Closed-form inverses for the mixed pixel/multipath interference problem in AMCW lidar

Abstract

We present two new closed-form methods for mixed pixel/multipath interference separation in AMCW lidar systems. The mixed pixel/multipath interference problem arises from the violation of a standard range-imaging assumption that each pixel integrates over only a single, discrete backscattering source. While a numerical inversion method has previously been proposed, no close-form inverses have previously been posited. The first new method models reflectivity as a Cauchy distribution over range and uses four measurements at different modulation frequencies to determine the amplitude, phase and reflectivity distribution of up to two component returns within each pixel. The second new method uses attenuation ratios to determine the amplitude and phase of up to two component returns within each pixel. The methods are tested on both simulated and real data and shown to produce a significant improvement in overall error. While this paper focusses on the AMCW mixed pixel/multipath interference problem, the algorithms contained herein have applicability to the reconstruction of a sparse one dimensional signal from an extremely limited number of discrete samples of its Fourier transform.

Coded time of flight cameras

“…work by finding a closed-form mathematical model method to mitigate multifrequency multipath [Godbaz et al 2012]. The work by Heide et al investigates multipath in the context of global illumination, which was proposed in [Raskar and Davis 2008].…”

A light transport model for mitigating multipath interference in Time-of-flight sensors

“…For comparison, O'Toole et al [40] use an involved, optical setup to filter out the scattered radiance. In addition, where most previous methods provide limited or no quantitative analysis of their results [11,12,32,40], we report errors for all scenes in the paper by evaluating our technique against ground truth results (e.g., from an accurate laser scanner).…”

“…Fuchs [11] and Jiminez et al [29] solve for the MPI-free phase using an optimization problem and a forward model for TOF imaging. Dorrington et al [8], Godbaz et al [12] and Kirmani et al [33] model MPI using two bounce approximation and propose iterative or closed form solutions using multiple frequency measurements (or 'frequency sweep'). Bhandari et al [4,3] and Freedman et al [10] propose iterative solutions for generalized MPI using multiple frequency measurements.…”

“…This simplification has been exploited in previous techniques for multipath interference correction (e.g., [11,8,12,33]). Under this assumption, the received signal can be modeled as…”

“…In this paper we cast the problem of MPI into the realm Paper Multipath Type Solution Type Quantitative Analysis Hardware Modifications Fuchs [11] Continuous Iterative Limited None Dorrington et al [8] 2-sparse Iterative Limited Frequency sweep Godbaz et al [12] 2-sparse Closed-form None Frequency sweep Kadambi et al [32] K-sparse Iterative None Custom code Kirmani et al [33] K-sparse Iterative Simulations only Frequency sweep Freedman et al [10] K-sparse Iterative Simulations only None Jiminez et al [29] K-sparse Iterative Limited None O'Toole et al [40] Continuous None None Extensive Gupta et al [19] Continuous Closed-form Yes Extensive This Paper Continuous Closed-form Yes External projector Table 1. This paper proposes a closed form approach to correcting multipath, by relying on light transport information.…”