Unlike inflexible structure of soft and hard threshold function, a unified linear matrix form with flexible structure for threshold function is proposed. Based on the unified linear flexible structure threshold function, both supervised and unsupervised subband adaptive denoising frameworks are established. To determine flexible coefficients, a direct meansquare error (MSE) minimization is conducted in supervised denoising while Stein’s unbiased risk estimate as a MSE estimate is minimized in unsupervised denoising. The SURE rule requires no hypotheses or a priori knowledge about clean signals. Furthermore, we discuss conditions to obtain optimal coefficients for both supervised and unsupervised subband adaptive denoising frameworks. Applying an OddTerm Reserving Polynomial (OTRP) function as concrete threshold function, simulations for polynomial order, denoising performance, and noise effect are conducted. Proper polynomial order and noise effect are analyzed. Both proposed methods are compared with soft and hard based denoising technologies—VisuShrink, SureShrink, MiniMaxShrink, and BayesShrink—in denoising performance simulation. Results show that the proposed approaches perform better in both MSE and signaltonoise ratio (SNR) sense.
Signals are usually corrupted by noise in capturing and transmission stages due to environment disturbance and device error. Signal denoising has become an important research topic for a long time and a wide variety of denoising methods have been proposed. Due to their effectiveness and good performance, wavelet threshold methods have become a powerful tool for denoising problems since Donoho and several others’ fundamental works. The main purpose of these methods is to estimate a wide class of functions in some smoothness spaces from their corrupted versions [
Of the various wavelet threshold schemes, soft and hard based threshold methods are the most popular technologies and have been theoretically verified by Donoho and Johnstone [
Soft and hard based threshold schemes suffer their own flaws due to inherent defects of soft and hard threshold functions. For soft threshold schemes, systematical biased estimation could happen, while hard threshold schemes are less biased but less sensitive to small perturbations in the data. In addition, the more important drawback is that soft and hard threshold functions do not have continuous derivatives. Various improvements had been proposed by exploring new threshold functions [
Most of mentioned methods above depend on a single parameter threshold, which makes those methods very sensitive to threshold value and lack of freedom. More flexible and convenient strategies had been proposed by Luisier et al. [
Our contributions can be summarized as follows:
The paper is organized as follows. In Section
In time domain, it is assumed that the clean signal
We only consider orthonormal wavelet transform (OWT), which keeps energy conservation. In OWT domain, the AGWN is still Gaussian. Relationship of wavelet coefficients in vector form is
A threelevel wavelet decomposition (each color represents a level). There are three detail subbands each with a length of
We measure the quality of denoising by meansquare error (MSE) defined as
Wavelet domain denoising generally consists of a threestage procedure. First, perform DWT on noisy signal. Then a threshold function is applied to wavelet domain coefficients. Finally, denoised signal is obtained through an inverse DWT (IDWT). We adopt this threestage procedure to perform our proposed denoising methods. Each subband is adaptively denoised by supervised or unsupervised denoising method as shown in Figure
Proposed wavelet domain denoising schemes.
The general problem can be formulated as a construction of threshold function that minimizes the MSE. By employing linear combination idea, an estimate of a clean signal can be represented by a linear combination of atoms; atoms mean the columns of a matrix. Involving wavelet transform, an estimate of clean wavelet coefficients can be represented by a linear combination of atoms which are decided by noisy wavelet coefficients. A general threshold function can be demonstrated as
In this paper, the adopted concrete threshold function is OTRP function with order
Visual comparison among soft and hard threshold function and a realization of a threeorder
In supervised denoising, a desired signal
In practice, when a desired signal is not available, a direct MSE minimization is impossible. Constructing an estimate of MSE seems a reasonable choice. A practical approach is Stein’s unbiased risk estimate [
This section mainly introduces Stein’s theorem stated in [
The proposed structure of threshold function does not strictly satisfy conditions of
Combining (
Proof is completed.
From (
As it can be seen from (
Optimal solution assurance conditions for both supervised and unsupervised denoising are discussed in this section. To guarantee optimal solution of
Standard experiment settings for latter simulations are given. The adopted signals in simulations are “blocks,” “quadchirp,” “multitone,” and “multiband” with AGWN. In order to explain the superiority, the proposed methods are also tested by real audio signal. All those clean signals are shown in Figure
Clean “blocks,” “quadchirp,” “multitone,” and “multiband,” each signal with length of 8192 and audio signal with a length of 65536.
In supervised denoising, coefficient
Applying the proposed OTRP function as concrete threshold function, supervised and unsupervised denoising methods are compared with several other available techniques, that is, VisuShrink, SureShrink, MiniMaxShrink, and BayesShrink. Wavelab 850 is applied to perform the other four techniques; their noise variance estimates are also provided by MAD. Denoising quality is measured by both MSE and SNR, which are defined as
Before carrying out the final denoising, proper polynomial order needs be explored. Due to the fact that polynomial order is equal to atom number of matrix
Signals 

