Validation of structure tensor analysis for orientation estimation in brain tissue microscopy

Abstract

Background:

Accurate localization of white matter pathways using diffusion MRI is critical to investigating brain connectivity, but the accuracy of current methods is not thoroughly understood. A fruitful approach to validating accuracy is to consider microscopy data that have been co-registered with MRI of post mortem samples. In this setting, structure tensor analysis is a standard approach to computing local orientations. However, structure tensor analysis itself has not been well-validated and is subject to uncertainty in its angular resolution, and selectivity to specific spatial scales.

New method:

Here, we conducted a simulation study to investigate the accuracy of using structure tensors to estimate the orientations of fibers arranged in configurations with and without crossings.

Results:

We examined a range of simulated conditions, with a focus on investigating the method’s behavior on images with anisotropic resolution, which is particularly common in microscopy data acquisition. We also analyzed 2D and 3D optical microscopy data.

Comparison with existing methods:

Our results show that parameter choice in structure tensor analysis has relatively little effect on accuracy for estimating single orientations, although accuracy decreases with increasing resolution anisotropy. On the other hand, when estimating the orientations of crossing fibers, the choice of parameters becomes critical, and poor choices result in orientation estimates that are essentially random.

Conclusions:

This work provides a set of recommendations for researchers seeking to apply structure tensor analysis effectively in the study of axonal orientations in brain imaging data and quantifies the method’s limitations, particularly in the case of anisotropic data.

1. Introduction

Over the past few decades, the scope of neuroscience research has broadened from a focus on mapping the functions of brain areas with increasingly fine resolution to investigating brain networks, which are believed to be mediated by white matter (WM), the myelinated portion of the brain that makes up the projections between cortical regions. This shift is supported by accumulated evidence showing that differences in WM correlate with a wide range of neurological and psychiatric diseases and disorders, including Alzheimer’s disease, autism spectrum disorder, major depressive disorder, bipolar disorder, and schizophrenia (Fields, 2008). Two recent large-scale research initiatives to map the brain’s connectome exemplify this change in research focus: the Human Connectome Project and the BRAIN Initiative Connectivity Across Scales (BRAIN CONNECTS) program. These studies rely heavily on diffusion magnetic resonance imaging (dMRI). By measuring the orientation of water diffusion, dMRI can be used to generate estimated reconstructions of the pathways traversed by WM bundles using any of the various computational tractography methods (Smith et al., 2020). Accurate reconstruction of WM pathways is also an important component of clinical research and practice. Studies have shown improved patient outcomes after brain surgery when dMRI tractography was used to locate WM bundles for surgical planning and navigation to avoid interference with critical pathways (Toescu et al., 2021). However, a scarcity of ground-truth data for validating the correspondence between diffusion models based on dMRI and true axon orientations presents a challenge for interpreting WM reconstructions derived from dMRI. In fact, a study that applied multiple methods to reconstruct simulated WM bundles observed that computational tractography studies may identify four times as many false positive connections as they do true positive connections (Maier-Hein et al., 2017). While tractography is a downstream processing task that includes many challenging components unrelated to orientation estimation, it is possible that orientation estimation may have some effect on the accuracy of this and other downstream processing tasks.

Several studies have used structure tensor (ST) analysis – an image processing method for extracting orientation information from images – to estimate the orientation of axons in brain tissue microscopy and compare these to orientation estimates derived from dMRI. Because ST analysis is applied to images with much higher resolutions than dMRI, some authors have considered it to be the ground truth in these comparisons. Budde and Frank (2012) and Budde and Annese (2013) first demonstrated the potential for ST analysis of histologically stained sections to validate dMRI. They showed the relationship between low-resolution imaging features and true fiber distributions, including examples of intersecting fibers and partial volume effect produced by orthogonal adjacent fiber populations. They also showed a high correlation between angular anisotropy derived from ST analysis and dMRI. Mitter et al. (2015) applied ST analysis to post mortem histology of human fetal brains to validate abnormalities found in utero with diffusion tensor imaging. However, these studies were limited to 2D histology to validate 3D dMRI. Other groups have extended ST analysis to three dimensions for more direct comparisons. Khan et al. (2015) describe a procedure for ST analysis of z-stacks of confocal images and show that the derived orientations correspond well with the diffusion tensors in three regions of tissue with mutually orthogonal fiber directions. Schilling et al. (2016) used a similar procedure with confocal microscopy data to compare dMRI orientation distributions with true fiber distributions. They observed angular errors of 6 degrees in regions with parallel fibers and 10–11 degrees in regions with crossing fibers. ST analysis was also considered to be the ground truth for quantifying and comparing the accuracy of several different high angular resolution diffusion imaging models (Schilling et al., 2018) and to measure the fiber response function for spherical deconvolution in dMRI orientation estimation (Schilling et al., 2019). These studies inform important decisions when modeling orientation in dMRI. Thus, the sensitivity to ST analysis parameter choices and confidence estimates of the derived orientations will likely be of interest in future studies.

