CN103578086B - A kind of interferogram data spectrum recovering method based on wavelet analysis - Google Patents

Abstract

A kind of interferogram data spectrum recovering method based on wavelet analysis of the present invention, by Fitting Interpolation Method based on wavelet analysis, applies to nonuniform sampling interferogram reconstruction, and carries out spectra inversion.First use fitting process based on wavelet analysis that nonuniform sampling interference data is fitted, obtain the interference data curve of matching, carry out uniform interpolation sampling the most again, obtain the interference data of uniform sampling.Interference data to uniform sampling, according to interference spectrum theory, carries out fast Fourier transform (FFT), obtains the spectrogram restored.According to the simulation experiment result, it can be seen that the method has well adapted to interference data and changed at zero optical path difference acutely, changes feature slowly at optical path difference larger part, has preferably simulated interference data curve.After uniform sampling, carry out FFT and obtain spectra inversion figure.Advantages of the present invention is: can be finer simulate interferogram data, preferably embody the feature of interferogram data, error of fitting is lower.

Description

A kind of interferogram data spectrum recovering method based on wavelet analysis

Technical field

The invention belongs to inteference imaging spectrometer data processing field, specifically, be a kind of based on wavelet analysis dry Relate to data spectrum recovering method.

Background technology

Interference type imaging spectrometer is owing to having multichannel, higher light compared to color dispersion-type, grating type imaging spectrometer The advantages such as flux, more preferable spectrum and spatial resolution, in the nearly more than ten years progressively by the attention of various countries full-fledged.But it is dry Relate to type imaging spectrometer detection data the non-immediate target optical spectral data that can use, but intermediate data interferogram number According to.It is therefore desirable to carry out data process, inverting target light spectrogram.

According to interference spectrum theory, the interference data that interference type spectral instrument obtains and the spectroscopic data of target, there is Fu In leaf transformation relation, i.e. interference data is carried out Fourier transformation, so that it may obtain the spectroscopic data of target.In common quick Fu It is uniform sampling data that leaf transformation (FFT) requires, and the interference data that interference type spectral instrument obtains, due to by various factors Impact, the problem that there is nonuniform sampling.Such as, inverting mirror interference spectroscope, its Sloped rotating reflecting mirror is at the uniform velocity to rotate , the sampling of detector is typically also constant duration, but due to the non-linear interference number making finally to obtain of optical path difference According to non-homogeneous.For space-time combined modulation spectrogrph, due to image-forming principle and the impact of environmental factors, there is also optical path difference non- Linearly cause interference data situation heterogeneous.In this case, directly carrying out FFT will be the most applicable.

For the non-homogeneous situation of interference data, (Yang Xiao permitted document, Zhou Sizhong, Xiangli are refined, rotary mirror type Fourier trasform spectroscopy The nonlinear fitting process of instrument optical path difference compensates. photon journal, volume 34 o. 11th in November, 2005) and Yang Xiaoxu, Zhou Sizhong,XiangLi Bin.Compensating Nonlinearity of Optical Path Difference of Rotary Fourier Transform Spectrometer with Fitting Interferogram.ACTA PHOTONICA SINICA, proposes disclosed in Vol.34No.11November2005 and utilizes based on carrying out many to interference data The interferogram double sampling method of item formula matching, the method utilizes the method for fitting of a polynomial to carry out nonuniform sampling interferogram Matching, carries out uniform sampling according to nyquist sampling law afterwards, can directly obtain the interferogram of uniform sampling, the most right The interferogram of uniform sampling carries out FFT；When nonuniform sampling interferogram is lack sampling, approximation compensates the information lost, so Can be obtained by recovered light modal data.It is multiple that this method can use other general-purpose algorithms and software to carry out spectrum easily Former；And it is possible to expand the choice of instrument parameter, engineering design has the biggest practice significance.

But, interferogram double sampling method based on fitting of a polynomial, need interferogram is blocked, segmentation is carried out Matching, thus introduce truncated error, and the improvement of visual effect for interference data matching is unsatisfactory；Inverting spectroscopic data is also There is certain error.

