HexaShrink Multiscale Representations

• HexaShrink Multiscale Representations are hierarchical, exactly reversible methods designed to encode structured hexahedral volume meshes while preserving critical discontinuities.
• The framework integrates specialized wavelet-like transforms for geometry, continuous, and categorical data, ensuring coherent multiresolution analysis.
• It enables efficient lossless compression and progressive visualization, benefiting applications in geosciences, biomedical engineering, and materials science.

HexaShrink Multiscale Representations refer to a class of hierarchical, exactly reversible multiresolution decompositions specifically developed for structured hexahedral volume meshes. The main objective is to efficiently encode and render high-resolution 3D meshes—such as those used in geosciences, biomedical engineering, and materials science—while preserving critical geometric features and discontinuities (faults) across scales. This framework integrates multiple “wavelet-like” transforms tailored to mesh geometry and attributes, allowing coherent handling of geometry, continuous, and categorical data, and ensuring exact reconstruction at every resolution level (Peyrot et al., 2019).

1. Architectural Principles of HexaShrink

HexaShrink is founded on the concept of dyadic multiresolution decomposition in structured hexahedral meshes, specifically pillar- or corner-point grids. The framework partitions the mesh into regular blocks (e.g., 2×2×2 cell clusters) and applies a combination of specialized transforms:

• Rounded integer-to-integer 1D wavelet (lifting scheme): Applied along vertical (pillar) directions and to pillar endpoint coordinates, using formulas that guarantee exact reversibility.
• Nonlinear, non-separable morphological 2D wavelet: Operates on the horizontal (x,y) plane to capture and propagate fault configurations using logical OR operations across grid nodes.
• Property-specific transforms: Continuous cell attributes are processed by piecewise linear wavelets, while categorical properties exploit the “modelet” transform—an attribute-specific scheme designed to handle non-numerical data types.
• Hierarchical analysis/synthesis: At each level, the mesh is decomposed into lower resolution (approximation) and distinct detail coefficients, enabling progressive mesh representations up and down the scale hierarchy.

The framework ensures that each transform phase (analysis and synthesis) is fully reversible, even in the presence of mesh faults, borders, and inactive cells—achieved by explicit constraint formulations for boundary cells and the "Actnum" Boolean indicator field.

2. Fault and Discontinuity Preservation

Preservation of geological or structural discontinuities (faults) is a critical requirement for many simulation and visualization tasks. HexaShrink addresses this through an explicit two-stage approach:

• Fault Segmentation: The 2D morphological wavelet extract and propagate fault configurations at each node based on possible orientations (up to 12 configurations per node).
• Prediction and Assignment: For each 2×2 block of nodes, the fault configuration at the next coarser scale is predicted by applying logical OR operations over cardinal directions. Geometry assignment at the coarser scale selects the parent block closest (in configuration distance) to the predicted fault layout.
• Propagation of Borders and Inactive Cells: The framework preserves external mesh constraints, ensuring borders and inactivated regions are retained during decomposition and synthesis.

Unlike standard upscaling techniques—which can "smear" fault lines or introduce border artifacts—HexaShrink preserves discrete jumps in geometry, maintaining the fidelity of high-resolution fault shapes across all scales.

3. Multiresolution Transforms for Mesh Attributes

HexaShrink handles both continuous and categorical mesh properties independently:

• Continuous Properties: For scalar fields such as porosity, linear wavelet transforms based on the lifting scheme are used. Vertical coordinates are processed by

$$d^{(l-1)}[n] = z^{(l)}[2n+1] - \Bigl\lfloor \frac{z^{(l)}[2n] + z^{(l)}[2n+2]}{2} \Bigr\rfloor$$

$$z^{(l-1)}[n] = z^{(l)}[2n] + \Bigl\lfloor \frac{d^{(l-1)}[n-1] + d^{(l-1)}[n]}{4} \Bigr\rfloor$$

where $z^{(l)}$ denotes values at grid level $l$; $\lfloor \cdot \rfloor$ is integer rounding for exact reversibility.

• Categorical Properties (“Modelet” Transform): Categorical data, such as lithofacies, are processed via a hierarchical transform that maintains the statistical and spatial distribution of categories during downsampling, supporting exact synthesis.

This separation enables efficient, lossless coding of mesh attributes regardless of data type.

4. Computational and Compression Performance

HexaShrink has been empirically evaluated on large geoscience meshes (up to millions of cells), demonstrating:

• Lossless compression: When combined with lossless coders such as LZMA, compression ratios between 3.6× and 13.3× have been observed compared to original mesh sizes.
• Hierarchical representation: Multiresolution decomposition yields a hierarchy of mesh approximations, supporting progressive visualization and multi-scale simulation workflows.
• Efficiency: Analysis (encoding) scales from seconds for small meshes to minutes for large volumes and is amenable to GPU or out-of-core implementations.
• Visual fidelity: Downsampled representations preserve faults, borders, and active cell regions more accurately than conventional upscaling or simulation software workflows.

5. Mathematical and Algorithmic Foundations

HexaShrink's reversibility and geometric consistency derive from modular application of transforms:

• Lifting scheme: Guarantees integer-to-integer processing and exact synthesis via rounded linear updates.
• Morphological processing: Fault segmentation and configuration propagation are strictly rule-based, using logical combinations over blockwise topological flags.
• Constraint enforcement: The system automatically preserves mesh boundaries and active regions via imposed conditions such as

$$z^{(l-1)}[0] = z^{(l)}[0], \quad d^{(l-1)}[-1] = -d^{(l-1)}[0]$$

A summary table of the main transforms:

Mesh Dimension	Transform Type	Role
Vertical (z)	Rounded 1D lifting	Geometry/attributes
Horizontal (x, y)	2D morphological wavelet	Fault segmentation
Continuous props	Piecewise linear wavelet	Attribute downsampling
Categorical props	Modelet	Category hierarchy

These design choices collectively guarantee that the original mesh can be reconstructed exactly from the collection of low-resolution mesh and detail coefficients produced at each analysis stage.

6. Applications and Implications

The HexaShrink framework supports numerous high-performance computing and big-data scenarios:

• Reservoir modeling: Coarse-to-fine mesh visualization with fault preservation critical for flow simulation and analysis.
• Medical and materials imaging: Multiscale representation of structured internal meshes supports interactive visualization and multi-scale analysis.
• Progressive transmission: Hierarchical mesh compression enables efficient data transfer, with on-demand detail extraction.
• Data archiving: Single-shot, lossless compression ("compress once, decompress many") with integrity of key mesh features.

Additionally, HexaShrink's property-specific transforms (notably, the modelet for categorical data) advance the state of the art in attribute hierarchy handling compared to prior methods focused solely on geometry.

7. Position in Broader Research Context

HexaShrink's approach is distinguished from both classical mesh compression and generalized wavelet decompositions (e.g., biorthogonal wavelets or geometric multiscale analysis for manifold data (Grohs et al., 2010)), by its explicit engineering for structured hexahedral meshes with discontinuities, exact reversibility, and attribute-specific transform design. Its fully reversible multi-attribute decomposition, morphological fault handling, and modular upscaling framework (applicable to continuous and categorical mesh data) are explicitly developed to address the demands of large-scale HPC visualization, simulation, and storage in fields characterized by immense, discontinuous data volumes (Peyrot et al., 2019).