ST analysis has been used for many years in different contexts, some early developers include (Nagel, 1983; Bigun and Granlund, 1987). The method uses partial derivative convolution filters to compute local image gradients that constitute the components of the structure tensors. A smoothing kernel is applied to the gradients to average over a local neighborhood. The eigenvectors corresponding to the smallest eigenvalues of the tensors point in the direction of the minimum gradient, i.e., parallel to the fiber directions. This method requires two parameters to be selected: the standard deviation for an isotropic first-derivative Gaussian filter $\left({\sigma }_g\right)$ or pre-smoothing kernel, which controls the size of the structure to which the kernel is sensitive, and the standard deviation of the window function $\left({\sigma }_w\right)$, which controls the size of the neighborhood that is blurred. These parameters were described as being analogous to the diffusion imaging time and the dMRI resolution, respectively (Khan et al., 2015). ST analysis is popular in part because of its computational efficiency relative to other methods such as higher-order derivatives, Fourier methods, directional wavelets and others as discussed in Püspöki et al. (2016). However, it has a weakness in that using a fixed pair of filter widths for the entire image limits its sensitivity to one spatial scale for an entire image (Püspöki et al., 2016). The filter output is highly sensitive to the filter parameters and the size of the structure being analyzed. If the kernel size is too small, it becomes oversensitive to noise and measures isotropic orientations inside fiber boundaries instead of characterizing the entire fiber by a single orientation. Conversely, if the kernel size is too large, it may blur orientation information across multiple fibers. For this reason, and because they observed a bias towards detecting two orientations in isotropic regions, Ning et al. (2021) recommended that care be taken when using structure tensors to identify fiber crossings. A separate study compared ST orientations with optic axis orientation measurements from optical coherence scanner images, finding agreement within 10 degrees in well-delineated fibers (Wang et al., 2015). A study by Schilling et al. (2016) included an analysis of the sensitivity of structure tensors to Gaussian filter standard deviations and in-plane resolution. They derived STs from 3D confocal microscopy with high resolutions (between 0.08 μm and 0.42 μm) and compared ST derived orientations to 100 manually traced fibers in three single-fiber regions. They reported accuracy within 5 degrees, which was relatively robust to the standard deviation of the spatial averaging filter but highly sensitive to the standard deviation of the derivative filter. While this accuracy appears relatively good, other authors Leuze et al. (2021) have found angular errors as high as 45 degrees in some regions. Schilling et al. (2016) observed that derivative filters with standard deviations close to the size of myelinated fibers (0.5 μm and 1.5 μm) produced the highest accuracy. However, the studies by Wang et al. (2015) and Schilling et al. (2016) were both performed at relatively high resolution and were limited to well-delineated fiber regions. It remains unclear how the standard deviation parameters affect the accuracy of the orientation estimates derived using ST analysis in regions with crossing fibers, or what the limits of accuracy are as the resolution decreases and resolution anisotropy increases. This is an important question to address because the point spread function of a confocal microscope is known to have a depth nearly three times its width (Pawley, 2006), and 2D microscopy imaging for whole brain mapping studies is often restricted to slices spaced more than 50 μm apart (Amato et al., 2016; Oh et al., 2014; Ragan et al., 2012).

We believe that a more thorough understanding of the effects of parameter choice and of accuracy limitations, both of which are currently unknown,s will be useful for researchers with data of differing dimensions and qualities and ascribe a confidence level to their ST analysis. We addressed these unknowns by producing digital phantoms that simulate brain histology images with different line patterns, including two and three crossing fiber orientations, different degrees of image resolution anisotropy, varying line widths, varying noise, and partial volume fiber orientations (image regions with mixtures of: single orientations, two orientations, and empty background). We then computed STs from these phantoms and quantified the accuracy of varying filter parameters. Additionally, we performed structure tensor analysis in 2D and 3D regions that we selected manually from mouse brain microscopy images that included intersecting axons. We visualized the resulting orientation distributions to provide a qualitative evaluation of the effects of parameter choice in real images.

2. Methods

We generated a series of 2D and 3D image data designed to simulate optical microscopy acquisitions of parallel or crossing fibers. Specifically, these digital phantoms consisted of grids of parallel lines, repeated at a specified period. These lines were rendered at a specific angle relative to one image axis for parallel fibers, or at two or three specific angles for crossing fibers. The image resolution along the first axis was varied to create anisotropic images of a fixed total size. We evaluated two widely-employed techniques for anisotropy correction: upsampling by linear interpolation to isotropic pixel sizes and upsampling with isotropic blur correction, where additional blur is introduced along the high-resolution directions to create equal blur along each dimension. The period (i.e., spacing between lines) and angles were also varied to simulate the variety of fiber arrangements found in tissue micrographs.

We analyzed our phantom images by computing structure tensors at every image voxel and deriving local image orientations from the eigenvectors of the structure tensors. Finally, we clustered the resulting orientations to extract the dominant estimated orientations from the image and compared them with the true orientations to quantify accuracy. We repeated this procedure for a range of ${\sigma }_g$ and ${\sigma }_w$ values. This method of aggregating orientations computed at high resolution to obtain the dominant orientations per image region simulates the procedure used in the studies described above to compare microscopy imaging data with lower-resolution dMRI (Budde and Frank, 2012; Budde and Annese, 2013; Mitter et al., 2015; Khan et al., 2015; Schilling et al., 2016, 2018, 2019). Accuracy results were averaged across all angles and periods in the results we show because, in general, fiber orientations will be unknown before a sample is placed in an imaging system. In the following sections, we describe in detail each component of our validation pipeline, which is summarized in Fig. 1.

Fig. 1.

Structure tensor validation pipeline. Example pipeline using a phantom with two angles and a resolution anisotropy ratio of 2.0. Each line phantom is specified by its angle(s) $\left(\theta \right)$, period $\left(p\right)$, and number of pixels in each dimension $\left(n\right)$. In this example, ST analysis is performed on the phantom after upsampling but not isotropic blur correction. ST analysis is parameterized by gradient and window standard deviations, ${\sigma }_g$ and ${\sigma }_w$. The boundaries of the images are cropped to remove boundary artifacts. The edges after cropping are shown in red. The orientation distribution (OD) represents the histogram of orientations as amplitude on a sphere. Computed means are represented by green lines through the OD with true orientations in purple. In the final step, we compute the median error over period and angles to form error as a function of standard deviations and anisotropy.

2.1. Phantom construction

Each phantom fiber was represented as an ideal line convolved with a Gaussian point spread function, $k$. This Gaussian has a diagonal covariance with one anisotropic direction. In 2D, we denote the standard deviation in the high-resolution direction as ${\sigma }_1$, and that in the low-resolution direction as ${\sigma }_2$. In 3D, we denote both standard deviations in the two high-resolution directions as ${\sigma }_1$ and the standard deviation in the low-resolution direction as ${\sigma }_2$. In our analyses, we chose these parameters equal to the pixel size, which was different in both directions, with the exception of the analysis of variable line widths in which the standard deviations are varied relative to pixel size. The idea here is that, generally, pixel sizes for sampling are chosen to be similar to the resolution of the imaging system along each axis.

