A Probabilistic Solution to the AX=XB Problem: Sensor Calibration Without Correspondence

28/08/2013
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A Probabilistic Solution to the AX=XB  Problem: Sensor Calibration Without  Correspondence

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A Probabilistic Solution to the AX=XB Problem: Sensor Calibration Without Correspondence M. Kendal Ackerman and Gregory Chirikjian Dept. Mech. Eng., Johns Hopkins University Baltimore, MD, USA Geometric Science of Information August, 29th 2013 Outline Motivations AX = XB is one of the most common mathematical formulations used in robot-sensor calibration problems. It can be found in a variety of applications including: •Camera calibration [1] •Cartesian robot hand calibration [1] •Robot eye-to-hand calibration [1] •Aerial vehicle sensor calibration [2] •Image guided therapy (IGT) sensor calibration [3] [1] Tsai, R., Lenz, R. (1989) [2] Mair, E., Fleps, M., Suppa, M., Burschka, D. (2011) [3] Boctor, E.M. (2006) Problem Notation Formulation Solution Conclusion An Example from Image Guided Therapy Problem Notation Formulation Solution Conclusion Definitions Problem Notation Formulation Solution Conclusion Solution Space It is well known that it is not possible to solve for a unique X from a single pair of exact (A, B)… One Parameter solution set Rank 2 + Two unspecified degrees of freedom …but if there are two instances of independent exact measurements, (A1, B1) and (A2, B2), then the problem can be solved. Problem Notation Formulation Solution Conclusion Solution Uniqueness • A unique solution is possible given two pairs with certain constraints[4,5] – SE(3) geometric invariants satisfied – Angle of the rotation axes is “sufficiently large” – (Ai, Bi) pairs have Correspondence • In the presence of noise, the traditional goal becomes one of finding an X with least-squared error given corresponding pairs (Ai, Bi) for i=1,2,...,n. [4] Chen (1991) [5] Ackerman, M.K., Cheng, A., Shiffman, B., Boctor, E., Chirikjian, G. (2014) Problem Notation Formulation Solution Conclusion What do We Mean by Correspondence Data Stream Data Stream A’s B’s A1 B1 A2 B2 An Bn … Problem Notation Formulation Solution Conclusion The Need for a New Solution Method • In practical problems, it is often the case that the data streams containing the A’s and B’s: – will present at different sample rates, – they will be asynchronous, – and each stream may contain gaps in information. • We therefore present a method for calculating the calibration transformation, X, that works for data without any a priori knowledge of the correspondence between the A’s and B’s. Problem Notation Formulation Solution Conclusion Probability Theory on where is the typical Lebesgue integration measure. In this context, the mean is defined And the covariance The mean and covariance of the convolution of two pdfs is Problem Notation Formulation Solution Conclusion Lie Groups and Rigid Body Motion is a Lie group and therefore the concept of integration exists: , where: The group of rigid body motions, SE(3), , Problem Notation Formulation Solution Conclusion Convolutions and Additional Notation on SE(3) Define a delta function as: Define the convolution of two functions as: And a shifted delta function as: The “v” operator is defined as: (x1, x2, … , xn)T Problem Notation Formulation Solution Conclusion Some Assumptions And given a distance metric we make the reasonable assumption that This is mainly due to the fact that the sensor reading frequency is “high” relative to the sensed motion. Given Problem Notation Formulation Solution Conclusion Given our assumptions, we can then define the mean and covariance of our probability density functions, f(H), as and Probability Theory on SE(3) For more information on defining the mean and covariance on SE(3), see Chirikjian, G., “Stochastic Models, Information Theory, and Lie Groups” Birkhauser, 2011 Problem Notation Formulation Solution Conclusion Where the Riemannian distance function is usually given as something akin to or - The Definition of Mean and Covariance The definition of mean used differs from that most often used in the literature when taking a Riemannian-geometric approach where The benefits of the definitions on the