CN103995973A - Signal sparse decomposition method based on set partitioning of over-complete dictionary - Google Patents

Abstract

The invention discloses a signal sparse decomposition method based on set partitioning of an over-complete dictionary and relates to a signal sparse decomposition method. The method is provided in order to solve the problem that when a classic MP method and an existing improvement method are used for decomposing signals, the operation time is too long. According to the method, the over-complete dictionary is subjected to set partitioning by selecting different factors, a huge redundant dictionary is divided into a plurality of sub-dictionaries, appropriate time frequency is selected from the sub-dictionaries through a matching pursuit algorithm to accurately and rapidly decompose signals, signal residuals are decomposed again according to the standard of actual requirements till reconstructed signals conforming to the standard are obtained, and the reconstructed signals can be shown in the form of the sum of the products of all stages of iteration residuals and corresponding matched atoms. The signal sparse decomposition method is applicable to the field of signal sparse decomposition.

Description

A kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division

Technical field

The present invention relates to a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division, be specifically related to the decomposition method of signal.

Background technology

Signal decomposition and be illustrated in signal and process in research field very important.Especially in to the processing of signal and analysis, signal decomposition is being played the part of vital role.In the method that signal is decomposed, traditional classical method is on being projected at the bottom of one group of complete orthogonal basis, such as Fourier transform or wavelet transformation.

At similar Fourier's base or wavelet basis, or the linearity of in the single substrate of other substrate, signal being carried out is decomposed flexible not enough.Some sparse characteristic of signal itself are not showed fully.Fourier's base only can be located good signal to time domain and be carried out expression to a certain degree, and its frequency domain polarization is poor.Wavelet basis is not suitable for for to having the signal of a close limit high-frequency band to carry out exploded representation at frequency domain after Fourier transform.Because the information of carrying on signal is scattered on whole substrate in the process of signal decomposition, on these two traditional orthogonal basiss, be difficult to utilize the expression coefficient of signal to detect unique forms and the characteristic having with judgment signal.On at the bottom of other orthogonal basis, also can run into same problem.

For an orientation range variation signal greatly on time domain and frequency domain, is olation plays an important role for this signal of statement flexibly.Signal demand is launched into the stacking pattern of some waveforms, and the time-frequency characteristic of these waveforms and the partial structurtes of this signal match, and these ripples are known as time-frequency atom.For instance, a class signal of similar impact signal should be broken down on the concentrated base of time domain, and the ripple being had compared with narrow-band with time-frequency domain part represents.When signal has above two constituents simultaneously, time-frequency atom must be made corresponding adjustment.Classical MP method in huge dictionary, find in matched atoms, expend time in long, thereby cause operation time long.Although existing, based on the more classical MP method of improving one's methods of shift factor, shortened the time to a certain extent, precision is lower, and the time is still long.Existingly based on improving one's methods of frequency factor, promoted on this basis precision, also further shortened the time of decomposition and reconstruction signal, but still the long period can be expended, when especially signal length is longer, expend time in still long, therefore must find under a kind of prerequisite that can guarantee enough accuracy, can from the dictionary by a large amount of time-frequency atomic buildings, pick out fast the method for matched atoms.

Summary of the invention

The present invention is in order to solve classical MP method and existing long problem operation time of improving one's methods and existing when signal is decomposed utilized, especially in order to solve in actual applications when signal is long, when dictionary is too huge, the problem that can further lengthen operation time.And then a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set and dividing proposed.

A kind of process based on crossing the signal Its Sparse Decomposition method of complete dictionary set division is:

Step 1: set up the complete dictionary D of different mistakes for the different signal f of characteristic, when Gauss's modulated window signal is analyzed, set up the complete dictionary D of mistake based on Gauss's modulation signal;

Step 2: according to the modulation correlativity of atom, the complete dictionary of mistake is divided into some mutually disjoint sub-dictionaries, making every sub-dictionary is all jointly to consist of the atom that meets relation of equivalence, and the atom by identical modulating characteristic forms jointly;

