The complete UKF algorithm [5], [6], [7], [8] consists of an initialization, the sigma-points calculation, a prediction step with constant weighting coefficients and an update step.

– Initialization (with *k* = 1):

As a prior information we assumed completly repolarized ventricles (cells in rest)

$${\underset{\xaf}{\widehat{\mu}}}_{0}=\left[\begin{array}{c}-80\\ \vdots \\ -80\end{array}\right].$$(9)But as the TMV may range between −80 mV and 20 mV the initial first moment had an uncertainty *ff* of 100 mV. Therefore the covariance ${\Sigma}_{{\underset{\xaf}{\widehat{\text{s}}}}_{\text{0}}}$ is

$${\Sigma}_{{\underset{\xaf}{\widehat{\text{s}}}}_{\text{0}}}={\sigma}^{2}\cdot \underset{\xaf}{I}=10000\cdot \underset{\xaf}{I}.$$(10)– Sigma-points:

For filtering the data at time step *k* = 1, the probability distribution of ${\underset{\xaf}{s}}_{k-1}$ had to be approximated with the UT

$${\underset{\xaf}{X}}_{k-1}\left[\begin{array}{c}{\widehat{\mu}}_{k-1}\\ {\widehat{\mu}}_{k-1}+\rho {\left(\sqrt{{\Sigma}_{{\underset{\xaf}{\widehat{\text{s}}}}_{k-1}}}\right)}_{i}\\ {\widehat{\mu}}_{k-1}-\rho {\left(\sqrt{{\Sigma}_{{\underset{\xaf}{\widehat{\text{s}}}}_{k-1}}}\right)}_{i}\end{array}\right].$$(11)– Prediction step:

The sigma-points ${\underset{\xaf}{X}}_{k-1}$ were propagated through $\underset{\xaf}{f}(\cdot )$

$${\underset{\xaf}{X}}_{k|k-1}=\underset{\xaf}{f}\left({\underset{\xaf}{X}}_{k-1}\right).$$(12)The first and second moment were predicted with

$${\underset{\xaf}{\mu}}_{k}^{-}={\displaystyle \sum _{i=0}^{2L}{W}_{i}^{m}}{\underset{\xaf}{X}}_{i,k|k-1}$$(13)$${\Sigma}_{\underset{\xaf}{\text{s}}{-}_{k}}={\displaystyle \sum _{i=0}^{2L}{w}_{i}^{c}}\left({\underset{\xaf}{\chi}}_{i,k|k-1}-{\mu}_{k}^{-}\right){\left({\underset{\xaf}{\chi}}_{i}-{\underset{\xaf}{\mu}}_{k}^{-}\right)}^{T}+\Sigma \underset{\xaf}{\text{q}}$$(14)and with constant weighting coefficients

$${W}_{0}^{m}=\frac{\delta}{L+\delta}$$(15)$${W}_{i}^{m}=\frac{\delta}{2(L+\delta )}$$(16)$${w}_{0}^{c}=\frac{\delta}{L+\delta}+(1-{\alpha}^{2}+\beta )$$(17)$${w}_{i}^{c}=\frac{1}{2(L+\delta )}.$$(18)The secondary scaling parameter *β* ∈ ℝ can be used to minimize approximation errors of higher probability distribution moments by using prior knowledge about $\underset{\xaf}{s}$ [7], [8]. We set to

$$\beta =2.$$(19)– Update step:

As a linear measurement model was used, see Equation 2, the update step was linear as well and could be implemented without an additional UT:

$${\underset{\xaf}{K}}_{k}={\Sigma}_{{\underset{\xaf}{s}}_{k}^{-}}{\underset{\xaf}{H}}^{T}{\left(\underset{\xaf}{H}{\Sigma}_{{\underset{\xaf}{s}}_{k}^{-}}{\underset{\xaf}{H}}^{T}+\Sigma {\underset{\xaf}{\text{n}}}_{k}\right)}^{-1}$$(20)$${\underset{\xaf}{\widehat{\mu}}}_{k}={\underset{\xaf}{\mu}}_{k}^{-}+{\underset{\xaf}{K}}_{k}\left({\underset{\xaf}{d}}_{k}-\underset{\xaf}{H}{\underset{\xaf}{\mu}}_{k}^{-}\right)$$(21)$${\Sigma}_{{\underset{\xaf}{\widehat{s}}}_{k}}=(\underset{\xaf}{I}-{\underset{\xaf}{K}}_{k}\underset{\xaf}{H}){\Sigma}_{{\underset{\xaf}{s}}_{k}^{-}}.$$(22)The considered measurement model in Equation2 did not need a forward solution. This part is already integrated in the calculation of the data vector *d*_{k} using Tikhonov reconstruction.

## 2.2.2.1 Process noise covariance ${\Sigma}_{\underset{\xaf}{\text{q}}}$

For calculating the process noise covariance, the ground truth $\underset{\xaf}{s}$ at time step *k* of simulated TMVs was propagated through $\underset{\xaf}{f}(\cdot )$. The simulation ${\underset{\xaf}{s}}_{k}$ was calculated with a cellular automaton as in [11]. The difference ${\underset{\xaf}{\in}}_{k+1}\in {\mathbb{R}}^{L\times 1}$ between the prediction output ${\underset{\xaf}{\mu}}_{k+1}^{-}$ and the ground truth ${\underset{\xaf}{s}}_{k+1}^{-}$ was built

$${\underset{\xaf}{\in}}_{k+1}{\underset{\xaf}{\mu}}_{k+1}^{-}-{\underset{\xaf}{s}}_{k+1}.$$(23)Therefore, the process noise covariance had the form

$${\underset{\xaf}{\rho}}_{k+1}=\left({\underset{\xaf}{\in}}_{k+1}-{\overline{\underset{\xaf}{\in}}}_{k+1}\right){\left({\underset{\xaf}{\in}}_{k+1}-{\overline{\underset{\xaf}{\in}}}_{k+1}\right)}^{T}$$(24)with ${\overline{\underset{\xaf}{\in}}}_{k+1}$: arithmetic mean of ${\overline{\underset{\xaf}{\in}}}_{k+1}$. Because ${\underset{\xaf}{\rho}}_{k+1}$ was close to singular for all *k* and has a contribution to the inversion in Eq.2 0, only the diagonal of ${\underset{\xaf}{\rho}}_{k+1}$ was used. The process noise in the UKF had the form

$$\Sigma {\underset{\xaf}{\text{q}}}_{k}=\left[\begin{array}{ccc}{\rho}_{k,11}& & \underset{\xaf}{0}\\ & \ddots & \\ \underset{\xaf}{0}& & {\rho}_{k,LL}\end{array}\right].$$(25)## 2.2.2.2 Measurement noise covariance $\Sigma {\underset{\xaf}{\text{n}}}_{k}$

With the output model in Equation 2, $\Sigma {\underset{\xaf}{\text{n}}}_{k}$ was built from the difference between the ground truth ${\underset{\xaf}{s}}_{k}$ and the data ${\underset{\xaf}{d}}_{k}$

$$\underset{\xaf}{v}={\underset{\xaf}{d}}_{k}-{\underset{\xaf}{s}}_{k}$$(26)$$\Sigma {\underset{\xaf}{\text{n}}}_{k}=({\underset{\xaf}{v}}_{k}-{\underset{\xaf}{\overline{v}}}_{k}){({\underset{\xaf}{v}}_{k}-{\underset{\xaf}{\overline{v}}}_{k})}^{T}.$$(27)${\underset{\xaf}{\overline{v}}}_{k}$ indicates the arithmetic mean of $\underset{\xaf}{v}$

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