# Signal restoration via a splitting approach

- Bushra Jalil
^{1}Email author, - Eric Fauvet
^{1}and - Olivier Laligant
^{1}

**2012**:38

https://doi.org/10.1186/1687-6180-2012-38

© Jalil et al; licensee Springer. 2012

**Received: **2 May 2011

**Accepted: **20 February 2012

**Published: **20 February 2012

## Abstract

In the present study, a novel signal restoration method from noisy data samples is presented and is termed as "signal split (SSplit)" approach. The new method utilizes Stein unbiased risk estimate estimator to split the signal, the Lipschitz exponents to identify noise elements and a heuristic approach for the signal reconstruction. However, unlike many noise removal techniques, the present method works only in the non-orthogonal domain. Signal restoration was performed on each individual part by finding the best compromise between the data samples and the smoothing criteria. Statistical results are quite promising and suggest better performance than the conventional shrinkage. Furthermore, the proposed method preserves the energy of the sharp peaks and edges which, is not however, the case for classical shrinkage methods.

### Keywords

continuous wavelet transform wavelet transform modulus maxima split or segmentation Stein unbiased risk estimate thresholding modulus maxima Lipschitz exponent## 1 Introduction

In the past two decades, wavelet transform has been used as significant non-parametric estimation tool to extract noise elements from the signal. Antoniadis et al provided an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimator developed to denoise data [1]. Among these denoising techniques, the modulus maxima approach proposed by Mallat et al. has received the most attention in continuous and non-orthogonal domains [2, 3]. Although many researchers have proposed different methods to estimate signals from the evolution of the wavelet transform modulus maxima (WTMM) across different scales [4–11], still the proposed reconstruction process are either complicated or computationally expensive. More recently, Chen at al. presented a neighboring coefficients based multi wavelets denoising method [12].

Another domain of denoising techniques studies on the principle of shrinkage [13]. The shrinkage method uses nonlinear thresholding approach to shrink the orthogonal wavelet coefficients as a denoising tool. This method relies on the idea that the energy of the function is often concentrated in a few wavelet coefficients while the energy of the noise is spread over all coefficients, therefore by selecting a suitable threshold value it is possible to reduce a significant amount of noise elements by using nonlinear thresholding in the wavelet domain [13]. However, the denoised signal in this case may contain spurious oscillations due to the translation variant property of the discrete wavelet transform. Afterwards, several approaches were proposed to overcome these problems of shrinkage but so far none of them guarantees the preservation of edges or sharp variations possibly due to their working in an orthogonal basis [13–17]. Fodor and Kamath presented a brief survey of some of these approaches [17]. More recently, Blu and Luisier presented the state of the art SURE-LET method for image denoising [18]. The performance of the SURE-LET method is quite outstanding in the presence of Gaussian noise. In the present study, we are addressing only 1D signal therefore it is not possible to compare with the SURE-LET method. However in the future study, the extension of the proposed method for 2D cases will give logical comparison of the two methods.

The main motivation of the current study is to overcome the existing problems in the previously proposed methods and to preserve the transitions at a relatively lower computational cost without introducing any oscillations. The new technique worked on the idea of splitting the signal. The method segments the signal based on the sharpness of its transitions. In this way, the main transitions presented inside the signal have been preserved at the initial stage. This segmentation is performed by Stein's unbiased risk estimate (SURE) based nonlinear thresholding of WTMM. Once the signal has been split into different sub signals, Lipschitz exponents were computed for all the significant transitions present inside each sub signal. These Lipschitz exponents permit to separate the regular or smooth points from the noise elements. At the final stage, signal reconstruction involves in finding the best compromise between the data points and the smoothness criterion. At the end, all sub signals were merged together to reconstruct the fully denoised signal.

