Two-dimensional signal time scale and scale estimation implementation method based on scale transformation
1. A two-dimensional signal time scale and scale estimation implementation method based on scale transformation is characterized by comprising the following steps:
step (I): for the original signal x (t)1,t2) Performing two-dimensional scale transformation (2D-ST) to obtain Dx(c1,c2) Wherein t is1、t2Is a two-dimensional time variable, c1、c2Is a two-dimensional scale variable;
step (II): according to a two-dimensional scale factor alpha1、α2To D, pairx(c1,c2) Performing phase correction to obtain phase-corrected conversion resultWherein the content of the first and second substances,
step (three): to pairTwo-dimensional inverse scale transformation (2D-IST) is carried out to obtain x (t)1,t2) Time-scaled signal y (t)1,t2)。
2. A two-dimensional signal time scale and scale estimation implementation method based on scale transformation is characterized by comprising the following steps:
step (I): the scale factor signal f (t) to be estimated1,t2) Template signal h (t)1,t2) Respectively performing two-dimensional scale transformation (2D-ST) to obtain Df(c1,c2) And Dh(c1,c2)；
Step (II): to Df(c1,c2) Conjugation with Dh(c1,c2) Performing dot multiplication to obtain
Step (three): to pairPerforming two-dimensional inverse scale transformation (2D-IST) to obtain f (t)1,t2) And h (t)1,t2) Two-dimensional scale cross-correlation function phifh(α1,α2)；
Step (IV): take phifh(α1,α2) Carrying out peak value search on the envelope, and obtaining f (t) according to the peak value coordinate1,t2) Two-dimensional scale factor estimationAnd
3. the method for realizing time scale and scale estimation of two-dimensional signals based on scale transformation according to claim 1, wherein the two-dimensional scale transformation (2D-ST) is represented by the following formula:
wherein S {. is 2D-ST denotes a symbol, Dz(c1,c2) Is a two-dimensional signal z (t)1,t2) 2D-ST of (1).
4. The method for realizing time scale and scale estimation of two-dimensional signals based on scale transformation according to claim 1, wherein the expression of two-dimensional inverse scale transformation (2D-IST) is:
wherein S is-1Is a 2D-IST symbol.
Background
Time-scaling (TS) and Scale-estimation (SE) are two novel signal processing methods, and are widely applied in the fields of radar signal processing, Linear Frequency Modulation (LFM) detection, Scale invariant system design, ultrasonic temperature compensation, and the like. TS is the quantitative temporal expansion or contraction of the original signal according to the known scale factor, and SE is the estimation of the scale factor when the original signal and the scale signal are known. For the continuous signal TS, the most direct method is to perform interpolation extraction on the original signal, for example, the scale factor is 0.5, the signal length is increased by 1 time, which can be realized by 2 times of interpolation, when the scale factor is 0.51, the interpolation needs to be performed by 100 times, and then the extraction needs to be performed by 51 times, which is obviously unacceptable. Aiming at the problem of quickly realizing TS and SE of one-dimensional signals, a method [ J ] for accumulating phase coding signals for a long time based on Keystone transformation, Beijing university of science and technology, 2009,29(1):55-57] proposes to realize the method by using Discrete Fourier Transform (DFT), but the method has lower execution precision, and a method [ ZHAO Y B, WANG J, HUANG L, et al. Low complex key transformation for directly outputting and transforming target [ C ]/IEEE CIE International Conference,2011:1745 transition 8] proposes to realize the method by using Chirp-Z transformation (Chirp-Z174form, CZT), but the method has the problems of inherent frequency aliasing and frequency point loss. For two-dimensional signals TS and SE, no effective quick implementation method is available at present.
