2.2.1. Synthetic data model

In order for a CNN to learn a binary segmentation of SCXI images into stagnation column and background, labelled training data must be provided. Data for this task are automatically generated using a static model for the deuterium plasma comprising the fuel column, which was originally developed to validate axially resolved Bayesian inferences of fuel conditions for MagLIF experiments (Knapp et al. Reference Knapp, Glinsky, Schaeuble, Jennings, Evans, Gunning, Awe, Chandler, Geissel and Gomez2022). The model can be roughly considered to be composed of a geometric portion, a physics portion and a synthetic diagnostic portion, all of which are described shortly along with details of parameter sampling for generating training data. We note that the model presented here allows not only for the creation of synthetic continuum emission SCXI data but can also be used to generate synthetic multi-channel pinhole camera images. We plan to utilise this model to address multi-channel data in a future publication. Furthermore, the use of this model allows us to automatically generate labelled datasets of arbitrary size for training. This avoids the need to provide hand-labelled experimental data and circumvents issues that would likely result from an experimental database of insufficient size to be used for training and validating such a model in any case, at least without significant data augmentation steps.

The geometric portion of the model provides for a parametrised spatial dependence of plasma conditions. The spatial structure allows for an arbitrary number, $s$, of helical strands, which, for simplicity, we fix to be $s=2$. The general spatial structure of the model is shown in the first column in figure 3. Spatial strand coordinates are defined parametrically via the axial coordinate, $z$, which is zero at the mid-plane of our helical strands, as

(2.1)\begin{gather} x_s(z) = A_s(z)\cos( k_s z + \phi_s(z) ), \end{gather}

(2.2)\begin{gather}y_s(z) = A_s(z)\sin( k_s z + \phi_s(z) ), \end{gather}

(2.3)\begin{gather}z_s(z) = z. \end{gather}

The helical wavenumber is given by $k_s = 2 {\rm \pi}/\lambda _s$, where $\lambda _s$ is the helical strand wavelength, specifying the helical pitch. In the above, $A_s(z)$ is a radial excursion amplitude allowed to vary over the axial extent according to the polynomial expansion

(2.4)\begin{equation} A_s(z) = A \times \sum_{i=0}^{\zeta} A_{\zeta,s}^i z^i, \end{equation}

with $A$ being an overall constant multiple setting the length scale for the radial excursions and $A^i_{s,\zeta }$ being polynomial coefficients that allow the radial excursion extent to vary slowly over the axial extent of the strand. The second column of figure 3 shows an example of the radial excursion amplitude vs axial coordinate. There is also a slowly varying phase factor, $\phi _s(z)$, that smoothly modulates the axial strand wavelength, which is expanded in Legendre polynomials as

(2.5)\begin{equation} \phi_s(z) = {\rm \pi}\sum_{i=0}^{\varGamma} p^i_{\varGamma,s} L_i(z/Z), \end{equation}

where $L_i(z/Z)$ is the $i{{\rm th}}$-order Legendre polynomial, with $Z$ setting a length scale over which we want the Legendre polynomials to evaluate in the interval $[-1,1]$. Here $Z = 5.75$ mm is chosen, corresponding to a domain slightly larger than half the strand sizes typically observed in our stagnation images. In (2.5), the $p^i_{\varGamma,s}$ are coefficients chosen so that $\phi _1(z)$ fluctuates approximately zero and $\phi _2(z)$ fluctuates approximately ${\rm \pi}$ as shown in the third column of figure 3. Note that, henceforth, we refer to the strand $s=1$ as the reference strand.

Figure 3.