We model an axon as an infinitely thin line $l$, oriented in the direction $a\in {ℝ}^d\left(d=1\text{or}\text{2}\right)$ with $\left|a\right|=1$, with offset $b\in {ℝ}^d$. We define

$$l\left(x\right)={\int }_{ℝ}\delta \left[x-\left(at+b\right)\right]dt$$ (1)

where $\delta$ is the Dirac delta distribution. We define our image $I$ by convolving with $k$ as

$$I\left(x\right)={\int }_{ℝ}k\left(x-\left(at+b\right)\right)dt.$$ (2)

In 2D, we can derive an expression for the standard deviation of the Gaussian normal to a straight line oriented along an angle $\theta :{\sigma }_{\perp }=\sqrt{{\sin }^2\left(\theta \right){\sigma }_1^2+{\cos }^2\left(\theta \right){\sigma }_2^2}$. In this case, we use an analytical formula to produce lines. In 3D, no such simplification exists. Instead, we define a Gaussian function analytically and evaluate it at a rotated set of points. In all cases, we use modular arithmetic on the pixel locations to create a periodic pattern efficiently. We constructed all phantoms by sampling these analytical functions on a regular voxel grid, with 96 pixels in the high-resolution direction in 2D or 72 pixels in the high-resolution directions in 3D. We varied the number of pixels in the low-resolution direction based on the anisotropy factor to give a square or cube. After line drawing, we resampled to an isotropic grid using one of two approaches. In the first approach, we upsampled the phantoms along the low-resolution dimension using linear interpolation to create isotropic voxel dimensions while maintaining an anisotropic blur. For the second approach, we created a second set of phantoms in the same way, including upsampling to isotropic voxel dimensions, but with the addition of Gaussian blur along the high-resolution dimension(s) with variance equal to the difference of the squared pixel sizes in the low- and high-resolution directions (resulting in isotropic pixel size and isotropic blur). This step has been applied in at least one previous study for anisotropy correction (Khan et al., 2015).

Line directions were specified using a single angle for the 2D case or two angles for the 3D case. For a single angle in two dimensions, we chose to use 100 angles spaced evenly over the interval $[0,\pi /2)$. We do not include angles outside this range because the symmetry of the phantom makes them redundant. In three dimensions, we chose 100 angles distributed on a hemisphere using an electrostatic dispersion algorithm (Jones et al., 1999), implemented in Python as the disperse_charges method in Dipy (Garyfallidis et al., 2014), using 2000 iterations and a step size constant of 0.1. For phantoms with two fiber directions, in both the 2D and 3D cases, given a set of 100 evenly spaced angles, $𝒮$, we defined the set of all possible pairs of angles, $𝒫=\left\{\left.\left(a,b\right)\right|a,b\in 𝒮\text{and}\text{a}\ne \text{b}\right\}$, and constructed a set of phantoms using a random sample without replacement of 100 angle pairs from $𝒫$. This method represents the full range of crossing fibers that may be encountered in a real brain. For each two angle phantom a three angle phantom was constructed by adding a third angle. The third angle was computed so that all three angles were equidistant. This choice was made to reduce a combinatorial explosion by producing results as a function of too many variables. Several examples of 2D crossing-line phantoms with varying resolution anisotropy and line angles are shown in Fig. 2. One important feature of our phantoms is that lines parallel to the high-resolution direction appear dimmer because their signal power is spread over a longer distance. Example 2D phantoms used for the analyses of noise, fiber width, and partial volume orientations are shown in Figs. 3–5.

Fig. 2.

Example Phantoms. Examples of 2D crossing-line phantoms showing the effect of varying resolution anisotropy and angle values.

Fig. 3.

2D phantoms with noise. A single line pattern with four levels of noise. Noise is relative to the maximum image intensity.

Fig. 5.

2D Phantoms with varying partial volume orientations. A single line pattern with four amounts of line masking in each direction. The masked amount is indicated above each plot as the percent of the image masked.

We recognized that the phantoms described above represent fluorescent microscopy images well, but the summation of crossing lines creates a pattern that could affect the measured orientation distribution and that this artifact would be less prominent in bright-field microscopy. We modeled bright-field images by modifying the original phantoms to represent exponential attenuation of photons according to the Beer–Lambert law as follows:

$$I^{\prime }=e^{-\alpha I},$$ (3)

where $I$ is the original phantom (a blurred bright line on a dark background), $I^{\prime }$ is the bright-field type phantom, and $\alpha =10$ (chosen empirically to produce realistic looking images). We performed the same analysis as described above on these modified phantoms but found only very minor differences in error compared to unmodified phantoms. Thus, to avoid redundancy, we only show results from the first analysis in the subsequent sections.

2.2. Estimating orientation

Structure tensor analysis begins by choosing standard deviations for the first derivative Gaussian filter and the window function. Gradients, $I_i$, are then computed along each image axis $i$ by applying a Gaussian filter to the image with degree one along $i$ and zero along the other axes, using the same standard deviation for each axis. Because of intensity inhomogeneity and a large variety of contrast mechanisms in microscopy, we normalize the gradient vector rather than weighing brighter regions more than darker regions. A 2D or 3D structure tensor is then constructed at each voxel

$$S\left[p\right]=\left[\begin{array}{ccc}{I}_0\left[p\right]I_0\left[p\right]& \cdots & I_0\left[p\right]I_n\left[p\right]\\ ⋮& \ddots & ⋮\\ I_n\left[p\right]I_0\left[p\right]& \cdots & I_n\left[p\right]I_n\left[p\right]\end{array}\right],$$ (4)

where $n$ is the index of the image dimension. A Gaussian window function, $w$, is applied to the resulting matrix for local gradient averaging,

$$S_w\left[p\right]=\sum _{r}w\left[r\right]S\left[p-r\right].$$ (5)

The local orientation is given by the principal eigenvector of the structure tensor. We instead acquire the direction corresponding to the smallest gradient by first subtracting the structure tensors from identity, $S\prime =Id-S_w$. In certain cases where $S_w$ was small, such as when we investigated without normalizing the gradient, we found that the eigenvalue solver we used, NumPy’s eigh function, was inaccurate. We thus opted to simply compute the eigenvalues of $-S_w$. With our normalization convention, this can be thought of as a diffusion tensor under a model where at a point particles may diffuse in any direction normal to the gradient with equal probability, and parallel to the gradient with probability zero. The full-rank macroscopic diffusion tensor is found by averaging gradients near the point with weights given by $w$.

