Unsupervised Trajectory Optimization for 3D Registration in Serial Section Electron Microscopy using Neural ODEs
Authors: Zhenbang Zhang
Jingtong Feng
Hongjia Li
Haythem El-Messiry
zhiqiang xu
Renmin Han
Venue: NeurIPS 2025 poster ()
Keywords: computer vision
biomedical image processing
ai for science
Primary area: machine_learning_for_sciences

Abstract: Series Section Electron Microscopy (ssEM) has emerged as a pivotal technology for deciphering nanoscale biological architectures. Three-dimensional (3D) registration is a critical step in ssEM, tasked with rectifying axial misalignments and nonlinear distortions introduced during serial sectioning. The core scientific challenge lies in achieving distortion mitigation without erasing the natural morphological deformations of biological tissues, thereby enabling faithful reconstruction of 3D ultrastructural organization. In this study, we present a paradigm-shifting optimization framework that rethinks 3D registration through the lens of manifold trajectory optimization. We propose the first continuous trajectory dynamics formulation for 3D registration and introduce a novel optimization strategy. Specifically, we introduce a dual optimization objective that inherently balances global trajectory smoothness with local structural preservation, while developing a solver that combines Gauss-Seidel iteration with Neural ODEs to systematically integrate biophysical priors with data-driven deformation compensation. A key strength of our method is its fully unsupervised training, which avoids reliance on ground truth and suits ssEM scenarios where annotations are difficult to obtain. Extensive experiments on multiple datasets spanning diverse tissue types demonstrate our method's superior performance in structural restoration accuracy and cross-tissue robustness.

Unsupervised Trajectory Optimization for 3D Registration in Serial Section Electron Microscopy using Neural ODEs
Zhenbang Zhang* MBZUAI: Shandong University 2hangzbebbang22211@maal
Jingtong Feng" Shandong University jagtongg4400ggaii.coom com
Hongjia Li
Haythem El-Messiry Canadian University Dubai haythem. elmessiry@cud ac
Beijing Institute of Technology lihongj 1a96egmai1....... com
Zhiqiang Xu MBZUAI
Renmin Han
Shandong University hanrenminesss edu.
zhiqiang. www.mzuuiiaccccommmmoc
Abstract
Series Section Electron Microscopy (ssEM) has emerged pivotal technol- ogy for deciphering nanoscale biological architectures. dinee-dimessionl (3D) registration is critical step in ssEM, tasked with rectifying axial misalignments and nonlinear distortions introduced during serial sectioning. The core scientific challenge lies in achieving distortion mitigation without erasing the natural morpho- logical deformations biological tissues, thereby enabling faithful reconstruction of 3D ultrastructural organization. this study, we present paradigm-shiftie optimization framework that rethinks 3D registration through the lens of mani- fold trajectory optimization. We propose the first continuous trajectory dynamics formulation for 3D registration and introduce novel optimization strategy. Specif- ically, we introduce a dual optimization objective that inherently balances global trajectory smoothness with local structural preservation, while developing solver ver that combines Gauss-Seided iteration with Neural ODEs to systematically inte- grate biophysical priors with data-driven deformation compensation. Extensive experiments on multiple datasets spanning diverse tissue types demonstrate our method's superior performance in structural restoration accuracy and cross-tissue robustness.
Introduction
Series Section Electron Microscopy (ssEM) has emerged as a powerful technology for nanoscale Thrree-dimensionl (3D) visualization of biological systems. Its impact spans diverse domains includ- ing neuroscience connectomics 84 79 developmental biology [19 43), and clinical diagnostics [29]. Recently, MICrONS project 13, utilizes ssEM as the fundamental method for neural circuit reconstruction. The ssEM workflow typically involves series of computational steps, in-
cluding two-dimensional (2D) stitching 21, 3D registration
[53 82, and 3D segmentation 44
"Co-First Authors.
'Coresponding Authors.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Previcus Mistho
Local Sequential
Rag Registration
Method
Global Global Trajectry V
Nony Imjjettcy
Smooothed Ingectrry
Teglocky toreston Ie Ie Dequatnttm
DIRCIING TINY
Figure Trajectory analysis and inspiration: Left: Previous methods is limited to adjacent registration, while our method achieves a paradigm shift by global trajectory optimization. Right: The upper-right demonstrates and noisy ID trajectories. The lower-right displays the side view of the original and distorted data. This inspires to explore the inner link between trajectory curvature mutation and nonlinear distortion from the perspective of biophysical motion.
comparisons between smoothed
Among these, 3D registration is particularly critical, as it corrects axial misalignments and nonlinear distortions, ultimately determining the accuracy and reliability of the final 3D structure.
The fundamental challenge stems from the topological entanglement of biological signal and technical noise specifically, the indistinguishability between natural cellular morphological deformation and nonlinear distortion induced by sample preparation [76] [5]] Furthermore, tfte-oofthhe-e equipment now generates TB-scale image data daily (77.. The massive data volume and error accumulation in long sequences (36] greatly complicate fast and accurate 3D registration.
82 However, these methods often fail to account for natural deformations between slices, and prioritizing pixel-level similarity may erase biologically meaningful changes, compromising the reconstructed structure. Recently, Zhang et al. [82] decouple natural deformation from nonlinear distortion by modeling them as low and high-frequency components. This insight motivated us to investigate the intrinsic characteristics of nonlinear distortion through the lens of biological motion coherence. We discovered that nonlinear distortion is essentially sudden change in local curvature. As shown in Figure b.)) anatomical structures inherently exhibit smooth manifold properties, maintaining evolutionary consistency in morphological changes. Mechanical sectioning, however, disrupts this continuity, introducing abrupt curvature mutations that contradict tissue biomechanics.
Several effective methods have been developed [56 80
76
Building upon this, we propose noval optimization framework, NeuroTrajReg, which redefines 3D registration as motion trajectory optimization problem. We establish a continuous trajectory dynamics formulation for ssEM, integrating spatiotemporal constraints effectively eliminate nonlinear distortions while preserving the natural evolution of tissue morphology. We introduce dual optimization objective that inherently balances global trajectory smoothness and local structural preservation, while developing a solver that combines Gauss-Seide iteration with Neural ODEs to systematically integrate biophysical priors with data-driven deformation compensation. Extensive experiments spanning diverse tissue types demonstrate Neuro'l Neuro'Trajie T'ajRRe'ss superior performance in structural restoration accuracy and robustness across different tissue types. The main contributions of this paper are summarized as follows:
We analyze the nonlinear distortions from the perspective of biological motion coherence. Based on this analysis, we redefine 3D registration a continuous manifold trajectory optimization problem and propose the first continuous trajectory dynamics formulation.
We propose novel framework that integrates biophysical priors with data-driven deforma- tion compensation, achieving a theory- guided yet data-corrected approach.
Extensive experiments demonstrate that NeuroTrajReg achieves superior performance in accurately and faithfully reconstructing biological 3D structures while maintaining cross- tissue robustness.

