An improved wavelet threshold function denoising method based on IGA optimization

2.1. The basic principle of wavelet threshold denoising

Wavelet threshold denoising decomposes a signal using the wavelet transform to target high-frequency noise. The wavelet coefficients are next treated using a threshold: hard thresholding puts coefficients below the threshold to zero, and soft thresholding decreases coefficients above the threshold while setting those below to zero. The treated coefficients are reassembled to generate the denoised signal. The flow chart for this procedure is depicted in Fig. 1 [7].

Fig. 1Flow chart of wavelet threshold denoising

2.2. Soft and hard threshold functions

Soft threshold functions and hard threshold functions as shown in Eq. (1) and Eq. (2). ${\omega }_{j,k}$ and ${\widehat{\omega }}_{j,k}$ respectively represent the coefficients before and after denoising, and $\lambda$ is the predetermined threshold:

The soft threshold function has the advantage of being smoothed and maintaining continuity at the threshold setting. However, there is a fixed discrepancy between the wavelet coefficient processed by the soft threshold function and the actual signal, resulting in distortion of the reconstructed signal.

The hard threshold function can effectively preserve the original signal's edge characteristics, yet it suffers from discontinuity. Specifically, discontinuities arise at threshold values of +$\lambda$ and –$\lambda$, leading to oscillations and significant variance in the reconstructed signal, ultimately causing signal deviation [8].

2.3. Improved wavelet threshold function

Eq. (3) demonstrates the proposed improved wavelet threshold function in this study, which integrates the advantages and limitations of the conventional wavelet threshold function:

In the Eq. (3), the wavelet coefficient after threshold processing is denoted as ${\widehat{\omega }}_{j,k}$, while the wavelet coefficient before threshold processing is denoted as ${\omega }_{j,k}$. The set threshold is represented by $\lambda$, and the adjustable parameter is denoted by $\alpha$. The three denoising functions are shown in Fig. 2.

The adjustable parameter α is introduced in the modified wavelet threshold function, which is based on the traditional threshold function. This addition can enhance the function’s flexibility to accommodate a range of signals, and the size of $\alpha$ can be suitably adjusted for different signal processing scenarios. The improved wavelet threshold function approaches the hard threshold function more closely the greater $\alpha$, and vice versa. Between the soft and hard threshold functions lies the improved wavelet threshold function. As ${\omega }_{j,k}$ approaches infinity, $\alpha +\mathrm{e}\mathrm{x}\mathrm{p}\left|{\omega }_{j,k}-\lambda \right|$ also approaches infinity, making ${\widehat{\omega }}_{j,k}$ approach ${\omega }_{j,k}$. This improved wavelet threshold algorithm addresses the consistent deviation and distortion in the reconstructed signal associated with the soft threshold technique. When ${\omega }_{j,k}$ approaches the threshold $\lambda$, $\alpha +\mathrm{e}\mathrm{x}\mathrm{p}\left|{\omega }_{j,k}-\lambda \right|$ approaches $\alpha +1$, this improvement tackles the signal discontinuity issue previously linked to the hard threshold function.

Due to the different selection of adjustable parameter $\alpha$, the denoising effect will be very different, so IGA is selected to optimize the adjustable parameter $\alpha$, which forms the denoising effect. To produce the best denoising effect, the wavelet threshold function with the higher SNR is built.

Fig. 2Soft threshold, hard threshold and improved wavelet threshold function

3.1. Improvement of fitness function

The fitness function evaluates individuals and affects their selection probability, influencing evolution and optimization. Normalizing the fitness function scales the values, improving selection and population evolution [9]. This paper uses normalization to enhance the fitness function.

Every individual’s fitness value is computed after the fitness function’s initial value range has been established. These values are then mapped to the interval [0, 1] using a linear normalization method, as shown in Eq. (4). Where min and max are the minimum and maximum fitness function values respectively, and $f^{\text{'}}\left(x\right)$ is the normalized fitness function value. Finally, the normalized fitness function value is calculated as the new fitness value of each individual in the genetic algorithm:

The improved genetic algorithm ensures that individual fitness values are on a uniform scale, avoiding instability from different fitness ranges, and enhances both convergence speed and optimization results.

3.2. Improvement of genetic strategies

Since the solutions obtained by the genetic algorithm are random, elite and roulette choices are used to avoid sub-optimal solutions. Elite selection keeps the most suitable individuals in the group, while roulette selection individuals based on their fitness ratio.

In each generation, the elite individuals are first retained and copied into the next generation to maintain their genetic traits. Roulette selection is then used to choose the remaining individuals based on their fitness values to form the parent population for crossover and mutation. Crossover creates new offspring from the selected parents, and mutation randomly alters the offspring’s genes with a set probability. Finally, the elite individuals are combined with the new offspring to form the next generation, which will continue to evolve.

Elite selection makes sure the optimal individual is not overlooked and enhances the algorithm’s capacity for worldwide search. Roulette selection preserves population diversity and prevents premature convergence to local optimality. Together, they balance population diversity and convergence rate.

3.3. IGA optimization parameter steps

(1) Initialize the population and determine the optimal interval through chromosome coding.

(2) The fitness function is determined to be the SNR, whose function expression is shown in Eq. (5). Where, $s\left(i\right)$ is the original signal, $\widehat{s}\left(i\right)$ is the signal processed by the wavelet threshold function, and $n$ is the signal length:

(3) The fitness value is calculated by the fitness function of the normalized treatment. According to the obtained fitness values, individuals are selected with an improved selection strategy.

(4) If the termination condition is satisfied, the optimal parameters can be output, and the improved wavelet threshold function can be determined to denoise the signal.