The article is organized as follows. The principle of the method is given in Section 2. Section 3 explains the proposed splitting approach. The proposed noise removal technique for the subs signals and the reconstruction method is given in Section 4. Results and comparative analysis of the SSplit method with the SURE Shrink method [13] is given in Section 5 and finally, Section 6 concludes the study. The summary of the method is given in Appendix section.

## 2 Principle of the method

*Y*such that:

*F*(

*t*

_{ i }) is the deterministic function with

*t*

_{ i }= (

*i*-1)/

*N, N*is the total number of samples and

*ϵ*

_{ i }is white gaussian noise

*N*(0,

*σ*

^{2}). The aim of the current study is to estimate the function

*F*with the minimum mean square error (MSE). The MSE of an estimator $\widehat{F}$ with respect to the estimated parameter

*F*. In this study, all MSE were computed between restored signal and original (noise free) signal. To estimate the function

*F*, the method proposes to split the function

*Y*into its subsets in spatial domain based on the sharpness of transitions or edges such that:

*Y*denotes the

*set*of all samples and each subset

*Y*

_{ i }, with

*i*= 1, ...,

*J*contains

*K*

_{ i }adjacent samples

*y*

_{ i,l }where

*l ϵ*1, ...

*K*

_{ i }.

*J*is the total number of subset and

*K*

_{ i }is the length of each respective subset. Piecewise analysis was performed on each subset of the function

*Y*. The selection of subsets is defined by SURE based nonlinear thresholding of WTMM. Once the signal has been split into subsets, the WTMM approach was applied on each subset individually to compute the Lipschitz exponents of the transitions in each subset. These Lipschitz exponents identify the regular or smooth points in the signal. The nonlinear function is then optimized between all the regular points using an iterative method to restore the signal from each subset. Finally, all subsets were merged together to reconstruct the fully denoised signal. Four different types of synthetic signals used in this study are taken from the work by Donoho et al.: blocks, bumps, heavisine and dopplers as shown in Figure 1.

## 3 Signal splitting method (SSplit)

*y*(

*t*) to compute the modulus maxima by using an integrable function [2, 3]:

where *WT*(*u, s*) is the wavelet coefficient of the function *y*(*t*), *Ψ*(*t*) is the analyzing wavelet, *s*(> 0) is the scale parameter and *u* is the position parameter. The Gaussian function (*θ*(*t*)) has an important property of continuous differentiability, which makes this function suitable for the analysis of most types of signals. Therefore, the derivative of the Gaussian function has been used as a wavelet analyzing function (*Ψ*(*t*)) for the splitting method. These WTMM computed by using the derivative of the Gaussian wavelet is defined as any point (*u*_{0}, *s*_{0}) such that ∥*Wf*(*u, s*_{0})∥ has a local maximum at *u* = *u*_{0} [2, 3].

*t*; minimizing this in

*t*gives a selection of the threshold level for that level

*j*(

*j*= 1, 2 ...,

*J*) [19]. SURE-based thresholding on modulus maxima results in splitting the signal into subsets based on the sharpness of transitions. Figure 2 shows the thresholding results and respective splitting points on decomposed signal. The first derivative of Gaussian function is used as a mother wavelet function on the block function with the sample size of 2048. Tracing these modulus maxima lines from coarser scales to finer scale gives the split edges at a convergence point on the finest scale. In short, a nonlinear soft thresholding at the decomposition level

*J*= 10 with the signal size of 2

^{ J }results in giving all the split points in block signal. In the present study, we termed the signal between two split points as a "subset". Figure 3 shows the resulting subsets along with the split points obtained in the original block signal by applying SURE based nonlinear soft thresholding. In this section, we had explained the proposed splitting process and the coming sections will explain the noise extraction method and the proposed reconstruction technique.

