Meshless Super-Resolution of Scattered Data via constrained RBFs and KNN-Driven Densification

TL;DR

The paper tackles the challenge of achieving high-spatial-resolution flow fields from scattered sensor data by proposing a fully meshless framework that blends KNN-PTV with meshless POD to identify locally similar realizations and enrich particle density, followed by constrained RBF regression to obtain an analytical, physically consistent velocity field. Ablation shows that penalties and divergence-free constraints crucially regularize the reconstruction, enabling accurate velocity, derivative, and pressure fields in a 3D jet flow. The approach delivers about $10\%$ average error compared with $12-13\%$ for competing methods, halved reduced-order errors, and broader spectral fidelity, illustrating significant gains in both accuracy and resolution. This meshless, grid-free method offers flexibility for complex geometries and moving sensors, with practical impact for high-fidelity turbulence statistics and instantaneous field reconstruction.

Abstract

We propose a novel meshless method to achieve super resolution from scattered data obtained from sparse, randomly positioned sensors such as the particle tracers of particle tracking velocimetry. The method combines K Nearest Neighbor Particle Tracking Velocimetry (KNN PTV, Tirelli et al. 2023) with meshless Proper Orthogonal Decomposition (meshless POD, Tirelli et al. 2025) and constrained Radial Basis Function regression (c RBFs, Sperotto et al. 2022). The main idea is to enhance the spatial resolution of flow fields by blending data from locally similar flow regions available in the time series. This similarity is assessed in terms of statistical coherency with leading features identified by meshless POD applied directly to scattered data, without interpolation onto a grid and relying instead on RBFs to compute the relevant inner products. The denser scattered distributions are then used within a constrained RBF framework to derive an analytical representation of the flow fields that incorporates physical constraints. The approach is fully meshless and does not require a grid at any stage, offering flexibility in complex geometries. An ablation study highlights the role of penalties and physical constraints in regularizing the regression and ensuring physically consistent reconstructions. The method is validated using three dimensional measurements of a jet flow in air. The assessment focuses on statistics, spectra, and modal analysis. Performance is evaluated against standard Particle Image Velocimetry, KNN PTV, and c RBFs. The results show improved accuracy, with an average error of about ten percent compared to twelve to thirteen percent for the other methods, nearly halved errors in reduced order reconstructions, and a higher frequency cutoff based on the noise floor.

Figures (14)

• Figure 1: Flowchart of the proposed algorithm. Step 1: extraction of information from particles (PTV or LPT); step 2: local meshless POD directly on the particles within the subdomains into which the domain is partitioned; step 3: computing optimal number of neighbours and then increasing particle density; step 4: weighted regression trough c-RBFs to achieve analytical high-resolution approximation of the velocity field.
• Figure 2: Snapshot enrichment process: first build the training set $\bm{\Theta} = \bm{\Psi}_r\bm{\Sigma}_r \in \mathbb{R}^{N_t \times r}$ for each subdomain using meshless POD (cyan box); then select the $k$ nearest neighbors using the KNN algorithm (yellow box) and merge them. The process is repeated for all subdomains. The snapshot indexes indicated in the yellow box are arbitrary.
• Figure 3: Sketch of the experimental setup. (1) jet nozzle; (2) Ng:Yag Quantel Evergreen laser; (3) ANDOR Zyla sCMOS $5.5$ MP camera.
• Figure 4: Mean velocity field (left column) and resultant standard deviation $\mathbf{\sigma}$ squared (right column): a) Ensemble averaging aguera2016ensemble with bin size $64$ voxels, b) PIV with interrogation window of $64$ voxels, c) PIV with interrogation window of $128$ voxels, d) KNN-PTV, e) c-RBFs and f) meshless KNN-PTV. Reference plane: $z/D = 0$. In black isolines for $\bar{U}/U_j = 0.95$.
• Figure 5: Instantaneous streamwise (first column), spanwise (second column, top) and crosswise (second column, bottom) velocity field contours for the middle planes: a) reference PIV with interrogation window of $64$ voxels, b) PIV with interrogation window of $128$ voxels, c) KNN-PTV, d) c- RBFs and e) meshless KNN-PTV.