Shift- and stretch-invariant non-negative matrix factorization with an application to brain tissue delineation in emission tomography data

Anders S. Olsen; Miriam L. Navarro; Claus Svarer; Jesper L. Hinrich; Morten Mørup; Gitte M. Knudsen

cs.LGApr 9, 2026

Shift- and stretch-invariant non-negative matrix factorization with an application to brain tissue delineation in emission tomography data

Authors: Anders S. Olsen, Miriam L. Navarro, Claus Svarer, Jesper L. Hinrich, Morten Mørup, Gitte M. Knudsen

Organizations: Neurobiology Research Unit, Copenhagen University Hospital Rigshospitalet, Copenhagen, Denmark · Department of Neuroscience, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark · Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark · Department of Clinical Medicine, Faculty of Health and Medical Sciences, University of Copenhagen, Copenhagen, Denmark

Abstract

Dynamic neuroimaging data, such as emission tomography measurements of radiotracer transport in blood or cerebrospinal fluid, often exhibit diffusion-like properties. These introduce distance-dependent temporal delays, scale-differences, and stretching effects that limit the effectiveness of conventional linear modeling and decomposition methods. To address this, we present the shift- and stretch-invariant non-negative matrix factorization framework. Our approach estimates both integer and non-integer temporal shifts as well as temporal stretching, all implemented in the frequency domain, where shifts correspond to phase modifications, and where stretching is handled via zero-padding or truncation. The model is implemented in PyTorch (https://github.com/anders-s-olsen/shiftstretchNMF). We demonstrate on synthetic data and brain emission tomography data that the model is able to account for stretching to provide more detailed characterization of brain tissue structure.

Figures & tables

Figure 1: SPECT data from an example pig after 99m Tc-DTPA radiotracer injection in the cisterna magna (CM). Time-activity curves in the time-series plot - from selected voxels and timepoints indicated by squares - were normalized to unit norm for visualization.

Figure 2: Two-component NMF models applied to two-component synthetic data with random shifts and stretches. Model performance is measured using matched correlation of the $\mathbf{A}$ -matrix to the ground truth design matrix (variance is over 25 repeated model estimates), while the learned profiles (bottom) correspond to the $\mathbf{S}$ -matrix of the estimate with the lowest loss.

Figure 3: Effect of model order $K$ on variance explained (shaded area is variance over pigs).

Figure 4: Extracted channel maps (left), temporal profiles (mid), and selected channel reconstructions (right) for pig-5 using $K=3$ components for the four tested models. CSF: cerebrospinal fluid.

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Abstract

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