Tracing Filaments in Simulated 3D Cryo-Electron Tomography Maps Using a Fast Dynamic Programming Algorithm

Abstract

We propose a fast, dynamic programming-based framework for tracing actin filaments in 3D maps of subcellular components in cryo-electron tomography. The approach can identify high-density filament segments in various orientations, but it takes advantage of the arrangement of actin filaments within cells into more or less tightly aligned bundles. Assuming that the tomogram can be rotated such that the filaments can be oriented to be directed in a dominant direction (i.e., the $X$, $Y$, or $Z$ axis), the proposed framework first identifies local seed points that form the origin of candidate filament segments (CFSs), which are then grown from the seeds using a fast dynamic programming algorithm. The CFS length $l$ can be tuned to the nominal resolution of the tomogram or the separation of desired features, or it can be used to restrict the curvature of filaments that deviate from the overall bundle direction. In subsequent steps, the CFSs are filtered based on backward tracing and path density analysis. Finally, neighboring CFSs are fused based on a collinearity criterion to bridge any noise artifacts in the 3D map that would otherwise fractionalize the tracing. We validate our proposed framework on simulated tomograms that closely mimic the features and appearance of experimental maps.

I. Introduction

Cryo-electron tomography (cryo-ET) of unstained frozen-hydrated cell samples is commonly used in biology to capture the 3D ultrastructure of supramolecular complexes in their native environment. In recent years, our work has focused on detecting and segmenting specific biomolecular shapes, such as cytoskeletal filaments [1]–[3], in such frozen cells. 3D tomograms exhibit low resolution (3–5 nm) and considerable anisotropic noise and artifacts because of the limited electron dose and the limited tilt series that masks out a wedge in Fourier space. Given the noise, missing wedge artifacts, and the enormous amount of information present in cryo-ET reconstructions, performing a manual tracing of cytoskeletal filaments is highly labor intensive [4]. For an objective analysis, automated approaches are needed.

In the last decade, we have developed several specialized approaches for filament detection. In Dictyostelium discoideum filopodia, individual actin filaments are well separated and randomly oriented. This enabled voltrac to find the seed locations of filaments using a genetic algorithm-based search of a population of cylindrical templates [1]. Once tagged, well-separated filaments can be traced using a bi-directional alignment of a moving template. In the shaft region of hair cell stereocilia, on the other hand, the filaments form dense hexagonally packed bundles that can be detected by bundletrac [2] using hexagonal filament bundle templates. However, the taper region of hair cell stereocilia [4] proves more challenging. Here, individual filaments can deviate from the dominant direction of a regular bundle, but they remain densely packed such that voltrac is prone to jumping tracks from one filament to the next. This led us to propose an initial correction of noise and missing wedge artifacts in such filamentous tomograms using template-based constrained deconvolution [3], which provided cleaner density maps for the tracing.

All our earlier approaches rely on computationally expensive convolution or deconvolution methods with rotating shape templates that require days of computing time when performed on a full tomogram. Furthermore, the earlier work flows all relied on separate denoising [3], [5] that was conceptionally unrelated to the tracing. The purpose of the present work was to test whether performing a significantly faster tracing of dense filaments by combining denoising with tracing in a single conceptual framework is possible in the most challenging scenario of the stereocilia taper region [4]. Instead of slow rotating templates, we take advantage of the dominant orientation of the filaments while allowing individual traces to deviate from the main direction.

Image filtering techniques can be broadly categorized based on the application domain. Spatial domain techniques operate directly on pixels or voxels, whereas frequency domain techniques operate on the Fourier transform. We use frequency domain masking in the modeling of the missing wedge in Section II. However, the path-based density filter introduced in Section III is a spatial domain technique that operates directly on the voxels of the 3D map.

II. Tomogram Simulation Method

To test the framework, we developed a method for the realistic simulation of tomograms based on an existing tracing of filament cores (Fig. 1). The ground truth tracing used for the simulation can be a manually obtained interpretation of an experimental tomogram or an automated tracing generated by the current (or another) computational approach. As described in more detail in [6], the simulation aims to mimic the noise and missing wedge artifacts found in experimental maps. The approach does not add (non-filamentous) biological features, such as membranes, to the simulated tomogram. Thereby, the simulated tomogram provides a known ground truth that is specifically designed to validate the accuracy of any filament tracing framework.

Fig. 1.

(Top) Manual annotation of filaments from the experimental map (yellow) [4] (Bottom) Simulated tomogram created from the manual annotation (gray).

