I Introduction
While deep learning has dominated much of the computer vision literature in recent years, “traditional” filtering methods still perform better in localization problems that use one or more cameras, such as Visual Odometry (VO) and Visual Inertial Odometry (VIO). This is because the filters hardcode known nonlinear kinematic models that are difficult to learn from data, although deep neural networks are often used to learn feature representations of measured data. The Extended Kalman Filter (EKF) is the most common because it is so simple, even though complex nonlinearities in the rotational dynamics and patent violation of the Gaussian assumption in the visual measurements remove all guarantees of convergence and accuracy. In practice, the EKF provides accurate state estimates
and overconfident covariance estimates [1]. To improve the covariance estimates of VO problems, [2] and [3] learn timedependent measurement noise covariances while [4] uses a deep convolutional network to correct and directly from images. On the other hand, [5, 6, 7, 8, 9, 10, 11] improve VIO covariance estimation by using filters specific to VIO. The accuracy of covariance estimates are evaluated by tabulating the percentage of timesteps where the estimation error is within the 1,2,3 bounds dictated by the estimated covariance .Our approach is most similar to [4]
in that we use supervised machine learning to improve the covariance estimates, but the input to our models only consist of
and instead of the entire input image; having only and as inputs allows us to use much smaller networks. We also create a statistical test similar to that used in [5, 11] to evaluate the calibration of estimates from a state estimator, except without the need for MonteCarlo experiments (Sect. II). Using that statistical test, we show that groundtruth covariance can be computed assuming ergodicity when MonteCarlo experiments are impractical (Section IID) and also count the number of timesteps that are within 1,2,3 bounds given by . In Section IV, we test our method on two simple examples as a quick validation. Section V contains our main experiment, where we test our method on a VIO system processing realworld data. We achieved significant calibration using both a stateindependent model and a statedependent model , which implies that the state contains very little information about the miscalibration of – most of the miscalibration of can be computed from itself.Ii Evaluating Calibration of Kalman Filters
Iia Background: Sources of Error in EKFs
Consider a nonlinear discretetime dynamical system and measurement model with state and measurement :
(1)  
Assume that and are white Gaussian noise processes, and the input is known, along with the dynamics and nominal measurement model . An Extended Kalman Filter (EKF) recursively computes an estimate of , , along with its covariance whenever it receives a new measurement by computing the quantities:
(2)  
The matrices and are the Jacobians of and with respect to evaluated at and . Estimates and
represent a posterior Gaussian distribution. If
and are both linear, then as increases, is guaranteed to converge to and the computation of the are completely separate from the computation of the . Moreover, the innovation should be zeromean and white in both components and time. These same guarantees do not apply when either or are nonlinear. In the nonlinear case, the mean and innovation are computed using the original nonlinear model and the covariance is updated using linearized models. These unaccounted linearization errors mean that is usually underestimated [1]. Finally, most implementations of VO and VIO treat and as constants, though they may be state and timedependent.IiB 1,2,3 Intervals in Multiple Dimensions
For a set of discrete samples , drawn from a 1D Gaussian distribution , about 68% lie in the interval , 95% in , and 99.7% in . The same can be done for a set of points , each from a potentially different multivariate Gaussian distribution . First, diagonalize each
with eigenvalue decomposition:
. Then, the columns of the matrix form an orthogonal, but not orthonormal, basis and(3) 
contains the coordinates of in the new coordinate system. Then, for each dimension of , 68% of samples are in the interval , 95% in and 99.7% in . By counting the value of for each dimension at each timestep, we can evaluate the filter’s calibration for each individual dimension, but not overall.
IiC Overall Calibration with MonteCarlo Simulations
Let be the estimation error and be the normalized estimation error squared (NEES), or square of the Mahalanobis distance at each timestep :
(4) 
, i.e. is a variable with degrees of freedom. If we run MonteCarlo simulations and compute a value of for every timestep in each run , then their sum is a variable with
degrees of freedom. Then, over the MonteCarlo runs, we can compute confidence intervals for values of
. Values for should remain within the confidence interval for all if the are wellcalibrated. If is consistently too high, then the covariance estimates are too optimistic. If is consistently too low, then the are conservative. This approach was used in [5, 11] to measure the accuracy of their covariance estimates.Unfortunately, MonteCarlo approximations are not scalable, and not practical in a realworld scenario where one would have to carefully place the sensor platform in the same precise position and orientation at every run to conduct repeated trials. Also, to perform repeated and sufficiently exciting motions required for VIO, one would need a precise actuation system, like a robot arm and not a quadcopter. Therefore, we need another approach to evaluate the covariance calibration.