Example of one possible 3D structure from the geometric helical model, with the $s=1$ ‘reference strand’ shown in blue. (a) A surface plot demonstrating the radial excursion amplitude, relative phase and strand radii of the two strands versus height. The light blue plane indicates the mid-plane $z=0$. (b) The radial excursion amplitude ($A_s(z)$ in (2.1)–(2.4)). (c) The axially varying phase shift of each strand. Note that the reference strand is centred around zero radians, whereas the $s=2$ strand is centred around $\phi _s(z) \approx {\rm \pi}$. (d) The radius of the helical strand versus height. Note that the difference between the two strand radii is parametrised by the strand radius ratio parameter $\bar {r}_s(z)$ described in the main text.

Before moving to the physics portion of the model, we define the remaining geometric parameters which are used in the physics portion. First, there is a function that will be applied later to physical and geometric conditions to constrain the axial extent of the fuel and provide additional axial fluctuations to these parameters according to a super-Gaussian profile given by

(2.6)\begin{equation} h^c(z) = \exp{\left[-\left( \frac{z^2}{2\sigma_H^2}\right)^\xi\right]}+\delta^c(z), \end{equation}

where $c$ denotes different ‘channels’, in particular, the strand radius, fuel temperature, fuel pressure, mix fraction and liner areal density (see, e.g., (2.12), (2.14) and (2.16)) to which the function h$^c(z)$ is applied as detailed shortly. In the above, $\delta ^c(z)$ is a function which randomly modulates the axial envelope, $\sigma = 2H/3$, where $H$ sets the axial scale length for the strand, and $\xi$ determines how rapidly the axial roll-off occurs. The factor of $2/3$ in the relation of $\sigma _H$ with $H$ is so that $H$ corresponds to a full width at half maximum when $\xi =1$. In the current implementation,

(2.7)\begin{equation} \delta^c(z) = \delta^c_0 \mathcal{U}^c_{[{-}1,1]}(z), \end{equation}

where $\delta ^c_0$ sets the scale for the amplitude of fluctuations and $\mathcal {U}_{[a,b]}(z)$ ($\text {log}\mathcal {U}_{[a,b]}(z)$) denotes a uniform (log-uniform) random number in the interval $[a,b]$ chosen at each axial position. There are two final geometric parameters. The first is a strand radius scale $r_0$ that sets the scale for the radial extent of temperature profiles at each axial position. The second is a strand radius ratio parameter $\bar {r}_s(z)$ which sets the difference in temperature profile radii of the strands. These last two parameters are only explicitly used in the physics model, and so are discussed in more detail in the following. For reference, the geometric model parameters along with a short physical description of their purpose are presented in table 1 along with the distribution from which those parameters are sampled for generating training data for the segmentation network.

Table 1.

Geometric parameter summary.

We now discuss the physics portion of the model which specifies the spatial dependence of the plasma conditions such as electron temperature, thermal pressure and mix. In what follows, the parametric dependence of the helical strand centres $x_s$ and $y_s$ on the axial coordinate $z$ is suppressed to save space. The electron temperature profile of the plasma is given by a sum of gaussian profiles for each of the strands as

(2.8)\begin{equation} T_e(x,y,z) =\sum_s T_{e,s}(x,y,z). \end{equation}

At each axial position $z$, the spatial temperature profile for strand $s$ is given by

(2.9)\begin{gather} T_{e,s}(x,y,z) = \frac{\bar{T}_s(z) \sum_s \bar{r}_s(z)}{ N_s \bar{r}_s(z) \sum_s{\bar{T}_s(z)}}T(z)\nonumber\\ \times \exp\left[\frac{-\big((x-x_s)^2+(y-y_s)^2\big)}{2 \sigma_{T,s}^2(z)}\right]. \end{gather}

In the above, $N_s$ is the number of helical strands (here $N_s=2$). The other functions appearing in (2.9) are

(2.10)\begin{gather} \bar{r}_s(z) = \begin{cases} 1 & s = 1,\\ \text{log}\mathcal{U}_{[\bar{r}_\text{min},\bar{r}_\text{max}]}(z) & s = 2 \end{cases}, \end{gather}