We crop the boundaries of the array of principal directions to remove boundary artifacts that are produced by interpolation and convolution. The upsampling interpolation introduces error onto one end of the low-resolution dimension of width equal to the upsampling factor minus one. We also cropped all edges by two-thirds of the radius of the largest convolution kernel. The two-thirds factor was chosen by visual inspection, because the error introduced by convolution varies by location and becomes negligible some distance away from the borders.

2.3. Clustering and error calculation

The resulting array of directions was processed using a k-means clustering procedure appropriate for antipodally symmetric directions (in this context, a direction is considered equal to its opposite) on a circle or sphere where the number of means was k = 1 for a single orientation, k = 2 for two orientations, or k = 3 for three orientations. The means were compared to the input angles used to generate the phantoms, and the reported error was the resultant angular difference in degrees, averaged over angles for crossing-fiber phantoms.

K-means clustering uses different modified distance and mean calculations for a circle and a sphere. For a circle, the input values are angles and the distance metric is modified for periodic boundary conditions:

$$d\left(a,b\right)=\min \left(\left|a-b\right|,T-\left|a-b\right|\right),a,b\in ℝ,$$ (6)

where $T$ is the period; in our case, $T={180}^{\circ }$. We use this distance metric to compute the Fréchet mean of the angles. That is, we find the angle, $m$, that minimizes the sum of squared distances between it and all angles in the sample:

$$m=\underset{p\in M}{argmin}\sum _{i=0}^N{d}^2\left(p,x_i\right).$$ (7)

In our case, $M$ is the set of whole number angles on a half-circle, $\left\{0^{\circ },\dots ,{179}^{\circ }\right\}$, so that $m$ is the Fréchet mean approximated to the nearest degree.

For a sphere, the input orientations are unit vectors in Cartesian coordinates. Distances are defined as

$$d\left(a,b\right)=arccos\left(\left|a\cdot b\right|\right),a,b\in {ℝ}^3.$$ (8)

This has the range $\left[0,\frac{\pi }{2}\right]$. The absolute value is used to account for antipodal symmetry, that is, $d\left(x,-x\right)=0$. The mean of the 3D direction vectors is their arithmetic mean normalized such that it lies on the unit sphere (Banerjee et al., 2005).

One modification must be made to the k-means clustering algorithm in the spherical case: if the cosine between a vector, $x$, and its assigned centroid is negative (i.e., their distance is greater than 90 degrees), the vector is reassigned to $-x$. This is necessary because the mean of two nearly antiparallel directions before normalizing is near zero even though they represent similar orientations because of symmetry. Flipping all vectors in the same cluster to the same hemisphere avoids instability from this effect. For both the 2D and 3D cases, the initial centroids are set to a random choice of $k$ data points without replacement, and the stopping criterion is reached when the new centroids are equal to the old centroids.

For 2D phantoms with a single angle, we computed a single mean using the periodic mean calculation given by Eq. (7). For the multiple angle case, means were computed using the periodic k-means algorithm with $k$ set to the number of orientations. For 3D phantoms, angle means were estimated using k-means clustering for all cases. The distances between the computed means and the true angles were calculated using Eq. (6) for 2D or Eq. (8) for 3D. In the two- and three-angle cases the error was the average of two or three distances, respectively: the distance between the nearest mean and true angle pair, the distance between the next nearest mean and true angle of the remaining angles, and, in the three-angle case, the distance between the remaining mean and true angle.

The resulting errors were tabulated using a Pandas data frame in Python. The error heat maps shown in Figs. 7–11 were created by computing the median of the errors over period and angles and grouping by the remaining variables: ${\sigma }_g$, ${\sigma }_w$, and anisotropy ratio (ratio of large to small pixel side lengths).

Fig. 7.

2D single angle. Left: Median errors in degrees with respect to ${\sigma }_g$ and ${\sigma }_w$ for 2D single-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

Fig. 11.

3D two angles. Left: Median errors in degrees with respect to ${\sigma }_g$ and ${\sigma }_w$ for 3D two-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

We validated the k-means and error calculations by computing errors using the method described above on samples from known orientation distributions. We drew samples from von Mises and von Mises–Fisher distributions using Scipy’s vonmises and vonmises_fisher methods for 2D and 3D, respectively, and compared the estimated means from our methods with the true sample means over a range of variances and over varying crossing angles for mixtures of two distributions (Virtanen et al., 2020). Fig. 6 shows a visual representation of a mixture of 3D von Mises-Fisher distributions and the resulting differences between k-means centers and true means.

Fig. 6.

K-Means error calculation. The estimated means (green lines) compared to true means (purple lines) shown with a sample orientation distribution represented as a 3D histogram. The error is the mean of $\text{Δ}{\alpha }_1$ and $\text{Δ}{\alpha }_2$.

