CN103142228A - Compressed sensing magnetic resonance fast imaging method - Google Patents

Abstract

A kind of compressed sensing magnetic resonance fast imaging method includes the following steps: to be sampled to obtain the space K undersampled signal y to the space K using quadruple sampling module; Fourier's undersampled signal of wavelet sub-band can be obtained by signal y, Fourier's undersampled signal of wavelet sub-band includes low frequency sub-band Fourier's undersampled signal uLL and high-frequency sub-band Fourier's undersampled signal un; Low frequency sub-band Fourier's undersampled signal uLL is rebuild to obtain low frequency sub-band Mathematical model is constructed, and high-frequency sub-band Fourier's undersampled signal un is rebuild to obtain high-frequency sub-band according to mathematical model N ∈ { HL, LH, HH }; To low frequency sub-band And high-frequency sub-band N ∈ { HL, LH, HH } does wavelet inverse transformation and obtains reconstruction image. Above-mentioned compressed sensing magnetic resonance fast imaging method improves signal sparsity by dictionary learning, and utilize the relationship of wavelet sub-band and the space K, it is smaller and can solve the problems, such as by parallel computation that image reconstruction problem in traditional compressed sensing magnetic resonance is optimized for a calculation scale, so as to rapidly reconstruct final image.

Description

Compressed sensing magnetic resonance fast imaging method

Technical field

The present invention relates to the magnetic resonance fast imaging method, relate in particular to a kind of compressed sensing magnetic resonance fast imaging method.

Background technology

Nuclear magnetic resonance (MRI, Magnetic Resonance Imaging) be to obtain the pictorial information of tissue by magnetic field, the molecule environment that shows the histiocyte core that it is inner intrinsic with pixel is for medical science provides the clinical diagnosis instrument more accurately of observing.It is collection physics, superconductor technology, strong magnetic, large-signal conveying and radiation, the reception of weak signal and processing, and many technology such as Digital Signal Processing, real time computer control, image reconstruction are in the integrated technology of one.

Compressive sensing theory is successfully applied in nuclear magnetic resonance.Compressive sensing theory utilizes the sparse property of signal under certain base, has realized a kind ofly under incoherent sampling matrix, only needs the method that a small amount of sampling (far fewer than the theoretical required sampling of nyquist sampling) can high-quality reconstruct primary signal.And if signal is more sparse under certain base, so needed sampling quantity is fewer.Traditional magnetic resonance fast imaging based on compressive sensing theory is mainly to utilize image in the sparse property of wavelet field, comes reconstructed image by K space undersampled signal.The reconstructed image formula is as follows:

$\hat{x}=\underset{x}{\arg \min }{\left|\right|y-F_{u}x\left|\right|}_2^2+\mathrm{\λ}{\left|\right|\mathrm{Wx}\left|\right|}_1$

Wherein, x is the unknown image that needs reconstruct, and y is known k space undersampled signal, F _uFor Fourier owes sampling matrix, W is wavelet transform matrix, and λ is a suitable constant, output Be desired reconstructed image.After completing magnetic resonance imaging and obtaining y, the process of solution formula (1) is exactly the restructuring procedure of magnetic resonance image (MRI).

Theoretical according to traditional compressed sensing nuclear magnetic resonance, although can realize the reconstruct of magnetic resonance image (MRI),, not taking full advantage of the sparse property of signal, the quantity of sampling is larger, and the magnetic resonance imaging time is long.

Summary of the invention

Based on this, being necessary provides a kind of compressed sensing magnetic resonance fast imaging that can shorten the magnetic resonance imaging time for the defective of above-mentioned compressed sensing magnetic resonance fast imaging existence.

