Topology Aware Neural Interpolation of Scalar Fields

TL;DR

This work tackles the ill-posed problem of inverting persistence diagrams to reconstruct time-varying scalar fields from sparse keyframes. It introduces TimeToScalarField, a neural generator that maps time to a scalar field, augmented with topology-aware losses derived from persistence diagrams and a differentiable persistence optimization framework. A two-phase training regime first learns plausible geometry and then enforces topology, yielding instant query-time interpolations that preserve topological features. Empirical results on 2D and 3D datasets show superior topological fidelity and competitive data fidelity compared to linear and neural baselines, with a public implementation provided for reproducibility.

Abstract

This paper presents a neural scheme for the topology-aware interpolation of time-varying scalar fields. Given a time-varying sequence of persistence diagrams, along with a sparse temporal sampling of the corresponding scalar fields, denoted as keyframes, our interpolation approach aims at "inverting" the non-keyframe diagrams to produce plausible estimations of the corresponding, missing data. For this, we rely on a neural architecture which learns the relation from a time value to the corresponding scalar field, based on the keyframe examples, and reliably extends this relation to the non-keyframe time steps. We show how augmenting this architecture with specific topological losses exploiting the input diagrams both improves the geometrical and topological reconstruction of the non-keyframe time steps. At query time, given an input time value for which an interpolation is desired, our approach instantaneously produces an output, via a single propagation of the time input through the network. Experiments interpolating 2D and 3D time-varying datasets show our approach superiority, both in terms of data and topological fitting, with regard to reference interpolation schemes. Our implementation is available at this GitHub link : https://github.com/MohamedKISSI/Topology-Aware-Neural-Interpolation-of-Scalar-Fields.git.

Figures (10)

• Figure 1: Filtration of the opposite elevation function (going downward, purple: low values, cyan: high values) on a toy terrain example (in transparent). From left to right, simplices are progressively added in the filtered simplicial complex for increasing values of opposite elevation. Each local minimum triggers the birth of a connected component in the complex (sub-figures a to d). Connected components are represented by growing bars of matching color in the diagram (bottom). When a component merges with an older one (sub-figures e, f and g), its corresponding bar terminates its growth in the diagram and the corresponding topological feature is said to die at the corresponding value.
• Figure 2: Scalar fields (height opposite) admitting a common persistence diagram (center). Each hill is encoded in the diagram by a vertical bar whose length encodes the persistence of the corresponding topological feature in the data (arrows indicate generators with infinite persistence). While the diagram encodes the list of topological features with their birth, death and persistence, it forgets their geometrical realization in the data.
• Figure 3: The Wasserstein distance $\pazocal{W}_{2}$ between $\pazocal{D}(f)$ (top) and $\pazocal{D}(g)$ (bottom) is obtained by assignment optimization (\ref{['eq_wasserstein']}) in the birth-death plane (right). The optimal assignment $\phi^*$ (arrows) defines a least-effort deformation of $\pazocal{D}(f)$ into $\pazocal{D}(g)$, by moving persistence pairs in the plane.
• Figure 4: TimeToScalarField: An input value $t$ is encoded using sinusoidal positional encoding and projected through a fully connected layer into a latent 3D tensor of size $C_0\times8\times8\times8$, where $C_0$ denotes the initial number of channels. This tensor is processed by a CNN composed of four sequential blocks, each consisting of: (i) trilinear upsampling (by a factor of 2), (ii) 3D convolution (with kernel size $3\times3\times3$), (iii) instance normalization, (iv) ReLU activation, and (v) a residual block (ResBlock3D, he2016deep). The spatial resolution is progressively increased while the number of channels is reduced. A final convolution followed by a Sigmoid activation produces the output volume (here, with resolution $128^3$).
• Figure 5: Loss evolution during scalar field training (phase $1$, top) and topology correction (phase $2$, bottom) for our datasets (from left to right: Gaussian mixture, Mixing vortices, Isabel, Asteroid impact).