Simply denoise: wavefield reconstruction via jittered undersampling

Fourier-domain undersampling artifacts

Undersampling artifacts in the Fourier domain are studied for two reasons. Firstly, several interpolation methods are based on the Fourier transform (Sacchi et al., 1998; Xu et al., 2005; Zwartjes and Sacchi, 2007). Secondly, the curvelet transform, a dyadic-parabolic partition of the Fourier domain, forms the basis of our recently-introduced recovery scheme (Herrmann and Hennenfent, 2007). Curvelets are in many situations to be preferred over Fourier because of their ability to sparsely represent complex seismic data. For a detailed discussion on this topic, we refer to Candès et al. (2005a) and Hennenfent and Herrmann (2006).

In the coming discussion, the sparsifying transform is defined as the Fourier transform, i.e., $\tensor{S}\; {\buildrel\rm def\over=}\; \tensor{F}$ . For this definition, the vector generating the Hermitian Toeplitz and circulant matrix $\tensor{A}^H\tensor{A}$ is the discrete Fourier transform of the (under)sampling pattern. This pattern has ones where samples are taken, zeros otherwise. Besides, the undersampling artifacts generated by the convolution operator $\tensor{L}$ are known as spectral leakage (Xu et al., 2005).