IiD Exploiting Residual Independence for Calibration
Here, we present a finergrained method for evaluating calibration in a multiple dimensions with a goodnessoffit test for unbiased estimators that does not require MonteCarlo simulations. First, we assume that the
are approximately independent. While not strictly satisfied, this assumption enables a practical procedure, which we will then validate empirically. Next, letbe the approximate probability density function of
, which can be computed with a normalized histogram of the . Next, since each if the system is wellcalibrated, we can then use in a goodnessoffit test with the distribution. For this work, we use the L2 divergence [12] between this approximate density and the density of as a comparison metric, :(5) 
is easy to compute and useful for comparing goodnessoffit of multiple sets of on the same dataset, but not as an absolute measure as one would use a pvalue. We will use it to compare methods of calibration to ground truth covariance in the rest of the paper.
Iii Computing a Calibrated Covariance
Our hypothesis is that the covariance estimates provided by an EKF present systematic errors and because of that, there exists a learnable map from the estimated value to a more calibrated value that can be executed in realtime. We consider maps of the following forms, in order of complexity:

A multiplicative scalar: All components of the estimated covariance are offset by a single scaling factor, .

, i.e. the map is a constant transformation of the covariance.

and , i.e. the map is an arbitrary function of the estimated covariance. This map, and the next, can be implemented by a feedforward neural network.

and , i.e. the map is an arbitrary function of the estimated state and the estimated covariance.
An even more general model would be a map represented with a recurrent neural network. Our experiments show, however, that the memoryless maps of hypotheses 3 and 4 are sufficient for uncertainty calibration.
Iiia Finding GroundTruth Covariance
Validating any of the hypotheses above requires a “groundtruth” value of covariance. In a MonteCarlo experiment, we can use the unbiased sample covariance at a given timestep given measurements :
(6) 
However, running many nearly identical tests on realworld equipment to measure groundtruth covariance is costly and timeconsuming. For realworld experiments, we can compute a pseudogroundtruth covariance with only one test if we additionally assume that that the motion state is approximately ergodic
, i.e. that the population statistics match the temporal statistics. Practically, assuming ergodicity means assuming that errors in the motion state vary slowly over time, which is often true for converged filters. For an oddsized time window
, we define pseudogroundtruth as(7) 
We can then use instead of in (4) to compute a new set of and a new value for . For simplicity, we discard timesteps for which we cannot compute a sample covariance, and , where is the total number of timesteps, from further analysis. Using these , we can verify the ergodic assumption: if the ergodic assumption is true, then should be small. In the experiments section, we visualize typical “small” and “large” values of .
The authors of [3] and [4] trained neural networks to predict covariances by training them with negative loglikelihood (NLL) losses. Unlike the typical maximum likelihood problem in which a few parameters are estimated from many data, in their NLL loss every datum is potentially drawn from a different distribution. We believe that they were nevertheless able to train a neural network to predict covariances because their training data was approximately ergodic and therefore not drawn from many different distributions.
IiiB Hypothesis 1: Constant Multiplicative Scalar
We solve for a constant factor over all sequences in a training dataset using the following optimization problem:
(8)  
subject to 
where denotes the th entry of the covariance matrix from the th timestep of the th sequence. This hypothesis is simplistic and not powerful enough, but it is an easily solvable quadratic program.
IiiC Hypothesis 2: Constant Linear Transformation
The second hypothesis is expressed as the optimization problem with decision variable
(9) 
This quartic and nonconvex optimization problem is the natural next step from Hypothesis 1. We implement Hypothesis 2 using IPOPT [13]. Theoretically, this adjustment should be at least as effective as the constant multiplicative scalar, since , where is the solution to (8), is a feasible solution to (9). However, because local nonconvex optimizers are sensitive to the initial guess and are not guaranteed to return the correct solution, the calibration of Hypothesis 2 is not uniformly better than that of Hypothesis 1. In Sections IV and V, it is always worse than Hypothesis 1.
