Simply denoise: wavefield reconstruction via jittered undersampling

Jittered undersampling ( $0<\xi <\gamma$ ):

In this regime, both coherent aliases and incoherent random undersampling noise are present. Depending on the choice for the jitter parameter, the energy either localizes or randomly spreads across the spectrum. Again, the reduction of the aliases is related to the locations of the zero crossings of the sinc function that move as a function of $\xi$ . As $\xi$ increases, the zeros move closer to the aliases. As expected, the matrix $\tensor{A}^H\tensor{A}$ , plotted in Figure 5(a), still contains the imprint of coherent off-diagonals, resulting in a kernel of $\tensor{L}$ , included in Figure 5(c), that is a superposition of coherent aliases and incoherent random noise. Although this regime reduces the aliases, coherent energy remains in the undersampling artifacts. This residue creates a situation that is less favorable for recovery. Depending on the relative strength of the aliases compared to the magnitude $n$ of the diagonal of $\tensor{A}^H\tensor{A}$ , recovery becomes increasingly more difficult, an observation that can be established experimentally.

In the next section, a series of controlled experiments is conducted to compare the recovery from regularly, randomly according to a discrete uniform distribution and optimally-jittered undersamplings.

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Figure 5. Jittered undersampling according to a discrete uniform distribution. (a) Suboptimal and (b) optimal jittered five-fold undersampling convolution matrices (in amplitude). The respective convolution kernels (in amplitude) that generate spectral leakage are plotted in (c) and (d). If the regular undersampling points are not shuffled enough, only part of the undersampling artifacts energy is spread, the rest of the energy remaining in weighted aliases. When there is just enough shuffling, all the undersampling artifacts energy is spread making jittered undersampling like random undersampling, yet controlling the size of the largest gap between two data points.

Simply denoise: wavefield reconstruction via jittered undersampling