previous slide [10,11] 1) Avoid the arbitrary choice of a weighting matrix 2) Mean behaves nicely under convolution 3) Equivariance of the mean under conjugation [10] Wang, Y., Chirikjian, G. (2008) [11] Chirikjian, G. (2011) Problem Notation Formulation Solution Conclusion Our “Batch” Method Because real-valued functions can be added and convolution is a linear operation on functions, all n instances can be written into a single equation of the form We can normalize the functions to be probability density functions (pdfs): Problem Notation Formulation Solution Conclusion Our “Batch” Method Given the convolution of two functions, we can write the evolution of the mean and covariance as: where and the “hat” operator is defined such that given and ≈ ≈ Problem Notation Formulation Solution Conclusion Our “Batch” Method Since the mean of is and its covariance is the zero matrix we can write our “Batch” method formulation Batch Method “AX=XB” Equations: (1) (2) Problem Notation Formulation Solution Conclusion Solving the Batch Equations From (2), if we decompose and into blocks Batch Method “AX=XB” Equations: we can write the equation of the first blocks as (1) (2) Problem Notation Formulation Solution Conclusion Using the Eigendecomposition, , we find that Solving the Batch Equations Batch Method “AX=XB” Equations: (1) (2) Therefore, Problem Notation Formulation Solution Conclusion Solving the Batch Equations From (1), we know that Batch Method “AX=XB” Equations: therefore we chose the Rx (from the set of 4) that minimizes Once we have Rx, we find tx, from the equation of the second blocks of (2) (1) (2) Problem Notation Formulation Solution Conclusion Numerical Solution of the Batch Equations With the discrete nature of our application, we can likewise define the mean and covariance in a discrete sense An iterative procedure can be used for computing MA which uses an initial estimate of the form Then a gradient descent procedure is used to update so as to minimize the cost The covariance can then be computed: Problem Notation Formulation Solution Conclusion Results The Batch Method solves the AX=XB problem regardless of “scrambled” correspondences. Problem Notation Formulation Solution Conclusion Conclusions: Contributions • We established that the AX = XB sensor calibration problem can be solved with a new “Batch Method” that does not require a priori knowledge of the A and B correspondence. • We showed both the formulation and solution of the probabilistic equations of the Batch Method. • The Batch Method is shown to solve for X for any level of mismatch between A’s and B’s, performing much better than traditional methods, which require correspondences, under similar circumstances. Problem Notation Formulation Solution Conclusion Future Work • Continue to optimize for the noise case • Continue to test with experimental data • Combine with addition data processing algorithms we have developed Problem Notation Formulation Solution Conclusion Acknowledgements • This work was supported by NSF Grant RI-Medium: 1162095 • Thank you to the organizers of GSI 2013 • Thank you for listening • Questions? Problem Notation Formulation Solution Conclusion Supplementary The AX=XB Problem: Traditional Solution Methods Require Known Correspondence [6] Mills, D. L. (1991) [7] Kang, H. J., Cheng, A., Boctor, E. M. (2012) There are a few instances in the literature when the sensor data used is temporally “unsynched” and methods are used to recover the correspondence: – time stamping the data[6] – dedicated software modules for syncing the data[7] – analyzing components of the sensor data stream to determine a correlation[8,9] [8] Mair, E., Fleps, M., Suppa, M., Burschka, D. (2011). [9] Ackerman, M.,K., Cheng, A., Shiffman, B., Boctor, E., Chirikjian, G. (2014) The AX=XB Problem: One Solution Method using the Kronecker Product re-write AX=XB as where, , for different pairs, (Ai, Bi), stack equations to obtain Jx = b where J is 12n x 6n and b is 6n-dimensional. Then compute the least squares solution , which is given as: kRx must then be projected back into SO(3) Using the fact that See Andreff, N. (2001) for more details…. Solving the Batch Equations: Results The Batch Method responds in a “nice” fashion to the introduction of noise.