The fourier transform is a classical integral transform, and in its turn, discloses another fundamental physical quantity, frequency, in nature, except for time. Cohn considers that Scale is a physical property like frequency, and the link linking time and Scale is Scale Transform (ST). ST is a special Mellin transform (the real part of a complex variable is 0.5), and the original signal and the scale signal ST have the same envelope and different phases due to the inherent scale invariance characteristic. The invention expands ST to two dimensions, and provides a rapid implementation scheme of TS and SE of two-dimensional signals by using the ST dimension invariance characteristic and combining Fast Scale Transform (FST) and Inverse Fast Scale Transform (IFST).
Disclosure of Invention
The invention aims to provide a method for realizing two-dimensional signal time scale and scale estimation based on scale transformation, so as to solve the problem of quick realization of the two-dimensional signal time scale and scale estimation. To illustrate the present invention, first, two-dimensional scale transformation and essential meaning are introduced; secondly, introducing a two-dimensional scale transformation numerical calculation method; thirdly, introducing a two-dimensional time scale and a scale estimation concept; then, the basic principle of the present invention is introduced; finally, the method steps of the invention are presented.
(1) Two dimensional scale transformation
The Scale Transform (ST) is a special mellin Transform, represented by the formula:
wherein, S {. is ST represents symbol, t is time, c is scale, Df(c) ST, γ (c, t) for signal f (t) is a characteristic function of ST, specifically:
the phase derivative is easily obtained with an instantaneous frequency c/t. γ (c, t) satisfies completeness and orthogonality, i.e.:
Df(c) it can be understood that the signal f (t) is at a set of different scale basis functions gamma*And (c, t) is a coefficient. The simplified form of ST is:
inverse Scale Transform (IST) is:
wherein S is-1{. is IST notation, simplified form:
expand ST to two dimensions to obtain signal f (t)1,t2) The Two-dimensional Scale Transform (2D-ST) of (1) is:
wherein D isf(c1,c2) Is the signal f (t)1,t2) 2D-ST, gamma (c)1,t1,c2,t2) As a 2D-ST characteristic function, c1、c2Is a two-dimensional scale independent variable, t1、t2Is a two-dimensional time independent variable. The Two-dimensional Inverse Scale Transform (2D-IST) is expressed as:
(2) two-dimensional scale transformation numerical calculation method
Let t be euAnd easily obtaining:
it can be seen that ST is actuallyThe fourier transform of (a) can be obtained by sampling f (t) exponentially, amplitude modifying,then, Fast Fourier Transform (FFT) is used to realize the Fast Scale Transform (FST). The following are easy to know:
similarly, IST can be implemented by Inverse Fast Fourier Transform (IFFT) to obtain Inverse Fast Scale Transform (IFST), and the numerical calculation process is opposite to FST, and D is first calculatedf(c) And then performing logarithmic sampling. In the same way, the following can be obtained:
it can be seen that 2D-ST and 2D-IST can be rapidly realized by using two-dimensional fast Fourier transform (2D-FFT) and two-dimensional inverse fast Fourier transform (2D-IFFT) to obtain two-dimensional fast scale transform (2D-FST) and two-dimensional inverse fast scale transform (2D-IFST), the calculation idea is the same as FST and IFST, and the dimension is increased to two dimensions.