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2
16). closely related to 3D Related works
3D registration for series section electron microscopy. Several software packages have been developed for 3D registration [39 38, [66. Among them, TrakEM2 [56] is one of the most widely used tools, performing 3D registration by iteratively optimizing a spring-connected particle system. Recently deep learning-based methods have emerged as promising alternatives for 3D registration SEAMLeSS [53] improves robustness through vector voting and achieves global registration via combined attenuation transformations. Recently, Zhang et al. [82] proposed a Gaussian filter-based 3D registration method that approaches the problem from frequency-domen perspective.
80
86 42
Medical image registration. Medical image registration [85
48
24
registration, has achieved notable progress [85 categorized by their transformation models. Dense models
52
5
83
20
22
Existing methods are typically estimate voxel-wise mappings 70 58 [1] approximate
2
g
to form dense deformation fields, whereas interpolation- based models 30
deformations using basis functions (e.g., B-splines) over spatial grids. Despite their success, these approaches mainly address image pairs. In contrast, 3D registration demands handling nonlinear distortions while maintaining axial continuity.
Neural ordinary differential equations. Neural Ordinary Differential Equations (Neural ODEs) 6 integrate differential equation solvers into neural networks, enabling continuous representations with adaptive computation and enhanced parameter efficiency. Formally, given hidden state h(t), Neural ODEs defines its dynamics through a parameterized function fe:
dh(() = (b()))) dt
and the final state is obtained by solving h(4)) h((6) + f. so (o()(), t)dt. Numerous variants
of Neural ODEs have emerged to capture more complex transformation [57
75
49
35
27).
Neural ODEs have been widely applied across various fields [50 55 68]: Vid-ODE 50 enables
continuous-time video synthesis, NODEO (74] adapts it for deformable image registration, and Latent
ODE (55] models continuous- sequences.
Reference line smoothing algorithms. In autonomous driving 81 665. the reference line smooth- ing algorithm [12 60 17 geometrically optimizes global paths to generate smooth trajectories with curvature continuity, kinematic feasibility, and minimal deviation from the original reference. Its primary role involves eliminating curvature discontinuities between discrete waypoints, constraining maximum curvature within vehicle kinematic limits. The loss function is formulated as:
=
P 2p: + Pi+1l +1 + Σ lp. -.11 +
5(Krg S Kmax),
Smoothness
Fidelity
Constraints
where the first term penalizes curvature variation via second-order differeneing. the second enforces proximity to original waypoints q: with adaptive weight X, and the third applies indicator function 8(-) to enforce maximum curvature Kmaxx- Optimization strategies include convex quadratic programming (QP) for real-time computation, sequential QP (SQP) for nonlinear constraint handling [12 17
spline curves 772 [4], and mass-spring physical simulation [34 23
Proposed method: NeuroTrajReg
For the problem of 3D registration, we propose dynamics-based based nonlinear distortion correction framework for ssEM image sequences. Given a stack of microscopic images {I,] J affected by complex nonlinear distortions, our objective is to reconstruct a 3D structure that faithfully preserves the biological morphology by establishing a registration model with biophysical priors. The core challenge lies in dynamically balancing two conflicting demands: on the one hand, eliminating nonlinear spatial distortions induced by external factors, and on the other hand, preserving the intrinsic morphological characteristics of the biological specimen, such as tissue deformation and developmental topological changes.
3

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a Pipeline of our method
(b) Trajectory tracking module
Upladd tralectory
hnw.nlls---seed based
(Sinajerri freyy Irneesien
and Inxae regletration
Mi U212-1
fraddag
(T 1
20 Undt
Exajectoa
Trajectory tradiigg module
Strp/3
Uz-12
go
p())
Step.22
(c) The network architecture of fo
P(L)
JP(t--2
6242
Rped
dP
pltaes)
p(leea)
dt
Figure (a) The overall pipeline of NeuroTrajReg. Our approach can divided into three main components: 1) Pixel trajectory tracking, 2) Dynamics-based trajectory optimization, and 3) Trajectory inversion and image registration. (b) Illustration of the trajectory tracking module. (c) The network architecture of fo (Neural ODEs) in Synamicc-basse trajectory optimization module.
3.1 Problem formulation
In this section, introduce the concept of pixel-wise motion trajectory tracking and smoothing. Drawing inspiration from the classical Laplace equation 7 and the reference line smoothing al- gorithms [1]]. we formulate the 3D registration problem. Specifically, each pixel E R22 is treated as a particle within a continuum medium, whose displacement along the slice axis E [zo, -N- generates parameterized trajectory P. While this trajectory ideally follows natural morphological variations, nonlinear distortions induce local curvature abruptions. To address this, we establish spatiotemporal smoothness constraints on pixel trajectories, transforming traditional 3D registration into physically interpretable continuous trajectory optimization problem. NeuroTrajReg is grounded in three key principles: 1) physical motion coherence in biological tissues; 2) Laplace-basse trajectory smoothing; 3) spatiotemporal continuity microscopy imaging. We model pixel-wise motion from individual dynamics to global trajectory constraints with strict mathematical consistency.
continuous manifold trajectory optimization
Single-particle dynamics. Let P; denote the parameterized trajectory. We formulate the single- particle optimization problem as follows:
min
[[(((2) P! (z)1 +1
[V) )
Data fidelity
Laplacian smoothness
where (00) denotes the observed origin trajectory. 2 is the axial Laplacian operator, and À 0 controls the regularization strength. The Laplacian term enforces second-order temporal smoothness by penalizing acceleration discontinuities, effectively filtering biologically implausible instantaneous acceleration changes.
Holistic trajectory constraint. For image stack {Le N-11 we formulate the holistic optimization:
(. (3)
[V) us²
where M is the total number of trajectories, corresponding to the number of pixels in image 1,, and the spatial coherence term S, is defined as:
5 = Σ
(5)
JEN(()