Step 3: utilize matching pursuit algorithm decomposed signal, according to the some sub-dictionary that complete dictionary is divided into excessively, from meeting the sub-dictionary of atomic building of relation of equivalence, each chooses an atom as representative, the formula of substitution matching judgment one by one compares, until find the atom with current input signal optimum matching, the sub-dictionary using the sub-dictionary at this atom place as optimum matching;

Step 4: in the sub-dictionary of the optimum matching of selecting, from first atom, start till last atom, atom substitution matching judgment one by one formula is compared, select the atom with current input signal optimum matching, utilize the atom of selecting to be reconstructed signal;

Step 5: contrast input signal f and reconstruction signal, picked up signal residual error, compare with require default standard according to realistic accuracy, when if signal residual error size reaches preset standard, stop whole computation process, with reconstruction signal described in step 4, represent input signal, if when signal residual error size does not reach preset standard, return to step 3 the remaining Rf of signal is decomposed again.

The present invention is deployed in signal linearity on one group of Non-orthogonal basis set, by choosing different factor pairs, cross complete dictionary and gather division, huge redundant dictionary is divided into some sub-dictionaries, utilizes matching pursuit algorithm from sub-dictionary, to choose suitable time-frequency atom and carry out decomposed signal.Meaning of the present invention is that it can under enough precision, can choose suitable time-frequency atom and carry out decomposed signal, thereby greatly cut down operation time fast guaranteeing from redundant dictionary.Signal is longer, the advantage of this method is more obvious, and when signal length N=1024, be 1.29% of classical match tracing method operation time the operation time of this method, for shift factor is improved one's methods 12.73% of operation time, for frequency factor is improved one's methods 23.37% of operation time.

Accompanying drawing explanation

Fig. 1 is the process flow diagram based on crossing the signal Its Sparse Decomposition method of complete dictionary set division;

Fig. 2 is original signal figure;

Fig. 3 is matched atoms signal graph;

Fig. 4 is the remaining figure of signal;

Fig. 5 is reconstruct signal graph.

Embodiment

Embodiment one: in conjunction with Fig. 1, present embodiment is described, a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division, it comprises the steps:

Step 1: set up the complete dictionary D of different mistakes for the different signal f of characteristic, when Gauss's modulated window signal is analyzed, set up the complete dictionary D of mistake based on Gauss's modulation signal.

Step 2: according to the modulation correlativity of atom, the complete dictionary of mistake is divided into some mutually disjoint sub-dictionaries, making every sub-dictionary is all jointly to consist of the atom that meets relation of equivalence, and the atom by identical modulating characteristic forms jointly.

Step 3: utilize matching pursuit algorithm decomposed signal, according to the some sub-dictionary that complete dictionary is divided into excessively, from meeting the sub-dictionary of atomic building of relation of equivalence, each chooses an atom as representative, the formula of substitution matching judgment one by one compares, until find the sub-dictionary with current input signal optimum matching, the sub-dictionary using the sub-dictionary at this atom place as optimum matching.

Step 4: in the sub-dictionary of the optimum matching of selecting, from first atom, start till last atom, atom substitution matching judgment one by one formula is compared, select the atom with current input signal optimum matching, utilize the atom of selecting to be reconstructed signal.

Step 5: contrast input signal f and reconstruction signal, picked up signal residual error, compare with require default standard according to realistic accuracy, when if signal residual error size reaches preset standard, stop whole computation process, with reconstruction signal described in step 4, represent input signal, if when signal residual error size does not reach preset standard, return to step 3 the remaining Rf of signal is decomposed again.

Embodiment two: the concrete operation step of the step 1 described in present embodiment is:

The Hilbert space H=L being formed by complex function ²(R), in, have

$\left|\right|f\left|\right|={\&Integral;}_{-\&infin;}^{+\&infin;}{\left|f\left(t\right)\right|}^2\mathrm{dt}<+\&infin;---\left(1\right)$

Wherein f represents input signal, and t represents the time.

