SphereFlow: Missing Modality Imputation via Geometric Transport on Hypersphere

SphereFlow. Missing Modality Imputation via Geometric Transport on Hypersphere. By Huihan Wang and eight others, from the Hong Kong Polytechnic University, Zhejiang University, and two other institutions. Published on the third of November.

Missing modality imputation means synthesizing the modalities you did not observe from the ones you did. Picture a set of medical scans where one contrast is absent and has to be filled in from the others. Existing methods treat each modality as living on its own latent manifold. That forces the generator to learn a long, unstructured jump from one manifold to another.

SphereFlow takes a different view. All modalities are projected into one shared latent space shaped like a hypersphere, using a frozen self-supervised encoder. Once every modality lives on the same sphere, imputing one from another stops being a long jump. It becomes short-range geometric transport between nearby points on that sphere.

Doing this with flow matching runs into two geometric conflicts. First, the standard Gaussian noise source is badly mismatched with data that sits on a unit-norm manifold. Second, the straight-line interpolation path leaves the surface of the sphere, entering a region where the decoder has never received a training signal. Both are resolved with simple, geometry-aware changes. The noise source is replaced by the latent of the observed modality, so the velocity field only has to learn the displacement from one modality to the other. And the decoder is trained with a time-weighted loss that extends its domain to the region just off the surface that the flow actually passes through.

There is a formal result behind the first change. The authors prove that starting from the data instead of from noise reduces the worst-case deviation off the manifold. Without it, that deviation grows with the latent dimension. With it, it is held to a small bound that does not depend on dimension and is governed by how similar the modalities are to each other.

Experiments cover multi-modal MRI imputation on BraTS, and translation between CT and MRI. Across all settings, SphereFlow is reported to reach state-of-the-art generation fidelity. The imputed data is also put to use downstream, in brain tumor segmentation. There, segmenting with the imputed scans closes roughly eighty percent of the gap between having a modality missing and the oracle that has every modality available.

The move here is a change of coordinates more than a change of model. Put every modality on one sphere with a shared encoder, start the flow from what you observed rather than from noise, and teach the decoder about the thin shell around the surface the flow travels through. The long jump becomes a short step, and the bound on how far it can stray no longer grows with dimension.