Background technology
The design of measurement matrix and the optimization of restructuring matrix are the key factors that concerns signal reconstruction in the compressed sensing.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, multinomial, 0-1 sparse matrix etc.).The 0-1 sparse matrix not only is easy to hardware and realizes but also the little fast operation of required memory space.But the ranks irrelevance of 0-1 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.And the ranks irrelevance of the restructuring matrix of 0-1 sparse matrix and sparse transform-based composition is also very poor, a little less than the signal reconstruction ability.
Mostly the optimization of current restructuring matrix be under the fixed situation of sparse transform-based, minimizes or on average turn to the objective optimization restructuring matrix with each row correlation, thereby obtain the measurement matrix that adapts with the restructuring matrix that can improve the signal reconstruction effect.But such measurement matrix design method is not only operated inconvenience, method complexity, and is unfavorable for hardware designs.At the measurement matrix that measuring phases adopts hardware to realize easily, character is relatively poor, adopt in reconstruction stage that hardware is difficult for realizing, character preferably Gauss's matrix be that measurement matrix design and the data that people expect are handled general layout.All to optimize restructuring matrix under the fixed situation also be the design processing scheme that people expect measuring matrix and sparse transform-based.
Summary of the invention
The present invention is low and measure the problem of matrix design in order to solve restructuring matrix signal reconstruction ability, and the spy provides a kind of two dimensional compaction perception image collection and reconstructing method based on DCT and DFT.
The present invention is achieved by following proposal: a kind of two dimensional compaction perception image collection and reconstructing method based on DCT and DFT, and the process of described method is:
Step 1: generate the 0-1 sparse matrix
,
,
Each row vector comprise that to be no less than 2 values be 1 element,
Each column vector comprise that to be no less than 1 value be 1 element.
With
It all is natural number.Generate restructuring matrix
, with optimizing matrix season
,
It is sparse transform-based.
Can be DCT matrix (discrete cosine matrix), also can be DFT matrix (discrete fourier matrix);
Step 2: set iterations
iInitial value be 0, set iteration error
Step 3: check calculating respectively with Ha Erke-Bei La (Jarque-Bera)
Each row and the real part of each row and the line number (line number of real part Gaussian distributed of imaginary part Gaussian distributed
, the line number of imaginary part Gaussian distributed
) and the columns (columns of real part Gaussian distributed
, the columns of imaginary part Gaussian distributed
), the DFT matrix is that complex matrix possesses real part and imaginary part simultaneously, the DCT matrix is that real number matrix does not have imaginary part; Calculate
Coefficient correlation between each column vector, the maximum of taking out its absolute value
Calculate the coefficient correlation between each row vector, the maximum of taking out its absolute value
Calculate
The mould of each row vector takes out its maximum
And minimum value
Step 4: quadrature standardization
Each row vector, each column vector of unitization makes then
i=
i+ 1, matrix just can be optimized
Calculate transition matrix simultaneously
, approximate matrix
, and order
Step 5: judgement optimization matrix
With
With
With
With
With
With ((
With
) or (
With
)), if execution in step six, otherwise return execution in step three;
Step 6: obtain the optimization matrix
, transition matrix
And approximate matrix
Step 7: gather by row with the method shown in the following formula by sparse matrix
Measurement data
,
(for the sparse conversion of one dimension
Or
),
:
Step 8: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to optimize measurement data with transition matrix to the measurement data that collects:
The sparse conversion of one dimension:
The sparse conversion of two dimension:
Step 9: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to find the solution by following formula to the measurement data of optimizing
With
, wherein
Be
Column vector,
Be
Column vector,
,
:
The sparse conversion of one dimension:
The sparse conversion of two dimension:
Step 10: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to pass through respectively
With
Restoring signal
The present invention is with right
The iterative cycles computing of the quadrature standardization of each row vector and the unitization of each column vector has realized the optimization of restructuring matrix.The measurement data that collects by transition matrix near-optimal conventional method then
And restructuring matrix
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 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, digital audio.
Embodiment
Embodiment one: specify present embodiment according to Figure of description 1.A kind of two dimensional compaction perception image collection and reconstructing method based on DCT and DFT, the process of described method is:
Step 1: generate the 0-1 sparse matrix
,
,
Each row vector comprise that to be no less than 2 values be 1 element,
Each column vector comprise that to be no less than 1 value be 1 element.
With
It all is natural number.Generate restructuring matrix
, with optimizing matrix season
,
It is sparse transform-based.
