Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Dynamic Inverse Problems in Imaging 2.1 Motion Model 2.2 Joint Image Reconstruction and Motion Estimation Methods 3.1 Numerical evaluation with Neural Fields 3.2 Numerical evaluation with grid-based representation Numerical Experiments 4.1 Synthetic experiments Conclusion, Acknowledgments, and References Dynamic Inverse Problems in Imaging 2.1 Motion Model 2.2 Joint Image Reconstruction and Motion Estimation Dynamic Inverse Problems in Imaging Dynamic Inverse Problems in Imaging 2.1 Motion Model 2.1 Motion Model 2.2 Joint Image Reconstruction and Motion Estimation 2.2 Joint Image Reconstruction and Motion Estimation Methods 3.1 Numerical evaluation with Neural Fields 3.2 Numerical evaluation with grid-based representation Methods Methods 3.1 Numerical evaluation with Neural Fields 3.1 Numerical evaluation with Neural Fields 3.2 Numerical evaluation with grid-based representation 3.2 Numerical evaluation with grid-based representation Numerical Experiments 4.1 Synthetic experiments Numerical Experiments Numerical Experiments 4.1 Synthetic experiments 4.1 Synthetic experiments Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References Conclusion, Acknowledgments, and References 4 Numerical Experiments 4.1 Synthetic experiments 4.1.1 Two-squares phantom 4.1.1 Two-squares phantom Remark 1 The second row in figure 1 represents the velocity field as follows: the coloured boundary frame indicates the direction of the velocity field. The intensities of the image indicate the magnitude of the vector. As an example, the square on the right moves constantly up and slightly to the right during the motion. Remark 1 The second row in figure 1 represents the velocity field as follows: the coloured boundary frame indicates the direction of the velocity field. The intensities of the image indicate the magnitude of the vector. As an example, the square on the right moves constantly up and slightly to the right during the motion. Measurements are obtained by sampling one random angle per frame and further corrupted with Gaussian noise with standard deviation σ = 0.01. See the third row in figure 1. To highlight the necessity of motion models, two naive reconstructions are shown in the fourth row of figure 1. The one on the left corresponds to a time-static reconstruction, i.e., assuming that the squares are not moving. The result is an image that blurs those regions where the squares moved. The one on the right is a frame-by-frame reconstruction which, as expected, cannot get a reliable reconstruction from one projection only. Effect of the motion regularization parameter γ. Effect of the motion regularization parameter γ. Neural fields versus Grid-based method. Neural fields versus Grid-based method. Generalization into higher resolution. Generalization into higher resolution. 5 Conclusion In this work we studied neural fields for dynamic inverse problems. We saw how to enhance neural fields reconstruction for dynamic inverse problems by making use of explicit PDE-based motion regularizers, namely, the optical flow equation. Constraining the neural field to this physically feasible motion meant a significant improvement with respect to the more widely used motionless implicitly regularized network. This opens the option for studying more motion models, e.g., continuity equation for 3D+time problems, for neural fields since most of the literature relies entirely on the implicit regularization of the network. We saw that the motion regularization parameter γ played a relevant role in the quality of the reconstruction however its choice is not clear, a small value of it led to a similar behaviour with the implicitly regularized neural field, while a very large value promotes no motion. Finally, we highlight that our goal was to improve the reconstruction of the image leaving the motion estimation as an auxiliary problem and not a goal. However, there are applications where the motion is a relevant quantity, for example it is used in cardiac imaging for clinical assessment of the heart. In such cases, it can be necessary to think of explicit regularizers for the motion as well. We have also studied the performance of neural fields against classical grid-based representations, in this case, an alternating scheme plus PDHG, and even for the choice of regularization parameters α, β, γ for which this approach performed the best, neural fields still were better in terms of PSNR. We conclude that neural fields with explicit regularizers can significantly improve the discovery of spatiotemporal quantities. Their mesh-free nature makes them suitable for such tasks since derivatives can be computed via automatic differentiation but also their memory consumption can remain controlled even for large-scale imaging tasks [33]. Acknowledgments Pablo Arratia is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/S022945/1. References [1] Jennifer A Steeden, Grzegorz T Kowalik, Oliver Tann, Marina Hughes, Kristian H Mortensen, and Vivek Muthurangu. 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Authors: (1) Pablo Arratia, University of Bath, Bath, UK (pial20@bath.ac.uk); (2) Matthias Ehrhardt, University of Bath, Bath, UK (me549@bath.ac.uk); (3) Lisa Kreusser, University of Bath, Bath, UK (lmk54@bath.ac.uk). Authors: Authors (1) Pablo Arratia, University of Bath, Bath, UK (pial20@bath.ac.uk); (2) Matthias Ehrhardt, University of Bath, Bath, UK (me549@bath.ac.uk); (3) Lisa Kreusser, University of Bath, Bath, UK (lmk54@bath.ac.uk). This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv
