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

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],

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],

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.

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

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

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

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:

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

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

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.