(3) Two-dimensional time scale and scale estimation
The Time-scaling (TS) mapping process of signals x (t) to y (t) can be expressed as:
wherein, TSα[·]For one-dimensional TS, the symbol is expressed, and alpha is belonged to R+Is a scale factor, and is a function of,to keep the signal energy before and after TS the same, namely:
it can be seen that TS on a signal results in time domain expansion (0 < α < 1) or contraction (0 < α < 1), as is known from the fourier transform scale property:
wherein X (f), Y (f) are Fourier transforms of x (t) and y (t), respectively. It can be seen that TS on the signal also results in spectral contraction (0 < α < 1) or expansion (0 < α < 1). Expanding the TS concept to two dimensions, wherein the expression of a two-dimensional signal TS is as follows:
wherein, x (t)1,t2) Is TS preamble, y (t)1,t2) Is a post TS signal, α1、α2Is a two-dimensional scale factor, alpha1∈R+、α2∈R+，Symbols are represented for a two-dimensional TS. x (t)1,t2)、y(t1,t2) The following are also satisfied:
so-called Scale-estimation (SE), i.e. the estimation of the Scale factor when x (t) and y (t) are known. To estimate the scale factor, a scale cross-correlation function of y (t) with x (t) is calculated:
where o is the scale cross correlation symbol and is the conjugate transpose. Phiyx(α) the scale factor corresponding to the maximum value is the scale factor estimation:
wherein the content of the first and second substances,is a scale factor estimate. The above expression can be understood as a search process for y (t) scale factors using x (t) as a template signal. Likewise, the SE concept is extended to a two-dimensional, two-dimensional signal y (t)1,t2) And x (t)1,t2) The scale cross correlation function of (a) is:
Φxy(α1,α2) The scale factor corresponding to the maximum value is the two-dimensional scale factor estimation:
wherein the content of the first and second substances,is a two-dimensional scale factor estimate.
(4) Basic principle of the invention
To solve the TS implementation problem, reference is made to ST scale invariance, and the signal y (t) is set to TSα[x(t)]And then:
Dy(c)＝eiclnαDx(c)
wherein D isx(c)、Dy(c) ST for x (t), y (t), respectively. When x (t), α are known, y (t) can be expressed as:
y(t)＝S-1{eiclnαDx(c)；t}
the time scale y (t) for x (t) can be obtained using the above formula. To estimate the scale factor, a scale cross-correlation function of y (t) and x (t), phi, is calculatedyx(α) can also be achieved with ST, i.e.:
is provided withIs easy to obtain:
the two-dimensional signal x (t) can be obtained by using the formula1,t2) Time scale y (t)1,t2). In the same way, y (t)1,t2) And x (t)1,t2) The scale cross-correlation function may be further expressed as:
peak search may yield y (t)1,t2) Two-dimensional scale estimation of (2).
(5) Method steps of the invention
1. The method for realizing the time scale of the two-dimensional signal mainly comprises the following steps:
step one, carrying out two-dimensional scale transformation on an original signal;
performing phase correction on a transformation result according to the two-dimensional scale factor;
and (III) performing two-dimensional inverse scale transformation on the phase-corrected result.
The steps are as follows:
step (I): for the original signal x (t)1,t2) Performing two-dimensional scale transformation (2D-ST) to obtain Dx(c1,c2) Wherein t is1、t2Is a two-dimensional time variable, c1、c2Is a two-dimensional scale variable;
step (II): according to a two-dimensional scale factor alpha1、α2To D, pairx(c1,c2) Performing phase correction to obtain phase-corrected conversion resultWherein the content of the first and second substances,
step (three): to pairTwo-dimensional inverse scale transformation (2D-IST) is carried out to obtain x (t)1,t2) Time-scaled signal y (t)1,t2)。
2. The two-dimensional signal scale estimation implementation method mainly comprises the following steps:
respectively carrying out two-dimensional scale transformation on a scale factor signal to be estimated and a template signal;
step two, conjugate is taken for the scale factor signal transformation result to be estimated, and dot multiplication is carried out on the result and the template signal transformation result;
performing two-dimensional inverse scale transformation on the dot product result to obtain a two-dimensional scale cross-correlation function of the scale factor signal to be estimated and the template signal;
and step four, performing peak value search on the cross-correlation function, and obtaining two-dimensional scale factor estimation according to the peak value coordinate.
