# Speckle Noise Detection and Removal for Laser Speech Measurement Systems
## 3 Automatic Noise Detection and Removal
### Speckle Noise Detection  
The average short-time energy and kurtosis are used as judgment indexes to distinguish the noise. The average short-time energy reflects the mean of the weighted sum of the squares of a frame sample values. The average short-term energy (\(E_{n}\)) of a speech signal at time \(n\) is expressed as:  
\[E_{n}=\frac{1}{N}\sum_{m=n-(N-1)}^{n}[x(m)\omega(n-m)]^{2} \tag{9}\]  
where \(N\) is the window length and each window length represents one frame. As the extracted signal enhances the energy ratio of the speckle noise, the average short-term energy can be used to locate the speckle noise with a long duration and large amplitude.  
On the contrary, the kurtosis is defined as the fourth central statistical moment normalized by the fourth power of the standard deviation. This describes the degree to which a sample deviates from the distribution:  
\[K_{i}=\frac{E\left[\left(X_{i}-\mu\right)^{4}\right]}{\sigma^{4}} \tag{10}\]  
where \(K_{i}\) is the kurtosis value of signal \(X\), \(X\) contains samples of the \(ith\) frame (\(n-(N-1)\)), \(x(n-N),\cdots x(n)\), \(\mu\) and \(\sigma\) are respectively the mean value and standard deviation of \(x\), and \(E\left[\ \right]\) stands for the overall average.  
Using the kurtosis as a method for locating statistical anomalies provides a strong sensitivity to sudden anomalies with a very short duration and a large numerical value.  
From the given definitions of average short-term energy and the kurtosis, the average short time energy is sensitive to high-amplitude noise with a long duration. However, it can easily miss speckle noise with a very short duration and a slightly smaller amplitude. However, the kurtosis is sensitive to sudden anomalies with a very short duration. Thus, the two methods are complementary when applied to the detection of different types of noise in laser-detected speech.  
Figure 6a,b show the short-time energy and kurtosis value of a raw detected speech signal, and the signal extracted by decorrelation, respectively. Looking at Figure 6a vertically and the long-duration speckle noise in the red dashed box, the average energy value is much greater than the voice. However, the average short-term energy value for the sharp speckle noise with a very short duration in the black box is very similar, or even lower than the voice signal. Simultaneously, the kurtosis value of the frame in the black box is very large. This confirms our proposition that the average short-term energy and kurtosis are complementary. If we then make a horizontal comparison between Figure 6a,b, it can be clearly seen that, by removing the influence of the correlated speech signal, the average short-term energy and kurtosis values of the speckle noise in the extracted signal are greater than they are in the raw signal, confirming that decorrelation can increase detectability of the noise.  
Traditional methods usually use a fixed threshold, but as the distance, detection objects and speaker volume can all change, the threshold needs to be reinitialized on each occasion, which reduces the detection efficiency. Therefore, we use the ratio of the present moment to the past average to detect the speckle noise. If the energy \(E_{i}\) or the kurtosis \(K_{i}\) of the current frame is \(T_{\rm E}\) times greater than the average energy \(E_{mean}\), or \(T_{\rm K}\) times greater than the average kurtosis \(K_{mean}\), the current frame is judged to contain speckle noise.