Uncertainty quantification and estimation in differential dynamic microscopy

Differential dynamic microscopy (DDM) is a form of video image analysis that combines the sensitivity of scattering and the direct visualization benefits of microscopy. DDM is broadly useful in determining dynamical properties including the intermediate scattering function for many spatiotemporally correlated systems. Despite its straightforward analysis, DDM has not been fully adopted as a routine characterization tool, largely due to computational cost and lack of algorithmic robustness. We present statistical analysis that quantifies the noise, reduces the computational order, and enhances the robustness of DDM analysis. We propagate the image noise through the Fourier analysis, which allows us to comprehensively study the bias in different estimators of model parameters, and we derive a different way to detect whether the bias is negligible. Furthermore, through use of Gaussian process regression (GPR), we find that predictive samples of the image structure function require only around 0.5%–5% of the Fourier transforms of the observed quantities. This vastly reduces computational cost, while preserving information of the quantities of interest, such as quantiles of the image scattering function, for subsequent analysis. The approach, which we call DDM with uncertainty quantification (DDM-UQ), is validated using both simulations and experiments with respect to accuracy and computational efficiency, as compared with conventional DDM and multiple particle tracking. Overall, we propose that DDM-UQ lays the foundation for important new applications of DDM, as well as to high-throughput characterization.

Figure 1

Schematic representation of DDM-UQ-based data reduction, sampling, and fitting procedure used to determine material constants. From the stack of images acquired by microscopy, around 0.5%–5% of the Fourier transforms are performed to obtain the observed image structure function $D_o(q,\mathrm{\Delta }t)$, from which a predictive sample is generated to estimate $D(q^{*},\mathrm{\Delta }{t}^{*})$ at unobserved $q^{*}$ and $\mathrm{\Delta }{t}^{*}$, given $D_o(q,\mathrm{\Delta }t)$. In the graphs of $D_o(q,\mathrm{\Delta }t)$ versus $\mathrm{\Delta }t$ and $f(q,\mathrm{\Delta }t)$ versus $\mathrm{\Delta }t$, at select $q$'s, the asterisks denote data selected for fitting, the lines denote values at all $\mathrm{\Delta }t$'s for a particular $q$. The shadow denotes 95% predictive interval, which is small compared to the range of the change in $D(q,\mathrm{\Delta }t)$ over the range of $\mathrm{\Delta }t$. The predictive samples preserve the quantiles of the distribution after transformation and are then used to find material quantities of interest.

Figure 2

Illustration of how four different approaches to estimating $B\left(q\right)$ influence the calculated MSD. For given $q$, the quantity $\mathrm{\Delta }{t}_{q,min}$ satisfies $D_o(q,\mathrm{\Delta }{t}_{q,min})=D_{min}\left(\mathrm{\Delta }{t}_{min}\right)$.

Figure 3

(a) The design points (blue circles) are selected along the logarithmically scaled coordinates $(\mathbf{q},\mathrm{\Delta }t)$, requiring only a fraction of the Fourier transformations to compute $D_o(q,\mathrm{\Delta }t)$. (b) Predictive median and predictive samples by GPR for representative values of $q$'s. The variance of noise is proportional to the inverse of the number of pixels in ensemble. 300 predictive samples for each $q$ were generated to obtain the 95% credible interval (gray shadow). The observations used for GPR are plotted as black dots. The full observations (black lines) overlap with the predictions (colored lines). (c) The predictive samples for the intermediate scattering function are plotted against the $\mathrm{\Delta }t$. (d) Predictive mean squared displacement is plotted against the $\mathrm{\Delta }t$ using all three methods. The $95\%$ predictive interval is shown with the error bar for DDM-UQ. The inset shows a snapshot of experiment with MPT trajectory overlay. Plots (b)–(d) are derived from a movie with $1\text{−}\mu \mathrm{m}$ probe particles diffusing in a viscous fluid.

Figure 4

The ensemble-averaged MSD calculated from the simulated 2D Brownian motion of 800 particles. Estimation of MSD by DDM-UQ with four different ways of estimating the mean of the noise. The error bars denote $95\%$ predictive interval of DDM-UQ with $B=2{\sigma }_0^2$. The truth of $\langle \mathrm{\Delta }{r}^2\left(\mathrm{\Delta }t\right)\rangle =8\mathrm{\Delta }t$ is denoted as the black line. The inset shows the initial position of particles, where particles are enlarged for better visualization. Note that it is not possible to perform MPT due to the large number of particles moving within the frame.

Figure 5

Comparison of different estimators (denoted by different color bars) for all simulation scenarios. Note that the true noise $2{\sigma }_0^2$ is kept constant in all simulations and denoted by the thick black line.

Figure 6

A close look at the estimators of noise $B_{\mathrm{est}}$ in the first and second scenarios of simple diffusion with identical underlying dynamics ${\sigma }_s=2$, above. The difference lies in that the second case includes $16\times$ as many particles while particle radius decreases by $4\times$ (filled circles). For the range of $q$ probed, $D_o(q,\mathrm{\Delta }t)$ does not decay to zero at $q_{max}$.