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and weighted by > 0 for preserving local topologies within each slice, where (2) denotes the set of spatial neighbors of point within the same slice. This dual regularization strategy embeds two fundamental biological principles:
Temporal smoothness. It preserves smooth and stable tissue morphology evolution by constraining axial acceleration discontinuities through Laplacian regularization, preventing unreliable instantancous trajectory changes.
Spatial consistency. maintains the local topology between slice pixels by ensuring the consistency of trajectory smoothing for neighboring pixels.
3.2 Trajectory optimization for 3D registration
Our approach, as shown in Figure Aa). mainly consists of three components: 1) pixel trajectory tracking. 2) dynamics-based trajectory optimization, and 3) trajectory inversion and image registration. Specifically, we perform trajectory tracking for each pixel the image stack. Then, we apply our proposed trajectory optimization algorithm to obtain smooth pixel trajectories. Afterward, we compute the trajectory displacements and invert them back to the pixel grid to generate deformation field, which is then used to register the image stack. We will now elaborate the implementation and contributions of each component.
Pixel trajectory tracking. Given image stack {1=) N-11 E IXNXIIHXW our goal is to track the trajectory of each pixel over time, accounting for subpixel displacements. We use the network shown in Figure b)) for estimating the displacement field 1g-1.a E Hx x2 in x-axis and y-axis between adjacent slices. To model the temporal trajectory of every pixel, we introduce trajectory volume PERNX NxHxxx2 x2, where (2)) stores the spatial coordinate in slice 2. This trajectory is initialized P E
P(O, v. = (v²,*") xx)))
indicating that the trajectory of each pixel starts from its position in the first slice. Here, P(z, v, denotes the displacement vector index (y. x)) in slice 2.
We recursively propagate the position using the displacement field from the previous slice. But the pixel location (p - 1, y, x) may not be aligned with the discrete grid of M.-1.2, accumulated displacements often lead to non-integer coordinates. As result, bilinear interpolation Ug-1,2 is
required to accurately estimate the flow at subpixel positions. Formally, the trajectory propagation is governed by the transport equation:
P(z, y, 2) P(z 1, + u..1, 1, 1,,,
where uz- (-) denotes the interpolated displacement field.
Dynamics- based trajectory optimization. According to Eq. 33 we model trajectory smoothing as the weighted minimization of discrete eurvature energy and deviations from the original trajectory. To solve this optimization objective, we employ the Gauss-Seidel method [18] to iteratively smooth the trajectory. The Seisel method is a classical algorithm for solving linear systems. Its key characteristic lies in accelerating convergence by immediately incorporating the most recently updated values. The corresponding Gauss Seidel iteration is given by:
p(++))(2)= 1((+1)(2 (z - 1) p(h) (2 + 1) + XP(() (2)
z= 1 N 1
(8)
2 + X
with boundary values fixed as (D() = (0) and (( p
p(0) (N 1). In each iteration
k, the internal positions z are updated from the to the N-1, where each p(++1) (z) is computed
using the most recent value of its predecessor () (2-1) and the previous value of its successor (2)) A detailed derivation provided in the supplementary material.
While the Gauss- method provides theoretically grounded approach for solving linear sys- tems through explicit point-wise updates with guaranteed convergence, practical effectiveness inherently limited by discrepancies between idealized theoretical models and real-world scenarios. Contrastingly, Neural ODEs (6] exeel in learning continuous dynamical systems through implicit dynamic modeling, effectively capturing complex nonlinear deviations.
5

