Research on Technical Support and Resource Sharing Mechanism Strategy of Mixed Teaching Mode in Piano Music Education

If the function ψ(x)∈L²(R) satisfies the condition: (1) Cψ=∫R|ψ^(ω)|2|ω|dω<∞ \[{{C}_{\psi }}=\int_{R}{\frac{|\hat{\psi }(\omega ){{|}^{2}}}{|\omega |}}d\omega <\infty \] where ψ^(ω)$\hat{\psi }(\omega )$ is the Fourier transform of ψ(x) such that: (2) ψa,b(x)=|a|−12ψ(x−ba) \[{{\psi }_{a,b}}(x)=|a{{|}^{-\frac{1}{2}}}\psi \left( \frac{x-b}{a} \right)\]

Then we have f(x)∈L²(R). The continuous wavelet transform of a function is defined as: (3) Wf(a,b)=〈 f,ψa,b 〉=|a|−12∫Rf(x)ψ(x−ba)dx \[{{W}_{f}}(a,b)=\left\langle f,{{\psi }_{a,b}} \right\rangle =|a{{|}^{-\frac{1}{2}}}\int_{R}{f}(x)\psi \left( \frac{x-b}{a} \right)dx\]

Its corresponding inverse transformation is: (4) f(x)=Cψ2−12∬R2Wf(a,b)ψa,b(x)dadba2 \[f(x)=C_{{{\psi }^{2}}}^{-\frac{1}{2}}\iint_{{{R}^{2}}}{{{W}_{f}}}(a,b){{\psi }_{a,b}}(x)\frac{dadb}{{{a}^{2}}}\]

In the above equation, ψ(x) is called the fundamental wavelet, and equation (2) is called the “tolerance condition”, and ψ_a,b(x) is also the result of shifting and stretching of the fundamental wavelet ψ(x). The reason why ψ(x) is called “wavelet” is that the function of the fundamental wavelet must satisfy ∫R| ψ(x) |dx<∞$\int_{R}{\left| \psi (x) \right|}dx<\infty $, i.e., ψ(x) has the attenuation property, and ψ(x) is a locally non-zero tightly branched function. ψ(x) is called a “wave” by the boundedness of the integral of the tolerance theorem, i.e., the Fourier transform of ψ(x) must be zero when ω = 0, i.e.,: (5) ∫−∞∞ψ(x)dx=0 \[\int_{-\infty }^{\infty }{\psi }(x)dx=0\]

Otherwise C_ψ would tend to infinity at ω = 0. This means that ψ(x) is volatile.

Since x,b is a continuous variable in equation (3), it is called continuous wavelet transform (CWT). The difference between continuous wavelet and Fourier transform is that the Fourier transform f^(ω)$\hat{f}(\omega )$ of f(x) represents the content of the signal f(x) at frequency ω, so the Fourier transform is a method of frequency domain analysis, whereas the Fourier inverse transform is the reconstruction of f(x) from f^(ω)$\hat{f}(\omega )$. For the continuous wavelet transform, it is the transformation of the signal f(x) into an information W_f(a,b) containing two parameters a,b, where a is the scaling factor so that its change represents the change of frequency and b is the translation factor, and its change represents the change in time. In this way, the continuous wavelet transform W_f(a,b) is a time-frequency domain transform.

In signal processing, continuous wavelets and their wavelet transforms need to be discretized in order to be realized by a computer. As a convenient form, computer implementation of the discrete wavelet is often binary discretization, and the discretized wavelet and the corresponding wavelet transform is called the discrete wavelet transform (DWT). The discrete wavelet transform is obtained by discretizing the scales and displacements of the continuous wavelet transform according to powers of 2, so it is also called the binary wavelet transform.

In continuous wavelets, consider the function (2), where b∈R,a∈R, and a ≠ 0,ψ are admissible, and for convenience, in discretization, such that a can only take positive values, so that the admissibility condition for discrete wavelets is: (6) Cψ=∫R+|ψ^(ω¯)|2|ω¯|dω¯<∞ \[{{C}_{\psi }}=\int_{{{R}^{+}}}{\frac{|\hat{\psi }(\bar{\omega }){{|}^{2}}}{|\bar{\omega }|}}d\bar{\omega }<\infty \]

The scale parameter a and the translation parameter b in the continuous wavelet transform are generally discretized as a=a0j$a=a_{0}^{j}$,b=ka0jb0$b=ka_{0}^{j}{{b}_{0}}$, where j∈Z, a₀ are fixed and not equal to 1. So the discrete wavelet function ψ_j,k(x) corresponding to the fundamental wavelet ψ(x) can be written: (7) ψj,k(x)=a0−j2ψ(x−ka0jb0a0j)=a0−j2ψ(a0−jx−kb0) \[{{\psi }_{j,k}}(x)=a_{0}^{-\frac{j}{2}}\psi \left( \frac{x-ka_{0}^{j}{{b}_{0}}}{a_{0}^{j}} \right)=a_{0}^{-\frac{j}{2}}\psi \left( a_{0}^{-j}x-k{{b}_{0}} \right)\]

And the discretized wavelet coefficients can be expressed as: (8) Cj,k=∫−∞∞f(x)ψj,k*(x)dx=<f,ψj,k> \[{{C}_{j,k}}=\underset{-\infty }{\overset{\infty }{\mathop \int }}\,f(x)\psi _{j,k}^{*}(x)dx=<f,{{\psi }_{j,k}}>\]

Its reconstruction formula is: (9) f(x)=C∑−∞∞∑−∞∞Cj,kψj,k(x) \[f(x)=C\sum\limits_{-\infty }^{\infty }{\sum\limits_{-\infty }^{\infty }{{{C}_{j,k}}}}{{\psi }_{j,k}}(x)\]

C is a signal-independent constant.