(2.11)\begin{gather}\bar{T}_s(z) = \begin{cases} 1 & s = 1,\\ \text{log}\mathcal{U}_{[\bar{T}_\text{min},\bar{T}_\text{max}]}(z) & s = 2 \end{cases}, \end{gather}

(2.12)\begin{gather}T(z) = T_0 h^T(z), \end{gather}

(2.13)\begin{gather}\sigma_{T,s}(z) = \dfrac{2}{3}\sqrt{\dfrac{N_s r(z) \bar{r}_s(z)}{\sum_s \bar{r}_s(z)}}, \end{gather}

where $\bar {\ast }_{\text {min}(\text {max})}$ indicates the minimum (maximum) value of the associated sampling distribution, $T_0$ is a parameter setting the overall temperature scale of the plasma and $\sigma _{T,s}(z)$ determines the radius for each strand's Gaussian profile. For (2.13), $r(z) = r_0 h^r(z)$, with an example of the resulting strand radii shown in the rightmost column of figure 3. We assume that each horizontal slice of the plasma is isobaric so that the pressure is

(2.14)\begin{equation} P(z) = P_{0} h^P(z).\end{equation}

The plasma is also allowed to have a uniform mix fraction of atomic species $i$ which varies axially as

(2.15)\begin{equation} f_i(z) = f_{0,i} h^f(z). \end{equation}

For our synthetic images, we specify the laser entrance foil material, Kapton, to constitute the mix species, with the remaining fraction of atomic species being the background deuterium. Finally, we assume a liner areal density that will contribute to filtering the continuum emission from this plasma when computing synthetic diagnostics. Currently, we only allow the liner areal density to vary along the axial coordinate according to

(2.16)\begin{equation} \rho R_\ell(z) = \rho R_{0,\ell}h^{\rho R_\ell}(z). \end{equation}

We summarise the interpretation of the different physics model parameters in table 2 for convenience. We note that, in generating image ensembles, thresholding is applied to the temperature, pressure, linear areal density and strand radius to ensure that none of those quantities falls below $10\,\%$ of the corresponding specified scale parameter. This is a simple fix to avoid issues with random sampling that can cause unphysical negative values of the corresponding thresholded quantities as well as removing the potential for division by zero, but may contribute to artifacts in the synthetic image data as discussed in § 2.2.2.

Table 2.

Physical parameter summary.

In order to compute synthetic x-ray images, we must evaluate the emissivity from this plasma. To do so, we compute the time-integrated emissivity of the plasma assumed to be emitting uniformly into a solid angle of $4 {\rm \pi}$ including liner opacity, which, after suppressing spatial dependence, is given by (Ma et al. Reference Ma, Patel, Izumi, Springer, Key, Atherton, Benedetti, Bradley, Callahan and Celliers2013; Epstein et al. Reference Epstein, Goncharov, Marshall, Betti, Nora, Christopherson, Golovkin and MacFarlane2015)

(2.17)\begin{equation} \epsilon_\nu = \tau_b \exp[-\rho R_\ell \kappa_\nu] g_\text{FF} \sum_i f_i j_i. \end{equation}

The factors appearing in (2.17) that have not already been described are the frequency-dependent liner opacity $\kappa _\nu$, the emissivity of each atomic species $j_i$, the burn duration here assumed to be uniform with $\tau _b = 2$ ns and the free–free Gaunt factor ($g_{\text {FF}}$). Evaluating the Gaunt factor in the high-temperature and high-photon-energy limit, $k_BT \gg \chi _H$ with $\chi _H =13.6$ eV being the hydrogen K-shell ionisation energy and $h\nu > k_BT$, respectively, where $k_B = 1.6022 \times 10^{-19}\,{\rm J}\,{\rm eV}^{-1}$, one may obtain (Epstein et al. Reference Epstein, Goncharov, Marshall, Betti, Nora, Christopherson, Golovkin and MacFarlane2015)