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Given the exceptional capability of Neural ODEs in modeling continuous dynamic systems [69 31 46
we use Neural ODEs to reformulate trajectory smoothing as the dynamic evolution of continuous trajectory. The network architecture of so in Neural ODEs is shown in Figure 2c). The
trajectory constraints are enforced through integration of velocity field V = =v== {
N-1 satisfying:
D dp(()) di fo (p()).t)) p(fe+t)= = p((t)) + 1 Dit
where So is neural network with trainable parameters 0.
Inspired by Universal Differential Equations (UDEs) [54],
propose a dynamically adaptive
optimization system that synergistically combines the explicit iterative convergence of Gauss-Seidel with the implicit dynamic modeling capabilities of Neural ODEs:
p(tx) (1 a)
((P(to))
+a ODESolver fo. p(ta), [to. tN]]
(10)
Gauss-Scidel Sere
Neural compensator
where ((-) denotes the Gauss-Seidel operator, which takes the initial trajectory at time ty as input and outputs smoothed trajectory at time (x. ODESolver[-] represents the neural dynamic correction over time interval [4o, Ax]. The adaptive weight a E [0, evolves through:
(3
A (fx)
a = B.
(11)
A (to)
where (-) denotes the sigmoid activation ensuring smooth transitions, / controls the adaptation rate sensitivity, and A (1x) = [[((p(())) - p((+)] quantifies theoretical operator progress.
Therefore, our architecture follows a theory-guided, data-corrected approach, incorporating an adaptive blending module that automatically balances theoretical components and corrective terms. It takes ad advantage of Gauss-Seide for rapid initial convergence, ensuring numerical stability and accelerating optimization. Neural ODEs are then employed to capture higher-order nonlinear effects and compensate for the gaps between the model and reality.
Trajectory inversion and image registration. After trajectory optimization, we obtain the resulting trajectories p(tw) E x x2, where each element (((t)) (z) E IR² represents the evolved
coordinate of the i-th trajectory p(tN) slice Based on this, the displacement NxHxx2xx2 can be computed by comparing with the observed original trajectory p(U);
p(tx) (()).
(12)
The displacement & encodes the offset information of p(tx). However, is important to note that this does not exactly correspond to the displacement of pixel grid points, as the coordinates in p(tx) are not necessarily integers. Therefore, inspired by the surface splatting [3] in computer graphics, we propose an efficient bilinear splatting to invert the trajectories and obtain accurate displacements pixel grid locations (detailed in the supplementary material):
(- In. (P((t)) +),
(13)
where Ths means bilinear splatting and ( N E R--xHxW* represents the deformation fields
corresponding the image stack N-1 -1 These deformation fields are subsequently used to register the image stack:
I. = t. I2, 2 == 0,...., where denotes the warping operation.
(14)
3.3
Loss functions
The trajectory tracking network is optimized using a composite unsupervised loss function comprising normalized cross-correlation (NCC) data fidelity term and a diffusion regularizer applied to the displacement fields, with balancing coefficient The loss function is formally expressed as:
Lusj NCC (L2, I--1 1221 + Al V
(15)

---

where indicates the spatial warping operator, and V computes first-order spatial gradients via finite difference approximation.
According to Eq. (4, the Neural ODEs training objective is formulated as a composite unsupervised loss function comprising three components:
M
PN
2N-1
NN-1
C
(P((2) P (
(2)1
E (v
+
Σ Σ (P((2)) (P())))
2g
A (EN((
Data fidelity term
Teajectory smoothness temm
Sputial consistency
(16)
where P(()) denotes the observed origin trajectory. The trajectory smoothness term, weighted by À >> 0, penalizes acceleration discontinuities through the temporal Laplacian operator V discretized with second-order differences. The spatial consistency term, weighted by μ > preserves local topological relationships within each slice.
4 Experiments
4.1
Datasets
We evaluate our algorithm on six publicly available datasets from the OpenOrganelle platform 77]. covering a variety of mouse tissues such as heart, kidney, liver, skin, and pancreas. These allow for comprehensive evaluation NeuroTrajReg's applicability and robustness across diverse tissue types. For real-world data, we utilized the female fruit fly brain neural dataset (FemFlyBrain) [63]. More details can be found in the supplementary material.
4.2 Implementation details
We implement our method using PyTorch, and all experiments are conducted on an NVIDIA A800 GPU with 80GB of memory. For the pixel trajectory tracking, we employ 2D U-Net architecture to capture the displacements between adjacent slices. Neural ODEs employ 3D U-Net architecture to parameterize So. The network takes pixel trajectories P as input and generates time-dependent velocity field More details can be found in the supplementary material.
4.3 Results
In Table 1 we compare NeuroTrajReg with advanced 3D registration techniques, including EMReg [4). EFSR (76), TrakEM2 [56]. SEMLeSS [53] and GaussReg [82], using synthetic datasets spanning six diverse tissue types. Our approach, along with the supervised GaussReg method, demonstrates superior performance due to consideration of natural deformations, which other methods typically overlook. Methods overly focused on pixel-level similarity can neglect biologically relevant movements such as cellular dynamics, as shown in Figure 7 Moreover, compared to the supervised- manner GaussReg, our method uniquely benefits from global trajectory optimization, enabling the computation of a single interpolation deformation across slices. Conversely. GaussReg is constrained by local receptive field, necessitating multiple interpolation steps. For the performance on real- world datasets, we selected several volumetric samples from the FemFlyBrain dataset [63). The registration results are visualized Figure where we employed the interpolation and visualization tools provided by FIJI (59] to display the side views of the image stacks. Although GaussReg reduces nonlinear distortions, it also crroneously removes some structural textures, leading to a noticcable deviation from the original data. In contrast, our method not only effectively corrects the nonlinear distortions in the image stacks but also better preserves local structural details.
To further evaluate our approach, Figure 3 compares the error accumulation across approximately 1000 slices for SEMLeSS (53) GaussReg [82]. and our method. SEMLeSS suffers significantly from етrоr accumulation over long sequences. Although GaussReg performs reasonably well, it exhibits lower precision and higher variance compared to our method, indicating less stable performance. Our method achieves superior average SSIM accuracy and demonstrates substantially greater stability maintaining consistent performance without noticeable degradation even after 1000 slices. Additional results are provided in the supplementary material.
7