Summary of the invention

For the problems referred to above, the interferogram double sampling method that document proposes the most above, in order to solve fitting of a polynomial Present in truncated error, more preferable matching interferogram data, the present invention propose utilize fitting based on wavelet analysis calculate Method is to interferogram data fitting, and the method carrying out spectrum recovering, can successively be decomposed according to frequency height composition by signal It is analyzed.Compared to Fourier transformation, it has good localization property in time domain and frequency domain simultaneously.Owing to it is to height Frequently composition uses the finest time-domain samples step-length, such that it is able to focus on any details of object.These features are the most fine Meet interference type spectral instrument interference data, near zero optical path difference, change is acutely, and in the change of optical path difference larger part slowly Feature, it is possible to preferably matching interferogram data, and carry out approximation and compensate interpolation.

Present invention interferogram based on wavelet analysis data spectrum recovering method, is realized by following step:

Step 1: shoot the extracting data same target point gray value obtained from interference type spectral instrument, obtain non-homogeneous adopting The interferogram data of sample；

Step 2: based on wavelet analysis, the interferogram data of matching nonuniform sampling；

Bring the interferogram data of nonuniform sampling into fitting formula of based on wavelet analysis, obtain the plan of interferogram data Close function, particularly as follows:

The interferogram data acquisition system making nonuniform sampling is:

f_s=[f(t₀)f(t₁)f(t₂)...f(t_i)...f(t_P-1)]^T(1)

Wherein, P is set f_sThe number of middle nonuniform sampling point；t₀、t₁、t₂、…、t_i、…、t_P-1For P nonuniform sampling Point；

Bring P nonuniform sampling point into fitting formula of based on wavelet analysis, obtain P unit system of linear equations:

$f\left(t_i\right)={\mathrm{\Σ}}_n{c}_{J,n}\mathrm{\Φ}(\frac{t_i}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{d}_{j,n}\mathrm{\Ψ}(\frac{t_i}{2^j}-n),i=0,...,P-1---\left(2\right)$

In formula (2), n is displacement；J is layers of resolution number of stages, J=1,2,3 ...；Φ(t_i) it is scaling function；ψ(t_i) For wavelet basis function；c_JFor the scaling function coefficient that j-th resolution level is corresponding；d_jFor sum be in J resolution level every The wavelet basis function coefficient that individual resolution level is corresponding；The span of n is determined by following manner:

Make Φ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^J}-n\≤N---\left(3\right)$

And then obtain:

-N+mint_i/2^J≤n≤maxt_i/2^J(4)

In formula (4), mint_iFor t_iIn minima；maxt_iFor t_iIn maximum；

Make ψ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^j}-n\≤N---\left(5\right)$

And then obtain:

-N+mint_i/2^j≤n≤maxt_i/2^j(6)

By different layers of resolution progression J=1,2,3 ... bring fitting formula respectively into, try to achieve fitting result (interferogram number According to curve)；It is calculated the square-error root of fitting result subsequently, simultaneously by fitting result inverting to spectral domain, then at spectrum Territory carries out square-error root calculating to spectrogram, obtains inverting spectrogram square-error root；Be respectively compared J=1,2,3 ... time intend Close the square-error root size of result, and inverting spectrogram square-error root size, obtain optimum resolution number of levels J；

C in said process, in formula (2)_JWith d_jObtained by following method:

Formula (2) is shown as by matrix table:

$f_s=G_J^s{c}_J+{\mathrm{\Σ}}_{j=1}^J{H}_j^s{d}_j---\left(7\right)$

Wherein,For scaling function sampled point transposed matrix in resolution level J, with each sampled point t_iRelevant；For wavelet basis function sampled point transposed matrix in each resolution level in J resolution level；

Ignoring signal f_sIn the case of detailed information, by f_sBeing expressed as of approximation:

$f_s\≈G_J^s{c}_{J_s}---\left(8\right)$

By method of least square, obtain coefficient c_JApproximate solution, be designated as

