GNGTS 2014 - Atti del 33° Convegno Nazionale

GNGTS 2014 S essione 3.1 observation tensor on their spanning subspace. The most time consuming step of this method is the computation of the tensor          that best approximate the residual            . This step requires to solve the maximization problem: In Zhang (2001) iterative methods, based on the Multilinear Rayleigh Quotients, are suggested to find a rank-one approximation of in an efficient way. Differently from the classical Orthogonal Matching Pursuit algorithm, that requires the storage of the entire vectors bases, the proposed algorithm estimates a basis tensor only once, allowing an efficient memory management. Orthogonal Rank-One Tensor Pursuit (OR1TP) • Input: and a tolerance parameter • Initialize: • Repeat • Find a pair of singular tensors of the observed residual        , set • Solve      reshape of    , reshape of • Set • Until • Output: Tab. 1. Orthogonal Rank-One Tensor Pursuit (OR1TP) solve the (OMP) by an orthogonal matching pursuit type greedy algorithm using rank-one tensors as the basis. Given a tensor  with missing values, finds a tensor which is a low-rank approximation of the observed tensor. Application to data interpolation problem. Recently, many trace interpolators based on low-rank tensor completion have been proposed. Our interpolation method considers 3D spatial data in frequency/space domain. We consider the case where seismic traces are attributed to a regular 3D grid through binning. In the case that more than one trace is assigned to a bin, we average them to retain only one observation in each bin. In real situations, mapping seismic traces from an irregular to a regular grid through the binning process, leads to a highly sparse volume with missing traces randomly disposed. Given a 3D seismic dataset in time-space domain, in order to recover missing traces, we first perform a 1D discrete Fourier transform on the time axis. A symmetric tensor is a tensor that is invariant under a permutation of its vector arguments             for any permutation σ . Symmetric tensors form a singular important class of tensors. Hankel tensors are symmetric tensors that originate from applications such as signal processing. Due to the fact that Hankel tensors are symmetric and then well structured, the low-rank assumption is enhanced by the geometrical structure of the data.