## 4 Singular points estimation and signal restoration

The second stage of the denoising algorithm deals with the extraction of noise elements from each individual subset. Mallat highlighted that the wavelet transform has a sequence of local maxima that converges to a point at a finer scale even though the function is regular at that point [2, 3]. Therefore, in order to detect the singularities it is not sufficient to follow the wavelet modulus maxima across scales. The Lipschitz exponents measure the regularity or singularity from the decay of these modulus maxima lines. By utilizing this property of Lipschitz exponents, the proposed method reconstructs the signal between all regular points using nonlinear functions.

### 4.1 Modulus maxima lines and Lipschitz exponent

*Y*

_{ i }(

*t*) individually to compute the modulus maxima line. The modulus maxima line is any connected curve

*S*(

*u*) in the scale-space plane (

*u, s*) along which all points are modulus maxima (Figure 4b). It has been shown by Mallat that the point wise singularities can be computed by measuring the decay of the slope of ∥

*Wf*(

*u, s*)∥ as a function of

*log*

_{2}(

*s*) and is termed as the Lipschitz exponent [2, 3].

**Definition 1**Suppose

*n*is an integer such that,

*n*<

*α*<

*n*+ 1, the signal

*y*(

*t*) has Lipschitz

*α*at

*t*

_{0}, if and only if there exists a constant A and

*t*

_{0}> 0 and a polynomial

*P*

_{ n }(

*t*) of order

*n*, such that for

*t*<

*t*

_{0}

*α*at the point of

*t*

_{0}is defined as the regularity of

*y*(

*t*) at the point of

*t*

_{0}. Lipschitz exponents can be computed from modulus maxima lines by using Eq. (4). Lipschitz exponents actually represent the nature of the signal in term of its differentiability and can be describe for a point or for interval as well. The scope of this study is limited to the study of point wise Lipschitz estimation. Figure 5 summarizes the results of Lipschitz estimation for different types of edges normally present inside the signal. In the case of Step function, Lipschitz exponent of 0 correspond to the discontinuity and also the function is non-differentiable as well. Similarly, impulse function is considered to be as a singular and have negative Lipschitz estimation. In many situations, noise is also considered as an impulse in nature. Therefore, in more generalized way negative Lipschitz correspond to the noise elements. The more smoother regions have higher Lipschitz order and hence are differentiable more than once [2, 3].

### 4.2 Reconstruction method

The restoration method between regular data samples utilizes all sampled points and the smoothness of each subset to estimate the best fit. In the final stage, merging of all reconstructed subsets result in giving fully denoised signal.

*Y*

_{ i }denotes

*i*= 1, ...,

*J*subsets with

*K*

_{ i }adjacent samples (

*y*

_{ i,l }with

*l ϵ*1, ...

*N*

_{ i }).

*k*define the iteration step. ${C}_{i}^{MSE}$ is the mean square error estimation of the restored subset with the original signal of respective subset (

*y*

_{ oi }) such that

*f'*(

*x*+

*dx*) = 0 therefore by simplifying Taylor series expansion:

*and*$f\to {C}_{i}^{MSE,k}$ the variables in

*eq.*9. can be replaced such that

*y*

_{ i }. It has been assumed for the initialization that the signal is most like the data points and has the least smoothness. In order to find the best compromise between the data samples and smoothing criteria, two optimization factors for data samples and smoothness (

*w*

^{ M }and

*w*

^{ S }, respectively) has been introduced such that Eq. (5) will become:

*w*

^{ M }or

*w*

^{ S }depends on the type of singularity. We define these two parameters in such a way that they should satisfy the stability conditions. The term corresponding to the data sample or mean square error is already stable due to the presence of the original signal. However, in term of smoothness, we rewrite Eq. (15) as:

*w*

^{ S }term in numerical form between 0 to $\frac{1}{64}$ as the signal will only be stable in this range. Therefore, based on our above finding we define two constant factor in term of an equation as:

*w*

^{ M }or

*w*

^{ S }) by taking an example of a peak signal with the addition of white Gaussian noise. Figure 7b shows the reconstruction result with the maximum smoothness (