IiiD Hypothesis 3 and 4: FullyConnected Neural Networks
Let be the output of the neural network and let the adjusted covariance be . We use a weighted elementwise difference between the upper triangles of and
as the loss function:
(10) 
Each timestep of each sequence serves as a training point. The inputs to the network are the upper triangle of a covariance matrix. In each of our experiments, we trained many simple feedforward neural networks using the Adam Optimizer in Tensorflow with varying architectures, L2 regularization weights, and epochs; very little thought was given to the architectures we tested. The outputs of these neural networks are wellcalibrated and the best performing architecture for each experiment is detailed in Sections
IV and V.Iv Two Contrasting Illustrations
Before presenting our main experiment in Section V, we illustrate concepts from the previous two sections with a couple of easytovisualize examples.
Iva Illustration 1: Linear Kalman Filter
A system that should have perfectly calibrated state estimates is a springmass damper system with mass , spring constant , and damping coefficient . After discretizing time into timesteps of length s, the discretetime dynamics and measurement equations are:
(11)  
where is the horizontal position, is the horizontal velocity, and is a forcing input. In our experiment, is the discretized version of . In both the simulation and the Kalman Filter, and each element of . Results for a single run from a set of 50 MonteCarlo trials are shown in Figure 1. The overlay of the density and the normalized histogram of the are a nearperfect fit. Additionally, using the transformation in (3), we find that in the first dimension 68.20, 95.42, and 99.74% of points are within , respectively. In the second dimension, the percentages are 68.12, 95.45, and 99.73%.
The state estimation and innovation of the linear Kalman Filter for a single run are shown in (a) and (b)  the state estimation is accurate and the innovation is essentially white noise. (c) plots
against the density for a single run. Visually, the histogram and the density are very close, showing that the independence assumption holds and that the covariance estimates are wellcalibrated. This is further verified in (d), which is the same plot as (c), except that the normalized histogram is computed using a groundtruth covariance from MonteCarlo trials. (6).IvB Illustration 2: EKF for 2D Localization
We repeat the same process above for an extended Kalman Filter for a 2D localization problem. The system is a Dubin’s car with state and acceleration and angular velocity inputs. It receives range and bearing measurements from a set of four known beacons located at , and . The discretetime dynamics with discretization s are:
(12)  
and the measurement equations for beacon located at are:
(13)  
In these experiments, we generate 11 training sequences and one test sequence of and . In all 12 sequences is a sinusoid with frequency 0.5Hz and is a constant. Figure 2 is the corresponding figure to Figure 1
for this localization problem. It is clear that although the state estimation errors are small, the covariance estimates are inaccurate, and we test Hypotheses 14 on this problem. The neural network in Hypothesis 3 is a fully connected network with hidden layers that have 512, 512, 256, 256, 128, and 64 nodes. The L2 regularization weight was 1e4 and it was trained for 50 epochs. The network for Hypothesis 4 is fully connected with five hidden layers that all have 128 nodes. Its L2 regularization weight was 1e3 and was trained for 150 epochs. Both networks use ReLU activations,
on the diagonals, and on the offdiagonals.Divergences for the unadjusted covariances, groundtruth, and the four hypotheses are shown in Table I with corresponding overlays are in Figures 2 and 3. In Table I
, the third column contains the mean and standard deviation of the divergences of 50 MonteCarlo runs. Groundtruth covariances are computed using (
