A Closed Form Solution for the Self-Calibration of Heterogeneous Sensors

Abstract

We present a novel closed-form solution for the joint self-calibration of video and range sensors. The approach single assumption is the availability of synchronous time of flight (i.e., range distances) measurements and visual position of the target on images acquired by a set of cameras. In such case, we make explicit a rank constraint that is valid for both image and range data. This rank property is used to find an initial and affine solution via bilinear factorization, which is then corrected by enforcing the metric constraints characteristic for both sensor modalities (i.e., camera and anchors constraints). The output of the algorithm is the identification of the target/range sensor position and the calibration of the cameras. The application extent of our approach is broad and versatile. In fact, with the same framework, we can deal with, but not restricted to, two very different applications. The first is aimed at calibrating cameras and microphones deployed in an unknown environment. The second uses a RGB-D device to reconstruct the 3D position of a set of keypoints using the camera and depth map images. Synthetic and real tests show the algorithm performance under different levels of noise and configurations of target locations, number of sensors and cameras.

FLORIS and CLORIS: Hybrid source and network localization based on ranges and video

“…These are corrupted by zero-mean additive white Gaussian noise with variance σ 2 ρij , where 20 ρ = r, φ, or α depending on the type of measurement. We note that neither the independence assumption between measured ranges and bearings underlying (18) nor the actual noise statistics match our joint generation model described at the start of Section 5. However, since deriving a ML formulation for our model is beyond the scope of this paper, and the mismatch with the assumptions of [12] does not seem severe under weak to moderate noise, we will directly feed that ML estimator with the same synthetic data that CLORIS operates on.…”

“…For the same 3D network configuration of Example 1, Figure 8 plots the RMSE as a function of the noise factor. In addition to CLORIS and the range-only algorithm of [34], the figure includes curves pertaining to direct refinement, using one of MATLAB 's generic unconstrained nonlinear minimization functions (fminsearch), of our non-relaxed cost function (2) (related to what is developed in [37] for range-only) and the likelihood function (18) proposed by Huang et al [12]. To complement the ML results, the figure also shows the Cramér-Rao Lower Bound (CRLB) computed as 1 N tr(F −1 ), where F is the Fisher Information Matrix (FIM) given in [12] for 3D collaborative positioning with ranges and bearings.…”

“…Iterative and ad-hoc methods for WSN localization that combine range and angle information have been proposed in [11,12,13,14,15] (the latter being specific for singlesource localization in underwater scenarios). In contrast with the previous iterative schemes, and to the best of our knowledge, only a few recent attempts were made to genuinely fuse range and orientation for hybrid localization [16,17,6,18]. The first two of these bases localization on received signal strength (RSSI) and angle measurements, whereas the last two are closer to our approach.…”

“…However, these methods impose severe limitations; the one in [6] is SDP-based, but the way it encodes angular constraints is specific for 2D and very different from the techniques used in FLORIS and CLORIS. The approach of [18] uses a bilinear matrix formulation inspired in computer vision techniques, but the assumption that the range and visual anchors overlap is too restrictive for many WSN scenarios of interest. In this work, we succeed in overcoming the limitations of both [6] and [18] by deriving a novel formulation based on a single optimization problem that jointly accounts for range and bearing data obtained from arbitrarily placed heterogeneous sensors in 2D or 3D, the latter being particularly relevant in indoor scenarios.…”

“…The approach of [18] uses a bilinear matrix formulation inspired in computer vision techniques, but the assumption that the range and visual anchors overlap is too restrictive for many WSN scenarios of interest. In this work, we succeed in overcoming the limitations of both [6] and [18] by deriving a novel formulation based on a single optimization problem that jointly accounts for range and bearing data obtained from arbitrarily placed heterogeneous sensors in 2D or 3D, the latter being particularly relevant in indoor scenarios. Optimal placement of nodes -either static [4,6,19] or mobile [20] -can significantly enhance the estimation/localization accuracy, but is beyond the scope of our work.…”