At Hilbert space H=L ²(R) the set D=(g of one group of vector of definition in _γ) _{γ ∈ Γ}for dictionary, and have || g _γ||=1, make g (t) for continuously differentiable real function, its high-order infinitesimal is O (1/ (t ²+ 1)), || g (t) || the integration of=1, g (t) is non-vanishing, and g (0) ≠ 0.Definition Gabor cluster expression formula is as shown in formula (2)

$g_{\mathrm{\&gamma;}}\left(t\right)=\frac{1}{\sqrt{s}}g\left(\frac{t-u}{s}\right)\cos (\mathrm{vt}+w)---\left(2\right)$

Γ=R ⁺* R ²represent two-dimentional arithmetic number territory, wherein R ⁺represent arithmetic number territory, R ²represent that the maximum dimensions of element in this real number field are two dimension, the two representative set Γ that multiplies each other possesses above-mentioned 2 requirements simultaneously.γ=(s, u, v, w) is time and frequency parameter index set, is Γ=R ⁺* R ²in an element.In formula, s is scale factor, and u is shift factor, and v is frequency factor, and w is phase factor, the factor in formula effect be to g _γ(t) be normalized.Gabor cluster g _γ(t) based on function g (t), build.

By time and frequency parameter index set γ=(s, u, v, w) discretize, obtain γ=(a ^j, pa ^j△ u, ka ^-j△ v, i △ w).A=2 wherein, △ u=1/2, △ v=π, △ w=π/6,0<j<log ₂n, 0≤p≤N2 ^-j+1, 0≤k≤2 ^j+1, 0≤i≤12.After discretize, the discrete parameter j in index set, p, k, i only allows value in the scope of definition, and each group discrete indicator collection parameter value is corresponding an atom all, and all atoms are organized in and obtain together a complete dictionary D of mistake.

Other step is identical with embodiment one.

Embodiment three: the concrete operation step of present embodiment step 2 is:

S in time and frequency parameter index set γ=(s, u, v, w), u factor is identical, v, the atom that the w factor is different are divided into a set, make β=(s, u) represent the index set of the sub-dictionary of equivalence, Γ _β={ β _i| i=1,2 ....Sub-dictionary is as shown in formula (3)

$D_{{\mathrm{\&beta;}}_i}=\left\{g_{\mathrm{\&gamma;}}\right|\mathrm{\&gamma;}=(s,u,v,w)\&Element;\mathrm{\&Gamma;},{(s,u)}_i{\mathrm{\&beta;}}_i\}---\left(3\right)$

Sub-dictionary is by atom g _γform, these atomic properties are close, i.e. (s, u) _i=β _i, each group discrete indicator collection γ=(s, u, v, w) parameter value is corresponding an atom g all _γ, discrete indicator integrates form as γ=(a ^j, pa ^j△ u, ka ^-j△ v, i △ w).

Cross complete dictionary D by some sub-dictionaries form, meet the following conditions,

The complete dictionary of whole mistake is equivalent to be segmented on a sub-dictionary form of two dimension, and in form, arbitrary atom and its adjacent atom around have akin character.Each sub-dictionary represents by an atom with the specific modulation factor.

Other step is identical with embodiment two.

Embodiment four: described in present embodiment, the concrete operation step of step 3 is:

By inner product <f, g> ∈ L ²(R) ²be defined as

$<f,g>={\&Integral;}_{-\&infin;}^{+\&infin;}f\left(t\right)\stackrel{\&OverBar;}{g}\left(t\right)\mathrm{dt}---\left(5\right)$

Wherein complex conjugate form for g (t).

In every sub-dictionary, choose one and represent Atomic Decomposition signal, signal f can be broken down into

f＝<f,g _γ0>g _γ0+Rf (6)

G wherein _{γ 0}∈ D, g _{γ 0}being certain atom participating in interative computation for the first time, is all atom g _γin a part, Rf is for utilizing atom g _{γ 0}remnants vector after signal f is approached.G _{γ 0}mutually orthogonal with Rf, have

||f|| ²|<f，g _γ0>| ²+||Rf|| ² (7)

Select suitable g _{γ 0}∈ D makes f, g _{γ 0}as far as possible large, signal is remaining || and Rf|| is as far as possible little.In all representative atoms of choosing, find one to make | <f, g _{γ 0}>| is maximum and meet the g of formula (8) _{γ 0}as optimum atom, and using the sub-dictionary at this optimum atom place as the sub-dictionary of optimum matching.