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Table 1: The performance of different 3D registration methods six synthetic datasets 77..
Mus Heart Mus Kidney
Mus Liver-3
Mus Liver
Mus Pancreas
Mus Skin
MI
0.73
+ 0.14
0.65
0.16
0.89 + 0.17
0.79
+ 0.18
0.76 # 0.14
0.66 + 0.18
EMReg 47
SSIM
0.55
0.03
0,44
0.52 0.4
0.43
0.3
0.43 0.02
0.50
NCC
0.90
+: 0.03
0.81
+ 0,05
0.86
# 0.05
0.78
# 0.08
0.81
+ 0.03
0.87 0.04
MI
1.15
+:
1.26
1.45
1.36
1.33
1.03
EFSR 76
SSIM
0.67
= 0.94
0.63 0.05
0.65 0.04
0.56 = 0.04
0.57
+ 0.04
0.60 0.07
NCC
0.98
+ 0:00
0.97
0.97
0.95
0.95 + 0.01
0.97
MI
1.01
=: 0.03
1.24 0.10
1,48 0,07
1.47 0.08
1,48 +: 0.06
1 1,08 0.15
TrakEM2 [6
SSIM
0.57
+ 0:02
0.61
+ 0.03
0.66 # 0.33
0.61 + 0.08
0.65 + 0:03
0.64 0.06
NCC
0.98
0.00
0.97 0.00
0.98 0.00
0.97 0.00
0.97 0.00
0.98
MI
1.00
= 0.12
1.07
= 0.14
1.35 = 0.17
1.29 0.19
1.18 + 0.15
0.89 1 0.18
SEMLeSS [53
SSIM
0.58
0.52
0.61
0.52
0.50 0.05
0.51 1
NCC
0.97
+: 0.01
0,96
# 0.01
0.97 # 0.01
0.95 # 0.01
0.95 +: 0.01
0,96 0.01
MI
1.43
+ 0:05
1.54
1.83 0,77
1.83
1.76 + 0.06
1 1.40
GaussReg 82
SSIM
0.83
+ 0.02
0.78
0.03
0.82
0.02
0,80 0.02
0.80 + 0.02
0.82 0.08
NCC
0.99
+ 0.00
0.99
0.00
0.99
1: 0.00
0.99 0.00
0.99 0.00
0.99 0.00
1.47
+ 0.08
1.67
= 0,14
1.88
:: 0.11
1.85
:: 0.21
1.81 + 0.12
1.52 + 0.17
Ours
SSIM
0.87
0.85
0.87
0.84
0.86
0.86
NCC
0.99
0.00
0.99 0.00
0.99 0.00
0.99 0.00
0.99 + 0.00
0.99 0.00
SSIM
0.90
0.85
0.80
Ours GaussReg
0.75
0.70
SEMLeSS
0.65
0.60
0.55
200
400
600
800
1000
Slice Number
Figure 3: Error accumulation comparison Mus Liver-3 dataset.
4.4
Ablation study
Loss Terms. We tested the impact of different components of the loss function (Eq, 16 on the results. The results are shown in Table [] where TS, DF, and SC correspond to the Trajectory Smoothness term, Data Fidelity term, and Spatial Consistency term, respectively. Removing the Data Fidelity term decreases SSIM, as this leads to the trajectory deviating excessively from the original, with a risk of over-smoothing. Removing the Spatial Consistency term results in a decrease in SSIM, as failing to preserve local topology can lead to local pixel distortions. As for the TS term, it is key to trajectory smoothing, as it imposes constraints on the trajectory's 's acceleration, preventing unreasonable instantancous variations. In conclusion, all components are essential and work together to ensure optimal performance.
Hyperparameters. We tested the impact of different hyperparameters À and in Eq. 16 on registration accuracy. À controls the smoothness of the trajectory to filter unreasonable instanta- neous velocity changes, while regulates the local consistency of the displacement field to ensure topological consistency before and after registration. performance; if it is too small, it fails to eliminate noise completely, while if it is too large, it may cause excessive deviation of the trajectory. In contrast, μ is more robust and has a lesser effect on registration performance.
has an important impact on registration

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GT
Neter
EMRrg
FFSR
Mouse Kidney
SEAML
TrakEM2
GuussReg
OLRS
EMRey
Mouse Heart
SEAMLeSS
Trakt:Me
GazeRay
Ours
Figure 4: 3D visualization of registration results on the Mus Kidney and Mus Heart datasets.
Table 2: The ablation study of different loss terms.
Table 3: The ablation study of different hyper- parameter in the loss.
TS Loes term
DF Loss team
Loes term
SSIM
SSIM
A/p
0.0001
0.001
0.01
0.1
x
0,001
0.775
0.782
0.771
0.762
Heart
x 0.855
Mus Heart
0.01
0.795
0.806
0.783
0.779
0.793
0.1
0.852
0.872
0.841
0.822
0.82
0.811
0.819
0.805
0.787
x
0.762
0.752
Kidncy
x 0.822
Mus Kidney
0.781
0.794
0.772
0.766
0.774
0.1
0.833
0.851
0.821
0.815
0851
0.768
0.773
0.758
0.743
Table 4: Ablation study the number of iterations in the Gauss-Seidel method.
SSIMAtteaatins
50
100
150
200
250
300
0.719
0.874
Mus Kidney
0.713
0.796
0.828
0.851
0.851
0.851
Liver
0.774
Mus Skin
0.682
0.788
0.842
0.864
0.868
0.866
Number of Iterations. Table 4 investigates the effect of varying the number of iterations in the Gauss-Seidel method. From to 200 iterations, the registration performance improves significantly and then gradually converges. Considering the trade-off between accuracy and efficiency, we select 200 iterations in practice achieve good balance.
Module Analysis. Our method can be decomposed into two main modules: the Gauss-Seidel method and Neural ODEs. The Seussl method achieves fast convergence through iterative updates, while Neural ODEs effectively capture complex nonlinear distortions by modeling implicit dynamics. As shown in Table[3] we analyze the contribution of these key components in the trajectory smoothing scheme, and the results demonstrate the necessity and effectiveness of both modules.
Different Solver. Table O further studies the effect of different solvers for Neural ODEs. To avoid additional computational overhead, we adopt the Fuler solver to balance performance and computational cost. More ablation experiments are provided in the supplemenatry material.

