Two-frame tilt-shift error estimation and phase demodulation algorithm.

We present an algorithm capable of performing fringe pattern phase demodulation from two frames with unknown, linearly nonuniform phase shift, i.e., under presence of the tilt-shift error. The method consists of intensity-based filtration of the tilt-shift component and subsequent two steps of a nonlinear error functional minimization. We verify the algorithm performance and robustness using both simulated and experimental data, indicating high accuracy of the presented method and its good numerical properties. Both small and large tilts can be treated. The Letter is complemented by numerical codes available online in Wielgus, "Two-frame tilt-shift error estimation and phase demodulation algorithm" (2015).

Continuous phase estimation from noisy fringe patterns based on the implicit smoothing splines.

We introduce the algorithm for the direct phase estimation from the single noisy interferometric pattern. The method, named implicit smoothing spline (ISS), can be regarded as a formal generalization of the smoothing spline interpolation for the case when the interpolated data is given implicitly. We derive the necessary equations, discuss the properties of the method and address its application for the direct estimation of the continuous phase in both classical interferometry and digital speckle pattern interferometry (DSPI). The numerical illustrations of the algorithm performance are provided to corroborate the high quality of the results.

Universal interferometric signatures of a black hole's photon ring.

The Event Horizon Telescope image of the supermassive black hole in the galaxy M87 is dominated by a bright, unresolved ring. General relativity predicts that embedded within this image lies a thin "photon ring," which is composed of an infinite sequence of self-similar subrings that are indexed by the number of photon orbits around the black hole. The subrings approach the edge of the black hole "shadow," becoming exponentially narrower but weaker with increasing orbit number, with seemingly negligible contributions from high-order subrings. Here, we show that these subrings produce strong and universal signatures on long interferometric baselines. These signatures offer the possibility of precise measurements of black hole mass and spin, as well as tests of general relativity, using only a sparse interferometric array.