To create the simulation, we start by interpolating the existing filaments onto a user-defined cubic grid (typically, the grid of an experimental map that corresponds to the existing tracing). In the following, cubic voxel indices $i$, $j$, and $k$ correspond to the $X$, $Y$, and $Z$ axes, respectively.

The filaments are then volumized by convolving the voxel densities $D(i,j,k)$ with a Gaussian shape kernel. Filtered colored noise is then added to match the radial power spectral density and the signal-to-noise ratio of an experimental reference tomogram (typically the experimental map that corresponds to the existing ground truth filaments). Finally, a user-defined missing wedge is masked out in Fourier space to emulate the artifact resulting from the limited tilt angle range of the electron microscope.

III. Filament Tracing Method

The proposed tracing framework operates in several stages, as shown in Fig. 2. The pre-processing step, which we also refer to as the path-based density modification of tomograms, strengthens the density values of the filaments to make them more prominent and distinguishable. Based on automatically determined seed points, candidate filament segments (CFSs) are then grown from the seeds, which are iteratively refined and fused to create the final filament segments (FSs).

Fig. 2.

The proposed filament tracing framework.

A. Pre-Processing: Path-Based Density Filtering

Because of the low radiation dose allowed for each image in a tilt series and the limited view directions, the reconstructed 3D cryo-ET maps typically exhibit a high noise level and missing Fourier wedge artifacts. The pre-processing step uses a path-based density filter to strengthen the underlying filamentous features so that they can be better visualized and automatically extracted. This idea was inspired by an earlier path density (PD)-based filter developed by Starosolski et al. [5]. In their work, however, a statistical discriminant analysis was needed to differentiate between isotropic random walks, whereas in the present study, we could take advantage of the nearly straight-line paths of the filaments.

The proposed dynamic programming algorithm assigns maximum PD values to the voxels $(i,j,k)$ in the 3D tomogram by considering a pyramidal search window in the forward (Fig. 3) and backward directions.

Fig. 3.

The diagram shows how the forward path density ($FPD$) of length $l$ is calculated from voxel $(i,j,k)$ (1 voxel = 0.947nm; Eqn. 1). For illustration purposes, a short path $l=3$ is used, although in practice, we use longer lengths for a more effective denoising effect. Each vertical slice $j^{\prime }$ of the pyramid consists of $\left(2j^{\prime }-2j+1\right)\times \left(2j^{\prime }-2j+1\right)$ voxels. The $FPD(i,j,k;l)$ is the maximum accumulated PD among the voxels of the front slice $j^{\prime }=j+l$ of the search pyramid (blue). The PD accumulation from the lower (red) slices $j\le j^{\prime }<j+l$ within the pyramid is described in Fig. 4.

The forward path density $FPD(i,j,k;l)$ (Fig. 3) and the backward path density $BPD(i,j,k;l)$ (similar, not shown) are computed for paths of fixed length $l$ that originate at each voxel $(i,j,k)$. The forward and backward directions are determined by the dominant coordinate system axis (in this work, the filaments are aligned mostly in the $Y$ direction [4], but generally, the dominant axis is assumed to be one of either the $X$, $Y$, or $Z$ coordinate axes). To allow for an up to a 45° deviation of filaments from the dominant directions, a pyramidal search window is considered (Fig. 3). Voxel densities $D(i,j,k)$ (normalized to a range from 0 to 1) are accumulated in the search cones in the forward (Fig. 4) and backward (similar, not shown) directions, yielding a $PD$ at the front ($j+l$; blue in Fig. 3) and rear ($j-l$; not shown) slices of the pyramids, from which maximum values are chosen:

$$FPD(i,j,k;l)=\underset{\begin{array}{c}i-l\le i^{\prime }\le i+l\\ k-l\le k^{\prime }\le k+l\end{array}}{\max }PD\left(i^{\prime },j+l,k^{\prime }\right),\text{and}$$ 1

$$BPD(i,j,k;l)=\underset{\begin{array}{c}i-l\le i^{\prime }\le i+l\\ k-l\le k^{\prime }\le k+l\end{array}}{\max }PD\left(i^{\prime },j-l,k^{\prime }\right),$$ 2

where $PD(i,j,k)$ is initialized as

$$PD(i,j,k)=D(i,j,k)$$ 3

and is iteratively accumulated in the pyramidal search windows, starting from the origin, as illustrated in Fig. 4 and described in the following.