(2.18)\begin{equation} g_\text{FF} = \frac{2 \sqrt{3} \times 0.87 }{\rm \pi} \sqrt{\dfrac{T}{h \nu}} . \end{equation}

Assuming an optically thin plasma with negligible line emission, we may further write the emissivity as a sum of free–free ($j_i^\text {ff}$) and bound–free ($j_i^\text {fb}$) emissivities

(2.19)\begin{gather} j_i = j_i^\text{ff} + j_i^\text{bf}, \end{gather}

(2.20)\begin{gather}j_i^\text{ff} = A_\text{ff} \dfrac{n_i^2 Z_i^2}{T^{1/2}}e^{{-}h \nu/T_e}, \end{gather}

(2.21)\begin{gather}j_i^\text{bf} = A_\text{bf} \dfrac{n_i^2 Z_i^4}{T^{3/2}}e^{(\text{Ry}Z_i^2-h \nu)/T_e}. \end{gather}

In the above, $Ry=13.6$ eV is the Rydberg constant, $A_\text {ff} = 2 \times 10^{-32}$ J$^{3/2}$ cm$^3$ s$^{-1}$ eV$^{-1}$ is the free–free emission coefficient and $A_\text {bf} = 4 \times 10^{-31}$ J$^{5/2}$ cm$^3$ s$^{-1}$ eV$^{-1}$ is the bound–free emission coefficient. Assuming equilibrated electron and ion populations ($T_e = T_{i} = T$) and an ideal gas equation of state provides

(2.22)\begin{equation} P = n_e k_{B} T_e + n_{i} k_{B} T_{i} = (1 +\langle Z \rangle) k_{B} T, \end{equation}

with $\langle Z \rangle = \sum _i Z_i f_i$ being the average charge of the plasma. Utilising (2.18)–(2.22) and assuming that the plasma is isobaric at each axial position, we may rewrite (2.17) as

(2.23)\begin{equation} \epsilon_\nu = A_\text{ff} P^2 \exp({-\rho R_\ell \kappa_\nu}) \dfrac{\langle Z \rangle g_\text{FF}}{(1+ \langle Z \rangle)^2} \sum_{i} f_i\kern0.09em \bar{j}_i \dfrac{{\rm e}^{{-}h \nu/T_e}}{T_e^{5/2}} ,\end{equation}

where

(2.24)\begin{equation} \bar{j} = \frac{j_i}{j_D} = Z_i^2 + \frac{A_{\text{bf}}Z_i^4}{A_\text{ff}T} e^{\text{Ry}Z_i^2/T} , \end{equation}

with $j_D$ being the emissivity of deuterium.

Using the emissivity, $\epsilon _\nu$, we are able to compute synthetic SCXI data by utilising a Radon transform along an axis perpendicular to the $z$-axis, which maps a function $f(x,y,z):\mathbb {R}^3 \to \mathbb {R}$ to a function $\mathcal {R}_{\hat {\alpha }}f(\bar {x}): \mathbb {R}^2 \to \mathbb {R}$ defined on a plane containing the unit vector on the two-dimensional sphere, $\hat {\alpha } \in S^2$ by integrating $\epsilon _\nu$ along lines perpendicular to $\hat {\alpha }$. The Radon transform of a function $f(x,y,z)$ is given by

(2.25)\begin{equation} \mathcal{R}_{\hat{\alpha}}f(\bar{x}) = \int_{\boldsymbol{x} \boldsymbol{\cdot} \hat{\alpha} = \bar{x}} f(x,y,z)\,{\rm d}A, \end{equation}

where for each $\bar {x}$ the integral runs over the solution locus of $\boldsymbol {x} \boldsymbol {\cdot } \hat {\alpha } = \bar {x}$, and ${\rm d}A$ is the surface area measure on a plane containing $\hat {\alpha }$. Note that here we are using $\bar {x}$ to represent the two-dimensional Euclidean coordinates on a plane containing $\hat {\alpha }$. Also note that for our purposes, $\hat {\alpha }$ will be restricted to the $x$–$y$ plane and may be parametrised as $\hat {\alpha } = (\cos \theta,\sin \theta,0)$ with $\theta$ specifying the view angle. In that case, we may write the intensity of the synthetic image along a particular view angle, $I_\theta (\bar {x})$ according to