PO1  PO2  PO3  PO4  PO5  PO6  

SB1  SB2  SB3  SB4  SB5  SB1  SB2  SB3  SB4  SB5  SB1  SB2  SB3  SB4  SB5  SB1  SB2  SB3  SB4  SB5  SB1  SB2  SB3  SB4  SB5  SB1  SB2  SB3  SB4  SB5  
Blocks  0.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100  100 

100  100  100 


0.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100  100 

100  100 




1.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100  100 

100  100 




1.2  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100  100 

100  100 




1.4  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100  100 







1.6  100  100  100  100  100  100  100 

100  100  100  100  100  100  100  100  100  100  100 

100  100  100 








1.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100 








2.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100 










Quadchirp  0.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 


0.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 



1.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 




1.2  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100 





1.4  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 







1.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 








1.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 








2.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 











Multitone  0.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 


100 
0.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 


100  
1.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 



 
1.2  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 



 
1.4  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100 






1.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 


100 






1.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 









2.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 











Multiband  0.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 



0.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 




1.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100  100  100 





1.2  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100 

100 





1.4  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 

100 







1.6  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 









1.8  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 










2.0  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100  100 









Six orders of PO are tested in Table
From Table
Visual comparisons of PO1 to PO3 in supervised denoising are shown in Figure
PO1 to PO3 comparisons in supervised denoising condition.
As mentioned above, comparisons are made among supervised and unsupervised denoising methods and VisuShrink, SureShrink, MiniMaxShrink, and BayesShrink. In both MSE and SNR sense, denoising performance comparisons in global and local views with different noise variance
Comparison of both MSE and SNR changing with noise variance
From global and local views of Figure
Our approaches are also validated for realworld signals. We test supervised and unsupervised denoising approaches by an AGWN corrupted real audio signal. In both MSE and SNR sense, comparisons with changing
Denoising performance comparisons in both MSE and SNR sense with changing noise variance
In this section, different noise levels of desired signal for supervised denoising are analyzed. It only aims at real audio signal. In simulation, desired signal noise levels are set in range
Different desired signal noise level and BayesShrink denoising performance comparisons in both MSE and SNR sense with changing noise variance
In order to utilize the effectiveness of wavelet domain denoising, two subband adaptive denoising schemes were proposed in this paper. For any subband, a unified linear matrix form with flexible structure of threshold function was proposed and OTRP function was proposed for a concrete realization. Based on the unified linear flexible structure threshold function, both supervised and unsupervised subband adaptive denoising frameworks were established. A direct MSE minimization was conducted in supervised denoising while Stein’s unbiased risk estimate as MSE estimate was minimized in unsupervised denoising for flexible coefficients determination. Conditions to obtain optimal coefficients were further discussed.
Applying the concrete OTRP function, simulations for polynomial order, denoising performance, and noise effect were conducted. A full rank probability statistical table was generated in polynomial order simulation. The table reflects that PO1–PO3 always ensure full rank of
The authors declare that there is no conflict of interests.
This work is fully supported by National Science and Technology Major Project of the Ministry of Science and Technology of China (no. Y6D8020801).