$|<f,g_{\mathrm{\&gamma;}0}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<f,g_{\mathrm{\&gamma;}}>|---\left(8\right)$

In formula, α is range factor, gets 1, sup|| representative and asks supremum.

Other step is identical with embodiment three.

Embodiment five: described in present embodiment, the concrete operation step of step 4 is:

In the sub-dictionary of the optimum matching of selecting, from first atom, start, till last atom compares atom substitution matching judgment one by one formula, in the sub-dictionary of optimum matching, to find one to make | <f, g' _{γ 0}>| is maximum and meet the g' of formula (9) _{γ 0}as optimum atom,

$|<f,{g^{\text{'}}}_{\mathrm{\&gamma;}0}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<f,g_{\mathrm{\&gamma;}}>|---\left(9\right)$

G' wherein _{γ 0}∈ D, g' _{γ 0}being certain atom in the sub-dictionary of optimum matching, is also all atom g _γin a part, select the atom with current input signal optimum matching, utilize the atom of selecting to be reconstructed signal.

Other step is identical with embodiment four.

Embodiment six: described in present embodiment, the concrete operation step of step 5 is:

According to the optimum matching atom of selecting, signal f has been broken down into f=<f, g' _{γ 0}>g' _{γ 0}+ Rf, by signal residual error with according to realistic accuracy, require default standard to compare, when if signal residual error size reaches preset standard, stop whole computation process, with this reconstruction signal, represent input signal, when if signal residual error size does not reach preset standard, return to step 3 the remaining Rf of signal is decomposed again; R during computing first ⁰f=f carries out iteration, the R in season of computing afterwards ^jf=f carries out iteration, 0≤j≤n, the final remaining R that obtains n rank by iterative computation ⁿf, n represents iterations, n>=0.The remaining R in n rank ⁿthe size of f can be used as the Rule of judgment that algorithm stops.

$|<R^{n}f,g_{\mathrm{\&gamma;n}}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<R^{n}f,g_{\mathrm{\&gamma;}}>|---\left(10\right)$

Utilize formula (10) from set in choose remaining R ⁿthe atom that f mates most.

Original signal f is decomposed on the base forming in some matched atoms, its reconstruction signal can be expressed as every one-level iteration residual error and corresponding matched atoms product and form.

Other step is identical with embodiment five.

Specific embodiment

According to embodiment two: the expression formula of the g described in test (t) continuously differentiable function is , be a Gauss function.According to embodiment three: first determine a frequency constant factor v and a phase constant factor w.According to test signal modulating characteristic, make the initial value that v=π and w=π/6 are sub-dictionary index set, initial value is added and subtracted to parameter increase △ v and △ w operation, obtain different sub-dictionaries.Initial parameter value needs to be converted into discrete form computing after setting in computer simulation program.According to embodiment four, embodiment five and embodiment six, signal is decomposed and reconstruct.Fig. 2 is that original signal figure, Fig. 3 are that matched atoms signal graph, Fig. 4 are that the remaining figure of signal, Fig. 5 are reconstruct signal graph, and from reconstruction signal result, this method is carried out Its Sparse Decomposition to original signal, and the quality of reconstruct is very good more afterwards.Table 1 is comparative effectiveness working time of this method and additive method.

Table 1

Can find out that the present invention is guaranteeing under enough precision, can from redundant dictionary, choose fast suitable time-frequency atom and carry out decomposed signal, thereby greatly cut down operation time, when signal length N=1024, be 1.29% of classical match tracing method operation time the operation time of this method, for shift factor is improved one's methods 12.73% of operation time, for frequency factor is improved one's methods 23.37% of operation time.In actual conditions, the length of signal can be quite large, and the advantage of this method can be more obvious.

