DOI: 10.5281/zenodo.18679860
title: Non-Intrusive Parametrized-Background Data-Weak Reconstruction of Cardiac Displacement Fields from Sparse MRI-like Observations

description: Dataset Documentation

Description

This dataset contains mesh data, Riesz representers associated with voxel-averaging functionals, and solution snapshots used for reduced-order modeling and PBDW reconstruction experiments on a three-dimensional ellipsoidal geometry.

The data are organized in an HDF5 file with several groups corresponding to:





geometric mesh information,




Riesz representers for different voxel definitions,




sparse matrices defining the underlying Hilbert space inner product,




voxel geometry metadata,




solution snapshots used for reconstruction experiments.



The dataset is designed to accompany a publication, with the same name, describing the construction of Riesz representers associated with voxel-average measurement functionals and their use in PBDW reconstruction frameworks.

Data Organization

Root Structure



/


├── mesh


├── riesz_reps_hz1_sz8

├── riesz_reps_hz2_sz8

├── riesz_reps_hz8_sz8

└── snapshots



Each group corresponds to a specific component of the computational workflow and is described below.

Mesh Data

/mesh

This group contains the geometric discretization of the ellipsoidal computational domain.It provides the boundary surfaces, vertex sets, and region labels required to reconstruct the finite-element mesh.

Boundary Element Files





ellipsoid.belem




ellipsoid.blon




ellipsoid.bpts



These datasets describe the discretization and connectivity, by providing the elements, the fibers and the points of the mesh.

Region-Specific Data

The ellipsoid is partitioned into multiple regions:





inside




outside




top boundary




coordinate axes



Each region contains:





.neubc — Neumann boundary condition information




.surf — surface elements




.vtx — vertex indices



Examples:







 



ellipsoid_inside.neubc
ellipsoid_inside.surf
ellipsoid_inside.vtx
ellipsoid_top.neubc
ellipsoid_xaxis.vtx
ellipsoid_yaxis.vtx



These files allow reconstruction of the mesh topology and boundary labels required for numerical simulations.

Riesz Representers

Overview

The dataset includes several groups containing Riesz representers, i.e., the solutions of the Riesz representation theorem associated with specific linear functionals.

The functionals considered here compute the mean value of a function over a 3-D voxel.Different groups correspond to different voxel geometries and spacings.

Each group contains:





the Riesz representers themselves,




a sparse matrix defining the inner product,




voxel midpoint coordinates,




voxel step sizes.



Functional Definitions

Three types of voxel-average functionals are provided:








Group
Functional Meaning




riesz_reps_hz1_sz8
voxel size (2,2,8), no spacing in z


riesz_reps_hz2_sz8
voxel size (2,2,2), spacing of 8 in z direction


riesz_reps_hz8_sz8
voxel size (2,2,1), spacing of 8 in z direction






These definitions follow the measurement operators introduced in the associated publication.

Group Structure

Example: /riesz_reps_hz1_sz8







 



h_step_size              (3,)
rieszreps                (2202, 148050)
sp_matrix/
    data                 (2081330,)
    indices              (2081330,)
    indptr               (148051,)
    shape                (2,)
xyz_midpoints_voxels     (2202, 3)



Description of Fields

rieszreps

Matrix whose rows contain the coefficients of the Riesz representers associated with each voxel-average functional.

h_step_size

Vector describing the voxel dimensions in physical coordinates.

xyz_midpoints_voxels

Coordinates of voxel midpoints defining the geometric location of each functional.

sp_matrix

Sparse matrix (CSR format) describing the inner product or discretized operator used to define the Riesz map.

Components:





data — non-zero entries




indices — column indices




indptr — row pointers




shape — matrix dimensions



Snapshots

/snapshots

This folder contains collections of solution snapshots used for reduced-order modeling and PBDW reconstruction experiments.

Each dataset has shape:







 



(number_of_snapshots, number_of_dofs)



Examples:





snapshots100 — 100 snapshots




snapshots200 — 200 snapshots




snapshots300 — 300 snapshots



Model-Error Dataset







 



snapshots84_bkerr



This dataset corresponds to a modified physical setting and is used to test the robustness of PBDW reconstructions in the presence of model mismatch.

Original Reference Data







 



snapshots150_original



Provides the original configuration prior to preprocessing.

Scientific Context

The Riesz representers correspond to measurement functionals that compute voxel averages of a field over the ellipsoidal domain.

Different voxel geometries lead to different families of representers, which are stored in the corresponding riesz_reps_* groups.

These data enable:





construction of observation operators,




testing of PBDW reconstruction methods,




analysis of measurement resolution effects.



Full mathematical details are provided in the associated publication.

File Format

The dataset is stored in HDF5 format.

Sparse matrices follow the CSR (Compressed Sparse Row) representation:





data




indices




indptr




shape



All arrays are stored as scalar datasets unless otherwise noted.