Fast algorithms for free-discontinuity problems on high-dimensional biomedical data
Final Report Abstract
The discontinuities in almost all types of biomedical data – such as the boundaries of cellular structures in microscopic images and tissue layers in tomographic images – encode signi cant information. Since classical methods for image enhancement destroy this important information, discontinuity preserving models, and in particular models based on free-discontinuity problems have been developed. Free-discontinuity models lead to algorithmically challenging nonsmooth and nonconvex problems which are computationally demanding even for low-dimensional data. However, processing higher dimensional data is important since the dimensionality of the acquired data increases tremendously. Further, it is important to deal with indirect data terms, since many applications involve imaging operator; examples are computed tomography (where frequently the Radon transform models the imaging operator) or microscopy (where the point-spread function of the microscope has to be considered). It is also important to deal with nonlinear data. Such data arise in modern imaging modalities such as Interferometric Synthetic Aperture Radar (InSAR) or Di usion Weighted Magnet Resonance Imaging (DWMRI), and in connection with color spaces and with registration problems, for instance. Based on previous work of the applicants, new algorithmic approaches to free-discontinuity problems – including Potts problems, Mumford-Shah problems, and their higher-order variants – were proposed. A strategy was to split the free-discontinuity problems into subproblems which can be solved non-iteratively, e ciently, and exactly. A particularly important design criterion was that the methods scale well with high-dimensional data. Scalability to high-dimensional codomains was achieved by using subproblem solvers that have linear computational complexity in the dimension of the codomain. Therefore it was possible to process multispectral images and images holding large feature vectors in reasonable times. To deal with high-dimensional image domains, solving the subproblems in parallel turned out to be key. Parallelization on GPU allowed to process even volumetric images in reasonable time. Most of the proposed new approaches could be extended in a way to deal with indirectly measured data such as multispectral computed tomography and confocal microscopic images, thus addressing the inverse problem setup. For non-linear data spaces, where many tools of vector spaces are not available, new types of algorithm were developed to deal with the discrete inverse problem setup. Further interpolatory multiscale transforms were studied and a fast, exact, and non-iterative algorithm for the Potts problem with circle-valued data was found. A software toolbox is provided to the scienti c community. The developed methods were evaluated on various types of biomedical data such data from confocal microscopy, DWMRI, and computed tomography. Practitioners from the life sciences already use the corresponding implementations to analyze their data.
Publications
- “Fast segmentation from blurred data in D uorescence microscopy”. In: IEEE Transactions on Image Processing
M. Storath, D. Rickert, M. Unser, and A. Weinmann
(See online at https://doi.org/10.1109/TIP.2017.2716843) - “Jump-penalized least absolute values estimation of scalar or circle-valued signals”. In: Information and Inference: A Journal of the IMA
M. Storath, A. Weinmann, and M. Unser
(See online at https://doi.org/10.1093/imaiai/iaw022) - “Model-based learning of local image features for unsupervised texture segmentation”. In: IEEE Transactions on Image Processing
M. Kiechle, M. Storath, A. Weinmann, and M. Kleinsteuber
(See online at https://doi.org/10.1109/TIP.2018.2792904) - “Smoothing for signals with discontinuities using higher order Mumford–Shah models”. In: Numerische Mathematik
M. Storath, L. Kiefer, and A. Weinmann
(See online at https://doi.org/10.1007/s00211-019-01052-8) - “An Algorithm for Second Order Mumford Shah Models Based on a Taylor Jet Formulation”. In: SIAM Journal on Imaging Sciences
L. Kiefer, M. Storath, and A. Weinmann
(See online at https://doi.org/10.1137/19M1300959) - “Iterative Potts minimization for the recovery of signals with discontinuities from indirect measurements: the multivariate case”. In: Foundations of Computational Mathematics
L. Kiefer, M. Storath, and A. Weinmann
(See online at https://doi.org/10.1007/s10208-020-09466-9) - “PALMS Image Partitioning-A New Parallel Algorithm for the Piecewise A ne-Linear Mumford-Shah Model”. In: Image Processing
L. Kiefer, M. Storath, and A. Weinmann
(See online at https://doi.org/10.5201/ipol.2020.295) - “Variational regularization of inverse problems for manifoldvalued data”. In: Information and Inference: A Journal of the IMA
M. Storath and A. Weinmann
(See online at https://doi.org/10.1093/imaiai/iaaa010) - “Wavelet sparse regularization for manifold-valued data”. In: Multiscale Modeling & Simulation
M. Storath and A. Weinmann
(See online at https://doi.org/10.1137/19M1249801)
