Metrological Fringe inpainting

Abstract

In optical metrology, important information is often carried by fringe patterns, which can be expressed as ) , ( cos ) , ( ) , ( y x y x b y x f (1) where ) , ( y x f , ) , ( y x b and ) , ( y x are the recorded intensity, fringe amplitude and phase distribution, respectively. We have assumed that the background intensity could be removed by, say, low-pass filtering, and hence is not shown in Eq. (1). Sometimes the fringe patterns have some undesired invalid areas due to the irregularity of the tested specimen, illumination shadows, or imperfection of the optical elements and detector. These invalid areas may introduce difficulties for further processing [1-3]. It is hence often required to “repair” the fringe patterns. This is very similar to the digital inpainting of artistic pieces [4,5]. Fringe extrapolation and interpolation usually carry the same meaning. Assume the intensity of the fringe pattern (usually noisy), f, is available at the entire image plane , except for some areas D. The goal of inpainting is to reconstruct the intensity at D (usually without noise), f0, as faithfully as possible. This task can be model by the Maximum A Posteriori (MAP) optimization, i.e., determining f0( ) that maximizes the posterior probability ) ( ) ( 0 D f f p .