
,
Figure 726836DEST_PATH_IMAGE008
and satisfy
Figure 880737DEST_PATH_IMAGE038
.
3. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1 is characterized in that computing formula
Figure 700925DEST_PATH_IMAGE023
and of transition matrix and approximate matrix in the described iterative process of step 4.
4. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1; It is characterized in that the described Rule of judgment of step 5
Figure 605744DEST_PATH_IMAGE026
;
Figure 43679DEST_PATH_IMAGE027
;
Figure 667558DEST_PATH_IMAGE030
,
Figure 874549DEST_PATH_IMAGE031
.
5. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1 is characterized in that computing formula and
Figure 850912DEST_PATH_IMAGE034
of described final transition matrix of step 6 and approximate matrix.
6. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1 is characterized in that the computing formula
Figure 9974DEST_PATH_IMAGE036
that the described measurement data that conventional method is collected of step 8 is optimized with transition matrix.
7. the near-optimal and the signals collecting reconstructing method of the sparse circular matrix of a kind of 0-1 according to claim 1, it is characterized in that step 9 described be the signal reconstruction model of core with
Figure 337050DEST_PATH_IMAGE039
and
Figure 559084DEST_PATH_IMAGE040
.
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