Source-linked AI summary
Precise Particle Tracking Against a Complicated Background: Polynomial Fitting with Gaussian Weight
Salman S. Rogers, Thomas A. Waigh, Xiubo Zhao, Jian R. Lu
TL;DR
The paper addresses particle tracking when complicated backgrounds and varied particle shapes undermine existing methods. It introduces Gaussian-weighted quartic polynomial fitting with subpixel centre refinement, evaluates accuracy and precision in simulations and cell images, and reports background-insensitive tracking with precision comparable to Gaussian fitting. The method is released as PolyParticleTracker for non-commercial Matlab use.
Problem
Existing Gaussian and centroid methods can fail for complicated backgrounds, larger or non-spherical particles, and non-uniform illumination.
Method
PFGW fits a fourth-order two-dimensional polynomial to each particle image with Gaussian distance weighting, then refines the centre from the fitted quadratic extremum.
Results
PFGW maintains precision comparable to Gaussian fitting across signal-to-noise ratios while remaining effective against complex backgrounds and in real cell images.
Takeaways & Limitations
PFGW is suited to tracking low-contrast intracellular particles and particles near complicated backgrounds or neighbouring particles.
Takeaways & Limitations
Frame-to-frame linking assumes particles move no farther than their own radii between successive images.
Abstract
from arXiv · showhide
We present a new particle tracking software algorithm designed to accurately track the motion of low-contrast particles against a background with large variations in light levels. The method is based on a polynomial fit of the intensity around each feature point, weighted by a Gaussian function of the distance from the centre, and is especially suitable for tracking endogeneous particles in the cell, imaged with bright field, phase contrast or fluorescence optical microscopy. Furthermore, the method can simultaneously track particles of all different sizes, and allows significant freedom in their shape. The algorithm is evaluated using the quantitative measures of accuracy and precision of previous authors, using simulated images at variable signal-to-noise ratios. To these we add a new test of the error due to a non-uniform background. Finally the tracking of particles in real cell images is demonstrated. The method is made freely available for non-commencial use as a software package with a graphical user-inferface, which can be run within the Matlab programming environment.
1 Introduction
The paper introduces PFGW to address tracking failures caused by complicated backgrounds and particle-shape variation, evaluates it against established accuracy and precision measures, and demonstrates it on real cell images.
- Motivation: Gaussian fitting is accurate for small spherical particles on uniform backgrounds but can fail with complicated backgrounds or larger, non-spherical particles.Centroid methods handle larger or non-spherical particles but also struggle when the background is non-uniform.
- Contribution: The proposed PFGW method fits each particle image with a fourth-order two-dimensional polynomial weighted by a Gaussian centered on the particle.The particle centre is determined from the polynomial’s local maximum.
- Contribution: PFGW is designed to combine insensitivity to particle edges, flexibility toward particle shape, and inherent insensitivity to image background.These properties target limitations identified for Gaussian-fitting and centroid-based methods.
- Evaluation: The method is evaluated using Cheezum et al.’s analysis and an added test of errors caused by a simple non-uniform background.Real cell images are also used to demonstrate tracking accuracy and precision.
- Availability: PFGW is distributed as PolyParticleTracker, a graphical-interface software package freely available for non-commercial use within Matlab 7.0 or higher.The package is intended to make the method directly usable for particle tracking analyses.
2 Particle tracking method
PolyParticleTracker processes images through detection, subpixel refinement, discrimination, and frame-to-frame linking; its novel refinement fits weighted quartic surfaces and iteratively locates particle centres.
- Pipeline: The tracking pipeline smooths images, identifies particles and coordinates, refines coordinates, discriminates particles, and links positions between frames.The subpixel refinement stage is the dominant determinant of tracking accuracy and precision.
- Illustration: Example cell-image fits show local quartic surfaces weighted by a Gaussian function at each feature.The figure illustrates the local fitting operation on particle features.
- Subpixel refinement: Each identified particle is locally fitted with a quartic polynomial to the image intensity map using estimated position and radius.The fit is performed in a least-squares sense.
- Subpixel refinement: A Gaussian pixel weight emphasizes the particle and excludes surrounding background, with α = 1 and a fitting domain extending 2R around the estimated centre.The correction is reported as fairly insensitive to the α and β parameters.
- Subpixel refinement: The refined centre is obtained from the extremum of the fitted quadratic component and updated iteratively when either subpixel correction exceeds Q = 0.5 pixels.Iterations stop after convergence or when the centre wanders beyond the allowed domain or the fit no longer resembles a particle.
- Trajectory linking: Frame-to-frame linking uses the previous position as the next initial estimate when particles move no farther than their own radii between frames.Polynomial coefficients also provide particle radius, eccentricity, rotation angle, and skewness.
3 Quantitative evaluation on simulated images
Simulated-image tests measure PFGW bias and precision across signal-to-noise ratios, pixel positions, particle sizes, and non-uniform backgrounds. The method remains effectively unbiased relative to CCD-pixel position, reaches nanometre precision for point particles, and is less disrupted by ridged backgrounds than boxcar-subtraction tracking.
- Evaluation design: The evaluation reproduces Cheezum et al.'s error analysis and adds a test of tracking against a simple non-uniform background.Simulations include particle images, optical broadening, CCD sampling, Poisson noise, and varied signal-to-noise ratios.
- Evaluation design: Bias B and precision σ are measured by comparing tracked coordinates with exact coordinates over 1000 successive images.The measures distinguish determinate position error from indeterminate variation.
- Pixel-position dependence: B is approximately 10 times smaller than σ at each S/N, while both measures are almost uncorrelated with position within CCD pixels.The result indicates effectively unbiased localization relative to pixel geometry.