*w*

^{ S }= 1/64). The maximum smoothness results in reducing the noise elements but at the cost of an offset on the slope towards the peak. It can be notice that, in this case RMSE gives worst response however visually the results appear to be best in terms of smoothness. On the other hand, Figure 7c shows the reconstruction with the minimum smoothness and maximum data sample (

*w*

^{ S }= 0), which result in removing an offset but at the cost of more noise. The best compromise for this type of signal is shown in Figure 7d. Therefore, the selection of these two parameters depends on the type of the singularity present in the signal and selected manually for different types of signal. After restoring each subset individually by using Eq. (14), final merging of all reconstructed subsets results in giving the fully denoised signal.

## 5 Results

*w*

^{ M }or

*w*

^{ S }is of extreme importance to get the best results as explained with an example of peak signal in the previous section.

The result on four different types of signals has been presented in this section. The main objective of the present study is to preserve these edges and then remove noise elements. Therefore, we first separate the sharp transition or edges from the signal by splitting and then performed smoothing operation. It can be seen from the results of block and heavisine signals that splitting preserve the main edges. After that we are left with the relatively homogeneous regions and we know that we need to perform smoothing along the data samples. However, this is not in the case of bumps signal because the only smoothing will result in generating an offset as explained with an example in Section 4.2. Therefore, we try to find the better compromise between data samples and smoothness. Apparently, it seems to be a random selection but in fact, this is not the case. It can be seen from the results that the selection of the parametric value (*w*^{
S
}) depends on the type of the signal.

### 5.1 Comparative analysis

#### 5.1.1 Test with the multiplicative noise

### 5.2 Generalization: extraction of small singularities

### 5.3 Application: electrocardiogram signal

- 1.
Signal Splitting.

- 2.
Reconstruction or Restoration.

#### 5.3.1 Electrocardiogram signal splitting

In order to split the signal multiscale analysis has been applied on ECG signal. 2^{
nd
}order Gaussian wavelet function has been used as an analyzing wavelet function for decomposition into succesive scales. The main reason for the selection of Gaussian wavelet function is its close similarity with the ECG signal and to ensure the evolution of modulus maxima on each scale as explained already in previous chapter. The CWT and the CWT based modulus maxima are two good tools for the analysis of ECG and both exhibit good performance even in the presence of noise. The low level of computational complexity of the modulus maxima makes it easier of the two to employ in practice. However, for the detection of small waveform features within the ECG, the CWT contains more detailed information and therefore has the potential to produce enhanced results.

*J*, if the total number of samples are 2

^{ J }. SURE-based thresholding on modulus maxima results in splitting the signal into subsets. It can be seen from the modulus maxima correspond to the coarsest scale of the multiscale analysis that the thresholding will result in giving only the R peaks point as the split points and after the thresholding this assumption was justified. The signal has been split from the corresponding R peaks and hence the reconstruction process was applied between these peaks.

#### 5.3.2 Electrocardiogram signal restoration

*w*

^{ S }or

*w*

^{ M }are manual inputs and depends on the nature of the signal. But for the given ECG signals, the weighing factors

*w*

^{ S }or

*w*

^{ M }are kept same as in Bumps signals (

*w*

^{ S }= 0.007) and the main reason for this selection is the close approximation of both signals. Figure 23 presents the denoising results. In order to illustrate the performance of the method, we have added known white Gaussian noise (10dB). It can be seen from the figure that the new method has successfully reduce the noise elements without introducing any spurious oscillations with the resulting MSE = 0.21. Not only that, it can be observed that the energy inside the peaks has also been preserved. The method has been tested with different types of signal given in the database and in term of statistics, the resulting MSE of almost 83% of the different cases of ECG is less than 0.23.

