Sparse Nystrom Approximation of Currents and Varifolds

TL;DR

An algorithm for compression of the currents and varifolds representations of shapes, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces is derived, which is faster than existing compression techniques, and comes with theoretical guarantees on the rate of convergence of the compressed approximation.

Abstract

We derive an algorithm for compression of the currents and varifolds representations of shapes, using the Nystrom approximation in Reproducing Kernel Hilbert Spaces. Our method is faster than existing compression techniques, and comes with theoretical guarantees on the rate of convergence of the compressed approximation, as a function of the smoothness of the associated shape representation. The obtained compression are shown to be useful for down-line tasks such as nonlinear shape registration in the Large Deformation Metric Mapping (LDDMM) framework, even for very high compression ratios. The performance of our algorithm is demonstrated on large-scale shape data from modern geometry processing datasets, and is shown to be fast and scalable with rapid error decay.

Figures (6)

• Figure 1: Numerical curves comparing error decay of RLS compression (black) to theory bound (red) and uniformly sampled compression (blue), on cat (top), head (middle) and flamingo (bottom) surfaces. Left: Error curves for currents. Right: Error curves for varifolds.
• Figure 2: Top left: spherical template. Top right: target mesh. Bottom left: Matching with full metrics taking $1$ hour and $41$ minutes, and Hausdorff metric error $d_{H} = 0.026$. Bottom right: Matching with $97\%$ compression of template and target taking only $14$ minutes, and Hausdorff metric error $d_{H} = 0.030$.
• Figure 3: Left: an example of LDDMM matching with Varifolds, from spherical template to brain without compression, taking $1$ hour and $49$ minutes, and Hausdorff metric error $d_{H} = 0.005$. Right: the same example matching but with $99\%$ compression taking only $11$ minutes, and Hausdorff metric error $d_{H} = 0.007$.
• Figure 4: Run-time in seconds (left column) and error comparison (right column) of RLS method vs Greedy method DURRLEMAN for compression of currents on example surface 'cat' (top row) and 'head' (bottom row).
• Figure 5: Run-time in seconds (left column) and error comparison (right column) of RLS method vs optimization method VarCompression for compression of varifolds, on example surface 'bunny' (top row) and 'brain' (bottom row).

Theorems & Definitions (17)

• Remark 3.1
• Theorem 4.1
• Theorem 4.2
• Theorem 4.3
• Theorem 4.4
• Theorem 5.1
• Corollary 5.2
• Lemma 6.1
• Proof 1
• Corollary 6.2