- Particle size and S/N: 1-nm precision is reached for point particles at S/N = 41.2, while 1-µm particles have slightly higher error but remain trackable.PFGW is more precise for point particles than for 1-µm particles and outperforms Gaussian fitting for point-particle tracking.
- Non-uniform background: Against ridged backgrounds, PFGW bias occurs only where particle and ridge images overlap, whereas Crocker–Grier bias extends approximately w pixels from the ridge.PFGW precision remains 0.02 pixels throughout, while boxcar subtraction becomes less precise near ridges.
4 Tracking of real particles
PFGW was demonstrated on simulated backgrounds, immobilised beads, and real cell images, tracking particles under varied imaging conditions. The method produced low static error at higher signal-to-noise and simultaneously tracked endogenous particles with varied sizes and shapes.
- Experimental demonstrations: PFGW tracking was tested on bright-field and fluorescence microscopy images, including immobilised beads and particles in living cells.The experiments used bright-field images of cells and fluorescence images of endocytosed latex beads.
- Non-uniform backgrounds: Against sharp background ridges, PFGW bias occurred only where particle and ridge images overlapped, whereas Crocker-Grier bias extended approximately w pixels from each ridge.For PFGW, standard deviation remained unaffected by the background even during overlap; Crocker-Grier bias and standard deviation were both large within approximately w pixels.
- Immobilised beads: 1.9 nm and 8.3 nm MSD static errors were measured for immobilised beads recorded at 10,000 and 45,000 fps, respectively.The faster acquisition reduced light per frame by a factor of 4.5 and correspondingly reduced signal-to-noise ratio.
- Immobilised beads: The measured static errors closely matched predicted standard deviations of 2.0 nm at 10,000 fps and 9.1 nm at 45,000 fps.The agreement indicates that brightness discretization limited precision in this experiment.
- Cell images: In bright-field cell images, PFGW tracked endogenous particles showing random and active motion, including abrupt changes in direction.In Amoeba proteus pseudopodia, the abundance of endogenous particles enabled high-resolution measurement of internal motion, with MSD(1 ms) = 4 nm.
- Cell images: PFGW successfully tracked particles of a variety of sizes and shapes simultaneously.This capability was demonstrated in real cell images containing numerous endogenous particles.
5 Conclusion
The PFGW method combines polynomial fitting with Gaussian weighting to track particles against complex backgrounds while retaining precision comparable to Gaussian fitting across signal-to-noise ratios. It is intended for challenging intracellular and dense-colloidal tracking and is available as a Matlab package.
- Conclusion: PFGW tracks particles against complex backgrounds while maintaining precision similar to Gaussian fitting across a range of signal-to-noise ratios.The method is based on polynomial fitting of feature points with a Gaussian weighting function.
- Conclusion: The method is suited to intracellular microrheology and tracking in dense colloidal situations where complicated backgrounds or neighbouring particles cause errors in other methods.This scope is stated as an example of situations where PFGW is useful.
- Conclusion: PolyParticleTracker is freely available for non-commercial use with a graphical user interface and requires Matlab version 7.0 or higher.The package can be downloaded from the authors’ website.
A.1 Image smoothing and noise reduction
Each image is smoothed by Gaussian convolution to reduce discretization noise, using a finite window that approximates an unbounded convolution. Unlike previous methods, the background is not subtracted.
- Image smoothing and noise reduction: The images are smoothed by convolution with a Gaussian function to correct for discretization noise.This follows the smoothing procedure of Crocker and Grier.
- Image smoothing and noise reduction: The correlation length is set to λ = 1, and a window width w = 3λ sufficiently approximates the unbounded convolution.The finite convolution window limits the calculation while preserving the intended smoothing.
- Image smoothing and noise reduction: PFGW does not subtract the background during image smoothing, unlike previous particle-tracking methods.The method instead retains the background for the subsequent weighted polynomial fit.
A.2 Identification of particles to track and estimation of particle coordinates
Particles are identified from local extrema, and their initial radius is estimated from nearby inflexion points before subpixel coordinate refinement.
- Particle identification: Particles are identified by local maxima or minima in the smoothed intensity I(x, y).For each extremum (x0, y0), adjacent inflexion points are located along the horizontal and vertical lines through it.
- Radius estimation: The initial radius estimate R0 is the mean distance from the extremum to four adjacent points of inflexion.The coordinates x0 and y0, together with R0, are passed to the subpixel correction routine.
A.3 Particle discrimination
Particle discrimination uses polynomial-fit features to characterize peak shape, orientation, size, and asymmetry, enabling acceptable particles to be selected for tracking.
- The fitted intensity map and polynomial coefficients also support calculation of eccentricity, rotation, radius, average brightness, and skewness.
- Quadratic terms define the particle peak as an ellipse, from which eccentricity e and rotation θ are obtained.
- The particle radius is estimated from distances between the fitted peak and four inflexion points along its major and minor axes.The four directions are θ, θ + π/2, θ + π, and θ + 3π/2.
- Cubic terms relative to quartic terms near the particle radius provide an approximate measure of peak skewness.
- Eccentricity, brightness, radius, and skewness can be used to determine whether a particle is acceptable for tracking.
A.4 Linking of particle postions between frames
Particle trajectories are linked between consecutive frames by using the prior position as the next initial estimate, with a fallback search when displacement exceeds the particle radius.
- The position in frame T initializes tracking in frame T + 1, reliably linking trajectories when displacement does not exceed the particle’s radius.
- For larger displacements, PolyParticleTracker searches for a nearby intensity extremum that is closer than the next-nearest particle and similarly bright.