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Computing Diffeomorphic Paths for Large Motion Interpolation.

Dohyung Seo1, Ho Jeffrey2, Baba C Vemuri2

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PubMed
Summary
This summary is machine-generated.

This study presents a new method for calculating paths between diffeomorphisms by utilizing a quotient space identified as a Hilbert space sphere. This approach enables smoother interpolation of video frames while preserving shapes and volumes.

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Area of Science:

  • Computational geometry
  • Image processing
  • Differential geometry

Background:

  • Computing geodesic paths in the infinite-dimensional space of diffeomorphisms (Diff(Ω)) is computationally challenging.
  • Existing methods like Large Deformation Diffeomorphic Metric Mapping (LDDMM) face difficulties in direct geodesic computation.
  • The space of diffeomorphisms modulo volume-preserving diffeomorphisms (Diff(M)/Diff(M)0) has recently been identified as a Hilbert space unit sphere.

Purpose of the Study:

  • To introduce a novel framework for computing paths of diffeomorphisms.
  • To bypass the computational difficulties of direct geodesic computation in Diff(Ω).
  • To enable shape and volume preservation during image interpolation.

Main Methods:

  • Utilizing the quotient map from Diff(Ω) to the Hilbert space unit sphere (Diff(M)/Diff(M)0).
  • Two-stage computation: projecting diffeomorphisms onto the sphere and computing geodesics there.
  • Lifting the geodesic path back to the space of diffeomorphisms by solving a constrained quadratic programming problem using augmented Lagrangian methods.

Main Results:

  • The proposed framework successfully estimates paths of diffeomorphisms.
  • The method preserves shapes and volumes in deformed images along the interpolated path.
  • Application to interpolating intermediate frames in frame-sub-sampled video sequences shows favorable comparison with LDDMM.

Conclusions:

  • The novel framework offers an effective approach to computing geodesic paths in the space of diffeomorphisms.
  • Leveraging the Hilbert space structure of the quotient space simplifies path computation.
  • The method demonstrates practical utility in video frame interpolation, outperforming existing techniques in certain aspects.