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Real-time quantitative Schlieren imaging by fast Fourier demodulation of a checkered backdrop

Sander Wildeman

arXiv:1712.05679v3physics.flu-dyn

TL;DR

Conventional DIC and OF have practical limitations in tracking distortions, while 1D stripe backgrounds recover only one displacement component. The paper introduces Fourier demodulation of a 2D periodic backdrop; FCD resolves large virtual strains on par with conventional methods, with a larger tested dynamic range and much higher processing speed.

  • Problem

    DIC and OF face accuracy and robustness trade-offs in single-pass distortion tracking, while 1D stripe backdrops recover only displacement perpendicular to the stripes.

    Source: §1 · p. 3

  • Method

    FCD uses a high-frequency 2D periodic background as a phase-modulated carrier and Fourier-demodulates it to recover the full vector displacement field.

    Source: §1 · p. 3, §2 · p. 4

  • Results

    FCD resolves large virtual strains on par with conventional correlation and optical-flow techniques, with about three times the tested DIC+OF dynamic range and 512 × 512-pixel processing at 50 FPS CPU and 190 FPS GPU versus 0.5 FPS and 1 FPS.

    Source: §4.1.3 · p. 14, §5 · p. 16

  • Takeaways & Limitations

    When the background can be chosen, a regular 2D periodic pattern makes FCD a flexible, accurate, and fast quantitative alternative to DIC and OF.

    Source: §5 · p. 17

    Faithful demodulation requires separated carrier peaks, constraining the smallest resolvable signal wavelength and the virtual strain that can be resolved.

    Source: §3.1 · p. 8

Abstract

from arXiv · show

A quantitative synthetic Schlieren imaging (SSI) method based on fast Fourier demodulation is presented. Instead of a random dot pattern (as usually employed in SSI), a 2D periodic pattern (such as a checkerboard) is used as a backdrop to the refractive object of interest. The range of validity and accuracy of this "Fast Checkerboard Demodulation" (FCD) method are assessed using both synthetic data and experimental recordings of patterns optically distorted by small waves on a water surface. It is found that the FCD method is at least as accurate as sophisticated, multi-stage, digital image correlation (DIC) or optical flow (OF) techniques used with random dot patterns, and it is significantly faster. Efficient, fully vectorized, implementations of both the FCD and DIC/OF schemes developed for this study are made available as Matlab scripts.

1 Introduction and motivation

Synthetic Schlieren imaging estimates optical inhomogeneities by recovering how a background pattern shifts through a refractive object. This section motivates replacing random-texture tracking with a periodic 2D pattern that can be Fourier-demodulated to recover the displacement field.

1 Introduction and motivation

  • SSI uses a camera to view a textured background through a Schlieren object, where optical inhomogeneities bend light rays and distort the pattern.The background may be random or periodic, with undistorted and distorted versions shown for the same wave-like disturbance.Source: §1 · p. 1, Figure 1 · p. 2
  • SSI reconstructs optical inhomogeneities by finding the displacement field that warps an undistorted reference image into the observed distorted image.To first order, displacement is proportional to local optical-path-length gradients.Source: §1 · p. 2
  • DIC and OF track pattern distortions; laboratory SSI commonly uses random dots, and the background must have texture at the smallest physical scale to be resolved.DIC matches sub-images by cross-correlation, while OF solves a linearized equation per pixel and requires regularization.Source: §1 · pp. 2–3
  • DIC and optical flow (OF) have limitations in single-pass reconstruction, including displacement-scale assumptions, sub-pixel interpolation constraints, regularization trade-offs, and difficulty with large distortions.Multiscale warping can alleviate some limitations, but at the expense of speed and simplicity and with a risk of accumulated errors.Source: §1 · p. 3
  • FCD uses a high-spatial-frequency 2D periodic pattern as a phase-modulated carrier for direct Fourier demodulation of the Schlieren signal.The method is intended to work under well-defined conditions discussed later in the paper.Source: §1 · p. 3
  • Unlike a 1D stripe backdrop, which recovers only displacement perpendicular to the stripes, a 2D periodic pattern allows complete vector-field reconstruction.The article frames FCD for general 2D periodic patterns and compares its practical speed and accuracy with hybrid DIC+OF for small water-surface waves.Source: §1 · pp. 3–4

2 Fourier demodulation of arbitrary 2D periodic patterns

Fast Fourier demodulation recovers displacement from a 2D periodic backdrop by filtering distortion-modulated carrier peaks. The method requires two linearly independent carrier vectors, and the illustrated synthetic reconstruction closely matches its ground truth.

2 Fourier demodulation of arbitrary 2D periodic patterns

  • A complete vector displacement field requires a periodic pattern with at least two linearly independent carrier wave vectors.The pattern represents two independent displacement components per pixel.Source: §2 · p. 4
  • A general 2D periodic backdrop can be represented as a harmonic series whose reciprocal lattice vectors specify the unit-cell spatial frequencies.The vectors k1 and k2 are non-collinear.Source: §2 · p. 4
  • Displacement modulates the phase of each harmonic, and when the distortion is sufficiently small and its wavelengths are not too short, its signal localizes around carrier peaks in k-space.Fourier filtering isolates complex signals around those peaks; signals from two independent reciprocal vectors provide two equations that can be solved for u(r).Source: §2 · pp. 4–5
  • The reconstructed displacement closely matches the ground truth in the synthetic example.The figure compares extracted displacement with ground truth along a line and shows their difference.Source: Figure 2 · p. 5, §2 · p. 6
  • Phase extraction can wrap at ±π, and the modulated carrier peaks must be sufficiently separated in k-space for unambiguous filtering.The section identifies peak-separation criteria as a practical design concern for the background pattern.Source: §2 · p. 6

3 Pattern design and measurement range

The section gives design criteria for faithful FCD reconstruction based on the signal’s frequency and amplitude and the camera’s resolution.