2.4. Example mouse brain microscopy

To illustrate the applicability of our model to real data analysis, we considered image regions from mouse brain microscopy datasets and qualitatively compared the performance of structure tensor analysis with the predictions of our phantom study. We used the ImageJ software package to view 2D and 3D brain microscopy images and select two regions each from 2D and 3D datasets that exhibited two predominant intersecting axon orientations (Schindelin et al., 2012). The first 2D image was selected from the area of the rostral Raphe nucleus and periaqueductal gray (Fig. 19). We chose a second region based on discussion of complex fiber architecture in Georgiadis et al. (2023), in the area of the supplementary somatosensory area, corpus callosum, and caudoputamen (Fig. 20). The 2D images were acquired from a mouse brain that was fixed using cardiac perfusion (4% paraformaldehyde in 1X PBS) and sliced into 50 μm thick coronal serial sections using a vibratome microtome (Precisionary Instruments LLC, Ashland, MA). The slices were stained for myelin using a proprietary stain under development in the Hongwei Dong lab, which has contrast similar to Luxol Fast Blue (LFB) staining, and is based on a gold chloride-based mechanism similar to Blackgold (Schmued, 1990) and imaged with an Olympus VS200 microscope under bright-field mode with 10X magnification at 1.1 μm × 1.1 μm pixel size. The 3D volume was taken from an optically cleared (SHIELD, LifeCanvas Technologies, Cambridge, MA) mouse spinal cord with fluorescently labeled axons (B6.Cg-Tg(Thy1-YFP)HJrs/J, The Jackson Laboratory, Bar Harbor, ME). YFP is a soluble protein and thus moves freely inside the cell body, effectively filling it and causing the entire axon as well as the cell body and dendrites to fluoresce yellow. The spinal cord was imaged on a TCS SP8 MP multiphoton microscope (Leica Microsystems Inc., Deerfield, IL) with a Leica HC FLUOTAR L 25×/1.0 IMM objective at 0.87 μm × 0.87 μm × 2.4 μm resolution. We selected a 100 × 100 pixel region from the 2D image and an 812 × 728 × 29 voxel region from the 3D volume for our analysis. As with the phantoms, we applied anisotropy correction to the volume region by linear interpolation along the low-resolution dimension to isotropic pixel size, yielding a final shape of 812 × 728 × 78 after resampling and cropping the interpolation artifact. The dominant axon orientations were manually estimated for each image.

Fig. 19.

2D example microscopy region 1. A bright-field image of myelin stained mouse brain is shown with dark neuron fibers visible on a light background as a whole slice in the top-left (A) and a local region (D). A subregion, shown as a white bounding box in (D), was chosen for analysis. The subregion with manually identified primary fiber orientations are shown as arrows with angles mapped to colors (E). The accuracy of ST derived orientations on the subregion are shown in the top-middle (B). ST analysis was also performed on a matching simulated phantom (F). The resulting errors are presented for comparison in the top-right (C). Three parameter settings, indicated by colored bounding boxes, were selected from the resulting heat map to compute orientations in the microscopy region. Derived orientations are shown as polar histograms and the computed means as arrows overlaid on the image for each parameter setting (J–L). The same regions are shown centered in their larger context in (G–I).

Fig. 20.

2D example microscopy region 2. A bright-field image of myelin stained mouse brain is shown with dark neuron fibers visible on a light background as a whole slice in the top-left (A) and a local region (D). A subregion, shown as a white bounding box in (D), was chosen for analysis. The subregion with manually identified primary fiber orientations are shown as arrows with angles mapped to colors (E). The accuracy of ST derived orientations on the subregion are shown in the top-middle (B). ST analysis was also performed on a matching simulated phantom (F). The resulting errors are presented for comparison in the top-right (C). Three parameter settings, indicated by colored bounding boxes, were selected from the resulting heat map to compute orientations in the microscopy region. Derived orientations are shown as polar histograms and the computed means as arrows overlaid on the image for each parameter setting (J–L). The same regions are shown centered in their larger context in (G-I).

We then generated two 2D phantoms and two 3D phantoms, each corresponding to an example microscopy regions. The axons in the example images were approximately 2 pixels wide for the 2D images and 4 pixels wide for the 3D volumes. These diameters are relatively large, likely due to tissue expansion during clearing, and the fact that these are myelinated axons that travel long distance through the spinal cord. For each phantom, we selected angles that matched the manually estimated axon orientations and periods appropriate to the spacing and width of neurons. We performed ST analysis on the phantoms using the same range of ${\sigma }_g$ and ${\sigma }_w$ as in the phantom experiments, and we used these results to select three parameter settings to use on the example microscopy. These included one setting that predicts high accuracy and two that predict low accuracy. Microscopy image orientations, mean orientations, and errors relative to the manually annotated orientations were computed as described in Sections 2.2 and 2.3 using the selected parameters.

2.5. Software availability

All software methods, example microscopy ROIs, and raw ST error results used in this paper are available under an open-source license and may be downloaded at https://github.com/BrysonGray/structure_tensor_validation.git.

3. Results

In the following section, we report the effects that filter parameters, image resolution anisotropy, line width, signal noise, and partial volume fiber orientations had on the accuracy of ST orientation estimates in our simulated phantoms. These results could be used to determine optimal filter widths for a given analysis task and as a guide for which suboptimal choices should be avoided. For all phantom experiments, errors are presented as a function of gradient and window standard deviations, ${\sigma }_g$ and ${\sigma }_w$. We also report error in relation to resolution anisotropy (for one, two, and three crossing fibers), line width, noise, and partial volume orientations. Furthermore, the optimal Gaussian filter standard deviations and corresponding best errors are reported for each experimental variable’s value. Note that we chose to report errors based on upsampled but not isotropically blurred phantoms because the latter step reduced accuracy in all experiments. Our results are the median over all line angles and periods.

3.1. Anisotropy experiments

3.1.1. 2D

The results of the 2D anisotropy experiments can be seen in Figs. 7–9. Note that in the single-angle case, accuracy is robust to parameter choice even up to relatively large resolution anisotropy. However, in the cases of two and three crossing angles a more dramatic difference in errors can be seen between good and bad parameter choices. These results suggest that for small anisotropy ratios, it is important to choose a ${\sigma }_g$ that is small relative to the pixel size and a ${\sigma }_w$ that is approximately equal to one pixel. For anisotropy ratios greater than two, accuracy becomes less sensitive to parameter choice, but decreases significantly. Also notable is the sharp boundary between low and high error in the two-angle isotropic case (Fig. 8, lower left corner of the top left heat map). This pattern is repeated for other experiments with 2D two-angle phantoms (see Figs. 13, 15 and 17). Possible causes of this specific failure case will be elaborated in the discussion in Section 4.

Fig. 9.

2D three angles. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 2D three-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

Fig. 8.

2D two angles. Left: Median errors in degrees with respect to ${\sigma }_g$ and ${\sigma }_w$ for 2D two-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

Fig. 13.

2D variable width. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 2D two-angle phantoms with isotropic pixel size, constant line period, and varying widths. Errors are represented in a heat map with the line width indicated above each plot represented as the standard deviation of the Gaussian point spread function used to draw the phantom lines. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The ${\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error as a function of width. Bottom right: The minimum angular error as a function of width.