Can be DCT matrix (discrete cosine matrix), also can be DFT matrix (discrete fourier matrix);
Step 2: set iterations
iInitial value be 0, set iteration error
Step 3: check calculating respectively with Ha Erke-Bei La (Jarque-Bera)
Each row and the real part of each row and the line number (line number of real part Gaussian distributed of imaginary part Gaussian distributed
, the line number of imaginary part Gaussian distributed
) and the columns (columns of real part Gaussian distributed
, the columns of imaginary part Gaussian distributed
), the DFT matrix is that complex matrix possesses real part and imaginary part simultaneously, the DCT matrix is that real number matrix does not have imaginary part; Calculate
Coefficient correlation between each column vector, the maximum of taking out its absolute value
Calculate the coefficient correlation between each row vector, the maximum of taking out its absolute value
Calculate
The mould of each row vector takes out its maximum
And minimum value
Step 4: quadrature standardization
Each row vector, each column vector of unitization makes then
i=
i+ 1, matrix just can be optimized
Calculate transition matrix simultaneously
, approximate matrix
, and order
Step 5: judgement optimization matrix
With
With
With
With
With
With ((
With
) or (
With
)), if execution in step six, otherwise return execution in step three;
Step 6: obtain the optimization matrix
, transition matrix
And approximate matrix
Step 7: gather by row with the method shown in the following formula by sparse matrix
Measurement data
,
(for the sparse conversion of one dimension
Or
),
:
Step 8: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to optimize measurement data with transition matrix to the measurement data that collects:
The sparse conversion of one dimension:
The sparse conversion of two dimension:
Step 9: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to find the solution by following formula to the measurement data of optimizing
With
, wherein
Be
Column vector,
Be
Column vector,
,
:
The sparse conversion of one dimension:
The sparse conversion of two dimension:
Step 10: divide the sparse conversion of one dimension and two kinds of situations of two-dimentional sparse conversion to pass through respectively
With
Restoring signal
Embodiment two: this embodiment is described a kind of based on the two dimensional compaction perception image collection of DCT and DFT and further specifying of reconstructing method to embodiment one, sets iteration error in the step 2
Err1 is
,
Err2 are
,
Err3 are
Embodiment three: this embodiment is described a kind of based on the two dimensional compaction perception image collection of DCT and DFT and further specifying of reconstructing method to embodiment one, the described quadrature standardization of step 4
Each row vector, the detailed process of each column vector of unitization is then: at first right
The vectorial orthogonalization of each row, each row vector, each column vector of unitization at last of unitization then.
Embodiment four: specify present embodiment below in conjunction with Fig. 2-Fig. 7.Present embodiment is to adopt gaussian signal and the 0-1 signal of different degree of rarefications to be applied to optimize matrix, approximate matrix and restructuring matrix respectively, relatively the reconstruct probability after each the 500 times experiments.And adopt lena figure to verify the reconstruct effect of optimizing matrix, approximate matrix and restructuring matrix respectively with 1D-DCT, 2D-DCT, 1D-DFT and 2D-DFT.Band among Fig. 2 "
" mark be the maximum curve; Band "
" mark be the minimum value curve; Band "
" mark be reference line.Band among Fig. 3-Fig. 4 "
" mark be the row curves; Band "
" mark be the row curve.(a) represents the optimization matrix that sparse matrix is the DFT matrix among Fig. 2-Fig. 3; (b) represent the approximate matrix that sparse matrix is the DFT matrix; (c) represent the optimization matrix that sparse matrix is the DCT matrix; (d) represent the approximate matrix that sparse matrix is the DCT matrix.(a) represents the real part that sparse matrix is the optimization matrix of DFT matrix among Fig. 4; (b) represent the real part that sparse matrix is the approximate matrix of DFT matrix; (c) represent the imaginary part that sparse matrix is the optimization matrix of DCT matrix; (d) represent the imaginary part that sparse matrix is the approximate matrix of DCT matrix; (e) represent the optimization matrix that sparse matrix is the DFT matrix; (f) represent the approximate matrix that sparse matrix is the DFT matrix.Band among Fig. 5 "
,
,
" curve of mark is respectively to adopt the reconstruct probability curve of optimizing matrix, approximate matrix and restructuring matrix.Among Fig. 5 (a) to represent sparse matrix be the DFT matrix, signal is Gauss's sparse signal; (b) representing sparse matrix is the DFT matrix, and signal is the 0-1 sparse signal; (c) representing sparse matrix is the DCT matrix, and signal is Gauss's sparse signal; (d) representing sparse matrix is the DCT matrix, and signal is the 0-1 sparse signal.Being respectively the Lena figure of 1D-DCT, 2D-DCT, 1D-DFT and 2D-DFT from top to bottom among Fig. 6, is respectively restructuring matrix, the reconstruct image of optimizing matrix and approximate matrix from left to right, and rightmost is original Lena figure.Fig. 7 is signal to noise ratio (snr) and the Y-PSNR (PSNR) of each figure among Fig. 6.
Experimental result such as Fig. 2-shown in Figure 7.As seen from Figure 2, restructuring matrix is optimized optimization matrix and the extreme difference of the corresponding vectorial mould of each row of the approximate matrix convergence that constantly diminishes with it in the iterative process; As seen from Figure 3, the maximum of optimizing matrix and each ranks coefficient correlation absolute value of the approximate matrix convergence that constantly diminishes; As seen from Figure 4, restructuring 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; As seen from Figure 5, optimize the reconstruct probability curve of matrix and approximate matrix very near similar, be positioned at the right side of the curve of restructuring matrix fully; As seen from Figure 7, signal to noise ratio and the Y-PSNR of optimization matrix and approximate matrix are very approaching.