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Orgin Data
GaussReg
Ours
Figure 5: Side views of the original data, the registration results of GaussReg 82 on two sampled volumes from the FemFlyBrain dataset [63
and our method
Table 5: Ablation study of the components our method.
Table 6: Ablation study of different solvers for Neural ODEs.
Gauss-Seide Seidel
Neural ODEs
SSIM
Fuler
Dopriš
RK4
x
0.847
Time
SSIM
Time
SSIM
Time
SSIM
Mus Heart
v
0.835
Mus Heart
2.45
0.872 8.33 0.877
7.68
0.876
0.872
Mus Kidney
2.41
0.851
8.52
0.855
7.66
0.854
x 0.833
Mus Kidney
0.827
Mus Liver
2.5
0,842
8.71
0.848
7.72
0.847
0.851
Mus Skin
2.46
0.864
8.39
0.869
7.25
0.868
5 Conclusion
In this paper, we present noval 3D registration method for series section electron microscopy. We analyze the inherent characteristics of nonlinear distortion from the perspective of biological motion coherence, and reconsider 3D registration from the viewpoint trajectory optimization. We introduce dual optimization objective that balances global trajectory smoothness and local structural preservation, and we develop solver that combines Gauss- iteration with Neural ODEs. Extensive experiments demonstrate that method excels in structural restoration accuracy and tross-tissue robustness.
6 Acknowledgements
This research was supported by the National Key Research and Development Program of China (2021YFFON) YFF0774300) Dubai Future Foundation (Award No. 2024CANAD-MES-061] and the Natural Science Foundation of Shandong Province [ZR2023YQ057].
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Appendix
A1
Dataset Details and Metrics
A1.1 Dataset Details
We selected six publicly available datasets from the OpenOrganelle platform 77 for simulation experiments. These datasets encompass high-resolution electron microscopy images of various mouse tissues, including the heart, kidney, liver, skin, and pancreas. The availability of ground-truth annotations in these datasets provides strong support for validating the applicability and robustness of our method across different types biological tissues. For real-world data, we utilized the female fruit fly brain neural dataset (FemFlyBrain) [63]. These extensive datasets enabled us to validate the robustness of our method against real-world data.
P7 Mouse Heart. This dataset consists of heart tissue extracted from a wild-type C57BL/6J mouse at postnatal day 7. The experimental procedure followed IACUC guidelines for animal anesthesia. The tissue was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. Subsequently, the sample underwent low-temperature reducing OTO staining, dehydration through a graded ethanol series, infiltration with Durcupan resin, and polymerization in an oven at 60"C.
Mouse Kidney. This dataset consists of kidney tissue extracted from an 8-week-old wild-type C57BL/6 mouse. The tissue was perfused with glutaraldchyde fixative and sectioned using vibratome, After staining with reducing OTO at room temperature, the sample underwent dehydration, infiltration with a graded ethanol series, and Durcupan resin. Finally, the sample was polymerized in an oven 60CC
P7 Mouse Liver. This dataset comprises liver tissue collected from a wild-type, postnatal day 7 C57BL/6J mouse. Following anesthesia in accordance with IACUC guidelines, the sample was perfused with glutaraldehyde fixative and sectioned using a vibratome. After low-temperature reducing OTO staining, the tissue underwent dehydration, infiltration with graded ethanol and Durcupan resin, and was subsequently polymerized in a 60 oven.
This dataset contains liver tissue from wild-type, 8-weck-old C57BL/6 mouse. The sample was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. Subsequently, reducing OTO staining was performed at room temperature. The sample was then dehydrated, infiltrated with graded ethanol and Durcupan resin, and polymerized a 60%C oven.
Mouse Liver.
P7 Mouse Skin. This dataset includes skin tissue from a wild-type, postnatal day 7 C57BL/6] mouse. Mouse anesthesia was performed in accordance with IACUC guidelines, followed by perfusion fixation using glutaraldehyde and vibratome sectioning. The sample underwent low-temperature reducing OTO staining, dehydration, infiltration with graded ethanol and Dureupan resin, and was polymerized in 60"C oven.
P7 Mouse Pancreas. This dataset consists of pancreas tissue from a wild-type, postnatal day C57BL/6 mouse. The sample was fixed via perfusion with glutaraldehyde and sectioned using a vibratome. It underwent low-temper reducing OTO staining, followed by dehydration, infiltration with graded ethanol and Durcupan resin, and polymerization in 60CC oven.
[53] consists of the right hemisphere of wild-type Oregon R female fruit fly brain. The brain was continuously sectioned at a thickness of 40 nanometers, covering regions including the medulla and downstream neuropils. The sections were imaged at magnification of 35,000. The connectome within the medulla includes 379 neurons and 8,637 chemical synaptic contacts.
Female Fruit Fly Brain Neural Dataset. FemFlyBrain
Full Adult Fly Brain Dataset. Full Adult Fly Brain Dataset (FAFB) 34 covers the entire depth of adult fruit fly brain (approximately 250 μu)) From the optimized sample, 7,062 consecutive sections of about 40 nm thickness were collected, optimized for high membrane contrast and fine ultrastructural preservation. A total of 7, 7,050 sections (99.8%) were successfully imaged.
Mouse Cortical Dataset. The Mouse Cortical Dataset 32 provides saturated reconstruction of mouse neocortical sub-volume, in which all cellular elements (axons, dendrites, and glia) and various subcellular components (synapses, vesicles, spines, postsynaptic densities, and mitochondria) are fully annotated.
23

