SSCD Metrics: Weighted Sobolev Geometry

• SSCD metrics are weighted Sobolev metrics defined on shape spaces that ensure scale invariance and positive geodesic distance.
• They integrate global volume-dependent weight functions with local differential operators to unify Sobolev, curvature-weighted, and almost local metrics.
• These metrics enable efficient numerical computation of geodesics, supporting applications in computer vision, medical imaging, and shape analysis.

SSCD metrics refer to a class of weighted Sobolev metrics defined over the space of immersions and shape spaces, characterizing geometry in infinite-dimensional manifolds of shapes (such as curves and surfaces). These metrics are structurally built to address scale invariance, provide positive geodesic distance, and unify previous Sobolev and almost local metrics through global volume-dependent weighting. Their mathematical formulation and implications extend to computational shape analysis, computer vision, and differential geometry, particularly where invariance and non-degenerate distance metrics are required.

1. Definition and Mathematical Construction

SSCD metrics take the form

$G^P_f(h, k) = \int_M \left[\sum_{i=0}^p \Phi_i(\Vol(f)) \cdot g((P_i)_f h, k)\right] \vol(f^*g)$

where:

• $M$ is a compact manifold representing the parameter domain of the shape,
• $N$ is a complete Riemannian manifold (the ambient space),
• $f : M \to N$ is an immersion (a smooth injection at the level of derivatives),
• $f^*g$ is the pullback metric on $M$,
• $h, k$ are tangent vectors to $f$ in the space of immersions,
• $\vol(f^*g)$ is the volume density induced by $f$,
• $\Vol(f) = \int_M \vol(f^*g)$ is the total immersed volume,
• Each $\Phi_i(\Vol(f))$ is a positive smooth function controlling global scaling effects,
• Each $(P_i)_f$ is a local (often Laplacian-based) symmetric, positive, elliptic pseudo-differential operator.

The metric integrates a weighted sum of inner products of operator-transformed tangents along the immersed manifold, and by varying $\Phi_i$ and $(P_i)_f$ it generalizes classical Sobolev, curvature-weighted, and conformal metrics.

The weights $\Phi_i$ are functions only of the global immersed volume, making the metric “almost local”: all nonlocality is confined to a simple global dependency, while local geometric processing is performed by $(P_i)_f$.

2. Geodesic Equations and Horizontal Bundles

The paper derives geodesic equations for these metrics in both immersion and shape spaces. The geodesic equation for evolution $f_t$ is governed by metric gradients $H$ and $K$: $\frac{d}{dt} f_t = H_f(f_t, f_t) - K_f(f_t, f_t)$ where

$p = G^P_f(f_t, \cdot) = P f_t \otimes \vol(f^*g)$

and $P = \sum_i \Phi_i(\Vol(f)) \cdot (P_i)_f$.

Adjoint computations (Adj$(V P_i)$) and explicit differentiation with respect to the immersion foot point are required to state $H$ and $K$ explicitly. Under ellipticity and regularity conditions, the equation is well-posed and the exponential map is a local diffeomorphism.

On shape space $B_i(M,N) = \Imm(M,N) / \mathrm{Diff}(M)$, vector fields orthogonal to the reparametrization orbit form the horizontal subbundle; for almost local metrics, horizontality reduces to normal vector fields—a major simplification for analysis and computation.

3. Scale Invariance and Geodesic Distance

A principal technical achievement is scale invariance via the weight functions $\Phi_i(\Vol(f))$, chosen (e.g., as powers of volume) to ensure

$G^P_{\lambda f} = G^P_f$

for any scaling $\lambda$.

This is central in applications where similarity up to scale is required (e.g., comparing anatomical structures of different sizes).

Another critical property enabled by the weighted (even order-zero) metrics is strictly positive geodesic distance:

• Classical $L^2$ metrics result in identically zero geodesic distance in shape space for curves and surfaces (degenerate metric topology).
• Weighted constructions allow for positive distances even when the operator order is low.

4. Special Classes of Weighted Sobolev Metrics

The paper identifies several important specializations:

• Almost Local Order-0 Metrics: Where $P(f) = \Phi(\Vol(f)) \cdot \mathrm{Id}$, optionally with curvature enhancement (GA-metric: $P = (1 + A|Tr(S)|^2) \cdot \mathrm{Id}$, $A \ge 0$).
  • Horizontal bundle reduces to normal vector fields.
  • Suitable for explicit numerical geodesic computation.
• Conformal Metrics: Weighting that depends only on total volume, leading to conformal structure for curves and hypersurfaces.
• Scale-Invariant Sobolev Metrics: Weighting as $Vol(f)^{-2(i-1)/m}$ (with $m = \dim M$) on operators of Laplacian-type order $2i$.
  • Retains both scale invariance and Sobolev smoothing.

5. Applications in Shape Analysis and Geometry Processing

Weighted Sobolev metrics provide a Riemannian structure that underpins many tasks in computational geometry:

• Computing intrinsic distances between shapes, surfaces for shape matching, registration, and quantification.
• Distances in shape space that are invariant under reparametrizations and scaling, necessary for robust comparison in computer vision, medical imaging, morphometry, and graphics.
• Framework supports conserved quantities and facilitates algorithmic computation of geodesics for continuous shape deformation.

The tractable horizontal bundle and explicit formulas allow efficient numerical implementation, supporting lifting of shape-space geodesics to immersion-space curves and enabling practical geodesic shooting or optimization.

6. Unification and Theoretical Implications

This family of metrics unifies previous work on Sobolev metrics, almost local metrics, and curvature-weighted metrics:

• Provides a parameterizable model, tunable for application-specific invariances.
• Admits theoretical investigation (e.g., sectional curvature, stability, completeness).
• Applies to both planar curves and higher-dimensional hypersurfaces in any complete Riemannian ambient manifold.

Conditions for well-posedness of the geodesic equation and nondegeneracy of the induced topology are explicitly characterized.

7. Summary Table of Core Metric Components

Component	Symbol	Role/Characteristic
Immersion	$f: M \to N$	Shape mapping
Tangent vectors	$h, k$	Variation directions
Pullback metric	$f^*g$	Induced by immersion
Volume density	$\vol(f^*g)$	Integration measure
Global volume	$\Vol(f)$	Scale and weight control
Weight functions	$\Phi_i(\Vol(f))$	Scale invariance / metric tuning
Operators	$(P_i)_f$	Differential, local action on tangents
Metric formula	$G^P_f(h, k)$	Weighted sum/integrated inner product

8. Concluding Perspective

Weighted Sobolev metrics (SSCD) offer a rigorous foundation for metric geometry of shape spaces. By addressing scale invariance and guaranteeing positive geodesic distance in infinite-dimensional shape manifolds, they resolve longstanding challenges in shape analysis. The explicit dependence on global volume, integration of higher-order differential operators, and reduction of nonlocality to minimal dependence render these metrics suitable both for theoretical exploration and practical implementation in computational geometry, vision, and allied disciplines (Bauer et al., 2011).