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Kappa (κ): Analysis of Curvature in Biological Image Data using B-splines

Hadrien MaryiD, Gary J. BrouhardiD

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Abstract

Curvature is a central morphological feature of tissues, cells, and sub-cellular structures. A challenge for computational biology is to measure the curvature of these structures from biological image data. We present an open-source Fiji plugin for measuring curvature using B-splines. The plugin is named Kappa after the Greek symbol for curvature,{kappa} . Kappa is semi-automated: users create an initialization curve by a point-click method, and the initialization curve is fit to the underlying data using an iterative minimization algorithm. We demonstrate Kappas applicability on images of cytoskeletal filaments in vitro, the cell wall of budding yeast, and whole worms moving in an agar dish. In order to verify the accuracy and precision of Kappa, we created a bank of synthetic images of known curvature using sine waves and golden spirals, which we digitized with different signal-to-noise ratios (SNR), pixel sizes, and point-spread functions (PSF). For synthetic images with characteristics similar to real data, the measured curvatures of those images show a high correlation with the theoretical curvatures. Our fitting algorithms perform better with higher SNR, smaller pixel sizes, and especially PSFs equivalent to super-resolution microscopy data (surprise, surprise). Kappa is freely available under the MIT license for simple integration into Fiji-based workflows. The source code and documentation can be found on GitHub at https://github.com/brouhardlab/Kappa.

In-depth summary

Claude Opus 5.5 · xhigh effort · 7 Nov 2026 · from the full text of version 1 (25 Nov 2019) · CC BY 4.0

Kappa, an open-source Fiji plugin, fits user-clicked cubic B-splines to image pixels to measure curvature; on synthetic sine waves, Pearson r ≃ 0.95 at signal-to-noise above 15 dB.

The curvature of membranes and filaments can direct protein recruitment, but it is hard to measure from noisy, crowded images of sub-diffraction structures that the optics blur and that are spread across pixels of about 100 nm. Kappa uses cubic B-splines, whose curvature is simple to evaluate at Bézier endpoints, and adds a control-point removal step against the over-fitting that Bicek et al. reported for cosine-sum and higher-order polynomial fits of filaments.

Key results

  1. ShownOn synthetic sine waves, curvature measured after fitting closely tracks the analytic curvature, and agreement improves with signal-to-noise ratio (SNR).

    At 0.16 μm/pixel, with SNR varied from 1 to 25 dB: Pearson r ≃ 0.95 for SNR > 15 dB and ≃ 0.75 for SNR < 10 dB. Mean-curvature error was around 0.022 μm^−1 below 5 dB and fell to 0.017 μm^−1 above 20 dB, against a true mean of 0.248 μm^−1 (30 initialization curves). · Kappa is accurate and precise; Variable SNR; Figure legends

  2. ShownPeak curvature at the sine-wave crests, usually the region of interest, came out close to the analytic value.

    Measured peak curvature was 0.98 ± 0.15 μm^−1 at 12 dB SNR vs a theoretical 1.05 μm^−1 at x = T/4 (period T = 15 μm, amplitude A = 6 μm). · Generating Synthetic Image Data; Variable SNR

  3. ShownGolden spirals were harder. The fit produced spurious curvature peaks in the tightly wound centre, which lowered correlation, though mean curvature stayed reasonably close to the true value.

    Pearson r ≃ 0.7 for SNR > 15 dB and ≃ 0.5 for SNR < 10 dB. Mean-curvature error was 0.04 μm^−1 below 10 dB, halving to 0.02 μm^−1 above 20 dB, against a true mean of 0.226 μm^−1. · Generating Synthetic Image Data; Variable SNR; Discussion

  4. SuggestedFits were largely insensitive to camera pixel size and to moderate differences in the hand-drawn initialization curve. They degraded only when pixels approached the width of the point-spread function (PSF) or when initial control points sat far from the object.