Fig. 4.

Iterative $PD$ accumulation step in Eqn. 4 (forward direction; the backward direction is similar and not shown). The updated $PD$ of voxel $\left(i^{\prime },j^{\prime },k^{\prime }\right)$ is the maximum $PD$ of its (up to) nine contributing neighbors in the previous $Y$ -slice plus its own voxel density value. For simplicity, the illustration shows all nine neighbors, but only neighbors that are also located within the search pyramid of Fig.3 are contributing.

Each voxel $\left(i^{\prime },j^{\prime },k^{\prime }\right)$ in the forward or backward pyramid accumulates its $PD$ from (up to) nine contributing neighbor voxels in the previous $Y$ -slice (i.e., $j^{\prime }-1$ for the forward direction or $j^{\prime }+1$ for the backward direction) according to

$$\begin{gathered} PD\left(i^{\prime },j^{\prime },k^{\prime }\right)=D\left(i^{\prime },j^{\prime },k^{\prime }\right)+ \\ \underset{\begin{array}{c}m,n\in \{-1,0,1\}\\ \left(\text{if}\text{contributing}\right)\end{array}}{\max }PD\left(i^{\prime }+m,j^{\prime }\mp 1,k^{\prime }+n\right), \end{gathered}$$ 4

where ∓ denotes minus for Eqn. 1 and plus for Eqn. 2, and only neighbors $m$, $n\in \{-1,0,1\}$ within the search pyramid of Fig.3 are contributing.

The purpose of this accumulation scheme (Fig. 4) is to ensure that the $PD\text{s}$ are large when filaments pass through the origin $(i,j,k)$, while providing some robustness against noise because individual voxel densities are replaced with $PS\text{s}$ in the final filtering.

Note that in both directions, density is accumulated from the origin (initialized in Eqn. 3). The purpose of this bottom-up approach is to avoid recursion in dynamic programming. In the initial prototype of the algorithm, dynamic programming was performed in a rectangular box that encloses the search pyramid ($i-l\le i^{\prime }\le i+l$; $j\le j^{\prime }\le j+l$ in the forward direction or $j-l\le j^{\prime }\le j$ in the backward direction; $k-l\le k^{\prime }\le k+l$). Voxels outside the search pyramid within this box were initialized to a large negative number, so they would be ignored by the max operations (Eqn. 4) in case any of the nine neighbors (Fig. 4) were non-contributing.

While $FPD$ or $BPD$ on their own are each directionally biased, a centered path density $CPD(i,j,k;l)$ can be obtained by blending the $FPD$ and $BPD$ values. Our first intuition was to use the arithmetic or geometric mean for such a blending. However, we found empirically that the filament contrast can be enhanced further when requiring both $FPD$ and $BPD$ to be simultaneously of high value. The simplest way to implement such a logical conjunction is by multiplying the densities $FPD$ and $BPD$ (essentially, using the square of the geometric mean):

$$CPD(i,j,k;l)=FPD(i,j,k;l)\times BPD((i,j,k;l).$$ 5

To compare with the original densities, $CPD$ values are also normalized to a range from 0 to 1. Fig. 5A shows that the density levels of filament and noise in the original map were very similar, and distinguishing them was difficult. In the $CPD$ map (Fig. 5B), however, filament voxels (FVs) are clearly more discernible from the noise, and the shapes of the filaments are more apparent. Furthermore, curved filaments are enhanced, as we designed the filter to allow up to a 45° deviation from the dominant direction.

Fig. 5.

A single $Z$-slice (1 voxel or 0.947nm thick) of (A) the original simulated map (B) the $CPD$-filtered map obtained from Eqn. 5 with $l=10$ voxels.

B. Candidate Seed Point Placement

Seed points play a pivotal role in the detection of filaments, as they act as entry points that initiate tracing. Existing tracing approaches often rely on manually selected seed points [2]. However, because of the toilsome annotation process, manual selection of seeds is only feasible when few filaments are present in the image. For loosely bundled actin filaments in cellular components, this manual seed point selection becomes very inefficient (and it is also subjective and not reproducible). Thus, in this work, we propose a novel method for seed point placement based on a spatial decomposition of the volume that is both automated and robust.

The candidate seed point (CSP) selection step involves identifying local high-density points. The $CPD$ map of Section III-A is subdivided into 3D cubes of a user-defined size. The voxel with the highest density value in each of the 3D cubes is considered a seed candidate voxel. Here, we used CSP cubes with a side length of 10 voxels or 9.47nm, which is the minimum filament separation observed in actin bundles [2]. (Our CSP cube side length was identical to the path length $l$.)

