# Volumetric calibration enhancements for single-camera light-field PIV

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### Abstract

This work presents a volumetric calibration method for single-camera light-field particle image velocimetry (light-field PIV or LF-PIV). The proposed technique makes use of the unique point-like feature of particle light-field images to accurately determine affected pixels for a spatial voxel over a relative large measurement volume. A calibration model is derived based on Gaussian optics, which relates a spatial point light source with its confusion circle produced on microlens array (MLA), and optical distortions are accounted for by introducing five calibration parameters. By taking lens defects and misalignment between MLA and image sensor into account, the calibration method can calculate weighting coefficient for particle image reconstruction more accurately than the theoretical ray-tracing method, especially for regions further away from focal plane where light ray deflections are significant due to optical distortions. The volumetric calibration method was validated by simulation tests using synthetic light-field images, and has been successfully applied to a classic vortex-ring LF-PIV measurement, where the measurable range in depth direction was successfully extended and the quality of reconstructed volumetric velocity field was greatly improved.

### Graphical abstract

## List of symbols

- \({S_{\text{o}}}\)
Object distance

- \({S_{\text{i}}}\)
Image distance

- \({f_{\text{m}}}\)
Focal length of the main lens

- \({f_\# }\)
*F*-number of the main lens- \({p_{\text{m}}}\)
Effective aperture of the main lens

- \(M\)
Magnification of the plenoptic camera, \(M= - \frac{{{S_{\text{i}}}}}{{{S_{\text{o}}}}}\)

- \({f_{\text{l}}}\)
Focal length of lenslet

- \({p_{\text{l}}}\)
Lenslet pitch

- \({S_{\text{y}}}\)
Lenslet centre (relative to optical axis)

- \({y_{\text{l}}}\)
Offset of a light ray from the lenslet centre

- \({y_{\text{p}}}\)
Intersection of a light ray made with the pixel plane

- \({y_{\text{c}}}\)
Intersection of a light ray made with the main lens, light ray passes lenslet centre

- \({y_{\text{m}}}\)
Intersection of a light ray made with the main lens

- \({C_{{\text{l}}(i)}}\)
Coordinate of the

*i*th lenslet centre in pixel plane- \({d_{\text{l}}}\)
Lenslet pitch (in pixel plane), \({d_{\text{l}}}={p_{\text{l}}}\frac{{{S_{\text{i}}}+{f_{\text{l}}}}}{{{S_{\text{i}}}}}=\left| {{C_{{\text{l}}(i)}} - {C_{{\text{l}}(i - 1)}}} \right|\)

- \(O\left( {X,Y,Z} \right)\)
3D coordinate of a point light source

- \(E\left( {X,Y,Z} \right)\)
Intensity of a voxel at \(\left( {X,Y,Z} \right)\)

- \(I({x_i},{y_i})\)
Pixel intensity of a light-field image at image coordinate \(({x_i},{y_i})\)

- \(w={w_1} \times {w_2}\)
Weighting coefficient for MART reconstruction

- \({w_1}\)
Lenslet weighting coefficient

- \({w_2}\)
Pixel weighting coefficient

- \(C{'_{df}}\)
Coordinate of the confusion circle centre (in MLA plane)

- \(D{'_{df}}\)
Diameter of the confusion circle (in MLA plane)

- \({C_{df}}\)
Coordinate of the confusion circle centre (in pixel plane), \({C_{df}}=C{'_{df}}\frac{{{S_{\text{i}}}+{f_{\text{l}}}}}{{{S_{\text{i}}}}}\)

- \({D_{df}}\)
Diameter of the confusion circle (in pixel plane), \({D_{df}}=D{'_{df}}\frac{{{S_{\text{i}}}+{f_{\text{l}}}}}{{{S_{\text{i}}}}}\)

- \({\mathcal{M}}_{4\times 3}\)
Mapping matrix that relates a spatial point \(O\left( {X,Y,Z} \right)\) with its corresponding \({C_{df}}\)

- \({p_{\text{c}}}\)
Centre of point-like feature beneath each lenslet

- ppm
Particle per microlens

## Notes

### Acknowledgements

Financial support provided by National Natural Science Foundation of China (Grant nos. 11472175, 11772197) and Singapore Ministry of Education AcRF Tier-2 grant (Grant no. MOE2014-T2-1-002) are gratefully acknowledged.

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