Claims (6)

1. the signal Its Sparse Decomposition method based on crossing complete dictionary set division, is characterized in that it comprises the steps:

Step 1: set up the complete dictionary D of different mistakes for the different signal f of characteristic, when Gauss's modulated window signal is analyzed, set up the complete dictionary D of mistake based on Gauss's modulation signal;

Step 2: according to the modulation correlativity of atom, the complete dictionary of mistake is divided into some mutually disjoint sub-dictionaries, making every sub-dictionary is all jointly to consist of the atom that meets relation of equivalence, and the atom by identical modulating characteristic forms jointly;

Step 3: utilize matching pursuit algorithm decomposed signal, according to the some sub-dictionary that complete dictionary is divided into excessively, from meeting the sub-dictionary of atomic building of relation of equivalence, each chooses an atom as representative, the formula of substitution matching judgment one by one compares, until find the atom with current input signal optimum matching, the sub-dictionary using the sub-dictionary at this atom place as optimum matching;

Step 4: in the sub-dictionary of the optimum matching of selecting, from first atom, start till last atom, atom substitution matching judgment one by one formula is compared, select the atom with current input signal optimum matching, the atom that utilization is selected is reconstructed signal, obtains reconstruction signal;

Step 5: contrast input signal f and reconstruction signal, picked up signal residual error, compares with require default standard according to realistic accuracy, if when signal residual error size reaches preset standard, stop whole computation process, with reconstruction signal described in step 4, represent input signal; When if signal residual error size does not reach preset standard, return to step 3 the remaining Rf of signal is decomposed again.

2. a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division according to claim 1, is characterized in that, the implementation procedure of step 1 is:

The Hilbert space H=L being formed by complex function ²(R), in, have

$\left|\right|f\left|\right|={\&Integral;}_{-\&infin;}^{+\&infin;}{\left|f\left(t\right)\right|}^2\mathrm{dt}<+\&infin;---\left(1\right)$

Wherein f represents input signal, and t represents the time;

At Hilbert space H=L ²(R) the set D=(g of one group of vector of definition in _γ) _{γ ∈ Γ}for dictionary, and have || g _γ||=1, make g (t) for continuously differentiable real function, its high-order infinitesimal is O (1/ (t ²+ 1)), || g (t) || the integration of=1, g (t) is non-vanishing, and g (0) ≠ 0; Definition Gabor cluster expression formula is as shown in formula (2)

$g_{\mathrm{\&gamma;}}\left(t\right)=\frac{1}{\sqrt{s}}g\left(\frac{t-u}{s}\right)\cos (\mathrm{vt}+w)---\left(2\right)$

Γ=R ⁺* R ²represent two-dimentional arithmetic number territory, wherein R ⁺represent arithmetic number territory, R ²represent that the maximum dimensions of element in this real number field are two dimension, the two representative set Γ that multiplies each other possesses above-mentioned 2 requirements simultaneously; γ=(s, u, v, w) is time and frequency parameter index set, is Γ=R ⁺* R ²in an element; In formula, s is scale factor, and u is shift factor, and v is frequency factor, and w is phase factor, the factor in formula effect be to g _γ(t) be normalized; Gabor cluster g _γ(t) based on function g (t), build;

By time and frequency parameter index set γ=(s, u, v, w) discretize, obtain γ=(a ^j, pa ^j△ u, ka ^-j△ v, i △ w); A=2 wherein, △ u=1/2, △ v=π, △ w=π/6,0<j<log ₂n, 0≤p≤N2 ^-j+1, 0≤k≤2 ^j+1, 0≤i≤12; After discretize, the discrete parameter j in index set, p, k, i only allows value in the scope of definition, and each group discrete indicator collection parameter value is corresponding an atom all, and all atoms are organized in and obtain together a complete dictionary D of mistake.

3. a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division according to claim 2, is characterized in that, the implementation procedure of step 2 is:

S in time and frequency parameter index set γ=(s, u, v, w), u factor is identical, and v, the atom that the w factor is different are divided into a set, make β=(s, u) represent the index set of the sub-dictionary of equivalence, Γ _β={ β _i| i=1,2 ...; Sub-dictionary is as shown in formula (3)

$D_{{\mathrm{\&beta;}}_i}=\left\{g_{\mathrm{\&gamma;}}\right|\mathrm{\&gamma;}=(s,u,v,w)\&Element;\mathrm{\&Gamma;},{(s,u)}_i{\mathrm{\&beta;}}_i\}---\left(3\right)$

Sub-dictionary is by atom g _γform, these atomic properties are close, i.e. (s, u) _i=β _i, each group discrete indicator collection γ=(s, u, v, w) parameter value is corresponding an atom g all _γ, discrete indicator integrates form as γ=(a ^j, pa ^j△ u, ka ^-j△ v, i △ w);

Cross complete dictionary D by some sub-dictionaries form, meet the following conditions,

The complete dictionary of whole mistake is equivalent to be segmented on a sub-dictionary form of two dimension, and in form, arbitrary atom and its adjacent atom around have akin character; Each sub-dictionary represents by an atom with the specific modulation factor.