$\widehat{f_0}=G_J\widehat{c_J}---\left(9\right)$

Wherein,For f_sApproximate fits result；Matrix G_JIt is scaling function transposed matrix in level J, displacement For integer sequence [0 ..., M-1], M is the uniform sampling value number of interferogram data；

As J=1, by formula (7), (9) available error signal e₀For:

$e_0=f_s-\widehat{f_0}{|}_{t=t_{k,k=0...p-1}}=f_s-G_J^s\widehat{c_J}\≈H_J^s{d}_J---\left(10\right)$

Wherein, e₀Comprise signal f_sRadio-frequency component；Coefficient d is obtained by formula (10)_JApproximate solution

$\widehat{f_1}=G_J\widehat{c_J}+H_J\widehat{d_J}---\left(11\right)$

For f_sApproximate fits result when J=1；Wherein, H_JIt is wavelet function transposed matrix in level J, displacement Amount for integer sequence [0 ..., M-1]；

As J=2, try to achieve according to said processAfterwards；Further according to available J-1 point of formula (7), (11) The wavelet basis function coefficient d that resolution level is corresponding_J-1Approximate solution；Concrete solution procedure is first to try to achieve error signal e₁For:

$e_1=f_s-\widehat{f_1}=f_s-G_J^s\widehat{c_J}-H_J^s\widehat{d_J}\≈H_{J-1}^s{d}_{J-1}---\left(12\right)$

The J-1 scaling function coefficient d corresponding to resolution level is obtained by formula (11)_J-1Approximate solution

$\widehat{f_2}=G_J\widehat{c_J}+H_J\widehat{d_J}+H_{J-1}{\hat{d}}_{J-1}---\left(13\right)$

For f_sApproximate fits result when J=2；Wherein, H_J-1It it is wavelet basis function displacement square in level J-1 Battle array, displacement be integer sequence [0 ..., M-1]；

As J=3, try to achieve according to said processAfter；Further according to available J-2 of formula (7), (13) The wavelet basis function coefficient d that resolution level is corresponding_J-2Approximate solution；Concrete solution procedure is first to try to achieve error signal e₂For:

$e_2=f_s-\widehat{f_2}=f_s-G_J^s\widehat{c_J}-H_J^s{\hat{d}}_J-H_{J-1}^s{\hat{d}}_{J-1}\≈H_{J-2}^s{d}_{J-2}---\left(14\right)$

Thus, the J-2 scaling function coefficient d corresponding to resolution level is obtained by formula (13)_J-2Approximate solution

$\widehat{f_3}=G_J\widehat{c_J}+H_J\widehat{d_J}+H_{J-1}{\hat{d}}_{J-1}+H_{J-2}{\hat{d}}_{J-2}---\left(15\right)$

For f_sApproximate fits result when J=3；Wherein, H_J-2It it is wavelet basis function displacement square in level J-2 Battle array, displacement be integer sequence [0 ..., M-1].

By that analogy, obtain the scaling function coefficient approximate solution that each quantity resolution level J is corresponding, and each quantity is differentiated The wavelet basis function coefficient approximate solution that under rate level J, each resolution level is corresponding, and bring in formula (2), obtain each number of layers Interference data f that level is corresponding_sThe result of approximate fits be:

$f_s\≈f\left(t\right)={\mathrm{\Σ}}_n{\hat{c}}_{J,n}\mathrm{\Φ}(\frac{t}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{\hat{d}}_{j,n}\mathrm{\Ψ}(\frac{t}{2^j}-n)---\left(16\right)$

Step 3: the interference data fitting result obtained in step 2 is carried out uniform sampling interpolation, obtains uniform sampling Interference data；

By uniform sampling point [0,1 ..., M-1] bring the approximate fits result of interference data f (t) obtained in step 2 into In, obtain the interference data of M uniform sampling:

f=[f(0)f(1)f(2)...f(M-1)]^T(17)

Step 4: the interference data of uniform sampling is carried out the spectroscopic data that fast Fourier transform (FFT) obtains restoring；

According to interference spectrum theory, the interference data of M uniform sampling that will obtain in step 3, carry out FFT, obtain The spectrogram restored.

