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where we employed the blockwise matrix inversion [8]. Under the concentrated Gaussian approximation, the top left block of P in (41) corresponds to P,j, 1. Thus, we obtain the following covariance prediction formula:

Figure 4 where we employed the blockwise matrix inversion [8]. Under the concentrated Gaussian approximation, the top left block of P in (41) corresponds to P,j, 1. Thus, we obtain the following covariance prediction formula:

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Fig. 1: Illustration of the geometry of a (matrix) Lie group
Fig. 1: Illustration of the geometry of a (matrix) Lie group
Fig. 2: Illustration of a concentrated Gaussian distribu- tion on SE (2). An arrow represents the position and the orientation of a robot in a 2D plane. First of all, let us define the minimizer of the cost func- tion ||¢(X) ||}:
Fig. 2: Illustration of a concentrated Gaussian distribu- tion on SE (2). An arrow represents the position and the orientation of a robot in a 2D plane. First of all, let us define the minimizer of the cost func- tion ||¢(X) ||}:
Thus, according to equation (20), the covariance P can be approximated by: By linearizing the error function inside the squared Mahalanobis norm in (34) around its minimizer, we ob- tain:
Thus, according to equation (20), the covariance P can be approximated by: By linearizing the error function inside the squared Mahalanobis norm in (34) around its minimizer, we ob- tain:
Fig. 3: Illustration of the relative motion averaging problem: dashed observations (Y;; with 7 > i+ 1) are used as measurements during the LG-IEKF update step. Temporally consecutive observations (Y;(;+1) for i = 1...N—1) are used as control inputs during the LG- IEKF prediction step in order to guide the estimation process.
Fig. 3: Illustration of the relative motion averaging problem: dashed observations (Y;; with 7 > i+ 1) are used as measurements during the LG-IEKF update step. Temporally consecutive observations (Y;(;+1) for i = 1...N—1) are used as control inputs during the LG- IEKF prediction step in order to guide the estimation process.
More precisely, the prediction step that we propose augments the size of the state by duplicating the global motion Tr and propagating it using the control input Yrtik = Ye tosti in order to predict T(,41)r. Thus we consider the following prediction model: where the notation (X;_1),, corresponds to extracting the global motion component Ty, from the state X,_1. bf tayE ~ Nea (Game Ogx1; Bt se) is a white Gaus- sian noise.
More precisely, the prediction step that we propose augments the size of the state by duplicating the global motion Tr and propagating it using the control input Yrtik = Ye tosti in order to predict T(,41)r. Thus we consider the following prediction model: where the notation (X;_1),, corresponds to extracting the global motion component Ty, from the state X,_1. bf tayE ~ Nea (Game Ogx1; Bt se) is a white Gaus- sian noise.
From Intrinsic Optimization to Iterated Extended Kalman Filtering on Lie Groups Fig. 4: Computational time and Residual errors for dif- ferent number of global motions (NV). The number of relative motions is fixed to N + n (in our experiments n = 30).
From Intrinsic Optimization to Iterated Extended Kalman Filtering on Lie Groups Fig. 4: Computational time and Residual errors for dif- ferent number of global motions (NV). The number of relative motions is fixed to N + n (in our experiments n = 30).
Fig. 5: Estimation error of the global motions for our approach, Chatterjee et al. [20] and Roberts et al. [55] on a camera pose estimation problem (A = a and num- ber of relative motions fixed to 5N +n with n = 60).
Fig. 5: Estimation error of the global motions for our approach, Chatterjee et al. [20] and Roberts et al. [55] on a camera pose estimation problem (A = a and num- ber of relative motions fixed to 5N +n with n = 60).
Fig. 6: Camera pose registration problem results, a cone represents a camera pose, a black line is an inlier mea- surement and a gray dashed line is an outlier
Fig. 6: Camera pose registration problem results, a cone represents a camera pose, a black line is an inlier mea- surement and a gray dashed line is an outlier
(d) [55] : (left) labeling matrix, (right) aligned camera poses of the video sequence
(d) [55] : (left) labeling matrix, (right) aligned camera poses of the video sequence
Fig. 7: Wearable camera experiment: in the labeling matrices a white pixel corresponds to an inlier, a black pixel corresponds to an unavailable measurement, a gray pixel corresponds to an outlier. In the results, a cone represents the pose of a camera. The first camera position of the video sequence is dark red and the last camera position is bright red.
Fig. 7: Wearable camera experiment: in the labeling matrices a white pixel corresponds to an inlier, a black pixel corresponds to an unavailable measurement, a gray pixel corresponds to an outlier. In the results, a cone represents the pose of a camera. The first camera position of the video sequence is dark red and the last camera position is bright red.
(b) Ground Truth : (left) labeling matrix manually an- notated, (right) scene overview
(b) Ground Truth : (left) labeling matrix manually an- notated, (right) scene overview
(d) This paper : (left) labeling matrix, (right) mosaic of input images (c) [55] : (left) labeling matrix, (right) mosaic of input images
(d) This paper : (left) labeling matrix, (right) mosaic of input images (c) [55] : (left) labeling matrix, (right) mosaic of input images
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