Estimating the derivative of modulo-mapped phases

The paper considers the problem of phase unwrapping which means generating absolute phase values from noisy, modulo-2π mapped phase ‘observations’. Phase unwrapping is the central key element in any kind of interferometric application. Nearly all known phase unwrapping techniques try to unwrap the mapped phases by a sequence of differentiating, taking the principal value of the discrete derivative and integrating again. This procedure, conceptually appealing as it may appear, however, yields strongly biased phase derivatives and thus strongly biased phase estimates. It can be shown mathematically, that computing the discrete derivative of noisy modulo-2π mapped phase yields estimates of the unambiguous discrete derivative, which are always biased towards lower absolute values. The bias clearly depends on the phase slope itself as well as on the coherence or on the signal to noise ratio (SNR), respectively. Considering the practical application of Synthetic Aperture Radar Interferometry, the paper presents the theoretical analysis, and gives some numerical results.