Quantifying Feature Association Error in Camera-based Positioning

Q: what is the covariance matrix of noise vector n?

Given a known landmark M consisting of mF feature points in the map (requiring mF ≥ 4 in general), the position and orientation of the camera can be estimated using corresponding 2D measurements { j = 1, ...,mF |µ j} by iteratively solving the following nonlinear optimization problemx̂ = argmin x ‖µ −π(x)‖2Σ−1n , (10)whereµ = µ 1... µ mF ,π(x) = π1(x)... πmF (x) ,n = n1... nmF . (11) Σn is the covariance matrix of noise vector n.

Q: What is the posterior estimate of a checkerboard landmark?

(18)The posterior estimate at time k can be updated using the following equationsx̂k = x̄k +Gkγ k, P̂k = (I−GkHk)P̄k, (19)in which Hk is the linearized matrix concatenated over all the landmarks i= 1, ...,NL, and Gk is the Kalman gain calculated using the equationGk = P̄kHTk (HkP̄kH T k +Vk) −1. (20)IV.

Q: What is the significance of the NL! permutation matrix?

In their implementation, the variables are expressed asΓi = ‖γ k,i‖2Σ−1i ∼ χ2(2mF NL +6). (27)Therefore, the authors choose the permutation with minimum weighted norm of the innovation as the feature association criteria, i.e.,î = arg min i=0,1,...,NL Γi. (28)V. INTEGRITY RISK FROM INCORRECT LANDMARK ASSOCIATIONFollowing the outline of fault analysis for data association error in the work by Joerger and Pervan [9], the authors also express the hazardous misleading information (HMI) for visual navigation at time epoch Nk asP(HMINk) =P(HMINk ,CA1...Nk)+P(HMINk , IA1...Nk) =P(HMINk |CA1...Nk)P(CA1...Nk) +P(HMINk |IA1...Nk) ( 1−P(CA1...Nk) ) , (29)where CA1...Nk is the event that all the data associations are correct from time 1 till time Nk, while IA1...Nk is the complement event that there can exist an incorrect association at any time between time 1 and time k.

Question

1. How many rads are used for the measurement noise?

2. What is the detection of a checkerboard landmark?

3. What is the standard deviation of measurement noise?

4. how can a camera be parametrized with a 6 degree of freedom?

Accepted Answer

5. The standard deviation for measurement noise is set to 1 [pixel], and the process noise standard deviation is 0.5 [m] for translation and 0.1 [rad] for rotation.

Accepted Answer

The detection can be based on models for designed pattern, or based on advanced machine learning methods for specific class of landmarks [15].

Accepted Answer

In their simulation, the standard deviation of measurement noise is set to 1 [pixel], and the process noise standard deviation is 0.5 [m] for translation.

Accepted Answer

the authors can parameterize the camera pose with a 6 degrees-of-freedom (DoF) vectorx = [ ~X(W )Cr] ∈ R6. (7)In order to simplify the notation, a function π j(·) : R6→ R2 is defined to project the 3D point ~X(W )j to the 2D coordinates on the image plane of a camera with 6DoF pose parameterized as x.

Accepted Answer

It can be seen that as the camera moves closer to the landmarks, the separability of the features increases, so the probability of correct association also increases.

Accepted Answer

It can be seen that by using landmarks with only 4 features each, the probability of incorrect association is smaller than 10−7 if the camera moves as close to the features as k= 10, while the landmarks with 35 feature pointshas reached P(IAk|CAk−1) < 10−7 from k ≥

Accepted Answer

Given a known landmark M consisting of mF feature points in the map (requiring mF ≥ 4 in general), the position and orientation of the camera can be estimated using corresponding 2D measurements { j = 1, ...,mF |µ j} by iteratively solving the following nonlinear optimization problemx̂ = argmin x ‖µ −π(x)‖2Σ−1n , (10)whereµ = µ 1... µ mF ,π(x) = π1(x)... πmF (x) ,n = n1... nmF . (11) Σn is the covariance matrix of noise vector n.

Accepted Answer

(18)The posterior estimate at time k can be updated using the following equationsx̂k = x̄k +Gkγ k, P̂k = (I−GkHk)P̄k, (19)in which Hk is the linearized matrix concatenated over all the landmarks i= 1, ...,NL, and Gk is the Kalman gain calculated using the equationGk = P̄kHTk (HkP̄kH T k +Vk) −1. (20)IV.

Accepted Answer

In their implementation, the variables are expressed asΓi = ‖γ k,i‖2Σ−1i ∼ χ2(2mF NL +6). (27)Therefore, the authors choose the permutation with minimum weighted norm of the innovation as the feature association criteria, i.e.,î = arg min i=0,1,...,NL Γi. (28)V. INTEGRITY RISK FROM INCORRECT LANDMARK ASSOCIATIONFollowing the outline of fault analysis for data association error in the work by Joerger and Pervan [9], the authors also express the hazardous misleading information (HMI) for visual navigation at time epoch Nk asP(HMINk) =P(HMINk ,CA1...Nk)+P(HMINk , IA1...Nk) =P(HMINk |CA1...Nk)P(CA1...Nk) +P(HMINk |IA1...Nk) ( 1−P(CA1...Nk) ) , (29)where CA1...Nk is the event that all the data associations are correct from time 1 till time Nk, while IA1...Nk is the complement event that there can exist an incorrect association at any time between time 1 and time k.

Quantifying Feature Association Error in Camera-based Positioning

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Most frequently asked questions

1. How many rads are used for the measurement noise?

2. What is the detection of a checkerboard landmark?

3. What is the standard deviation of measurement noise?

4. how can a camera be parametrized with a 6 degree of freedom?

5. How does the probability of correct association increase?

6. How many features are there in the map?

7. what is the covariance matrix of noise vector n?

8. What is the posterior estimate of a checkerboard landmark?

9. What is the significance of the NL! permutation matrix?

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References

A Mathematical Introduction to Robotic Manipulation

EPnP: An Accurate O(n) Solution to the PnP Problem

Scalable Object Detection Using Deep Neural Networks

Two years of Visual Odometry on the Mars Exploration Rovers

Two years of Visual Odometry on the Mars Exploration Rovers: Field Reports

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