Note that all these local high-density points are initially considered CSPs, but whether they are actually part of a filament is not yet known because the final traces will be determined later (see Sections III-D and E below). An automatic placement of seeds is typically quite challenging because of the low signal-to-noise ratio and missing wedge artifacts present in tomograms [1]. However, our local CSPs undergo another level of screening when they are validated globally by the final FSs, so the eventual placement of filaments is robust.

C. Candidate Filament Segment Generation

From each CSP, we trace a path of length $l$ (same length as in Section III-A) in the dominant forward direction ($+Y$ axis here) to generate a set of CFSs. The CFSs are fragments of filaments that will be fused later to form a complete FS. A CFS can be represented by $\left(\left(i_s,j_s,k_s\right)\to \left(i_e,j_e,k_e\right)\right)$, where $\left(i_s,j_s,k_s\right)$ is the start voxel (i.e., the CSP), $\left(i_e,j_e,k_e\right)$ is the end voxel (which is determined after tracing), and $l=j_e-j_s$.

The CFS generation step utilizes similar forward processing algorithms used in the filtering stage (Eqns. 1, 3, and 4). Specifically, we compute $FPD\left(i_s,j_s,k_s;l\right)$, and we also identify the voxel $\left(i_e,j_e,k_e\right)$ that maximizes the $PD$ according to Eqn. 1. The $FPD\left(i_s,j_s,k_s;l\right)$ has a range from 0 to $l$ (as the densities $D(i,j,k)$ were normalized). We divide $FPD$ by $l$ to obtain $NPD$ (normalized $FPD$) scores between 0 and 1. In this work, we set the length of $l$ to 10 voxels, considering the trade-off between the noise present in the image and the shape and distance characteristics of filaments. If the length of the CFS is too small, it may be affected by the presence of noise in the image. On the other hand, a very long CFS may not be able to follow the curvature of the actin filaments, as each CFS represents a straight line. Besides, a large $l$ increases the risk of inadvertently jumping across the gap between two parallel filaments.

D. Candidate Filament Screening

The CFSs are placed into 10 bins based on their $NPD$ scores. Each bin has a width of 0.1. A higher bin number represents CFSs with higher $NPD$ values (e.g., bin 10 contains CFSs with $NPD$ scores ranging from 0.9 to 1.0]). Fig 6 shows CFSs belonging to bins 9, 8, 7, 6, and 5. As shown in Fig 6, CFSs belonging to bins 6–9 are primarily true CFSs, whereas bin 5 introduces many non-true CFSs.

Fig. 6.

CFS sorted by $NPD$ value bins (see text). (A) bin 9. (B) bin 8. (C) bin 7. (D) bin 6. (E) bin 5.

To identify the bin that starts introducing non-true CFSs, which we refer to as threshold bin, we note that the location of the CFSs in the threshold bin spreads to the full volume (as seen in Fig. 6E), even into the noise outside the filament region. By iterating from high to low bins, we automatically detect at which bin value the CFSs are no longer localized and spread to the full volume. Here, we subdivided the tomogram into 100 × 100 × 100 voxel cubes, a level of detail that is intermediate between the fine CSP grid and the global map dimensions. If we find that at least 15% of these cubes contain less than 10 CFS mid-points, we deem that the CFSs do not yet occupy the entire volume, and we proceed to test the next lower bin. This way, the approach selects the top bins that represent mostly true CFSs. In our simulated tomogram, we observe that bin 5 is the threshold bin, so we deem bins 6 and above to represent mostly true CFSs.

The CFSs of the selected bins above the threshold bin are refined further based on backward retracing. To determine whether a CFS is a true one, we extract its endpoint and retrace it in the opposite directions of the original forward tracing used to detect the CFS. Differences in the path of both original and backward tracing are considered to exclude non-true CFSs. We note that the algorithm is sufficiently fast in processing all CFSs, so this test could be applied to filaments within the threshold bin and below (although we have not done this here to improve efficiency).