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Table 7: Details of the synthetic datasets for testing phase.
Datasets
Shape
Numbers
URL
P7 Mouse Heart
1184x1184
1000
https///opooogaaallllllllllaaaas mus-heart-
Mouse Kidney
1184x1184
1000
http://oopppuaaaaallllluraaaoooc mus--idneyy
P7 Mouse Liver
1184x1184
1127
Mouse Liver
1184x1184
558
http:///gponngaaaallliiiillooooo lasps//jjjj_____iiii liver
P7 Mouse Pancreas
1184x1184
898
P7 Mouse Skin
1184x1184
1231
https///ppeoogaaeelleeeeellllolp http:///po....tatttttoooaaaaoaaa mus-skin-1
FemFlyBrain
1024x1024
1299
FAFB
1024x1024
700
https//temmcaallataaaryy
Mouse Cortex
1024x1024
700
http://meemmdaaaiiidddaaaasstutu
A1.2 Metrics
To quantitatively evaluate the performance of our method, we calculated several metrics, including Normalized
(NCC), Mutual Information (MI), and Structural Similarity Index (SSIM) between the registrated image series and the ground truth (GT). These metries provide comprehensive evaluation of the accuracy of our alignment and registration results from different perspectives. Specifically, NCC measures the similarity between the result images and the GT, MI reflects the mutual information between the two, and SSIM evaluates the accuracy based on structural information. Dice is set similarity metric commonly used to measure the similarity between two samples. Here, we use the Dice score to evaluate the segmentation accuracy of 3D segmentation results on the registered data.
A2 Experimental Details
A2.1 Dataset Setup
For training the trajectory tracking module, we split the original image data into 3000 image pairs (To, J. where In and /i represent adjacent images. These images are cropped to size of 1184 x 1184. During training, the data is normalized to the range of [-1, 1]. The training-validation split is set to 0.95. Additionally, random clastic deformation to simulate nonlinear distortions. Specifically, we first generate a random deformation field Drand, which indicates the pixel displacement matrix. This matrix is then smoothed using a Gaussian filter, and the resulting deformation is applied to create the deformed images. The deformation process is described by the following formulas:
randomly select one of the images from {1o, 41 and apply
{ Φ a size(L:) Gauss Drand. a size(L4)), 1:
(17)
di li,
where o; represents the random deformation field applied to image 4, o denotes the deformation operation, Drand is the generated random displacement field, Gauss(-) is the 2D Gaussian filter operator, a controls the displacement magnitude, and a determines the smoothing extent. In our experiments, we set 0.08 and 1.0.
During the testing phase, we cropped all six datasets into image stacks size 1184 x 1184 x N. where N denotes the number of slices. Additionally, we applied random elastic deformations to all images except the first one to simulate nonlinear distortions. The deformation parameters were set to 0.08 and a 1.0. Detailed information regarding the datasets can be found in Table 7 Our test data consists of several hundreds to around 1000 samples, which fully demonstrates our method's capability in handling long sequences and addressing the challenge of error accumulation.
24

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A2.2 Network Architecture and Training Details
The trajectory tracking network adopts a 2D U-Net architecture with residual multi-kernel fusion, consisting of 1) a cascaded feature encoder with hybrid convolutional blocks, and 2) a dense feature decoder with transposed upsampling. The encoder progressively reduces spatial resolution through four strided convolutions (stride=2), doubling the number of channels from an initial 8 to 64. Specifically, the encoder is composed of four convolutional layers with output channels of 8, 16, 32, and 64, respectively. Each convolutional layer includes two convolution operations: the first convolution uses a 3 x 3 kernel, followed by a second convolution with a larger 7 X 7 kernel to expand the receptive field, similar to the approach in LKUnet (28). The decoder consists of four layers and reconstructs dense predictions using four transposed convolutions (kernel size = 2, stride = 2), halving the number of channels from 64 to 2. Skip connections concatenate the encoder features with the corresponding decoder layers to facilitate hierarchical feature fusion. The final displacement field is generated through dual 3 x 3 convolutions with a Softsign activation. To enable spatial warping. Spatial Transformer Network (STN) 25 is employed for image registration. The model is optimized using the ADAM optimizer with parameters 81 0.5 and B2 0.999. The learning rate is set to 1 x 10 and training is performed for 500 epochs with a batch size of 16. For the regularization term in the loss function, we set À 1.5 encourage smoothness in displacement field.
Neural ODEs employ a 3D U-Net architecture to parameterize fa- The number of network layers and the number of channels per layer are kept consistent with the 2D U-Net used in the trajectory tracking module, but large convolutional kernels are avoided to reduce memory consumption. For training, use the ADAM optimizer with parameters 81 = 0.5 and B2 = 0.999. The learning rate is set to 1 x 10-1, and training is conducted for 500 epochs with a batch size of 4. For the loss function, we set À = 0.1 and , 0.005. For the Gauss-Seidel method used in trajectory smoothing, we set the number of iterations to 200 and À = 0.1. For Neural ODEs, we use the simple Euler solver, with all other parameters kept at their default settings.
A2.3 Training and Testing on Real Data
A key advantage of our method lies in its unsupervised learning paradigm, which is particularly valuable for real-world datasets that lack reliable ground-truth correspondences. The training process on real data is divided into two stages: training of the trajectory tracking module and training of the trajectory smoothing module. We begin by cropping a small spatiotemporal block of size W T from the real dataset. Training samples are then constructed following the procedure described in Section | except that no additional elastic deformation is applied. The trajectory tracking network is subsequently trained in unsupervised manner, identical to the approach used for synthetic data. For training the trajectory smoothing module, we first apply the previously trained network to obtain estimated trajectories. These are then used for the unsupervised training of the smoothing module, again following the same procedure as used for synthetic data.
For testing on real data, the networks trained on cropped data blocks can be directly applied to the full image stack. We adopt sliding-window registration strategy to perform registration across the entire long image sequence. Specifically, given the full real image stack {I, N=- we define a receptive field of moderate size (about 100 slices). Within each windowed region, the trajectory tracking network is used to estimate the displacement trajectories. These trajectories are then refined by the trajectory smoothing module. Subsequently, inverse warping is applied based on the smoothed trajectories to perform image registration. The registered image block then placed back into its corresponding location in the full stack. The window is shifted along the sequence, and the procedure is repeated until the entire image stack has been registered.
A3
Supplementary Mathematical Derivations
A3.11 Background of the Gauss-Seidel Method
The Gauss-Seide method [8). classical iterative solver for linear systems Ax = b, that leverages the most recent updates to accelerate convergence, particularly in sparse, tridiagonal systems. This system can be equivalently written as a symmetric tridiagonal linear system:
Ap b,
(18)
25

