Degrees of Freedom Sampling

I. INTRODUCTION

• Global DoF counts do not describe how spatial information is distributed, while uniform grids can be redundant because they ignore source extent and geometry.Source: §I · p. 1
• Existing reduced-sampling approaches use geometry adaptations, operator properties, or sparsity, but generally depend on particular parameterizations, geometries, or prior spectral information.Source: §I · p. 1
• The framework uses the mutual-shadow DoF distribution to determine where to sample and evaluates reconstruction with spectral- and Frobenius-norm errors.The spectral norm measures worst-case error, while the normalized Frobenius norm reflects average reconstruction performance over spatial modes.Source: §I · p. 1
• In the far field, projected source shadow area defines directional DoF density, and the sphere is partitioned into cells with approximately equal mutual-shadow contribution.Sampling directions are also treated as beam-steering directions to examine beam coverage and spatial selectivity.Source: §I · p. 1
• In the near field, mutual-shadow density sets sampling-cell area, while a DoF density direction sets cell orientation and aspect ratio.Source: §I · p. 1
• Numerical singular-value, reconstruction-error, and beamforming analyses are used to test whether geometry-derived distributions retain dominant spatial information.Source: §I · p. 2

II. MODELLING AND ERROR FORMULATION

• The model considers a continuous source on a convex surface and a field observed on a general surface; the exposition uses a scalar field but says the method extends to vector fields.Source: §II · p. 2
• Sampling is treated as an inverse source problem, with the selected observation points restricting the densely sampled radiation operator to form the sampled operator.Source: §II · p. 2
• The spectral-norm sampling error measures the worst-case reconstruction error over unit-norm source distributions and is characterized by σN+1.Source: §II · p. 2
• The Frobenius error measures average reconstruction performance over source coefficients and can be expressed through the singular modes of the densely sampled operator.The spectral and Frobenius norms are complementary; the latter is particularly informative when the post-knee singular-value tail decays slowly.Source: §II · p. 2
• The stated lower bound follows from the Eckart–Young theorem because the sampled measurements project onto a source subspace of dimension at most N.Source: §II · p. 2

III. FAR-FIELD SAMPLING

• Directional sampling density is proportional to the source’s projected shadow area divided by λ^2, assigning denser sampling to higher-gain directions.The total far-field DoF scales with total shadow area and inversely with wavelength squared.Source: §III · p. 3
• For bodies of revolution, polar layers are divided uniformly in azimuth, with layer boundaries chosen so each layer contains approximately its DoF count in samples.Equal-area coordinates (ϕ, cos θ) express the construction as uniform sampling in transformed spherical area; layer sample counts follow the DoF increment.Source: §III · p. 3, Figure 2 · p. 3
• For a spherical source, the proposed distribution is nearly uniform in spherical area, whereas nonredundant angular sampling clusters samples near the poles.DoF, DEIM, and Lebedev have similar global coverage; the DoF method arranges points in latitude layers suited to spherical scanning.Source: §III · p. 4, Figure 3 · p. 4
• DoF sampling, DEIM, and Lebedev show similar convergence and closely follow reconstruction lower bounds, especially under the Frobenius norm.Nonredundant sampling converges more slowly; near its nominal critical count of Nrs ≃202, both errors remain significantly larger than those of DoF sampling and DEIM.Source: Figure 4 · p. 4, §III · pp. 4–5
• Across cylinders, spheres, and prolate ellipsoids, reconstruction error decreases with N/Na, with faster decay and a sharper knee at larger Na.At fixed Na, geometry-specific error curves remain close but do not collapse: shadow density and projected aperture widths affect cell area and aspect ratio, especially for prolate ellipsoids.Source: §III · p. 5, Figure 5 · p. 5
• MRT beam regions correspond to the DoF sampling cells: spherical-source beams are near-isotropic, while cylindrical-source beams are elongated in a direction-dependent way.The cylinder’s beam stretching is consistent with the local DoF-cell aspect ratio.Source: §III · p. 5, Figure 6 · p. 5

IV. NEAR-FIELD SURFACE SAMPLING

• Tilting the receive aperture by 60° doubles spacing along the orthogonal tangent and reduces the approximate DoF count from 64 to 32.The general construction interprets intersections of DoF-density field lines with the observation surface as sampling locations.Source: §IV · pp. 6–7
• The DoF density sets each local cell’s area, while the projected DoF density direction sets its orientation and aspect ratio.The in-plane sampling directions are chosen relative to the surface normal and the tangential projection of the DoF density direction; locally normal propagation yields isotropic cells.Source: §IV · p. 7
• Sampling density follows the local DoF distribution: samples are closer near the source and spread farther apart as density decreases.The construction expands from high-density regions, and its finite-grid partition may deviate slightly from one DoF per cell without materially affecting the distribution.Source: §IV · pp. 7–8
• Beyond the DoF estimate ⌈Na⌉=157, proposed sampling follows the reconstruction-error trend but has higher error than DEIM at larger sample counts for offset square apertures.The remaining gap to the lower bound is mainly attributed to finite-region edge effects not explicitly modeled in the construction.Source: Figure 11 · p. 8, §IV · p. 8
• MRT beams form compact, spatially separated focal regions, and their tangent-direction stretching ratio agrees with the sampling-cell ratio h1/h2.Each beam is normalized to its own maximum, with field magnitude decaying rapidly away from focal regions.Source: §IV · p. 8, Figure 12 · p. 8
• For the hemisphere, proposed DoF sampling provides stable convergence beyond the DoF knee when compared with DEIM.The compared setup uses a 5λ × 10λ planar source and a hemisphere of radius 10λ, with ⌈Na⌉=158 samples.Source: Figure 14 · p. 9, §IV · p. 9
• The method uses leading-order mutual-shadow DoF and neglects finite-size and boundary effects; strongly multidirectional fields may require a higher-order angular description.The stated scope is electrically large systems.Source: §IV · p. 9

• Near-field DoF density sets sampling-cell area, while its direction sets cell orientation and aspect ratio.Source: §V · p. 9
• In the far field, source shadow area determines sampling density through an approximately equal-DoF partition of the observation sphere.Source: §V · p. 9
• Numerical results show that estimated DoF identifies the characteristic reconstruction-error transition and that DoF sampling preserves dominant spatial information.Source: §V · p. 9
• The framework unifies geometry-based sampling across far- and near-field configurations; future edge correction may further improve reconstruction accuracy.Source: §V · p. 9