3.1 Preventing overlap of modulated carrier peaks

  • Faithful Fourier filtering requires modulated carrier peaks to remain separated, including significant harmonics of the signal.When the criteria fail, signal components can be filtered out and leakage from neighboring peaks can cause aliasing artifacts.Source: §3.1 · pp. 6, 8, Figure 3 · p. 7
  • The first criterion requires background lattice spacing below the smallest signal wavelength; the second limits resolvable displacement gradients independently of lattice spacing.In practice, reduce the distance between the background and Schlieren object until sensitivity is low enough to satisfy the gradient criterion.Source: §3.1 · p. 8

3.2 Phase wrapping

  • Phase wrapping creates displacement-field discontinuities when distortion exceeds the phase limit; for a square lattice, distortions must stay below half the lattice spacing to avoid it.Larger distortions require phase unwrapping, which can be performed spatially or, for video, across time or between successive frames.Source: §3.2 · pp. 8–9

3.3 Limits imposed by finite image resolution

  • Finite camera sampling limits pattern wavelength: although an ideal sine-wave pattern needs at least about 3.4 pixels, wavelengths above about 6 pixels work well in practice.Together with optical magnification and pixel size, pattern wavelength sets the smallest physical wavelength that can be resolved.Source: §3.3 · p. 9

4 FCD in practice

The water-wave comparison tests FCD against DIC+OF using checkerboard and random-dot backdrops. FCD runs much faster and recovers small capillary waves, while occlusion artifacts can be reduced with mask treatment.

4.1.1 Experimental procedure

  • The experiment compared checkerboard-based FCD with random-dot DIC+OF on synchronized, aligned recordings of reproducible waves generated by popping soap bubbles.The recordings were made at 500 Hz; the checkerboard’s binary pattern was practical to print at high resolution.Source: §4.1 · p. 9, §4.1.1 · p. 10

4.1.2 Implementation of the distortion tracking algorithms

  • Both schemes were implemented as fully vectorized Matlab scripts; DIC+OF iteratively warped the reference before a Horn–Schunck optical-flow refinement.Fourier integration was faster in general, but in these localized bubble disturbances it gave practically the same result as real-space inversion.Source: §4.1.2 · pp. 10, 12, 13

4.1.3 Measurement results

  • 50 FPS (CPU) and 190 FPS (GPU), versus 0.5 FPS and 1 FPS for DIC+OF; FCD’s signal-to-noise ratio was roughly three times higher.Fig. 4 compares the methods on 512 × 512 images; the SNR comparison is for this SSI setting.Source: Figure 4 · p. 11, §4.1.3 · p. 14
  • FCD recovered the smallest capillary waves, although both methods produced similarly low noise in the integrated height fields.Before the bubble popped, neither method recovered the theoretical height below the bubble reliably because the surface became vertical or multi-valued, disconnecting the integration domains.Source: §4.1.3 · p. 14

4.2 Occlusions and masking

  • Untreated masks produced local fringe artifacts near occlusion borders; low-pass filtering the wave-maker mask and filling the hand’s mask with the reference pattern removed most artifacts.Mask-spectrum frequencies above approximately kc/√2 are not properly recovered, producing aliasing.Source: §4.2 · pp. 15–16

5 Conclusion and outlook

FCD extends spatial Fourier demodulation to 2D periodic backgrounds and provides quantitative SSI with a full displacement field. The conclusion reports accuracy on par with conventional DIC/OF, a larger tested dynamic range, and real-time processing, while outlining broader pattern and reference-image uses.

5 Conclusion and outlook

  • FCD resolves large virtual strains on par with conventional image correlation and optical flow, with a dynamic range about three times larger than the tested DIC+OF scheme.It typically processes 512 × 512 images at 50 FPS on a standard desktop PC.Source: §5 · p. 16
  • The Fourier-domain signal can indicate whether a good reconstruction is expected, supporting data validation.Source: §5 · p. 16
  • Although this study used square-lattice backgrounds, the general procedure can accommodate rectangular grids or multiple frequency components for directional bandwidth or signal unwrapping.Source: §5 · p. 17
  • Comparing a digitally generated ideal reference with the physical background can reveal pattern alignment and lens distortions for in situ calibration, or help when an undistorted reference is unavailable.Source: §5 · p. 17
  • When the background pattern can be chosen, a regular 2D periodic pattern with FCD offers a flexible, accurate, and fast quantitative SSI alternative to conventional DIC and OF.Source: §5 · p. 17

Appendix: Comparison to PIVlab

The appendix compares PIVlab with DIC+OF and FCD on soap-bubble distortions, noting that PIV-optimized software may not suit SSI's large virtual strains. PIVlab misses strong distortions and small waves, while FCD is reported to outperform both correlation methods in speed and accuracy.

Appendix: Comparison to PIVlab

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