Non-intrusive parametrized-background data-weak reconstruction of cardiac displacement fields from sparse MRI-like observations

Abstract Personalized cardiac diagnostics requires accurate reconstruction of myocardial displacement fields from sparse clinical imaging data. In this work, we apply the Parametrized-Background Data-Weak (PBDW) approach to three-dimensional (3D) cardiac displacement field reconstruction from limited magnetic resonance image-like observations. We introduce two methodological enhancements: (i) an H -size minibatch worst-case orthogonal matching pursuit algorithm that improves Sensor Selection (SSEL) while maintaining reconstruction accuracy, and (ii) memory optimisation techniques exploiting block matrix structures in vectorial problems. We demonstrate the effectiveness of the method through validation on a three-dimensional left ventricular model with simulated scar tissue. Starting with noise-free reconstruction, we systematically incorporate Gaussian noise and spatial sparsity mimicking realistic Magnetic Resonance Image acquisition protocols. Results show exceptional accuracy in noise-free conditions with relative $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> error of $${1e-5}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> , robust performance with 10% noise achieving relative $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> error of $${1e-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and effective reconstruction from sparse measurements with relative $$L_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> error of $${1e-2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>e</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The online reconstruction achieves sub-second computation times for a given patient geometry, enabling rapid clinical feedback and parameter studies that would be prohibitive with full Finite Element simulations, demonstrating significant potential for integration into clinical cardiac modelling workflows.