Simply denoise: wavefield reconstruction via jittered undersampling

Optimally-jittered undersampling ( $\xi =\gamma$ ):

Now the sampling points are perturbed within contiguous windows, as depicted in the third row of Figure 3, and equation 6 reduces to

$$\left\{\ensuremath{\mathbf{a}}[k]\right\} \approx \left\{ \begin{array}{cl} n, & \mbox{for}\quad k=1\\ 0, & \text{otherwise.} \end{array} \right.$$ (8)

In this special case, the cause of the aliases is removed by the zeros of the sinc function. As with random undersampling according to a discrete uniform distribution, the off-diagonals of the matrix $\tensor{A}^H\tensor{A}$ (cf. Figure 5(b) and 2(c)) are random, turning aliases into noise. Again, the kernel of $\tensor{L}$ does not contain coherent energy, as observed in Figure 5(d), for a five-fold undersampling ($\gamma =5$ ) and a jitter parameter of $\xi=5$ . In that sense, this specific relation between the jitter parameter and the undersampling factor is optimal because it creates the most favorable conditions for recovery with a localized transform.

Simply denoise: wavefield reconstruction via jittered undersampling