A new approach to the intracardiac inverse problem using Laplacian distance kernel

Figures

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  Fig. 1. Section of two-dimensional cardiac tissue model. The upper plane corresponds to the region where the catheters are catching the measurements, whereas the lower plane corresponds to the cardiac tissue acting as bioelectric sources

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  Fig. 2. Construction scheme of transfer matrix, by using a two-dimensional uniform impulse response \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_{e}(\overline{r})$$\end{document}he(r¯). Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$vect(h_e(\overline{r}{-}\overline{\tau }))$$\end{document}vect(he(r¯-τ¯)) denotes a vectorization of a two-dimensional impulse response shifted from the origin by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\tau }$$\end{document}τ¯, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs}…

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  Fig. 3. Scheme of the one-dimensional cardiac tissue model of source and catchment elements

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  Fig. 4. Ill-conditioning of the inverse problem in electrophysiology. A delta function is used as a source, and it is estimated in space (down) by different methods and for different SNR. The corresponding frequency responses are calculated (up): a SNR=15 dB; b SNR=25dB; c SNR=35 dB. Note that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\lambda }(\omega )$$\end{document}Hλ(ω) denotes the frequency response after tuning the free parameters on each method, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h_\lambda (x)$$\end{document}hλ(x) denotes the impulse response of each method

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  Fig. 5. Unipolar EGM waveforms obtained with different values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0$$\end{document}z0, normalized to its maximum amplitude for visualization purposes (a), and with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$z_0 = 0.02$$\end{document}z0=0.02 cm (b) on the one-dimensional tissue model

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  Fig. 6. Examples of free-parameter search for DSM−SVR. Blue points indicate the set of free-parameter values that were scrutinized in the search grid. Red circles indicate the obtained optimum combination for these values, to be used in the final solution

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  Fig. 7. Results on noise robustness of the methods in the one-dimensional model. MAE has been calculated for all the tested methods as a function of SNR (a, b) and of HNR (c, d). We also compared the classically used L-Curve (a, c) versus the LOO (b, d) as validation criterion to tune the free parameters

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  Fig. 8. Estimated potentials in the one-dimensional source, for all the tested methods, when tuning the free parameters in the classical algorithms with L-Curve (a, b) and with LOO (c, d), without noise (a, c) and with HNR = 45 dB (b, d)

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  Fig. 9. Details on the simulation results in the two-dimensional model, for SNR = 40 db. a Transmembrane potentials as a function of spatial coordinates, for the ideal and the estimated versions. b Absolute residuals for the estimated transmembrane potentials in a

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  Fig. 10. Details on the simulation results in the two-dimensional model, for HNR = 40 db. a Transmembrane potentials as a function of spatial coordinates, for the ideal and the estimated versions. b Absolute residuals for the estimated transmembrane potentials in (a)