Two-Tensor Tractography Using a Constrained Filter

. Author manuscript; available in PMC: 2010 Jun 28.

Published in final edited form as: Med Image Comput Comput Assist Interv. 2009;12(Pt 1):894–902. doi: 10.1007/978-3-642-04268-3_110

Algorithm 1. Unscented Kalman Filter

1: Form weighted sigma points

X_{t} = {w_{i}, x_{i}}_{i = 0}^{2 n}

around current mean x_t and covariance P_t with scaling factor ζ

x_{0} = x_{t} x_{i} = x_{t} + {[\sqrt{ζ P_{t}}]}_{i} x_{i + n} = x_{t} - {[\sqrt{ζ P_{t}}]}_{i}

2: Predict the new sigma points and observations

X_{t + 1 | t} = f [X_{t}] Y_{t + 1 | t} = h [X_{t + 1 | t}]

3: Compute weighted means and covariances, e.g.

{\bar{x}}_{t + 1 | t} = ∑_{i} w_{i} x_{i} P_{x y} = ∑_{i} w_{i} (x_{i} - {\bar{x}}_{t + 1 | t}) {(y_{i} - {\bar{y}}_{t + 1 | t})}^{T}

4: Update estimate using Kalman gain K and scanner measurement y_t

x_{t + 1} = {\bar{x}}_{t + 1 | t} + K (y_{t} - {\bar{y}}_{t + 1 | t}) P_{t + 1} = P_{x x} - K P_{y y} K^{T} K = P_{x y} P_{y y}^{- 1}