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test fusion
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test projective
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manifold of positive
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manifold for estimating covariances
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manifold SE3 right
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HARD Epipolar constraint between projected points in two perspective views, see Roberto Tron's page
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MAYVE Symmetric, positive definite matrices
A point X on the manifold is represented as a symmetric positive definite matrix X (nxn). Tangent vectors are symmetric matrices of the same size (but not necessarily definite).
The Riemannian metric is the bi-invariant metric, described notably in Chapter 6 of the 2007 book "Positive definite matrices" by Rajendra Bhatia, Princeton University Press.
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example spd vs wishart
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examples
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smoothing http://becs.aalto.fi/en/research/bayes/ekfukf/documentation.pdf eq. 3.61
input: all the states output: use X(k, kalman) and f to computer X(k+1,pred) D = C(k+1 pred, k) / P(k+1,pred) P(k,smoothed) = P(k,kalman) D (P(k+1,smoothed)-P(k+1,pred)) D' mu(k,smoothed) = mu(k,smoothed) boxplus D (mu(k+1,smoothed) boxminus mu(k+1,pred))
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what about other variables/parameters? e.g. a common set of parameters? See slides about total variance and uncoditioning we aim at: (Xt|t|Xt+1|t = xt+1) but we don't know xt+1 so we use law of total exèectatopm and variance: E(X) = EZ( E(X|Y = Z) ) Var(X) = EZ( Var(X|Y = Z) ) + VarZ( E(X|Y = Z) ) so we obtain X(t|T) noting that Xt|t | Xt+1|t=Xt+1|T
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if we save both the X(k,kalman) and the X(k,pred) then we can skip the f
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svdsqrt reduction if too small