It is an advantage of the current invention that:

1, interferogram data spectrum recovering method of the present invention, makes full use of fitting algorithm based on wavelet analysis excellent Point, i.e. has good localization property in time domain and frequency domain simultaneously, and radio-frequency component is used the finest time-domain samples Step-length, such that it is able to focus on any details etc. of object；Can be finer simulate interferogram data, preferably embody dry Relating to the feature of diagram data, error of fitting is lower；

2, interferogram data spectrum recovering method of the present invention, compared to polynomial fitting method, the method need not dry Relate to diagram data and carry out piecewise fitting, do not introduce truncated error；

3, interferogram data spectrum recovering method of the present invention, by inverting spectrogram, spectral domain compares, and can see Going out, the inverting spectroscopic data of Fitting Interpolation Method based on wavelet analysis is lower compared to fitting of a polynomial interpolation method error.

Accompanying drawing explanation

Fig. 1 is interferogram data spectrum recovering method of the present invention.

Fig. 2 is nonuniform sampling interference data schematic diagram；

Fig. 3 is small echo matching interferogram result schematic diagram；

Fig. 4 is the interference data schematic diagram after uniform sampling；

Fig. 5 is the recovered light spectrogram that uniform sampling interference data carries out after FFT.

Detailed description of the invention

Below in conjunction with the accompanying drawings the present invention is further illustrated.

Present invention interferogram based on wavelet analysis data spectrum recovering method, as it is shown in figure 1, real by following step Existing:

Step 1: shoot the extracting data same target point gray value obtained from interference type spectral instrument, obtain non-homogeneous adopting The interferogram data of sample；

Step 2: based on wavelet analysis, the interferogram data of matching nonuniform sampling；

As in figure 2 it is shown, bring the interferogram data of nonuniform sampling into fitting formula of based on wavelet analysis, interfered The fitting function of diagram data, particularly as follows:

The interferogram data acquisition system making nonuniform sampling is:

f_s=[f(t₀)f(t₁)f(t₂)...f(t_i)...f(t_P-1)]^T(1)

Wherein, P is set f_sThe number of middle nonuniform sampling point；t₀、t₁、t₂、…、t_i、…、t_P-1For P nonuniform sampling Point.

Bring P nonuniform sampling point into fitting formula of based on wavelet analysis, obtain P unit system of linear equations:

$f\left(t_i\right)={\mathrm{\Σ}}_n{c}_{J,n}\mathrm{\Φ}(\frac{t_i}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{d}_{j,n}\mathrm{\Ψ}(\frac{t_i}{2^j}-n),i=0,...,P-1---\left(2\right)$

In formula (2), n is displacement；J is layers of resolution number of stages, J=1,2,3 ...；Φ(t_i) it is scaling function；Ψ(t_i) For wavelet basis function；c_JFor the scaling function coefficient that j-th resolution level is corresponding；d_jFor sum be in J resolution level every The wavelet basis function coefficient that individual resolution level is corresponding；The span of n is determined by following manner:

Make Φ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^J}-n\≤N---\left(3\right)$

And then obtain:

-N+mint_i/2^J≤n≤maxt_i/2^J(4)

In formula (4), mint_iFor t_iIn minima；maxt_iFor t_iIn maximum；

Make Ψ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^j}-n\≤N---\left(5\right)$

And then obtain:

-N+mint_i/2^j≤n≤maxt_i/2^j(6)

By different layers of resolution progression J=1,2,3 ... bring fitting formula respectively into, try to achieve fitting result (interferogram number According to curve)；It is calculated the square-error root of fitting result subsequently, simultaneously by fitting result inverting to spectral domain, then at spectrum Territory carries out square-error root calculating to spectrogram, obtains inverting spectrogram square-error root；Be respectively compared J=1,2,3 ... time intend Close the square-error root size of result, and inverting spectrogram square-error root size, obtain optimum resolution number of levels J.