    Qualitative, from figures only. At 20 dB SNR, 0.1 to 0.4 μm/pixel gave similar Pearson r, dropping only once pixel size reached the PSF σ. Adding 0.1 to 1 μm Gaussian noise to initial control points eventually broke the fit, though less than low SNR did. The 30 initialization curves gave non-identical results, read as many close local minima. · Variable Pixel Size; Variable Initialization Curve; Figure legends

  5. SuggestedShrinking the PSF to super-resolution scales gave the largest accuracy gain of any parameter tested.

    Gaussian PSF σ was reduced from 300 nm (conventional GFP-like imaging) to 10 nm. At σ similar to stochastic optical reconstruction microscopy (STORM; 20 to 50 nm), the sine fits were described as nearly perfect, the peak x-offset almost vanished, and spiral fits converged faster to smooth curves. The text gives no error values for this test. · Variable Point Spread Function; Figure legends

Methods, data and reproducibility

In Fiji, a user-clicked cubic B-spline is fit to thresholded, intensity-weighted pixels by iterative least squares (point or squared distance minimization), and densely spaced control points are merged to curb over-fitting. Curvature is read at Bézier endpoints after recursive de Casteljau subdivision. Accuracy was tested on synthetic sine waves and golden spirals of known curvature.

  • Synthetic sine-wave images (T = 15 μm, A = 6 μm) rendered at 10 nm, blurred with a Gaussian PSF (σ typically 300 nm), given additive Gaussian noise and binned to 0.1 to 0.4 μm pixels · n = 30 initialization curves
  • Synthetic golden-spiral images generated the same way (10 nm rendering, Gaussian PSF, added noise, binning) · n = 10 initialization curves (S11); figure legends give n = 30
  • Fluorescence images of rhodamine-labeled microtubules attached to cover glass (in vitro)
  • Bright-field images of budding yeast cells · Saccharomyces cerevisiae (named in the Figure 2 legend)
  • Dissection-microscope images of C. elegans moving on an agar plate · C. elegans
  • Outline drawing of a Citroën 2CV (non-biological demonstration)
Methods & tools
Kappa, Fiji / ImageJ1, ImageJ Roi Manager, pykappa, Python, Jupyter notebooks, point distance minimization (PDM), squared distance minimization (SDM), de Casteljau algorithm, Böhm algorithm
Software & environment
An ImageJ1 plugin for Fiji, installable through an ImageJ update site; validation analyses use the pykappa Python library and Jupyter notebooks. No software versions or requirement files are stated.
Reproducibility gaps
No datasets are deposited. The synthetic images are regenerated from notebooks in the repository, and the paper does not say the microtubule, yeast or C. elegans example images are shared.
Limitations (6)
  • Both fitting algorithms struggle with the innermost region of the golden spiral, where curvature changes rapidly, producing curvature spikes and higher errors. The authors suggest other algorithms, such as tangent distance minimization, might do better. · stated by the authors
  • Measured sine-wave curvature is shifted along x relative to theory, which the authors attribute to PSF convolution; Pearson r is sensitive to this offset. · stated by the authors
  • The fitting landscape has many close local minima, so different initialization curves give different results, and performance breaks down when the initialization strays far from the object. · stated by the authors
  • As an ImageJ1 plugin, Kappa slows down when curves and control points become numerous (> 40), and scripting still requires the user interface to be loaded in memory. · stated by the authors
  • Accuracy is quantified only on author-generated sine waves and golden spirals with a Gaussian PSF and additive Gaussian noise. The microtubule, yeast and C. elegans examples are shown visually, without ground truth or error estimates. · noted in this summary
  • No comparison with other curvature-measurement tools is reported. The text also describes the spiral test with 10 initialization curves, but the figure legends give n = 30. · noted in this summary

Read this if you need to measure the curvature of filaments, cell walls or whole animals in light-microscopy images within Fiji, or to relate curvature to protein intensity in a second fluorescence channel.

Citations
67

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