A kind of compressed sensing magnetic resonance fast imaging method comprises the steps:

Adopt the quadruple sampling module to be sampled in the K space and obtain K space undersampled signal y;

Owe to adopt signal y to obtain wavelet sub-band Fourier undersampled signal based on described, Fourier's undersampled signal of described wavelet sub-band comprises low frequency sub-band Fourier undersampled signal u _LLAnd high-frequency sub-band Fourier undersampled signal u _n, wherein, n={HL, LH, HH};

Adopt parallel imaging method to described low frequency sub-band Fourier undersampled signal u _LLRebuild and obtain low frequency sub-band

Build mathematical model, and according to described mathematical model to described high-frequency sub-band Fourier's undersampled signal u _nRebuild and obtain high-frequency sub-band

N ∈ HL, and LH, HH}, wherein, described mathematical model is:

${\hat{w}}_n=\underset{w_n,D,\mathrm{\Γ}}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^2$

$s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

Wherein, u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ _sF, Φ _sThe template of sampling is owed in expression at random, and F is Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Be the degree of rarefication of required coefficient, Γ is sparse coefficient set, Γ={ α _{N, 1}, α _{N, 2}..., α _{N, i}..., α _{N, m};

To above-mentioned low frequency sub-band And high-frequency sub-band

{ HH} does wavelet inverse transformation to n ∈ for HL, LH.

In the present embodiment, described quadruple sampling module obtains by following step: the template Φ that samples is owed in definition at random _s, described the first sampling module is of a size of 1/4 of K space; The described template of owing at random to sample is carried out quadruple copy, obtain the second sampling template, described the second sampling template is consistent with the K space size; Replace described the second middle stochastical sampling of sampling template with the periodic sampling template, obtain the 3rd sampling template; Add full sampling template at the center of described the 3rd sampling template and obtain the quadruple sampling module.

In the present embodiment, wherein, build mathematical model, and according to described mathematical model to described high-frequency sub-band Fourier's undersampled signal u _nRebuild and obtain high-frequency sub-band

{ HH} comprises the steps: to adopt the duty Optimization method to calculate described mathematical model, and obtains described high-frequency sub-band n ∈ for HL, LH N ∈ { HL, LH, HH}.

In the present embodiment, wherein, adopt the duty Optimization method to calculate described mathematical model, and obtain described high-frequency sub-band

{ HH} comprises the steps: n ∈ for HL, LH

Step S41: objective definition function:

N ∈ LH, and HL, HH},

Wherein, w _nBe the subband that needs reconstruct of the unknown, F _uFor Fourier owes sampling matrix: F _u=Φ _sF,

The expression inverse Fourier transform, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S42: definition iteration K step Known;

Step S43: based on above-mentioned

Described mathematical model is converted to the first mathematical model, and wherein said the first mathematical model is:

$\{D_n^k,{\mathrm{\Γ}}^k\}=\underset{D,\mathrm{\Γ}}{\min }\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n^{k-1}-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^{2}s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\},$

In above-mentioned formula, R _iFor image block extracts operator, w _nBe the subband that needs reconstruct of the unknown, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S44: use the K-SVD computational methods to find the solution described the first mathematical model, and obtain

And Γ ^K

Step S45: based on above-mentioned

And Γ ^K, change mathematical model into second mathematical model, and obtain

Wherein said the second mathematical model is:

$w_n^{\left(k\right)}=\underset{w_n}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n^{\left(k\right)}{\mathrm{\α}}_{n,i}^{\left(k\right)}\left|\right|}_2^2,n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

In above-mentioned formula, u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ _sF, F are Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S46: judge that above-mentioned iterations K is whether more than or equal to the iterations of actual set, if "Yes" is carried out next step; If "No" is returned to step S43;

Step S47: export described high-frequency sub-band

Above-mentioned compressed sensing magnetic resonance fast imaging method improves the sparse property of signal by dictionary learning, and utilize the relation in wavelet sub-band and K space, image reconstruction problem in traditional compressed sensing magnetic resonance is optimized for the problem that calculation scale is less and can solve by parallel computation, thereby can reconstructs rapidly final image.

Description of drawings

The flow chart of steps of the compressed sensing magnetic resonance fast imaging method that Fig. 1 provides for the embodiment of the present invention.

The step schematic diagram of the quadruple sampling module that Fig. 2 provides for the embodiment of the present invention.

The flow chart of steps of the quadruple sampling module that Fig. 3 provides for the embodiment of the present invention.