C in said process, in formula (2)_JWith d_jThe unknown, can be obtained by following method:

Formula (2) is shown as by matrix table:

$f_s=G_J^s{c}_J+{\mathrm{\Σ}}_{j=1}^J{H}_j^s{d}_j---\left(7\right)$

Wherein,For scaling function sampled point transposed matrix in resolution level J, with each sampled point t_iRelevant；For wavelet basis function sampled point transposed matrix in each resolution level in J resolution level.

Ignoring signal f_sIn the case of detailed information, can be by f_sBeing expressed as of approximation:

$f_s\≈G_J^s{c}_{J_s}---\left(8\right)$

Due to nonuniform sampling point t_iNumber is limited, then formula (8) is over-determined systems, it is impossible to accurately draw c_J, therefore this By method of least square in bright, obtain coefficient c_JApproximate solution, be designated as

$\widehat{f_0}=G_J\widehat{c_J}---\left(9\right)$

Wherein,For f_sApproximate fits result, i.e. the interferogram data containing only low-frequency component；Matrix G_JIt it is scaling function Transposed matrix in level J, displacement be integer sequence [0 ..., M-1], M is the uniform sampling value number of interferogram data；

As J=1, by formula (7), (9) available error signal e₀For:

$e_0=f_s-\widehat{f_0}{|}_{t=t_{k,k=0...p-1}}=f_s-G_J^s\widehat{c_J}\≈H_J^s{d}_J---\left(10\right)$

Wherein, e₀Comprise signal f_sRadio-frequency component, these radio-frequency components can not be showed by scaling function, but permissible With the wavelet basis function approximate representation e in J level₀.Owing to being similarly over-determined systems, then obtain coefficient d by formula (10)_J's Approximate solution

$\widehat{f_1}=G_J\widehat{c_J}+H_J\widehat{d_J}---\left(11\right)$

For f_sApproximate fits result when J=1.Wherein, H_JIt is wavelet function transposed matrix in level J, displacement Amount for integer sequence [0 ..., M-1].

As J=2, try to achieve according to said processAfterwards.Further according to available J-1 point of formula (7), (11) The wavelet basis function coefficient d that resolution level is corresponding_J-1Approximate solution.Concrete solution procedure is first to try to achieve error signal e₁For:

$e_1=f_s-{\hat{f}}_1=f_s-G_J^s\widehat{c_J}-H_J^s\widehat{d_J}\≈H_{J-1}^s{d}_{J-1}---\left(12\right)$

Thus, owing to being still overdetermined equation, the J-1 scaling function system corresponding to resolution level is obtained by formula (11) Number d_J-1Approximate solution

$\widehat{f_2}=G_J\widehat{c_J}+H_J\widehat{d_J}+H_{J-1}{\hat{d}}_{J-1}---\left(13\right)$

For f_sApproximate fits result when J=2.Wherein, H_J-1It it is wavelet basis function displacement square in level J-1 Battle array, displacement be integer sequence [0 ..., M-1].

As J=3, try to achieve according to said processAfter.Further according to available J-2 of formula (7), (13) The wavelet basis function coefficient d that resolution level is corresponding_J-2Approximate solution.Concrete solution procedure is first to try to achieve error signal e₂For:

$e_2=f_s-\widehat{f_2}=f_s-G_J^s\widehat{c_J}-H_J^s{\hat{d}}_J-H_{J-1}^s{\hat{d}}_{J-1}\≈H_{J-2}^s{d}_{J-2}---\left(14\right)$

Thus, the J-2 scaling function coefficient d corresponding to resolution level is obtained by formula (13)_J-2Approximate solution

$\widehat{f_3}=G_J\widehat{c_J}+H_J\widehat{d_J}+H_{J-1}{\hat{d}}_{J-1}+H_{J-2}{\hat{d}}_{J-2}---\left(15\right)$

For f_sApproximate fits result when J=3.Wherein, H_J-2It it is wavelet basis function displacement square in level J-2 Battle array, displacement be integer sequence [0 ..., M-1].