### 5.4 Discussions

Although in the present form, the method performed very well on signals where small information about the nature of the signals are apriori known, e.g., if it is electrocardiogram (ECG) signal, speech signal or mechanical vibrations. With this information, it is possible to roughly estimate the parametric value of *w*^{
S
}. This is due to the fact that if system is design for the denoising of ECG signal then all the ECG signals have approximately similar structure, same goes for the other types of the signals. In this context, we say that the proposed method is semi automatic in nature. With a very little information, the method effectively denoise any type of the signal in the case of either additive or multiplicative noise elements. In addition to that, If we know that the smoothness is of high priority then we can assign higher weighage to *w*^{
S
}. It has already been explained in the Section 4.2, that the maximum smoothness gives visually best results with the splitting but higher statistical error in term of MSE.

At the current stage, we are working on the possible solutions to completely automate the method. One approach could be to relate the smoothness of the signal with the type of singularities present inside the signal (shown in Figure 5). However, further experiments are needed to relate the type of singularity (Lipschitz value) with the choice of *w*^{
S
}and will be presented in the future study.

## 6 Conclusion

In the present study, a novel approach for the restoration of signals from noisy data samples has been presented. The most useful aspect of the proposed method is the separation of sharp edges or transitions from the suspicious noise elements. Based on the SURE estimation, thresholding is performed on modulus maxima across selected scales to split the signal from the edges. The wavelet transform has shown to be a useful tool to extract noise elements by locating modulus maxima. Lipschitz exponents computed from modulus maxima lines can be used to identify the noise elements. The reconstruction process involves in finding the best compromise between the data samples in terms of MSE and the smoothing criteria. The trade off between mean square error and the smoothing criteria can be optimized and it depends on the type of singularity present inside the signal. It has been shown with an example (peak signal in Figure 6) that it is possible to have bad results in terms of MSE even though the results visually appears to be good in terms of smoothness. Moreover due to the detection of singularities, an over smoothing will not really change the shape of the signal and main information about the shape of the signal has been preserved. Graphical results shown in the figures demonstrate that the proposed method performs equally well as compared to the conventional shrinkage methods at different variance of noise and sample sizes. The present method is particularly useful for the extraction of small singularities hidden inside the signal. Perspective will be focus on such small edges which cannot be detected due to the mutual influence of their adjacent strong transitions. However, splitting the signal into significant subsets allows such small hidden singularities to be identified. Furthermore, the proposed method proved that the denoising in spatial domain work equally well as in transform domains. By looking at the statistical and graphical results, it could be quite logical to extend the method for 2D cases. We will present the extension of this method in 2D cases and our findings in future study.

## Appendix

- 1.
At the first step, the signal has been divided into different subsets or sub signal. The selection of these subsets is defined on the basis of SURE based nonlinear thresholding of WTMM.

- 2.
The second step is to identify Lipschitz exponents computed from each subsets individually. These Lipschitz exponent results in identifying the regular points present in the subset. In the present study, all the differentiable points (

*α*≥ 1) are considered as regular points. - 3.
The nonlinear function is then optimized between all regular points to restore each subset individually by following reconstruction algorithm as follows:

*Step* 1 : *Initialization k* = 1, ${Y}_{i}^{1}={Y}_{oi}^{1}$

*Step* 2 : ${Y}_{i}^{k+1}={Y}_{i}^{k}+{w}^{M}{\lambda}_{i}^{k}\left(-\frac{\partial {C}_{i}^{MSE,k}}{\partial {Y}_{i}}\right)+{w}^{S}{\gamma}_{i}^{k}\left(-\frac{\partial {C}_{i}^{MSO,k}}{\partial {Y}_{i}}\right)$

*Step* 3 : *if* $\frac{1}{N}\sum _{l=0}^{N-1}{\left({y}_{i,l}^{k}-{y}_{i,l}^{k+1}\right)}^{2}<\epsilon $

*Stop*

*else k* = *k* + 1

*ε* is the minimum root mean square error and *l* is the index of data sample.

## Declarations

## Authors’ Affiliations

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## Copyright

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