(2.26)\begin{equation} I_\theta(\bar{x}) = \left( \frac{\varOmega}{4 {\rm \pi}} \int_0^\infty \mathcal{R}_{\hat{\alpha}} \epsilon_\nu(\bar{x}) {R}_\nu d(h\nu) \right) \ast \varPi(\bar{x}). \end{equation}

Note that the expression contains a term, $\textit{R}_\nu$, accounting for the reflectivity of the spherical crystal, and $\varPi (\bar {x})$ representing the point-spread function for the optical system, assumed here to be Gaussian.

Before proceeding to use this model to train a convolutional network to segment the strand from the background, we pause to make several observations that may be of use in future work. First, note that the terms $\delta ^c(z)$ allow for axial variation of various parameters. However, the amplitudes are selected uniformly at random for each height from $[-\delta ^c_0,\delta _0^c]$ as specified by (2.7). It is possible to impose smoothness in a variety of ways, such as using a Gaussian process to sample a variation function along the axial coordinate with amplitude determined by $\delta ^c_0$ and a specified correlation length scale $\ell _0^c$. Extending this further, one could enforce correlations between different quantities. In our implementation, the point-spread function $\varPi (\bar {x})$ will smooth out to some extent the noisy structure resulting from uniform random sampling in $\delta ^c(z)$, and for the purposes of this paper, the precise sampling approach is unimportant as we will see from the good performance of the segmentation network developed in the next subsection.

2.2.2. Segmentation CNN

We now apply the synthetic image model from § 2.2.1 in order to create a dataset to train a CNN to segment an image into strand and background. To do so, we begin by generating $N_\text {base} =1000$ images using the sampling distributions for each parameter as listed in tables 1 and 2. Provided the rather large parameter space indicated by tables 1 and 2, this might appear to be a small number of training samples. However, we apply a data augmentation step to obtain a factor of $10$ more training samples. Furthermore, the task of segmenting an image is much simpler than other tasks one could train a machine learning algorithm to perform (e.g. inverting images to obtain the image geometric and physical parameters as in Glinsky et al. Reference Glinsky, Moore, Lewis, Weis, Jennings, Ampleford, Knapp, Harding, Gomez and Harvey-Thompson2020). More complex tasks typically require more training data. As we show, the obtained network does not overfit on validation data and performs well on experiments, justifying the use of this relatively small training dataset.

The masks indicating pixels corresponding to the strand are computed as the set of pixels taking on value greater than or equal to $20\,\%$ of the peak intensity, with a 2-pixel-wide expansion of the mask along each axis after thresholding. As indicated briefly in the previous section, thresholding physical parameters of the modelled plasma in conjunction with computing the Radon transform for synthetic image generation can result in an artificial striping structure in some of the computed images and masks demonstrated for a particular instance in figure 4. For our purposes, we consider this to be useful, because it may actually serve to help the convolutional network to ignore any off-strand features such as the cold-liner emission structures that are sometimes observable in our SCXI data as long as we are careful to remove the corresponding structures from the training data masks. Thus, we do not attempt to address the underlying issue here, but instead apply a cleaning algorithm to the resulting masks. This is achieved via a simple line-by-line scan that searches for the start of a horizontal stripe at the image edges and uses a nearby stripe-free region to compute the new mask edge.

Figure 4.

(a) The training data from the synthetic model discussed in § 2.2.1, which can exhibit artificial horizontal banding. (b) Computing a simple mask by thresholding retains these artificial structures, and may result in poor performance on our experimental data. (c) After applying a simple cleaning procedure, the artificial structures are removed from the mask.