4. a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division according to claim 3, is characterized in that, the implementation procedure of step 3 is:

By inner product <f, g> ∈ L ²(R) ²be defined as

$<f,g>={\&Integral;}_{-\&infin;}^{+\&infin;}f\left(t\right)\stackrel{\&OverBar;}{g}\left(t\right)\mathrm{dt}---\left(5\right)$

Wherein complex conjugate form for g (t);

Signal f can be broken down into

f＝<f,g _γ0>g _γ0+Rf (6)

G wherein _{γ 0}∈ D, g _{γ 0}being certain atom participating in interative computation for the first time, is all atom g _γin a part, Rf is for utilizing atom g _{γ 0}remnants vector after signal f is approached; g _{γ 0}mutually orthogonal with Rf, have

||f|| ²=|<f，g _γ0>| ²+||Rf|| ² (7)

Select suitable g _{γ 0}∈ D makes | <f, g _{γ 0}>| is as far as possible large, and signal is remaining || and Rf|| is as far as possible little; In sub-dictionary, finding one makes | <f, g _{γ 0}>| is maximum and meet the g of formula (8) _{γ 0}as optimum atom, and using the sub-dictionary at this optimum atom place as the sub-dictionary of optimum matching;

$|<f,g_{\mathrm{\&gamma;}0}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<f,g_{\mathrm{\&gamma;}}>|---\left(8\right)$

In formula, α is range factor, gets 1, sup|| representative and asks supremum.

5. a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division according to claim 4, is characterized in that, the implementation procedure of step 4 is:

In the sub-dictionary of the optimum matching of selecting, from first atom, start, till last atom compares atom substitution matching judgment one by one formula, in the sub-dictionary of optimum matching, to find one to make f, g' _{γ 0}maximum and meet the g' of formula (9) _{γ 0}as optimum atom,

$|<f,{g^{\text{'}}}_{\mathrm{\&gamma;}0}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<f,g_{\mathrm{\&gamma;}}>|---\left(9\right)$

G' wherein _{γ 0}∈ D, g' _{γ 0}being certain atom in the sub-dictionary of optimum matching, is also all atom g _γin a part, select the atom with current input signal optimum matching, utilize the atom of selecting to be reconstructed signal.

6. a kind of signal Its Sparse Decomposition method based on crossing complete dictionary set division according to claim 5, is characterized in that, the implementation procedure of step 5 is:

According to the optimum matching atom of selecting, signal f has been broken down into f=f, g' _{γ 0}g' _{γ 0}+ Rf, by signal residual error with according to realistic accuracy, require default standard to compare, when if signal residual error size reaches preset standard, stop whole computation process, with this reconstruction signal, represent input signal, when if signal residual error size does not reach preset standard, return to step 3 the remaining Rf of signal is decomposed again; R during computing first ⁰f=f carries out iteration, the R in season of computing afterwards ^jf=f carries out iteration, 0≤j≤n, the final remaining R that obtains n rank by iterative computation ⁿf, n represents iterations, n>=0; The remaining R in n rank ⁿthe size of f can be used as the Rule of judgment that algorithm stops;

$|<R^{n}f,g_{\mathrm{\&gamma;n}}>|\&GreaterEqual;\mathrm{\&alpha;}\underset{\mathrm{\&gamma;}\&Element;\mathrm{\&Gamma;}}{\sup }|<R^{n}f,g_{\mathrm{\&gamma;}}>|---\left(10\right)$

Utilize formula (10) from set in choose remaining R ⁿthe atom that f mates most;

Original signal f is decomposed on the base forming in some matched atoms, its reconstruction signal can be expressed as every one-level iteration residual error and corresponding matched atoms product and form.