Furthermore, the approach automatically removes spatially isolated CFSs from the selected bins. Because of the presence of noise in the tomogram, detecting non-true CFSs with moderate to high $NPD$ scores (i.e., belonging to high bins) is occasionally possible. However, if they are indeed caused by erratic, local noise and not by nearby filaments (that are populated by many CFSs), such spurious CFSs can be detected and excluded by considering the mutual separation of CFSs in a local region (as shown in Fig. 7). Specifically, the mid-point of a CFS is computed, and then within a distance cutoff of 20 voxel radius from its mid-point, the number of other CFSs present (considering the Euclidean distance of their midpoints) is counted. If the number of these neighbor CFSs is less than three, the CFS is considered a false one and excluded accordingly.

Fig. 7.

Examples of isolated CFSs (inside blue circles) that can be removed considering the frequency of CFSs in the local neighborhood.

E. Individual Filament Generation

The filament generation phase uses multiple refinement steps to identify and exclude spurious CFSs (i.e., CFSs that do not belong to any filament). Finally, it fuses the large number of surviving true CFS fragments into fewer but longer individual filaments.

Connecting CFS by collinearity:

This step considers the collinearity between neighboring CFSs to join them. A pair of CFS that are collinear or nearly collinear (0 to 6° angle) and are very close (or connected) along the primary axis of the filament (separated by 0 to 10 voxels) are deemed to represent the same filament and are fused to create a single longer CFS.

Extending and fusing the CFS:

This step aims to fill any noise-induced gaps between the remaining CFSs (which represent the same filament) by extending them automatically and then using a flood fill algorithm to fuse them. Each CFS of length $l$ is gradually extended in the forward direction as follows. Initially, a segment of length $l$ is extended from the CFS. Then, whether this new segment has an $NPD$ score of at least that of the threshold bin is checked. If it has, then a new segment of length $l$ is created, and the same check is performed. This process continues until the $NPD$ value of the newly generated segment falls below that of the threshold bin or until a maximum length $5l$ is reached (this empirical cap on the filament length keeps the number of overlapping traces in check, which will need to be reduced in a subsequent step, but the exact multiplier of $l$ has little effect on the final results).

Flood fill is a well-known recursive algorithm for labeling nodes based on predefined criteria. Starting from a particular node, it iteratively searches for neighbors with the same criteria and gradually grows the labeled region until no adjacent node that fulfills the same criteria is found. Here, we use flood fill to fuse short CFSs belonging to the same individual filament in order to obtain the final long FSs. Specifically, we label all voxels on the CFSs that survived the previous steps as FVs. Then, we start the flood fill algorithm from one FV as a seed. Any other FV reached from the seed FV belongs to the same FS. This process is performed recursively and continues until no neighbor FV is found, which indicates the end of one FS. Then, we initiate flood fill again from another FV, which does not belong to any of the existing FSs yet,and follow the same procedure. This process continues until all the FVs are assigned to their final FSs.

Excluding overlapping filament segments:

This final refinement step excludes short FSs that overlap with a longer FS along the dominant axis of filaments. By discarding the spurious FS, this step can also help distinguish true filaments from noise artifacts. FSs that have more than 90% voxels in common with a longer FS are automatically discarded, as shown in Fig. 8. The final FS results are compared to the manual tracing in Fig. 9.

Fig. 8.

Examples of short CFSs (red) that overlap with one or more longer CFSs (green) and are therefore excluded.

Fig. 9.

Automatically detected FS (red; this work) overlaid with manually traced filaments (yellow, [4]).

IV. Results and Discussion

The persistence length of undecorated actin filaments is on the order of 10 micrometers, three orders of magnitude longer than $l=10$ voxels (9.47 nm). Nevertheless, some of our traces appear curved on the shorter scales when CFSs are joined. Although curved filaments on this scale have also been detected by manual tracing [4]), and there may be a biological interpretation, for example, because of the crosslinking of filaments, ruling out any artifacts introduced by the positional and directional granularity of our approach is important. (Given the short CFS length $l$ and the restriction of CFS end points to voxel positions on the 3D grid, our approach is limited to about 6° directional and 1 voxel = 0.947 nm positional granularity.)

To validate the proposed tracing framework, a statistical F1 score similar to that used by [7] is considered. The ground truth FVs (from a manual annotation [4]) is compared with the automatically traced filaments (in this work). True positive (TP), false positive (FP), and false negative (FN) voxels are defined as follows:

True Positive: A predicted FV is classified as a TP voxel if within a 3 × 3 × 3 voxel neighborhood, a true FV exists.

False Positive: A predicted FV is classified as an FP voxel if within a 3 × 3 × 3 neighborhood, no true FV exists.

False Negative: An FV in the ground truth map is considered an FN if within a 3×3×3 neighborhood, no voxel is predicted as an FV.