Fig. 15.

2D variable noise. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 2D two-angle phantoms with isotropic pixel size and varying noise levels. Errors are represented in a heat map with the noise indicated above each plot. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error as a function of noise. Bottom right: The minimum angular error as a function of noise.

Fig. 17.

2D variable partial volume orientations. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 2D two-angle phantoms with isotropic pixel size and varying percent masked. Errors are represented in a heat map with the percent masked indicated above each plot. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The $g{\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error as a function of percent masked. Bottom right: The minimum angular error as a function of percent masked.

3.1.2. 3D

Results from the 3D anisotropy experiments can be seen in Figs. 10–12. These results show trends that are similar to the 2D experiments, but with some notable differences. First, for all 3D structure tensor analysis, it is important to use ${\sigma }_w$ greater than zero. This is because structure tensors are rank one prior to spatial averaging, meaning that the two smaller eigenvalues are degenerate and their eigenvectors are ambiguous. Only after spatial averaging by the window function does the tensor become full rank, and the two smaller eigenvectors become unambiguous. Note that we ignore ${\sigma }_w$ equal to zero when reporting the maximum error.

Fig. 10.

3D single angle. Left: Median errors in degrees with respect to ${\sigma }_g$ and ${\sigma }_w$ for 3D single-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

Fig. 12.

3D three angles. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 3D three-angle phantoms represented in a heat map with the anisotropy ratio indicated above each plot. Top right: The ${\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error from each heat map. Bottom right: The minimum and maximum errors corresponding to each heat map.

Similar to the 2D case, for single angles in 3D, errors remain near zero even up to large resolution anisotropy with low sensitivity to parameter selection. Conversely, in the case of 3D crossing-line phantoms, there is greater sensitivity to parameter choice, accompanied by a marked rise in error as anisotropy increases. For small anisotropy ratios, best accuracy was achieved overall with $0.5\text{px}\le {\sigma }_g\le 1.0\text{px}$ and $0.5\text{px}\le {\sigma }_w\le 1.0\text{px}$. Interestingly, the optimal ${\sigma }_w$ is smaller for 3D data compared to the 2D results.

In both 2D and 3D, the difference between two and three crossing angles is small, with the three-angle case having a slightly steeper increase in error with increase in resolution anisotropy.

3.2. Fiber width experiments

Figs. 13 and 14 present results of an analysis on multiple fiber widths. We created 2D and 3D sets of phantoms with two fiber orientations and constant line periods. We varied the widths of the fibers by adjusting the standard deviation of the Gaussian point spread function used to construct the lines. We repeated the experiment with isotropic pixel sizes and anisotropic pixel sizes with anisotropy ratio of 3.0.

Fig. 14.

3D variable width. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 3D two-angle phantoms with isotropic pixel size, constant line period, and varying widths. Errors are represented in a heat map with the line width indicated above each plot represented as the standard deviation of the Gaussian point spread function used to draw the phantom lines. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The ${\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error as a function of width. Bottom right: The minimum angular error as a function of width.

The figures show the angular errors as a function parameter choice as in the anisotropy experiment results above. However, here the heat maps show only the errors at isotropic resolution, with different axes showing different widths. Plots to the right show best ${\sigma }_g$, ${\sigma }_w$ and error as functions of width for isotropic phantoms and anisotropy ratio of 3.0.

For the 2D case, as with the 2D two-angle of the anisotropy experiment case shown in Fig. 8, there is a sharp boundary in the parameter space separating good parameter choice from failure. Aside from this region of catastrophic failure in 2D phantoms, the heat maps reveal a smooth trend in errors across parameters, with the best ${\sigma }_g$ and ${\sigma }_w$ close to one pixel, that remains relatively constant as the line width increases (best ${\sigma }_g$ tends to decrease slightly with increased width). This is true for both the 2D and 3D setting.

3.2.1. 2D

See Fig. 13.

3.2.2. 3D

See Fig. 14.

3.3. Noise experiments

Figs. 15 and 16 present results of an analysis on random background noise. We created 2D and 3D sets of phantoms with two fiber orientations and varied the background noise by adding Gaussian random noise to the image (prior to anisotropy correction when applied). The magnitude of noise is given by the standard deviation of a normally distributed random variable and is shown as a fraction of the image maximum intensity. We repeated the experiment with isotropic pixel sizes and anisotropic pixel sizes with anisotropy ratio of 3.0.

Fig. 16.

3D variable noise. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 3D two-angle phantoms with isotropic pixel size and varying noise levels. Errors are represented in a heat map with the noise indicated above each plot. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The ${\sigma }_g$ and ${\sigma }_w$ corresponding to the minimum error as a function of noise. Bottom right: The minimum angular error as a function of noise.

Increases in noise appears to have little effect on error or best parameters in isotropic phantoms. In phantoms with anisotropic resolution, increased noise generally results in a greater increase in error and a trend towards larger best ${\sigma }_g$ and ${\sigma }_w$. It should be noted that noise in anisotropic data acquires an orientation biased towards the anisotropic axis after anisotropy correction due to resampling. This can be seen in Figs. 21 and 22 as a high error (yellow) band towards the top of each heat map. This may be the cause for a difference in trends between the isotropic and anisotropic settings.

Fig. 21.

3D example microscopy region 1. A small fluorescently labeled 3D volume from mouse spinal cord is shown with neurons visible as bright objects on a dark background. The volume is represented as maximum intensity projections from three orthogonal views (rows 2–4). Primary orientations are represented by arrows with angles mapped to colors and length proportional to the in-plane component of the orientation. Manually identified primary orientations are shown in (G) over the selected subregion. A matching simulated phantom is shown on the left (F). The errors of ST derived orientations on the phantom and microscopy subregion are shown above each as a function of ${\sigma }_g$ and ${\sigma }_w$ (A and B). Three sigma settings, indicated by colored bounding boxes, were selected from (A) to compute orientations in the microscopy region. Derived orientations are shown as polar histograms marginalized over out-of-plane components (colored using only red and blue as in the 2D figures), and the computed means as arrows overlaid on the image for each sigma setting (H-J). The same regions are shown centered in their larger context in the x-y view in (C-E).