6) and the divergences in Table I are the mean and standard deviation of divergences of the 50 runs. The second column contains the reduction in divergence of the means in the second column as a percentage of the reduction from the unadjusted covariances to the groundtruth covariances. We observe that the calibration of the groundtruth covariances are within one standard deviations of the calibrations of both Hypothesis 3 and 4. The main conclusion of the results is that both Hypotheses 3 and 4 can achieve calibration, but that the statedependence in Hypothesis 4 only yields a modest benefit. Note that in Figures 2 and 3, we did not plot the density over the histogram when there was very little overlap between the two.% Dec.  Test Set Sampling  % 1  % 2  % 3  
Original Estimated  0%  0.3394 0.0145  16.2, 17.6, 8.9, 58.5  30.0, 34.1, 17.1, 86.8  42.5, 48.3, 27.3, 94.8 
MC GroundTruth  100%  0.1839 0.0547  60.8, 67.5, 68.5, 68.5  98.7, 96.1, 95.4, 95.9  99.7, 99.9, 99.5, 99.7 
Global Scalar  31.6%  0.2902 0.0510  30.2, 34.3, 17.3, 87.0  53.3, 60.3, 39.2, 96.7  69.3, 75.9, 60.8, 97.5 
Global Matrix  9.1%  0.3535 2.4e06  5.6, 2.1, 0.7, 0.4  11.2, 4.1, 1.0, 0.5  16.2, 6.1, 1.4, 0.5 
Neural Network  75.4%  0.2222 0.0884  89.6, 62.0, 64.1, 45.6  98.0, 88.3, 88.0, 75.2  99.8, 96.3, 94.4, 89.0 
StateDependent NN  90.9%  0.1981 0.0553  95.7, 56.4, 51.1, 64.8  100.0, 95.3, 80.7, 91.4  100.0, 99.8, 92.7, 98.0 
V Calibration of Visual Inertial Odometry
% Dec.  Test Set Sampling  % 1  % 2  % 3  

No Adjustment  0%  0.2697 5.61e08  11.4, 9.5, 10.2, 6.2, 5.1, 6.0, 3.3, 2.4, 2.1  21.8, 19.1, 20.5, 12.0, 9.6, 12.9, 6.2, 4.7, 4.7  32.6, 29.1, 30.4, 18.2, 14.5, 19.2, 9.2, 7.3, 7.0 
“GroundTruth”  100%  0.1103 0.0223  60.4, 65.3, 67.4, 70.9, 71.7, 70.5, 67.1, 67.3, 70.3  98.2, 95.1, 96.1, 96.7, 96.9, 95.6, 95.8, 95.4, 95.1  99.9, 99.5, 99.5, 99.6, 99.7, 99.8, 99.7, 99.6, 99.4 
Global Scalar  42.7%  0.1937 0.0088  96.7, 93.4, 90.8, 74.7, 62.5, 79.7, 42.9, 30.1, 30.3  100.0, 97.8, 97.6, 95.5, 90.6, 96.8, 72.8, 50.3, 50.8  100.0, 98.6, 99.0, 98.8, 96.9, 99.0, 86.3, 65.1, 64.4 
Global Matrix  16.1%  0.2433 0.0037  98.1, 92.9, 78.1, 60.9, 66.5, 71.1, 60.0, 56.0, 74.2  99.7, 98.1, 83.0, 69.9, 77.0, 85.2, 71.1, 70.6, 90.3  99.9, 98.3, 83.9, 74.9, 82.8, 88.2, 75.3, 75.9, 94.9 
Neural Network  97.8%  0.0987 0.0159  76.4, 64.7, 72.8, 70.6, 70.6, 70.9, 69.8, 67.8, 65.7  96.5, 91.0, 93.3, 93.8, 94.0, 94.8, 93.2, 90.6, 90.1  99.6, 99.6, 98.5, 98.5, 98.8, 99.3, 98.4, 98.3, 98.1 
NN with State  105.6%  0.1028 0.0219  69.9, 65.4, 70.5, 70.2, 71.7, 71.3, 71.1, 70.6, 71.3  95.8, 91.2, 94.9, 95.0, 95.9, 95.6, 94.2, 94.2, 94.8  99.1, 99.2, 99.6, 99.3, 99.2, 99.4, 99.0, 99.2, 99.3 
In the VIO problem, the state
consists of the orientation, position, velocity, the map states, and any autocalibration states. Since we often do not have ground truth for the map, alignment, and autocalibration states, we will only analyze the localization states. Orientation is represented as a rotation vector, so in this work, the state dimension is
. A detailed description and derivation of the equations of motion for a VIO system can be found in [14].We evaluated XIVO, a reimplementation of the EKF SLAM system described in [14], on the TUM Visual Inertial Dataset [15], a benchmark that features sequences of large, fast, and aggressive motions of a rig containing a stereo camera pair and an IMU. The dataset includes six sequences, named room1room6, with “groundtruth” position and orientation collected using a motion capture system. Since XIVO’s algorithm only uses monocular images, and not stereo images, we effectively have twelve sequences, totalling 32,470 timesteps. XIVO’s estimate of the position and orientation is comparable with other stateoftheart VIO systems.^{1}^{1}1See the table at https://github.com/uclavision/xivo/blob/devel/wiki.md for XIVO’s absolute trajectory error (ATE) and relative pose error (RPE). Note that these errors are not the same as the mean error in Section VA.