Multi-resolution analysis is established by a scale function, so the establishment of multi-resolution analysis is equivalent to finding the nature of the scale function in the framework of multi-resolution analysis, and the following scale function equation relation is established according to V_j⊆V_j+1 as well as V_j+1 = V_j⊕V_j.

Assuming that {V_n;n∈Z} is an orthogonal multiresolution analysis with a scale function φ , the following scale relation holds: (10) φ(x)=∑k∈Zhkφ(2x−k) \[\varphi (x)=\sum\limits_{k\in Z}{{{h}_{k}}}\varphi (2x-k)\]

Among them: (11) hk=2∫−∞+∞φ(x)φ(2x−k)¯dx \[{{h}_{k}}=2\int_{-\infty }^{+\infty }{\varphi }(x)\overline{\varphi (2x-k)}dx\]

And there is: (12) φ(2j−1x−l)=∑k∈Zhk−2iφ(2jx−k) \[\varphi \left( {{2}^{j-1}}x-l \right)=\sum\limits_{k\in Z}{{{h}_{k-2i}}}\varphi \left( {{2}^{j}}x-k \right)\]

Eq. (12) can sometimes be equivalently expressed as: (13) φj−i,l(x)=22∑k∈Zhk−2lφj,k(x) \[{{\varphi }_{j-i,l}}(x)=\frac{\sqrt{2}}{2}\sum\limits_{k\in Z}{{{h}_{k-2l}}}{{\varphi }_{j,k}}(x)\] φj,l(x)=2j2φ(2jx−k)${{\varphi }_{j,l}}(x)={{2}^{\frac{j}{2}}}\varphi \left( {{2}^{j}}x-k \right)$ of them.

The concept of multiresolution analysis can be viewed as a mathematical description of target awareness from visual observation. As the scale increases, the closer to the target, the more information it contains. As the scale decreases, the further away from the target, the less information it contains.

Wavelet denoising is essentially a function of the approximation process, is composed of wavelet function translation and expansion of the function space to find the best approximation of the image, in order to achieve the purpose of distinguishing between the noise and the original image.

Threshold denoising method is to set a threshold T by calculation, then the wavelet coefficients obtained from decomposition are thresholded, and finally the denoising process is completed by inverse transform.

Set y is the original wavelet coefficients, T is the threshold value, and T(y) is the thresholded wavelet coefficients. There are two commonly used threshold functions:

Hard threshold function: (14) Th(y)={ 0|y|<Ty|y|≥T \[{{T}_{h}}(y)=\left\{ \begin{array}{*{35}{l}} 0 & |y|<T \\ y & |y|\ge T \\ \end{array} \right.\]

Soft Threshold Functions: (15) Ts(y)={ 0|y|<Tsgn(y)(|y|−T)|y|≥T \[{{T}_{s}}(y)=\left\{ \begin{matrix} 0 & |y|<T \\ sgn (y)(|y|-T) & |y|\ge T \\ \end{matrix} \right.\]

Wavelet threshold denoising method is the most widely used, and the algorithm is computationally small and simple to implement. Therefore, this paper chooses wavelet threshold denoising method as the denoising method of this design.

Before regression analysis, it is necessary to determine whether there is a certain correlation between the variables, so this study used Pearson product-difference correlation analysis with two-tailed test for the research object to detect whether there is a correlation between the variables. If the correlation coefficient takes values between 0 and 1, it means that there is a positive correlation between the variables.If the correlation coefficient is between -1 and 0, it indicates that the variables are negatively correlated. The study used SPSS22.0 software to analyze the correlation between the three dimensions of blended learning and learning effectiveness, and the results of the correlation analysis between blended learning and learning effectiveness are shown in Table 2.

The mean values of the five-variable measurements of blended learning, cognitive dimension, skill dimension, affective attitude, and learning effect were 3.224, 15.986, 30.021, 21.447, and 21.438, respectively, and the standard deviations of the five variables were 0.618, 3.315, 6.728, 5.483, and 5.012, respectively.

From the Pearson coefficient, it can be seen that the p-value of cognitive dimensions, skill dimensions, affective attitudes, and learning effects in the students’ learning effects are less than 0.01, and have a significant positive correlation with the blended learning, which can be considered that the variables have the conditions for regression analysis.

Introducing control variables (grade, gender, major) and dependent variables cognitive, skill, and affective attitude learning effects for regression analysis, the results are shown in model 1. Then the independent variable blended learning is introduced and the results are shown in Model 2. The impact of blended learning on students’ learning outcomes is analyzed as shown in Table 3.

Cognitive model 1 shows that the grade, gender, and major control variables do not have a significant effect on the dependent variable cognitive learning effect, and at this time the 1 of the regression model is 0.003. When the independent variable blended learning is added to model R², the R² of the regression model is significantly increased to 0.268, indicating that blended learning explains 37.8% of the variance variance of the cognitive learning effect, and the regression coefficient of the blended learning on cognitive learning effect is 0.587, with a p-value significant at the 0.001 level. The regression coefficient of blended learning is 0.587, with a p-value significant at the 0.001 level, and the DW value is 1.961, indicating a positive correlation between blended learning and cognitive learning effects. Similarly, there is a positive correlation between blended learning and the learning effects of skills and affective attitudes.