Fig. 22.

3D example microscopy region 2. A small fluorescently labeled 3D volume from mouse spinal cord is shown with neurons visible as bright objects on a dark background. The volume is represented as maximum intensity projections from three orthogonal views (rows 2–4). Primary orientations are represented by arrows with angles mapped to colors and length proportional to the in-plane component of the orientation. Manually identified primary orientations are shown in (G) over the selected subregion. A matching simulated phantom is shown on the left (F). The errors of ST derived orientations on the phantom and microscopy subregion are shown above each as a function of ${\sigma }_g$ and ${\sigma }_w$ (A and B). Three $\sigma$ settings, indicated by colored bounding boxes, were selected from (A) to compute orientations in the microscopy region. Derived orientations are shown as polar histograms marginalized over out-of-plane components (colored using only red and blue as in the 2D figures), and the computed means as arrows overlaid on the image for each sigma setting (H–J). The same regions are shown centered in their larger context in the x-y view in (C–E).

3.3.1. 2D

See Fig. 15.

3.3.2. 3D

See Fig. 16.

3.4. Partial volume orientations experiments

Figs. 17 and 18 present results of an analysis on partial volume fiber orientations (image regions with mixtures of: single orientations, two orientations, and empty background). We created 2D and 3D sets of phantoms with two fiber orientations and varied the fraction of the image from which fibers are masked in each direction. We repeated the experiment with isotropic pixel sizes and anisotropic pixel sizes with anisotropy ratio of 3.0.

Fig. 18.

3D variable partial volume orientations. Left: Median errors in degrees with respect to the ${\sigma }_g$ and ${\sigma }_w$ for 3D two-angle phantoms with isotropic pixel size and varying percent masked. Errors are represented in a heat map with the percent masked indicated above each plot. Plots on the right are shown for isotropic pixel size (blue) and for anisotropy ratio of 3.0 (orange). Top right: The ${\sigma }_g$ and ${\sigma }_w$ values corresponding to the minimum error as a function of percent masked. Bottom right: The minimum angular error as a function of percent masked.

The results indicate that for 2D, the trend is towards greater error with a greater proportion of lines masked. Between 20% and 40% masked, the optimal parameters shift suddenly towards large values. Interestingly, for 3D the error and sensitivity to parameters slightly decreases up to about 60% masking before increasing again.

3.4.1. 2D

See Fig. 17.

3.4.2. 3D

See Fig. 18.

3.5. Example mouse brain microscopy results

Figs. 19–22 present the results of ST analysis on the example mouse brain microscopy regions, including error heat maps over sigmas and visualizations of angles computed on the image regions using three different parameter settings. Phantoms were constructed for each microscopy region to mimic the line patterns in the microscopy as closely as possible and ST analysis was performed on them. The error heat maps for the phantoms are presented side-by-side with those of the microscopy for comparison. The sigma settings selected to visualize angle distributions are indicated by colored boxes on the phantom error heat maps.

In both the 2D and 3D cases, the pattern of errors predicted by ST analysis of the phantoms resembles that of the microscopy (i.e. comparing panels B and C in the 2D cases, and comparing panels A and B in the 3D cases) with some variations. Most notably, analysis of the 2D phantoms shows a trend towards higher error at large ${\sigma }_w$ which is absent in the microscopy errors. Additionally, large errors predicted in the bottom left corners (large ${\sigma }_g$, small ${\sigma }_w$), a phenomenon noted previously in Section 3.1, appear exaggerated compared to microscopy errors.

The errors predicted by phantom analysis in the 3D microscopy (Figs. 21 and 22) correspond closely to actual microscopy error. In both cases the ST analysis fails at small ${\sigma }_g$ (shown as a yellow band at the top of the heat maps). This effect is absent when the phantom is made isotropic, or when no noise is added, indicating that it is likely caused by random noise that acquires an orientation as a result of anisotropy correction along the low-resolution axis as described in our noise experiment in Section 3.3. The effect of the directional noise can be reduced by selecting a large ${\sigma }_g$. However, even when optimal parameters are selected, the computed mean angles are biased somewhat towards the low-resolution axis as seen in the figures.

4. Discussion

In this work, we used computational phantoms to study the accuracy of ST analysis for measuring crossing and single fiber orientations in well-characterized settings. We produced a series of 2D and 3D phantoms at different anisotropy ratios, studied accuracy with different ST filter parameters, and compared them to real microscopy data. We also studied the effects of fiber width, noise, and partial volume orientations. The results lead to two general conclusions. First, structure tensor analysis is potentially very accurate. For both 2D and 3D cases in regions with homogeneous orientations, structure tensors can capture angles with accuracy within 0.01 degrees in isotropic images and maintains accuracy close to one degree up to large anisotropy ratios. In regions with crossing fibers, the best accuracy for isotropic images was approximately one degree in 3D and approximately three degrees in 2D. However, it is important to be aware that this accuracy was not sustained and deteriorated as the anisotropy ratio is increased.

Second, the accuracy of structure tensor analysis in regions with crossing fibers is very sensitive to the standard deviations of the Gaussian filters used. For example, in 2D data with crossing fibers, median errors below about ten degrees can be achieved only in a range of about one pixel in either ${\sigma }_g$ or ${\sigma }_w$.

Based on these results we can make a few recommendations for researchers planning to employ structure tensor analysis. First, assuming the region of interest includes a significant amount of overlapping fibers along with regions with single orientations, our experiments suggest that high accuracy can be achieved in both regions by selecting $0.1\text{px}\le {\sigma }_g\le 0.5\text{px}$ and $1.0\text{px}\le {\sigma }_w\le 2.0\text{px}$ for 2D images and $0.5\text{px}\le {\sigma }_g\le 1.0\text{px}$ and $0.5\text{px}\le {\sigma }_w\le 1.0\text{px}$ for 3D images. In the case of overlapping fibers, we recommend caution when using structure tensor analysis on images with any dimension having resolution lower than the structure being analyzed because even the best accuracy at anisotropy ratios of two or greater is below what could reasonably be considered ground truth. In addition, we do not recommend introducing blur in the x-y plane to create isotropic blur, as was implemented in Khan et al. (2015). This only decreased the accuracy in our experiments, as shown in Fig. 23. We also found that normalizing the gradients before constructing tensors had a negligible effect on accuracy.

