Random noise attenuation using local signal-and-noise orthogonalization

Local orthogonalization

Based on the assumption that final estimated signal $\mathbf{s}$ and noise $\mathbf{n}$ should be orthogonal to each other, we can orthogonalize the estimated signal and estimated noise by

$$\hat{\mathbf{n}} = \mathbf{n}_0 - w\mathbf{s}_0,$$ (3)

$$\hat{\mathbf{s}} = \mathbf{s}_0 + w\mathbf{s}_0,$$ (4)

where $w$ is the global orthogonalization weight (GOW) defined by

$$w=\frac{\mathbf{n}_0^T\mathbf{s}_0}{\mathbf{s}_0^T\mathbf{s}_0}.$$ (5)

Here $[\cdot]^T$ denotes transpose. Appendix A provides a demonstration and proof for the global orthogonalization process as denoted by equations 3, 4 and 5. The orthogonality assumption is similar as assuming that the signal and noise do not correlate with each other, which implies the kinds of noise that do not correlate with the useful signal, e.g., random noise. The orthogonality assumption is assumed to be valid in the original time-space domain, but also has the possibility of being applied in other transformed domains. Instead of using GOW, we propose to use nonstationary local orthogonalization weight (LOW). One possible definition of LOW is:

$$w_m(t) = \frac{\displaystyle\sum_{i=t-m/2}^{t+m/2} s_0(i) n_0(i)}{\displaystyle\sum_{i=t-m/2}^{t+m/2} s_0^2(i)},$$ (6)

where $w_m(t)$ denotes the LOW for each temporal point $t$ with a local window length $m$ . $s_0(t)$ and $n_0(t)$ here denotes the initially estimated signal and noise for each point $t$ .

Random noise attenuation using local signal-and-noise orthogonalization