Phase ambiguity resolution and mixed pixel detection in EDM with multiple modulation wavelengths

Distance measurement based on the accumulated phase of intensity-modulated continuous-wave lasers is an established approach for pointwise geometry acquisition in uncontrolled conditions, enabling some of the most accurate Lidar/laser-scanning solutions. Inherently affected by cycle ambiguities due to the modulo 2 pi phase observations, several modulation wavelengths are typically used in practical implementations to extend the measurement range. One of the main limitations of these systems arises from their inability to decouple mixed signals affected by different delays. This is especially relevant when the laser beam illuminates simultaneously two or more surfaces at significantly different distances, producing mixed-pixel measurements in which the estimated distance does not reliably correspond to any of the involved scatterers. Algorithms for detection and filtering of mixed pixels within point-cloud data have been proposed based on analyzing consistency across local neighborhoods. The resolution of multiple scatterers on individual measurements, however, has not been yet reported nor is provided by undisclosed commercial implementations. In this work, we analyze the problem of signal mixing in multi-wavelength phase-based distance estimation, and propose an algorithm for joint phase ambiguity resolution and mixed pixel detection based on mixed integer linear programming. We first analyze theoretically the performance limits of the proposed solution, demonstrating unbiasedness in the presence of noise and robustness in the presence of multiple scatterers. In addition, we investigate empirically the impact of noise, mixed measurements and choice of modulation wavelengths on the performance of the proposed algorithm. The results demonstrate the capacity of the presented approach, given an adequate wavelength selection, to provide unambiguous distance estimates under realistic noise conditions and to identify measurements affected by mixed pixels while approximating the position of the dominant scatterer. Aside from contributing to improving the reliability and resolution capability of Lidar and laser-scanning data, the proposed solution sets a promising step towards fully resolved multi-target distance measurement.