ABSTRACT

A 1-D and 2-D Daubechies 5 (db5) discrete wavelet shrinkage methods using a 10 level decomposition was applied to white light lidar data particularly at 350 nm and 550 nm backscattered signal. At 350 nm, the backscattered signal is very weak as compared to 550 nm backscattered signal because of the spectral intensity distribution of the generated white light. The 1-D and 2-D wavelet shrinkage method gave a much better result as compared with the moving average method. However, the 2-D wavelet shrinkage method produced a much better denoised lidar signal compared with the 1-D wavelet shrinkage method. This is indicated by the 142% increase in correlation coefficient between the 2-D denoised lidar signal and the 800 nm original lidar signal as compared with only 12% increase in correlation coefficient for the 1-D denoised lidar signal. The 2-D wavelet shrinkage method also gave a much higher SNR value of 65.9 compared to 1-D which is 38.8.

1. Introduction

Supercontinuum generation on air and other gas media using high peak power femtosecond lasers opened the way for multispectral atmospheric remote sensing using a white light lidar. Because of its broad spectrum ranging from UV to IR, the technique offers several applications [1,2]. We have demonstrated that the coherent white light continuum can be used for depolarization and multiwavelength measurement in the same way as the conventional lidar [3]. However, multi-wavelength lidar observations for conventional lidar often use at least two laser sources. The multi-wavelength lidar measurements using a coherent white light continuum have the capability of obtaining the wavelength dependence of the backscatter coefficients of aerosols, which can be used to evaluate the particle size distribution using one laser source [1]. However, the present experiment does not fully utilize the potential of a broadband white light continuum. Lidar applications using the infrared region of the white light remains a challenge, because of the rapid decrease of the infrared content of the white light. Furthermore, the transmitted intensity of the white light was very weak for short wavelength (350 nm and 450 nm) as compared with the fundamental wavelength (800 nm). The signals are usually buried in noise, depending on the power of the laser and the observed altitude. In general, lidar signals with noise can be improved by moving average method. However, the moving average method only smoothen the signals and does not remove specky values especially the negative values produced by noises [4]. Lidar signals are often presented in single profile representing one acquisition (one dimension), or in terms of time-height-intensity (THI) display (two dimensions) to represent one observation period. In this paper, we propose a method to improve the lidar data by means of one dimensional (1-D) and two dimensional (2-D) wavelet shrinkage method since the wavelet function is of a localized property and has sensitivity to the transient signals such as lidar signal. In addition, since the WT has different resolutions on noise and signal, it can perform denoising process on lidar signal.

2. Denoising Algorithm

In our previous paper [5], we applied several wavelets to find the most suitable wavelet for white light lidar system described in [3]. These wavelets were Haar, Daubechies 2 (db2), 5 (db5), and 8 (db8), Symlets 2 (sym2), 5 (sym5), and 8 (sym8), and Coiflets 2 (coif2) and 5 (coif5). The result showed that db5 was the most suitable for our application, henceforth it is the one used for denoising the lidar signal discussed in this paper.

2.1. 1-D Wavelet Shrinkage

In the noise reduction based on WT, we have used the discrete WT (DWT) over the continuous WT (CWT) because CWT is often redundant and computationally expensive. The DWT involves transforming a given signal with wavelet basis functions by dilating and translating it in discrete steps [6].

The wavelet shrinkage [7] is a signal denoising technique based on the idea of thresholding the wavelet coefficients. The wavelet shrinkage method is shown in Figure 1 and can be summarized as follows:

1) Apply the DWT to the signal.

2) Estimate a threshold value.

3) Remove the coefficients that are smaller than the threshold.

4) Perform an inverse DWT and reconstruct the signal.

An algorithm for calculating discrete wavelet decom-

ositions and reconstructions is the Mallat algorithm [8].

The noisy experimental signal f(t) is considered as, called a “scaling coefficient” at the 0 level of signal decomposition. Then, as shown in Figure 2, s^(j) is successively decomposed into both s^(j-1) and w^(j-1) by the following formulas:

and  (1)

where w^(j) is called the DWT coefficient, {p_n} and {q_n} are the sequence of coefficient.

The sequences p_n of Daubechies’ wavelet is given in Table 1 where q_n is given by

(2)

The level of noise in lidar data is unknown and must be estimated from the noisy data. In this algorithm we have used the universal threshold as suggested in [9],

(3)

where N is the dimensionality of the input data vector and σ is the standard deviation of the noise. The σ is often estimated from the median value of the DWT coefficients at the first level of signal decomposition [4],

(4)

Once the threshold value has been calculated, we can apply a soft threshold to reduce the noise in signal. If the magnitudes of the DWT coefficients, w_k^(j), are smaller than this threshold value, the DWT coefficients are replaced by zero, while the rest of them are calculated as w_k^(j) - T. Signal reconstruction can be presented as follows:

Table 1. Daubechies 5 wavelet sequence Daubechies compactly supported wavelet for N = 5.

(5)

The denoised signal s^(j-1) can be successively obtained from w^(j) and s^(j) as shown in Figure 3.

2.2. 2-D Wavelet Shrinkage

Figure 4 gives the flowchart for the 2-D wavelet shrinkage method (2-D). As shown in Figure 4, the noisy experimental image which is represented by the time-height-intensity (THI) display of the lidar signal is considered as in the same way as 1-D wavelet shrinkage (1-D). DWT is applied first to in the horizontal direction, represented by the time component of the lidar return signal. The coefficients applied in the

horizontal direction of the scaling function (s_m,n^(j+1,x)) and the wavelet function (w_m,n^(j+1,x)) are given by

(6a)

(6b)

Secondly, DWT is also applied in the vertical direction to the obtained coefficients, and these are given by:

(7a)

(7b)

(7c)

(7d)

where is the coefficient which is applied to the scaling function in the horizontal direction and to the wavelet function in the vertical direction; is the coefficient which is applied to the wavelet function in the horizontal direction and to the scaling function in the vertical direction; and is the coefficient which is applied to the wavelet function in both directions. The vertical direction represents the height component of the lidar return signal.

Then, the algorithm for the computation of the can be summarized by the following four equations:

(8a)

(8b)

(8c)

(8d)

The same procedure is done for and successively decomposed in 2-D. Finally, the de-noised image can be reconstructed by

(9)

Figure 5 shows an example of one level decomposition 2-D wavelet transform.

3. Experimental Results and Discussion

In this section, we apply the 1-D and 2-D wavelet de-

noising algorithms for reducing noise in previous observational results found in [3]. We have taken a data size of 1024 (= 2¹⁰) points and decomposed into 10 levels for 1-D and 2-D wavelet denoising.

3.1. 1-D wavelet Signal Denoising

Figure 6 shows the denoised lidar signal based on the above 1-D denoising procedure with soft threshold. For comparison, the real signal and moving average signal are presented in Figure 6. In order to check whether our method can filter the noise out and also extract the cloud signals, the channel with the weakest backscattered signal, 350 nm, is compared to the channel which relatively has strong backscattered signal, 550 nm. Figure 6(a) shows the backscattered signal at 550 nm. Cloud peaks can be seen at about 0.5 km and 1 km. For the strong 550 nm backscattered signals, the 1-D wavelet denoising method applied to the signal caused no significant difference. However, it can be found that cloud signals which were buried in noise in the weaker channel at 350 nm became noticeable after the denoising procedure by comparing Figure 6(b) with Figure 6(a). Thus, this method can effectively detect the lidar signal buried in noise and thus reduce the noise.