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. 2008 Dec 1;80(3):317-336.
doi: 10.1007/s11263-008-0141-9.

Large Deformation Diffeomorphic Metric Curve Mapping

Joan Glaunès  1 , Anqi Qiu, Michael I Miller, Laurent Younes
Affiliations

Large Deformation Diffeomorphic Metric Curve Mapping

Joan Glaunès et al. Int J Comput Vis. .

Abstract

We present a matching criterion for curves and integrate it into the large deformation diffeomorphic metric mapping (LDDMM) scheme for computing an optimal transformation between two curves embedded in Euclidean space ℝ(d). Curves are first represented as vector-valued measures, which incorporate both location and the first order geometric structure of the curves. Then, a Hilbert space structure is imposed on the measures to build the norm for quantifying the closeness between two curves. We describe a discretized version of this, in which discrete sequences of points along the curve are represented by vector-valued functionals. This gives a convenient and practical way to define a matching functional for curves. We derive and implement the curve matching in the large deformation framework and demonstrate mapping results of curves in ℝ(2) and ℝ(3). Behaviors of the curve mapping are discussed using 2D curves. The applications to shape classification is shown and experiments with 3D curves extracted from brain cortical surfaces are presented.

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Figures

Fig. 1
Fig. 1
Evaluation of a vector field (blue arrows), w(x), along a curve (in red), C. Left: Directions of vectors match those of the tangents to the curve. Middle: Directions are perpendicular to the curve. Right: Directions match, but norm of vectors are close to zero along the curve
Fig. 2
Fig. 2
Two parallel segments C and S, separated by a small distance ε
Fig. 3
Fig. 3
Examples for plane curve matching. Bone, bird, and hand examples are respectively shown in rows. The first column shows target shapes. The second and third columns show source shapes. Blue curves are source shapes, while green curves are deformed source shapes
Fig. 4
Fig. 4
Panels from the left to the right depict the sequence of geodesic mappings connecting the source hand to the target hand at time t = 0, 0.2, 0.4, 0.6, 0.8, 1. The source shape and target shape are respectively represented in green and red
Fig. 5
Fig. 5
Comparison between the landmark and curve matchings. In each example (A) or (B), panels (a, b) show the source and target curves. Panels (c, d) show results from the curve and landmark matchings, respectively. The target and the deformed source shapes are respectively shown in red and blue
Fig. 6
Fig. 6
Box bump experiments. Each row shows one mapping from a template (cyan curve) to a target (red curve) with evolution of φtυ^ at different times t ∈ [0, 1] denoted by green curves. Top row: first experiment when two bumps on the template and target curves are close to each other. Bottom row: second experiment when two bumps on the template and target curves are further from each other
Fig. 7
Fig. 7
2-dimensional scaling plot of the distance matrix computed from pairwise matchings of 17 box bumps shapes
Fig. 8
Fig. 8
Rows (a–d) respectively show hand, dude, turtle, and fish
Fig. 9
Fig. 9
A scatter plot of feature dimensions from multidimensional scaling. 1: hand, 2: turtle, 3: fish, 4: dude
Fig. 10
Fig. 10
Panel (a) shows the curve defined as boundary curve for planum temporale cortical surface. Panel (b) depicts six curves describing the main shape of cingulate gyrus. Each curve is colored differently with indices in the same color scheme. The surfaces are colored by the curvature information
Fig. 11
Fig. 11
Three-dimensional deformation of planum temporale cortical surface. Panels (a)–(d) depict the deformation applied to planum temporale cortical surfaces. From the left to the right, panels respectively show the source and deformed source surfaces as well as the source and deformed source surfaces overlaying with the target surface in red. Panel (e) shows the target of planum temporale surface
Fig. 12
Fig. 12
Three-dimensional deformation of cingulate cortical surface. Panels (a)–(c) depict the deformation applied to cingulate cortical surfaces. In each panel, the top row respectively shows the source and deformed source surfaces from the left to the right; the bottom shows the source and deformed source surfaces overlaying with the target surface in red, which is shown on panel (d)
Fig. 13
Fig. 13
Panels (a) and (b) depict the cumulative distance distributions for planum temporale and cingulate surfaces. The cumulative distance distributions for the source surfaces are in gray, while the distributions for the deformed source surface are in black. Red curves are the mean distributions for the source and deformed source surfaces
Fig. 14
Fig. 14
Comparison of the cumulative distance distributions between the landmark and curve matchings. Solid, dashdot, and dash lines represent average cumulative distance distributions over the original source, deformed source surfaces via the landmark and curve matchings, respectively
Fig. 15
Fig. 15
Comparison of deformation fields between the landmark and curve matchings. Panels (a–d) show the target, source, deformed source surface after the landmark matching, deformed source surface after the curve matching. Red dots represents paired landmarks on the target and source surfaces. Panels (e–g) show triangulated meshes respectively corresponding to the surfaces in panels (b–d)
Fig. 16
Fig. 16
Comparison of deformation fields between the landmark and curve matchings. Panels (a–d) show the target, source, deformed source surface after the landmark matching, deformed source surface after the curve matching. Red dots represents paired landmarks on the target and source surfaces. Panels (e–g) show triangulated meshes respectively corresponding to the surfaces in panels (b–d)
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References

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