The specific embodiment

In order to make purpose of the present invention, technical scheme and advantage clearer, below in conjunction with drawings and the specific embodiments, the present invention is further elaborated.Should be appreciated that specific embodiment described herein only in order to explain the present invention, is not intended to limit the present invention.

See also Fig. 1, the flow chart of steps of the compressed sensing magnetic resonance fast imaging method that Fig. 1 provides for the embodiment of the present invention comprises the steps:

Step S1: adopt the quadruple sampling module to be sampled in the K space and obtain K space undersampled signal y.

See also Fig. 2 and Fig. 3, the flow chart of steps of the quadruple sampling module that the step schematic diagram of the quadruple sampling module that Fig. 2 provides for the embodiment of the present invention, Fig. 3 provide for the embodiment of the present invention comprises the steps:

Step S11: the template Φ that samples is owed in definition at random _s, the first sampling module is of a size of 1/4 of K space.

Step S12: the above-mentioned template of owing at random to sample is carried out quadruple copy, obtain the second sampling template 200.Be appreciated that the second sampling template 200 is the same with the K space size.

Step S13: replace the second middle stochastical sampling of sampling 200 templates with the periodic sampling template, obtain the 3rd sampling template 300.In embodiment provided by the invention, the periodic sampling template is one dimension Cartesian sampling (Descartes's sampling).

Step S14: add full sampling template at the center of the 3rd sampling template 300 and obtain quadruple sampling module 100.

Be appreciated that through above-mentioned steps and can obtain quadruple sampling module 100.Sample based on 100 pairs of quadruple sampling modules K space and obtain K space undersampled signal y, as shown in Figure 2, white point is the point that samples, and black region represents not sample.

Step S2: the Fourier's undersampled signal that obtains wavelet sub-band based on undersampled signal y.Fourier's undersampled signal of wavelet sub-band comprises low frequency sub-band Fourier undersampled signal u _LLAnd high-frequency sub-band Fourier undersampled signal u _n, wherein, n={HL, LH, HH}.In embodiment provided by the invention, according to the corresponding relation of 2D signal wavelet transformation on time domain and frequency domain, with y difference dot product small echo two dimension resolution filter D _n, n={LL, HL, LH, then HH} does twice down-sampling on frequency domain to result, can obtain Fourier's undersampled signal u corresponding to subband _n, n={LL, HL, LH, HH}.

Step S3: adopt parallel imaging method to above-mentioned low frequency sub-band Fourier undersampled signal u _LLRebuild and obtain low frequency sub-band

Be appreciated that by existing parallel imaging method and can realize above-mentioned low frequency sub-band Fourier undersampled signal u _LLReconstruction.

Step S4: build mathematical model, and according to mathematical model to described high-frequency sub-band Fourier's undersampled signal u _nRebuild and obtain high-frequency sub-band N ∈ { HL, LH, HH}.Wherein, mathematical model is:

${\hat{w}}_n=\underset{w_n,D,\mathrm{\Γ}}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^2$

$s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

Wherein, u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ, F, the template of sampling is owed in Φ, expression at random, and F is Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator, R _iPicture breakdown is become the m piece, and each image block is become vector; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Be the degree of rarefication of required coefficient, Γ is sparse coefficient set, Γ={ α _{N, 1}, α _{N, 2}, α _{N, i}..., α _{N, m}.

In embodiment provided by the invention, build mathematical model, and according to mathematical model to high-frequency sub-band Fourier undersampled signal u _nRebuild and obtain high-frequency sub-band

{ HH} adopts duty Optimization method computational mathematics model, and obtains high-frequency sub-band n ∈ for HL, LH { HH} specifically comprises the steps: n ∈ for HL, LH

Step S41: objective definition function

N ∈ LH, and HL, HH}, wherein, w _nBe the subband that needs reconstruct of the unknown, F _uFor Fourier owes sampling matrix: F _u=Φ _sF,

The expression inverse Fourier transform, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient.