By that analogy, obtain the scaling function coefficient approximate solution that each quantity resolution level J is corresponding, and each quantity is differentiated The wavelet basis function coefficient approximate solution that under rate level J, each resolution level is corresponding, and bring in formula (2), obtain each number of layers Interference data f that level is corresponding_sResult f (t) of approximate fits be:

$f_s\≈f\left(t\right)={\mathrm{\Σ}}_n{\hat{c}}_{J,n}\mathrm{\Φ}(\frac{t}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{\hat{d}}_{j,n}\mathrm{\Ψ}(\frac{t}{2^j}-n)---\left(16\right)$

To in interferogram data matching, according to data characteristics, suitable wavelet basis function can be selected.The present invention is medium and small Ripple basic function can select Daubechies4, Symlets4, Symlets8 etc.；And as J=3, the effect of matching is best.

Step 3: the interference data fitting result obtained in step 2 is carried out uniform sampling interpolation, obtains uniform sampling Interference data；

By uniform sampling point [0,1 ..., M-1] bring the approximate fits result of interference data f (t) obtained in step 2 into In, as shown in Figure 4, obtain the interference data of M uniform sampling:

f=[f(0)f(1)f(2)...f(M-1)]^T(17)

Step 4: the interference data of uniform sampling is carried out the spectroscopic data that fast Fourier transform (FFT) obtains restoring；

According to interference spectrum theory, the interference data of M uniform sampling that will obtain in step 3, carry out FFT, obtain The spectrogram restored, as shown in Figure 5.

Claims (2)

1. an interferogram data spectrum recovering method based on wavelet analysis, it is characterised in that: realized by following step:

Step 1: shoot the extracting data same target point gray value obtained from interference type spectral instrument, obtain nonuniform sampling Interferogram data；

Step 2: based on wavelet analysis, the interferogram data of matching nonuniform sampling；

Bring the interferogram data of nonuniform sampling into fitting formula of based on wavelet analysis, obtain the matching letter of interferogram data Number, particularly as follows:

The interferogram data acquisition system making nonuniform sampling is:

f_s=[f (t₀)f(t₁)f(t₂)…f(t_i)…f(t_P-1)]^T (1)

Wherein, P is set f_sThe number of middle nonuniform sampling point；t₀、t₁、t₂、…、t_i、…、t_P-1For P nonuniform sampling point；

Bring P nonuniform sampling point into fitting formula of based on wavelet analysis, obtain P unit system of linear equations:

$f\left(t_i\right)={\mathrm{\Σ}}_n{c}_{J,n}\mathrm{\Φ}(\frac{t_i}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{d}_{j,n}\mathrm{\Ψ}(\frac{t_i}{2^j}-n),i=0,...,P-1---\left(2\right)$

In formula (2), n is displacement；J is layers of resolution number of stages, J=1,2,3 ...；Φ(t_i) it is scaling function；Ψ(t_i) it is Wavelet basis function；c_JFor the scaling function coefficient that j-th resolution level is corresponding；d_jFor sum be in J resolution level each The wavelet basis function coefficient that resolution level is corresponding；The span of n is determined by following manner:

Make Φ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^J}-n\≤N---\left(3\right)$

And then obtain:

-N+mint_i/2^J≤n≤maxt_i/2^J (4)

In formula (4), mint_iFor t_iIn minima；maxt_iFor t_iIn maximum；

Make Ψ (t_i) compactly support be [0, N], N is natural number, then have:

$0\≤\frac{t_i}{2^j}-n\≤N---\left(5\right)$

And then obtain:

-N+mint_i/2^j≤n≤maxt_i/2^j (6)

C in said process, in formula (2)_JWith d_jObtained by following method:

