Random noise attenuation using local signal-and-noise orthogonalization

Solving LOW by shaping regularization

In order to better control the locality and smoothness of LOW, we follow the local-attribute scheme introduced by Fomel (2007a). Simplifing equation 2, and adding a local-smooth constraint to the optimization problem, we obtain:

$$\mathbf{w} = \arg\min_{\mathbf{w}} \parallel \mathbf{n}_0 - \math... ...}_0\mathbf{w}\parallel_2^2 + \mathbf{R}(\mathbf{w}). %+ \mathbf{R}(\mathbf{w})$$ (7)

Here, $\mathbf{w}$ is the LOW, $\mathbf{S}_0$ is a diagonal matrix composed of the initial estimated signal $\mathbf{s}_0$ : $\mathbf{S}_0=diag(\mathbf{s}_0)$ . $\mathbf{R}$ denotes a smoothing regularization operator. Then we solve the least-squares problem 7 with the help of shaping regularization (Fomel, 2007b) using a local-smoothness constraint:

$$\mathbf{w} = [\lambda^2\mathbf{I} + \mathcal{T}(\mathbf{S}_0^T\mathbf{S}_0-\lambda^2\mathbf{I})]^{-1}\mathcal{T}\mathbf{S}_0^T\mathbf{n}_0,$$ (8)

where $\mathcal{T}$ is a triangle smoothing operator and $\lambda$ is a scaling parameter set as $\lambda = \Arrowvert\mathbf{S}_0^T\mathbf{S}_0\Arrowvert_2$ . Inserting $\mathbf{w}$ into equation 1, we can obtain a new denoising approach:

$$\mathbf{s} =\mathbf{s}_0+\mathbf{s}_1=\mathbf{P}[\mathbf{d}]+\mathbf{w}\circ\mathbf{P}[\mathbf{d}]=(\mathbf{I}+diag(\mathbf{w}))\mathbf{P}[\mathbf{d}],$$ (9)

$$\mathbf{n} =\mathbf{n}_0-\mathbf{s}_1=\mathbf{d}-\mathbf{P}[\mathbf{d}]-\mat... ...{P}[\mathbf{d}]=\mathbf{d}-(\mathbf{I}+diag(\mathbf{w}))\mathbf{P}[\mathbf{d}].$$ (10)

Here, $\mathbf{s}$ and $\mathbf{n}$ are the final estimated signal and noise.

low1,low2,low3,low4
Figure 1. LOWs using different smoothing radii for the example as shown in Figure 3. (a) Vertical and lateral smoothing radii are 25 samples. (b) Vertical and lateral smoothing radii are 20 samples. (c) Vertical and lateral smoothing radii are 10 samples. (d) Vertical and lateral smoothing radii are 5 samples.

The controlling parameter for calculating LOW is the width of the triangle smoother, or the smoothing radius (SR). Figure 1 shows the calculated LOWs that correspond to the second synthetic example (Figure 3) with different smoothing radii. The LOW are calculated by solving the equation 7 with the initial denoised signal and initial removed noise set as Figures 3d and 3e, respectively, using the approach shown in equation 8. As we can see from the comparison, LOW is robust with respect to the choice of the smoothing radius (e.g., from 25 samples to 20 samples, or from 10 samples to 5 samples), but may differ a lot for significantly different smoothing radii (e.g., from 25 samples to 5 samples). We can also conclude from the comparison that as the smoothing radius increases, the temporal and spatial resolutions decrease, but the anti-noise ability increases. Note that the orthogonalization equations 2 and 7 can also be understood as one-point nonstationary matching filtering (Fomel, 2009).

Random noise attenuation using local signal-and-noise orthogonalization