🍩 Database of Original & Non-Theoretical Uses of Topology

(found 2 matches in 0.000893s)

  1. TILT: Topological Interface Recovery in Limited-Angle Tomography (2024)

    Elli Karvonen, Matti Lassas, Pekka Pankka, Samuli Siltanen
    Abstract A wavelet-based sparsity-promoting reconstruction method is studied in the context of tomography with severely limited projection data. Such imaging problems are ill-posed inverse problems, or very sensitive to measurement and modeling errors. The reconstruction method is based on minimizing a sum of a data discrepancy term based on an \$\ell\textasciicircum2\$-norm and another term containing an \$\ell\textasciicircum1\$-norm of a wavelet coefficient vector. Depending on the viewpoint, the method can be considered (i) as finding the Bayesian maximum a posteriori (MAP) estimate using a Besov-space \$B_\11\\textasciicircum\1\(\\mathbb T\\textasciicircum\2\)\$ prior, or (ii) as deterministic regularization with a Besov-norm penalty. The minimization is performed using a tailored primal-dual path following interior-point method, which is applicable to problems larger in scale than commercially available general-purpose optimization package algorithms. The choice of “regularization parameter” is done by a novel technique called the S-curve method, which can be used to incorporate a priori information on the sparsity of the unknown target to the reconstruction process. Numerical results are presented, focusing on uniformly sampled sparse-angle data. Both simulated and measured data are considered, and noise-robust and edge-preserving multiresolution reconstructions are achieved. In sparse-angle cases with simulated data the proposed method offers a significant improvement in reconstruction quality (measured in relative square norm error) over filtered back-projection (FBP) and Tikhonov regularization.

    Community Resources

  2. Homology-Based Morphometry of Brain Atrophy: Methods and Applications (2026)

    Donato Quiccione, Mariam Pirashvili, Nathan Broomhead, Sean J. Fallon
    Abstract Understanding the structure of the brain, and how it changes with time and disease, is a core goal of structural neuroimaging. Contemporary approaches to structural brain analysis are dominated by voxel-wise, mass-univariate methods such as voxel-based morphometry (VBM). However, these techniques require images to be normalized to a standard template, which can obscure subject-specific geometric features. Normalization to a common stereotactic space can also be problematic when comparing groups with substantial brain pathology, lesions, or other anatomical abnormalities. Here, we introduce two complementary pipelines based on persistent homology (PH), a tool from topological data analysis, to quantify multiscale geometric features of structural T1-weighted MRI scans. Pipeline 1 quantifies regional thinning by applying the Euclidean distance transform to tissue masks in a slice-wise manner. Pipeline 2 uses \(\alpha\)-filtrations to measure structural similarity between pairs of scans, capturing sulcal widening and ventricular enlargement. Synthetic experiments with controlled induced lesions showed that Pipeline 1 is best suited to between-subject analyses, whereas Pipeline 2 is better suited to within-subject designs. Applied to real-world data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), Pipeline 1 separated Alzheimer's disease (AD) from cognitively normal (CN) participants using single-modality T1-weighted MRI without nonlinear registration (ROC-AUC = 0.895), with peak effects localized to medial temporal regions. Pipeline 2 captured disease-related longitudinal change, with follow-up scans remaining closest to their own baselines and AD subjects showing greater short-interval change than CN subjects. Together, these pipelines provide interpretable topological biomarkers for cross-sectional group comparisons and longitudinal tracking.

    Community Resources