Fig. 23.

Effect of isotropic blur correction. Comparison of error distributions between phantoms with isotropic blur correction and those without.

The effect of line width and noise on accuracy and optimal parameters are relatively minor, though the experiments do reveal some points of interest. First, the fiber width appears to have little effect except to expand the sharply bounded bright failure-region in the lower left (large ${\sigma }_g$, small ${\sigma }_w$) of the 2D heat maps seen in Fig. 13. This is a phenomenon unique to the 2D phantoms and can also be seen in the top left of Fig. 8. The phantoms used in the 2D anisotropy experiment shown in Fig. 8 varied in their line period between seven and thirteen pixels, but the period of the phantoms used for Fig. 13 was fixed to a much larger period of twenty pixels. This, and the growth of this failure-region as fiber width is increased, would suggest that the failure-region is a consequence of closely packed crossing fibers, perhaps causing the appearance of a new predominant orientation in the points of overlap and spaces between the fibers. An example of this can be seen in the largest width phantom shown in Fig. 4. This may not be predominant in real microscopy, since the irregularity of fiber spacing and curvature would likely prevent the emergence of these second order orientations. This regularity is one potential shortcoming of our simulation studies. In 3D, fibers in different orientations intersect much less frequently, reducing this effect. The lack of any major difference in optimal parameters, aside from this failure-region, between phantoms with the smaller range of periods and the large period of twenty pixels also reveals that line period, one variable we did not isolate for study, does not significantly impact parameter selection.

Fig. 4.

2D phantoms with varying width. A single line pattern with four line widths. Widths shown are the standard deviation of the Gaussian point spread function used to draw the lines.

Second, while the addition of noise had no perceptible effect in isotropic phantoms, it did substantially increase the value of the optimal parameters in the phantoms with anisotropy ratio of 3.0 and increase the error in 2D and 3D. As stated in Section 3.2, uncorrelated noise in anisotropic data acquires an orientation biased towards the anisotropic axis after anisotropy correction due to resampling. These results suggest that, for anisotropic 3D image data, ${\sigma }_g$ should be chosen between 2.0 px and 2.5 px, and ${\sigma }_w$ should be chosen between 2.0 px and 4.0 px for best results.

The experiment in partial volume orientations shows a general increase in error associated with more heterogeneity in orientations within an image region for 2D phantoms. Counterintuitively, for 3D phantoms the error decreases with a larger proportion of fibers masked, before again increasing. This may have to do with the increased distance between crossing fibers in 3D compared to 2D data, however the exact cause is unknown. In general, these results indicate that including different orientations or partial volumes with no orientation will produce less reliable orientation distributions.

While our phantom experiments have the benefit of having known true orientations and line spacing, they come with some inherent limitations. Phantoms are idealizations of real images, which have curving lines or densely packed homogeneous neurons without distinct edges. With this in mind, we made an effort to identify the limits of this method and make it generalize as much as possible. We modeled noise, partial volume effects, multiple widths, and multiple angles. We also modeled both fluorescent images and bright-field images. It is notable that we observed a negligible difference between summed lines (fluorescent-type) phantoms and exponentially attenuated (brightfield-type) phantoms. This suggests that the artifact caused by summing intersecting lines does not significantly affect the errors. In our example microscopy data, both the curving of axons and densely packed neurons blurring together are apparent. Nevertheless, our model is able to predict example parameter choices that may lead to poor performance, and example parameter choices that lead to good performance. That being said, the phantom study results we reported were averaged across all simulated angles, whereas the microscopy datasets only have a few primary orientations. We demonstrated in our study that the angle of fibers relative to the low-resolution axis affects the accuracy of the predicted angle. As seen in Fig. 24, angle has relatively little effect on accuracy for low resolution anisotropy up to about anisotropy ratio of 4.0, however a pronounced effect can be seen for larger anisotropy ratios where error is low when the fibers align with either the low-resolution axis or the high-resolution plane but increases for intermediate angles.

Fig. 24.

ST errors vs. Angle relative to low-resolution dimension. Angular error in ST derived orientations in 3D single-angle phantoms is plotted as a function of the true orientation’s angle relative to the low-resolution dimension. Data was aggregated into five degree increments then smoothed using a rolling window mean to reduce noise.

The method used to aggregate structure tensors and characterize orientation in a local region is also relevant. K-means clustering is known to be sensitive to initial centers, so it is common when employing k-means clustering to try several different initial centers and use the estimate that achieves the smallest sum of squared distance of points from their cluster centers. When testing our k-means on known distributions, we found small differences from the known means. Increasing the number of initializations did not significantly change the difference between estimated and true means so we chose to use one initialization to reduce computation time.

Finally, while the lines we drew were sufficiently anti-aliased, the details disappeared and the pattern blurred away as the anisotropy ratio approached the period of our line pattern. The smallest period we used was 7.0 and the greatest anisotropy ratio we used was 8.0, thus there was significant error due to this artifact. This can be seen in Fig. 7. This demonstrates how substantially fiber orientation accuracy is compromised even with moderate anisotropy levels. The resolution anisotropy is an important consideration for researchers intending to apply ST analysis to their datasets, and good accuracy could likely be achieved by following our recommendations above.

Overall, the studies presented in this work can give confidence to investigators who consider applying ST analysis by clarifying the range of data and parameters appropriate for its use and by demonstrating the high degree of accuracy that can be achieved under the right conditions. This work will help give researchers certainty in the accuracy of their diffusion MRI images and downstream processing. We expect this will have the potential to impact development of tractography algorithms and analysis of brain connectivity. This may lead to improvements in our understanding of basic neuroscience (cell types are often understood based on their connectivity) (Hawrylycz et al., 2023) and medical applications of connectomics (Toescu et al., 2021).

Data availability

The link to our data/code is https://github.com/BrysonGray/structure_tensor_validation.git.