Figure 7

Comparisons of MSDs calculated using DDM-UQ, DDM, and MPT for various simulated scenarios: (a)–(c) Simple diffusion with (a) ${\sigma }_s=2$, (b) ${\sigma }_s=0.5$, and (c) ${\sigma }_s=4$, corresponding to intermediate, low, and high step sizes. (d), (e) Diffusion with drift in (d) optically dilute and (e) optically dense samples, and (f) diffusion of a particle within a harmonic potential well (Ornstein-Uhlenbeck process) subjected to drift. The pink diamonds and the cyan triangles indicate mean values obtained by estimating $A\left(q\right)$ from the plateau in $D(q,\mathrm{\Delta }t)$, or from $\langle |{\widehat{I}}_o(q,t){{|}^2\rangle }_t$, respectively, using DDM algorithm based on all values of $D(q,\mathrm{\Delta }t)$. The blue circles and error bars depict the mean and $95\%$ predictive interval estimated by DDM-UQ based on $1\%$ of the $D(q,\mathrm{\Delta }t)$, respectively. The golden squares and black solid line represent the output of MPT analysis and the true values as directly calculated from the known particle positions, respectively. The generated particle trajectories are shown in the insets, in a simulation box of $480\times 480$ pixels.

Figure 8

Mean squared displacements estimated from the motions of 100 nm diameter probe particles in a 30 wt.% sucrose solution, which serves as a model Newtonian fluid. The pink diamonds and the cyan triangles indicate MSD obtained by using $A\left(q\right)$ determined from the plateau in $D_o(q,\mathrm{\Delta }t)$, or from $\langle |{\widehat{I}}_o(q,t){{|}^2\rangle }_t$, respectively, based on all values of $D(q,\mathrm{\Delta }t)$. Blue circles denote results obtained using the DDM-UQ analysis by a small fraction of the data; in this case $A\left(q\right)$ was only estimated using $\langle |{\widehat{I}}_o(q,t){{|}^2\rangle }_t$. The black solid line denotes reference values and is calculated using the known viscosity (at $20{}^{\circ }\mathrm{C}$) and the Stokes Einstein equation. The inset shows a single experimental image of the movie, where particles appear to be grainy and cannot be individually resolved.

Figure 9

Results of microrheology and bulk rheology measurements of solutions wormlike micelles, which form viscoelastic fluids. (a) Mean squared displacements obtained from full $D_o(q,\mathrm{\Delta }t)$ with $A\left(q\right)$ estimated from the plateau in $D(q,\mathrm{\Delta }t)$ (pink diamonds), from $\langle |{\widehat{I}}_o(q,t){{|}^2\rangle }_t$ (cyan triangles), DDM-UQ (blue circles), and MPT (golden squares). The inset shows an experimental snapshot of the microrheology experiment. (b)–(d) Comparison of the frequency-dependent linear viscoelastic moduli obtained either from bulk rheology experiments (black symbols) or calculated from the MSDs obtained by either using (b) DDM with $A\left(q\right)$ estimated from $\langle |{\widehat{I}}_o(q_{max},t){|}^2\rangle$ (pink diamonds), (c) DDM-UQ (blue circles), or (d) MPT (golden squares). Solid symbols denote the storage [elastic, $G^{\prime }\left(\omega \right)$] modulus while open symbols denote the loss [viscous, $G^{\prime \prime }\left(\omega \right)$] modulus. MSDs estimated by MPT and by DDM-UQ nearly overlap.

Figure 10

Pair bar graphs demonstrating the computation efficiency of DDM-UQ scheme (blue) over current DDM approaches (pink). The size of each image stack is labeled underneath the data set, in terms of frame size $\times$ time points ($L_x\times L_y\times T$).

Figure 11

Example of analysis of an actively driven system. Experimental data extracted from a fluorescently labeled actin-myosin-microtubule composite network system reported in [20], processed using DDM (red symbols) or DDM-UQ (blue symbols), and fit to a stretched exponential model to describe the system dynamics [Eq. (28)]. (a) Comparing DDM and DDM DQ fits to the original $D_o(q,\mathrm{\Delta }t)$ matrix, where the uncertainty is denoted by the gray shaded area, which is very small as the uncertainty is low. The solid line denotes full $D_o(q,\mathrm{\Delta }t)$ data, while the solid dots denotes $D_o(q,\mathrm{\Delta }t)$ selected to obtain the predictive distribution. (b) Fits to $f(q,\mathrm{\Delta }t)$ shown at several different $q$'s. (c) Parameter $\tau$, which describes the system relaxation time, plotted as a function of $q$. Different symbols represent $\tau$ extracted from six different data sets. From this, the system velocity is obtained (Fig. 12).

Figure 12

Velocity estimates and confidence intervals extracted from fitting the stretched exponential model to either the full observations (DDM) or to reconstructed $D(q,\mathrm{\Delta }t)$ from selected design points (DDM-UQ). The error bars denote 95% confidence intervals.