In order to increase our training dataset size and encourage the segmentation network to be robust to features that might arise in experimental data, we apply a set of several random data augmentation steps with $10$ different permutations applied to each of our original $1000$ images, resulting in a total of $N_\text {augmented}=10\,000$ images in our final dataset. The augmentation proceeds as follows. First, with probability $p_{\text {fid}} = 0.25$, a multiplicative ‘fiducial’ is applied

(2.27)\begin{equation} f(x,z) = 1 - \exp\left({\frac{-(z-z_\text{fid})^2}{2 \sigma_\text{fid}^2}}\right). \end{equation}

The synthetic fiducial is placed at a random height, $z_\text {fid}$, between $10\,\%$ and $85\,\%$ of the image height, and serves to suppress pixels in a horizontal Gaussian band of random width, $\sigma _\text {fid}$, between $1\,\%$ and $10\,\%$ of the image height. This is meant to approximate the effect of fielding a spatial fiducial in experiment, which obstructs part of the emission, thereby creating an identifiable artifact that can be used to obtain an absolute position of the emission structure. Unfortunately, this can result in the mask to become disconnected, which may bias the registration algorithm discussed in § 2.3. The above augmentation and additionally serves to make our segmentation less sensitive to axial brightness variation. After (possibly) applying a synthetic fiducial, with probability $p = 0.25$ we include a term to minimise the effect of large-scale brightness variation of the strand via multiplication by a step-like function

(2.28)\begin{equation} g(x,z) = (1 - g_0) \frac{1}{\exp({-\boldsymbol{\aleph}\boldsymbol{\cdot}\boldsymbol{x}})} + g_0, \end{equation}

where $\boldsymbol {x} = (x,z)$, $g_0$ sets the minimum of the step-like function and $\boldsymbol {\aleph }$ sets the direction and width over which the step-like function drops from $1$ down to $g_0$. Subsequently, the image and mask are simultaneously rotated by the same random angle between $[-5^{\circ },5^{\circ }]$ and then translated identically by a random amount between $\pm 25\,\%$ in the $x$ and $\pm 5\,\%$ in the $z$ directions, respectively. Experimentally, such translations and rotations occur due to both the original orientation of the image plate as fielded in the experiment as well as its orientation in the scanner.

In our experiments, we often see a slowly varying background, which, lacking coherence with respect to the signal arising from the strand, is expected to be additive (i.e. no optical interference effects will occur). As an additional augmentation step, we add a slowly varying background given by

(2.29)\begin{equation} b(x,z) = b_0\sum_{n+m \leq \text{order}} b_{n,m}x^nz^m,\end{equation}

to the image. In (2.29), $x$ and $z$ are normalised by the corresponding image dimension, and allowed to take values in $[-1,1]$, $b_{n,m}$ are sampled from $\mathcal {U}[-1,1]$ and $b_0$ sampled from $\mathcal {U}[0.01,0.3]\times \max _{\bar {x}}(I_\theta (\bar {x}) )$, whereas the order is chosen at random from $\text {order} \in \{1,2,3\}$ with probability of selection for each order given by $p_\text {order} = [0.25,0.5,0.25]$. Note that a priori, we do not have quantitative information about the expected form or distribution of the background in experiment. Such an analysis is the primary goal in § 3. That said, low-degree polynomials are an obvious first choice to approximate the background, and the value of $b_0$ was chosen to allow the background to range from between $1\,\%$ and $30\,\%$ of the peak intensity, which appears sufficient for the segmentation task to perform well on experimental images. Finally, normally distributed random noise is added to the synthetic image with a standard deviation equal to $10\,\%$ of the original image peak amplitude. The resulting images are min–max scaled (scaled so that pixel values lie within $[0,1]$) before use in training and validating the segmentation network described in the next paragraph.