Based on the TP, FP, and FN values, the recall ($R$), precision ($P$), and F1 ($F1$) scores are calculated as follows:

$$R=\frac{TP}{TP+FN},$$ 6

$$P=\frac{TP}{TP+FP},$$ 7

$$F1=\frac{2\ast R\ast P}{R+P}.$$ 8

Table I shows the precision, recall, and F1 scores of the proposed framework when it is used in the simulated tomogram. As seen in Table I, the proposed framework provides a very high recall score of 0.95, which suggests that the framework identifies most of the filaments present in the simulated tomogram. The precision score is slightly lower than the recall score because of remaining FPs, as can be expected given the noisy nature of the tomogram. Overall, we achieved a high F1 score of 0.89. This is a high value for a density-based structure prediction. For comparison, in a recent state-of-the-art deep learning prediction of secondary structure features in cryo-electron microscopy maps, we achieved F1 scores of 0.72 for helices and 0.65 for beta sheets [8].

TABLE I.

Performance of the proposed framework for identifying actin filaments in the simulated tomogram.

Precision	Recalll	F1 score
0.83	0.95	0.89

In addition to its demonstrated accuracy, the proposed tracing (Sections III-B,C,D, and E) is also very efficient. On an Apple MacBook Pro with a 2.9 GHz Intel Core i7 processor, it took around 3 minutes to detect filaments in a simulated tomogram with a size of 283 × 664 × 269 voxels. The denoising of all voxels (Section III-A) in the pre-processing was slower and took about 7 hours. The tracing of the filaments is much more efficient than their denoising because of the coarse-grained selection of the CSPs (1 voxel out of 10 × 10 × 10 = 1,000 was selected, whereas the denoising was performed on all voxels). For denoising, each voxel is traced twice: once in the forward direction and once in the backward direction.

V. Summary and Conclusions

This study presents a fully automatic and fast method for tracing filaments in an actin bundle. As a welcome byproduct, the proposed approach provides a new bi-directional denoising filter to improve the visibility of actin filaments so that they can be detected better. We note that there is no physical meaning to the product of the two densities used in density filtering (Eqn. 5). When the filtered map $CPD$ is used in this manner (without any normalization), it is understood that the voxel intensities of the filtered map no longer correspond to the density of the biological specimen. The $CPD$ in this work is merely a heuristic score to ensure a logical conjunction (and gate); only if both $FPD$ and $BPD$ are large is the $CPD$ large as well. We initially envisioned an averaging of $FPD$ and $BPD$, but that would be similar to a logical disjunction (or gate) and would provide for a less discriminating filter. The key advantage (shown in Fig. 5) is that voxels representing the signal (i.e., the filament in the simulated data) carry a significantly higher $CPD$ when traced along the dominant direction (i.e., along $Y$ in this work). Even if a signal voxel originally held a low density because of the noisy nature of the cryo-ET map, any adjacent high-density voxels (in the dominant direction) help it obtain a high $CPD$ in the filtered map.

There are many existing denoising filters in tomography. Many are edge preserving [5], [9], [10], but generally, such filters make no assumptions about the shape of the biological structures. Our bi-directional filter is uniquely optimized for filaments that are mainly oriented in the dominant direction.

The proposed tracing algorithm is based on the same approach and assumes that filaments are aligned and bundled, although individual filaments may stray up to 45° from the main direction. This scenario often holds for cytoskeletal filaments imaged in cryo-ET. The main advantage of the current approach is its speed and applicability to dense filament bundles while still being able to follow individual curved filaments.

Although the tracing itself is up to three orders of magnitude faster than our earlier methods and only takes minutes (facilitated by the spatial coarse graining of the CSP), the denoising of all the voxels still takes several hours on a standard computer. We will explore multi-threaded or GPU-accelerated implementations in the future. We also note that our dynamic programming could be optimized further by shrinking the current rectangular computation box to the pyramid embedded in it (Fig. 3); this would require only slightly more than a third of the current number of voxels to be computed. Furthermore, we have not explored a memoization schedule to avoid repeat calculations of Eqn. 4. Memoization is widely used in dynamic programming, but it is not trivial here because of the overlap of the pyramid (Fig. 3) with the contributing voxel neighborhood (Fig. 4). The potential optimizations do call for a deeper refinement of our codes in future work (the pre-release source can be downloaded at https://situs.biomachina.org/fflavors.html ).