Step S42: definition iteration K step

Known.In embodiment provided by the invention, adopt duty Optimization method computational mathematics model, the definition iteration K step

Step S43: based on above-mentioned

Mathematical model is converted to the first mathematical model, and wherein the first mathematical model is:

$\{D_n^k,{\mathrm{\Γ}}^k\}=\underset{D,\mathrm{\Γ}}{\min }\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n^{k-1}-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^{2}s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\},$

In above-mentioned formula, R _iFor image block extracts operator, w _nBe the subband that needs reconstruct of the unknown, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient.Be appreciated that due to this moment

Known, mathematical model is out of shape obtains the first mathematical model.

Step S44: use the K-SVD computational methods to find the solution the first mathematical model, and obtain

And Γ ^KIn embodiment provided by the invention, adopt existing K-SVD computational methods to find the solution the first mathematical model, obtain above-mentioned

And Γ ^K

Step S45: based on above-mentioned

And Γ ^K, change mathematical model into second mathematical model, and obtain

Wherein the second mathematical model is:

$w_n^{\left(k\right)}=\underset{w_n}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n^{\left(k\right)}{\mathrm{\α}}_{n,i}^{\left(k\right)}\left|\right|}_2^2,n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

In above-mentioned formula, u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ _sF, Φ _sThe template of sampling is owed in expression at random, and F is Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient.Be appreciated that due to above-mentioned

To find the solution be a least square problem, therefore, the form that can directly write out its analytic solutions is as follows:

$w_n^{\left(k\right)}={(\underset{i}{\mathrm{\Σ}}{R}_i^T{R}_i+v{F}_u^H{F}_u)}^{-1}(\underset{i}{\mathrm{\Σ}}{R}_i^T{D}_n^{\left(k\right)}{\mathrm{\α}}_i^{\left(k\right)}+v{F}_u^H{u}_n)$

Wherein,

Represent respectively R _i, F _uAdjoint operator.

Step S46: judge that above-mentioned iterations K is whether more than or equal to the iterations of actual set, if "Yes" is carried out next step; If "No" is returned to step S43.In embodiment provided by the invention, choosing generally of iterations carried out Rational choice according to operator's experience.

Step S47: output high-frequency sub-band

N ∈ { HL, LH, HH}.Be appreciated that by above-mentioned steps, and obtain high-frequency sub-band

N ∈ { HL, LH, HH}.

Step S5: to low frequency sub-band

And high-frequency sub-band

{ HH} does wavelet inverse transformation to n ∈ for HL, LH.Can be to low frequency sub-band obtained above

And high-frequency sub-band

{ HH} does wavelet inverse transformation to n ∈ for HL, LH, thereby obtains required reconstructed image.

Above-mentioned compressed sensing magnetic resonance fast imaging method improves the sparse property of signal by dictionary learning, and utilize the relation in wavelet sub-band and K space, image reconstruction problem in traditional compressed sensing magnetic resonance is optimized for the problem that calculation scale is less and can solve by parallel computation, thereby can reconstructs rapidly final image.

the above, it is only preferred embodiment of the present invention, be not that the present invention is done any pro forma restriction, although the present invention discloses as above with preferred embodiment, yet be not to limit the present invention, any those skilled in the art, within not breaking away from the technical solution of the present invention scope, when the technology contents that can utilize above-mentioned announcement is made a little change or is modified to the equivalent embodiment of equivalent variations, in every case be not break away from the technical solution of the present invention content, any simple modification that foundation technical spirit of the present invention is done above embodiment, equivalent variations and modification, all still belong in the scope of technical solution of the present invention.

Claims (4)

1. a compressed sensing magnetic resonance fast imaging method, is characterized in that, comprises the steps:

Adopt the quadruple sampling module to be sampled in the K space and obtain K space undersampled signal y;

Obtain Fourier's undersampled signal of wavelet sub-band based on described undersampled signal y, Fourier's undersampled signal of described wavelet sub-band comprises low frequency sub-band Fourier undersampled signal u _LLAnd high-frequency sub-band Fourier undersampled signal u _n, wherein, n={HL, LH, HH};

Adopt parallel imaging method to described low frequency sub-band Fourier undersampled signal u _LLRebuild and obtain low frequency sub-band

Build mathematical model, and according to described mathematical model to described high-frequency sub-band Fourier's undersampled signal u _nRebuild and obtain high-frequency sub-band

N ∈ HL, and LH, HH}, wherein, described mathematical model is:

${\hat{w}}_n=\underset{w_n,D,\mathrm{\Γ}}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^2$

$s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

Wherein, v is positive weight factor; u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ _sF, Φ _sThe template of sampling is owed in expression at random, and F is Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Be the degree of rarefication of required coefficient, Γ is sparse coefficient set, Γ={ α _{N, 1}, α _{N, 2}..., α _{N, i}..., α _{N, m};

To above-mentioned low frequency sub-band

And high-frequency sub-band

{ HH} does wavelet inverse transformation to n ∈ for HL, LH.