The steps are as follows:
step (I): the scale factor signal f (t) to be estimated1,t2) Template signal h (t)1,t2) Respectively performing two-dimensional scale transformation (2D-ST) to obtain Df(c1,c2) And Dh(c1,c2)；
Step (II): to Df(c1,c2) Conjugation with Dh(c1,c2) Performing dot multiplication to obtain
Step (three): to pairPerforming two-dimensional inverse scale transformation (2D-IST) to obtain f (t)1,t2) And h (t)1,t2) Two-dimensional scale cross-correlation function phifh(α1,α2)；
Step (IV): take phifh(α1,α2) Carrying out peak value search on the envelope, and obtaining f (t) according to the peak value coordinate1,t2) Two-dimensional scale factor estimationAnd
the beneficial effects of the invention are illustrated as follows:
the invention expands the TS and SE concepts to two dimensions, and can solve the problem of quick realization of time scale and scale estimation of two-dimensional signals by utilizing the ST scale invariance characteristic.
Drawings
FIG. 1 is a flow of implementing two-dimensional signal time scale and scale estimation;
FIG. 2 is a digital image and two-dimensional scale transformation result;
FIG. 3 is a digital image time scale processing result;
FIG. 4 is a template signal and a scale factor signal to be estimated;
FIG. 5 is a two-dimensional scale transformation result of a template signal and a scale factor signal to be estimated;
FIG. 6 is a two-dimensional scale cross-correlation function of a template signal and a scale factor signal to be estimated;
fig. 7 is a graph of two-dimensional signal time scale and scale estimation processing accuracy.
Detailed description of the invention
The present invention will be described in detail below with reference to the accompanying drawings. FIG. 1 is a flow of implementing two-dimensional signal time scale and scale estimation; firstly, a ship image of 300 x 800 is used as a two-dimensional signal to verify the feasibility of the two-dimensional signal time scale implementation method. The digital image and the two-dimensional scale transformation result are shown in figure 2; setting scale factors as 0.8 and 1.5 respectively, and performing time scale on the ship image according to the flow of the attached figure 1(a), wherein the result is shown in the attached figure 3; secondly, scale estimation is carried out on the scale factor signals to be estimated (the real scale factors are respectively 1.5 and 0.8) by taking two-dimensional LFM signals (the time widths are respectively 25 mus and 20 mus, the bandwidths are both 5MHz and the sampling frequency is 20MHz) as template signals, and feasibility of the two-dimensional signal scale estimation implementation method is verified. The two-dimensional scale transformation results of the template signal and the scale factor signal to be estimated are shown in figure 5; FIG. 6 is a two-dimensional scale cross-correlation function; and finally, evaluating the processing precision of the method by taking the Pearson Correlation Coefficient (PCC) of the signal after the time scale and the real signal and Mean Relative Error (MRE) of the real scale factor estimation as indexes. The two-dimensional scale factors are 0.5-1.5, the interval is 0.01, the change trend of PCC along with the two-dimensional scale factors is shown in figure 7(a), and the change trend of scale factor estimation MRE along with the two-dimensional scale factors is shown in figure 7 (b); as can be seen from fig. 1, in order to calculate the time scale of the two-dimensional signal, 1 time of 2D-FST, 1 time of complex product, and 1 time of 2D-IFST are required, and 2D-FST, 1 time of complex product, and 1 time of 2D-IFST are required to calculate the two-dimensional scale cross-correlation function between the scale factor signal to be estimated and the template signal; as can be seen from the attached figure 2, the two-dimensional scale transformation result of the ship image has an obvious single peak; as can be seen from fig. 3, the invention can effectively realize two-dimensional signal time scale; as can be seen from fig. 4, the scale factor signal to be estimated is a two-dimensional scale version of the template signal; as can be seen from fig. 5, the two-dimensional scale transformation envelopes of the template signal and the scale factor signal to be estimated are relatively similar; as can be seen from fig. 6, the scale factors corresponding to the maximum peak of the correlation function are 1.5 and 0.8024, respectively, and are substantially consistent with the true scale factors 1.5 and 0.8; as can be seen from FIG. 7, the PCC is greater than 0.89 under different scale factors, and the MRE estimated by the scale factors is less than 1%, which indicates that the method has higher processing precision.