By different layers of resolution progression J=1,2,3 ... bring fitting formula respectively into, try to achieve fitting result；It is calculated subsequently The square-error root of fitting result, simultaneously by fitting result inverting to spectral domain, then carries out error at spectral domain and puts down spectrogram Root calculates, and obtains inverting spectrogram square-error root；Be respectively compared J=1,2,3 ... time fitting result square-error root big Little, and inverting spectrogram square-error root size, obtain optimum resolution number of levels J；

Formula (2) is shown as by matrix table:

$f_s=G_J^s{c}_J+{\mathrm{\Σ}}_{j=1}^J{H}_j^s{d}_j---\left(7\right)$

Wherein,For scaling function sampled point transposed matrix in resolution level J, with each sampled point t_iRelevant；For Wavelet basis function sampled point is the transposed matrix in each resolution level in J resolution level；

Ignoring signal f_sIn the case of detailed information, by f_sBeing expressed as of approximation:

$f_s\≈G_J^s{c}_J---\left(8\right)$

By method of least square, obtain coefficient c_JApproximate solution, be designated as

Wherein,For f_sApproximate fits result；Matrix G_JBeing scaling function transposed matrix in level J, displacement is integer Sequence [0 ..., M-1], M is the uniform sampling value number of interferogram data；

As J=1, by formula (7), (9) available error signal e₀For:

Wherein, e₀Comprise signal f_sRadio-frequency component；Coefficient d is obtained by formula (10)_JApproximate solution

For f_sApproximate fits result when J=1；Wherein, H_JBeing wavelet function transposed matrix in level J, displacement is Integer sequence [0 ..., M-1]；

As J=2, try to achieve according to said processAfterwards；Further according to formula (7), (11) available the J-1 resolution The wavelet basis function coefficient d that level is corresponding_J-1Approximate solution；Concrete solution procedure is first to try to achieve error signal e₁For:

The J-1 scaling function coefficient d corresponding to resolution level is obtained by formula (11)_J-1Approximate solution

For f_sApproximate fits result when J=2；Wherein, H_J-1It is wavelet basis function transposed matrix in level J-1, position Shifting amount be integer sequence [0 ..., M-1]；

As J=3, try to achieve according to said processAfter；Further according to formula (7), (13) available the J-2 resolution The wavelet basis function coefficient d that rate level is corresponding_J-2Approximate solution；Concrete solution procedure is first to try to achieve error signal e₂For:

Thus, the J-2 scaling function coefficient d corresponding to resolution level is obtained by formula (13)_J-2Approximate solution

For f_sApproximate fits result when J=3；Wherein, H_J-2It is wavelet basis function transposed matrix in level J-2, position Shifting amount be integer sequence [0 ..., M-1]；

By that analogy, the scaling function coefficient approximate solution that each quantity resolution level J is corresponding, and each quantity resolution layer are obtained The wavelet basis function coefficient approximate solution that under level J, each resolution level is corresponding, and bring in formula (2), obtain each quantity level pair Interference data f answered_sResult f (t) of approximate fits be:

$f_s\≈f\left(t\right)={\mathrm{\Σ}}_n{\hat{c}}_{J,n}\mathrm{\Φ}(\frac{t}{2^J}-n)+{\mathrm{\Σ}}_{j=1}^J{\mathrm{\Σ}}_n{\hat{d}}_{j,n}\mathrm{\Ψ}(\frac{t}{2^j}-n)---\left(16\right)$

Step 3: the interference data fitting result obtained in step 2 is carried out uniform sampling interpolation, obtains the interference of uniform sampling Data；

By uniform sampling point [0,1 ..., M-1] bring in the approximate fits result of interference data f (t) obtained in step 2, Interference data to M uniform sampling:

F=[f (0) f (1) f (2) ... f (M-1)]^T (17)

Step 4: the interference data of uniform sampling is carried out the spectroscopic data that fast Fourier transform (FFT) obtains restoring；

According to interference spectrum theory, the interference data of M uniform sampling that will obtain in step 3, carry out FFT, restored Spectrogram.

A kind of interferogram data spectrum recovering method based on wavelet analysis, it is characterised in that: described Wavelet basis function selects Daubechies4 or Symlets4 or Symlets8.