2. compressed sensing magnetic resonance fast imaging method according to claim 1, is characterized in that, described quadruple sampling module obtains by following step:

The template Φ that samples is owed in definition at random _s, described the first sampling module is of a size of 1/4 of K space;

The described template of owing at random to sample is carried out quadruple copy, obtain the second sampling template, described the second sampling template is consistent with the K space size;

Replace described the second middle stochastical sampling of sampling template with the periodic sampling template, obtain the 3rd sampling template;

Add full sampling template at the center of described the 3rd sampling template and obtain the quadruple sampling module.

3. compressed sensing magnetic resonance fast imaging method according to claim 1, is characterized in that, wherein, builds mathematical model, and according to described mathematical model to described high-frequency sub-band Fourier's undersampled signal u _nRebuild and obtain high-frequency sub-band

{ HH} comprises the steps: n ∈ for HL, LH

Adopt the duty Optimization method to calculate described mathematical model, and obtain described high-frequency sub-band

N ∈ { HL, LH, HH}.

4. compressed sensing magnetic resonance fast imaging method according to claim 3, is characterized in that, wherein, adopts the duty Optimization method to calculate described mathematical model, and obtain described high-frequency sub-band

{ HH} comprises the steps: n ∈ for HL, LH

Step S41: objective definition function:

N ∈ LH, and HL, HH},

Wherein, w _nBe the subband that needs reconstruct of the unknown, F _uFor Fourier owes sampling matrix: F _u=Φ _sF, The expression inverse Fourier transform, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S42: definition iteration K step

Known;

Step S43: based on above-mentioned

Described mathematical model is converted to the first mathematical model, and wherein said the first mathematical model is:

$\{D_n^k,{\mathrm{\Γ}}^k\}=\underset{D,\mathrm{\Γ}}{\min }\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n^{k-1}-D_n{\mathrm{\α}}_{n,i}\left|\right|}_2^{2}s.t.{\left|\right|{\mathrm{\α}}_{n,i}\left|\right|}_0\≤T_0,\∀i;n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\},$

In above-mentioned formula, R _iFor image block extracts operator, w _nBe the subband that needs reconstruct of the unknown, D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S44: use the K-SVD computational methods to find the solution described the first mathematical model, and obtain

And Γ ^K

Step S45: based on above-mentioned And Γ ^K, change mathematical model into second mathematical model, and obtain

Wherein said the second mathematical model is:

$w_n^{\left(k\right)}=\underset{w_n}{\arg \min }v{\left|\right|u_n-F_u{w}_n\left|\right|}_2^2+\underset{i}{\mathrm{\Σ}}{\left|\right|R_i{w}_n-D_n^{\left(k\right)}{\mathrm{\α}}_{n,i}^{\left(k\right)}\left|\right|}_2^2,n\∈\{\mathrm{LH},\mathrm{HL},\mathrm{HH}\}$

In above-mentioned formula, u _nBe high frequency Fourier undersampled signal; F _uFor Fourier owes sampling matrix: F _u=Φ _sF, F are Fourier's matrix; w _nThe subband that needs reconstruct for the unknown; R _iFor image block extracts operator; D _nDictionary for needs study; α _{N, i}Be sparse coefficient, its nonzero value number can not be greater than T ₀, T ₀Degree of rarefication for required coefficient;

Step S46: judge that above-mentioned iterations K is whether more than or equal to the iterations of actual set, if "Yes" is carried out next step; If "No" is returned to step S43;

Step S47: export described high-frequency sub-band

N ∈ { HL, LH, HH}.

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