We now train a CNN to produce a pixel probability map indicating the probability $p \in [0,1]$ that a given pixel belongs to the stagnation strand. The architecture of this network is shown in figure 5. In performing this study, alternate sets of hyperparameters (e.g. learnable filter bank sizes and number of layers) were explored, with filter bank size having the largest effect. It was found that $7\times 7$ filters were the smallest size that resulted in drastically outperforming smaller $3 \times 3$ filters we initialised the architecture with on both training and validation datasets. For learning network parameters, the augmented dataset is split into 70 %, 15 % and 15 % for training, validation and additional behaviour and performance testing, respectively. The network is constructed using the Keras package in Python with a tensorflow backend. The network is trained over 30 epochs with a batch size of 64 using Keras’ default ‘adam’ gradient descent optimiser (Kingma & Ba Reference Kingma and Ba2017) with a binary cross-entropy loss function (Bishop Reference Bishop2006) and Glorot uniform initialisation (Glorot & Bengio Reference Glorot and Bengio2010) of the network weights. We note that the network contains $N_\text {params} \approx 110\,000$ trainable parameters, significantly larger than the number of training samples. However, the convolutional structure significantly regularises the network, and we do not see evidence of over-fitting as indicated by the comparison of the loss function on the training and validation data shown in figure 6.

Figure 5.

Diagram of the image segmentation network architecture.

Figure 6.

The binary cross-entropy loss function versus training epoch computed over the training data (red) and a hold-out validation dataset (blue). The close agreement between loss computed on the training and validation datasets is a good indicator that the network generalises well to unseen data and that over- and under-fitting have been effectively avoided.

The output of the convolutional network is a probability map, with each pixel taking on a value in $[0,1]$. Denoting this output probability at the pixel in the $i$–$j$ position as $P_{i,j}$, we can convert this into a mask, $\mathcal {M}_{i,j}$, by thresholding at a fixed probability value

(2.30)\begin{equation} \mathcal{M}_{i,j} = \begin{cases} 1, & \text{if}\ P_{i,j} \geq p_{\text{thresh}},\\ 0, & \text{otherwise}. \end{cases} \end{equation}

We note that in our datasets, roughly $10\unicode{x2013}20\,\%$ of the pixels in a given image will belong to the stagnation column. As a result, pixel classification is inherently an imbalanced-class problem (Johnson & Khoshgoftaar Reference Johnson and Khoshgoftaar2019). In order to encourage the mask to include the entire strand, we set the threshold probability to be a fairly low value, $p_\text {thres} = 0.1$. In addition, as noted in § 2.1, the experimental images are sigma-clipped and then min–max scaled before passing to the network to be segmented. Several randomly selected examples of the generated pixel probabilities and segmented images from experiment are demonstrated in figure 7. The results generally indicate excellent performance as seen in figures 7(a)–7(c), with only the most extreme cases of excessive background or image plate damage causing artifacts in the segmentation as demonstrated in figure 7(d). It is, in principle, possible to either modify our synthetic training data to capture these types of features, or to use transfer learning with labelled experimental data to make our network less sensitive to these anomalies. However, the volume of our experimental datasets being roughly 100 images rather than tens or hundreds of thousands or more, we chose to simply allow for a minimal human-in-the-loop presence during the refinement step described in § 2.3, recording all relevant metadata for reproducibility, to mitigate the influence of imperfect masking in the small number of extreme cases.

Figure 7.

Computation of mask probabilities works exceptionally well when transferring the network trained on synthetic data to our experimental images. Even in the small number of cases where significant plate damage was observed as indicated by the red ellipses in the left column of (b,d), the strand is successfully identified. The inclusion of additional off-strand probability indicated by red ellipses in the centre column of (b,d) does somewhat reduce the effectiveness of the automated refinement procedure discussed in § 2.3. However, as long as the strand is effectively contained within the mask, slowly varying background subtraction performed in § 2.4 is expected to work effectively even if some background is falsely identified as belonging to the strand.