Since the motion capture system did not measure velocity, we backdifferenced the position ground truth in order to compute a velocity ground truth as well. Next, we used the method of Horn [16] to compute the transformation (a rotation and a translation) between the groundtruth points and the estimated points, since the two sets of points were not recorded in the same coordinate frame. The two room6 sequences were set aside as a test set while the other ten sequences were used for training. The trajectory of the test set, the innovation of the translation states, and overlay with the distribution are shown in Figure 4.
Va Validating the ZeroMean Assumption
The analysis in Section IIC
assumes that the errors are zeromean. We compute the errors for all timesteps of the twelve sequences in the TUM VI dataset and find the mean of all of them. After interpolation and alignment of the groundtruth data, the mean translation, rotation, and velocity errors are 5.18e17m, 0.0064rad, and 0.0017m/s, respectively while the mean Euclidean norms of groundtruth translation, rotation, and velocity across all 12 sequences in the dataset are 1.15m, 1.50rad, and 0.902m/s. The translation error is zero because the method of Horn optimizes translation error when computing the alignment between the coordinate frames from the groundtruth measurements and the estimated states. Although technically nonzero, the rotation and velocity errors are small when compared to the motions in the dataset. Therefore, we will consider them negligible and consider the mean of the errors to be zero.
VB Validating the Ergodicity Assumption
The sample covariance in (7) is computed for each timestep using a window of states centered around . In order to find the best possible window size for each state of interest, we computed using oddnumbered window sizes between 27 and 601 for the ten training sequences. A window size of 275 produced low divergences for both the ten training sequences and the two test sequences. The divergence on the test set was 0.1105. The overlay with the distribution is in Figure 4. Because these numbers are relatively small when compared to the divergences computed with the sampled covariances, and because of the relative visual fit compared to the overlay generated with the unadjusted covariance, we consider the ergodicity assumption validated.
VC Experimental Results
We run our ten sequences of training data through Hypotheses 14. For Hypotheses 3 and 4, we use ReLU activations, a L2 regularization weight of 0.001 in the loss function, and set along the diagonals, for offdiagonals in the same state’s 3 x 3 block, and 0.5 otherwise. The neural network for Hypothesis 3 had hidden layers with widths 1024, 512, 256, 128, 64 and was trained for 25 epochs. The best network for Hypothesis 4 had hidden layers with widths 256, 256, 256, 128, 128 and was trained for 50 epochs.
Divergences for the combined two test sequences are shown in Table II and visualized in Figure 5. The “% Dec.” column in Table II displays the total decrease in divergence from the unadjusted covariances as a percentage of the reduction achieved using ergodic groundtruth. Numbers in the “Test Set Sampling” column are means and standard deviations of divergences computed from 50 groups of 200 points each from the test sequences. In the interest of space, the last three columns contain percentages for the position and orientations only. The trends are the same as those shown for the 2D localization experiment despite using ergodicity rather than MonteCarlo simulations to compute groundtruth covariances: the calibration of the groundtruth covariance is within one standard deviation of the calibration of both neural networks and the calibrations of Hypotheses 1 and 2 are inadequate. Once again, when there was very little overlap between a histogram and the density, we did not plot the density.
Vi Conclusions and Future Work
We have shown that there exists a learnable map between the uncalibrated estimates of a typical Extended Kalman Filter for VIO and the true, calibrated values. Another conclusion is that the ergodicity assumption is a reasonable way to compute a “groundtruth” value for covariance for suitable groundtruth motion when MonteCarlo trials are not possible. Our next step will be to investigate how generalizable these results are. The datasets in this work consisted of several sequences all collected using the same sensor in the same environment. We suspect that the calibration described in this work can be performed once for each sensor and VIO implementation, but generalize across multiple environments. A second direction of research would include using information from a neural network to modify an EKF in the loop in an endtoend fashion instead of simply adjusting the covariances posthoc. In other words, the final line of equation (2) would take the form:
(14) 
where is a recurrent network. This raises questions not only of prediction accuracy, but of stability, since the online optimization would create closedloop dependencies.
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