---

where: p .P( P(N -1)"" is the unknown trajectory, b = (1), AN 1))²,- -ndd A E R(N-1)x(N-1) is tridiagonal matrix with structure:
(N
4 A -2
-2
4+ X
-2
A =
(19)
-2
4 4+
-2
-2
4+
Applying the Gauss-Seides Seidel method to this system, the general update rule for the i-th variable is:
(k+1) 5
1 Aa
b;
Σ
Agjay
(k+1)
(20)
J>1
For our tridiagonal matrix, each row z only has nonzero entries at z - 1, 2, and + Substituting these into the general Gauss-Seidel formula, the update rule for P(z) becomes:
(x) 12.:-1P(ktt) 1) A p(u) (2+1) ((k+1) A22
(21)
where: - -A2: + X, A3,--1 A2,2+1 = -2
Substituting these in yields:
((++1) 1) (=)= 4+
1
11 1) (=))
(22)
For simplicity, divide both numerator and denominator by 2, resulting in the final explicit Gauss-Seidel update:
(()+)()) (((+))(2 1) P()) + +
p (2)
2 = I - -11.
(23)
2+ A
A3.2 Derivation of the Gauss-Seidel Iteration Update Formula
For our trajectory smoothing method, the fidelity constraint. Below is the detailed derivation of the iteration formula. Specifically, minimize the energy function:
iteration can be adapted by introducing a
N-1 Σ (u
N-2 (i,-1
E(u)
(2)22
Σ
2u; + u4+1)2
(24)
take the partial derivative of E with respect each and set it obtain the optimality condition.
The fidelity term contributes derivative of:
a Ou;
(x())- (0)))== = 22(u; -n(()) (0)).
(25)
The smoothing term consists of sum of squared second-order differences, where each point u, appears in multiple overlapping terms, together with 14-1 and M:+1- For clarity, we focus on representative term where u; is the center, to illustrate how the smoothness constraint acts locally on the trajectory:
(u-1
2u, u2+1)² -( 2u; + 14+1).
(26)
ou;
Combining the derivatives both terms, we obtain the total gradient:
aF
2X(4: - ,,0)) 4(u4-1 2u; + u;-1) = 0.
(27)
Ou;
26

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Sample
Staliee Interpolation
Grid points
Scattered points
Scattered points
Grid points
(a) Grid Sample
(b) Bilinear Splatting
Figure 6: Regular grid sampling and Bilinear splatting. Grid sampling is used to obtain trajectory coordinates at sub-pixel resolution. For trajectory inversion, bilinear splatting is used to estimate the displacement at pixel grid locations.
Expanding and rearranging terms gives:
2X(u; u, ( - 4u; + S1; 4u441 0.
(28)
Solving for ui we derive the following linear equation:
(2X + 8)u; 2Au; (0)
+ 44--1 + 44++
(29)
This leads to the Gauss-Seidel iteration update formula:
(k+1)
22xu (0) + 4u + du;t1) 2A +
(6+1) (6)
(30)
Simplifying further, we obtain
(k+
1
(Au
(k+1) (k) (0)
+ u;-1 u(+1)
(31)
À +-2
where À is halved from its original value. We use the updated w
from the current iteration and the old u4 from the previous iteration update (k+1)
(k-1)
(k)
A4 Algorithmic Details of Bilinear Splatting
The goal of bilinear splatting is to map discrete feature points onto a regular grid using bilinear interpolation. Given a set of normalized coordinates and their associated feature vectors, we need to compute the grid values using the surrounding grid points. The coordinates are normalized to the range [o, H) and [0, W). where H and W are the height and width of the target grid. The feature vectors are associated with these coordinates and are projected to the grid using bilinear interpolation, as shown in Figure 6
Let the coordinates of the discrete points be represented as coords (111,1)),... (yN, F.v }. where each (ys, 224) is a 2D coordinate within the normalized range. The corresponding feature vectors for each point are denoted by values = (FI...., uN where E RC is the feature vector for point and C is the feature dimension.
To perform bilinear splatting, we begin by identifying the four nearest grid points that surround each continuous coordinate (Wi, These grid points form the corners of the smallest axis-aligned square in the discrete grid that encloses the given coordinate. Specifically, we obtain them by applying the floor and ceiling operations to both the y- and z-coordinates:
{ YAooor [ya],
TBoor [x;]] [vt] + 1, Feeil = [x, +1
(32)
Veeill
These points form the basis for distributing the value (yr, 2:)) across the surrounding pixels based on their bilinear interpolation weights.
27

---

Next, the weights for bilinear splatting are calculated based on the fractional parts of the coordinates.
Let Ay = y/ L3b1 and Ar 2, [æj] represent the fractional parts of the coordinates. The four interpolation weights are given by:
wi - Az) - Ay), w2 = Ax)Ay, Az(1 - Ay). W4 ArAy.
(33)
These weights represent the contribution of each neighboring grid point to the interpolated value at the target coordinate (yi, z;). Now, we distribute the feature values to the corresponding grid locations based these weights. For each coordinate (y;, update the feature grid by adding
the weighted value the point to the grid locations corresponding to the four neighboring points. This is done using the following scatter operation:
Grid Filoor]
= Grid Uitoors Fooor
+ Uj W1,
Grid[ycel Fioorr
Griddyeeit Thoor
U; U22
(34)
Grid Yiooor Feeil]
Grid Tesil
+ Uj W/3,
Grid[yeeit Freit
Grid GriddUeeit Meeit Teeil B3 W4-
The grid values are accumulated the corresponding grid positions, and the sum of the weights at each grid location is also accumulated for normalization purposes.
Finally, ensure proper averaging when multiple points contribute to the same grid position, normalize the resulting grid by the sum of the weights:
Ee 0/ Wi
Grid y, x]]
(35)
Σ 10
To avoid division by zero in case of contribution grid location, small epsilon 1e is added to the denominator during the normalization step. Thus, the bilinear splatting operation effi- ciently interpolates the feature values from discrete points to the target grid, with smooth interpolation properties ensured by the bilinear weights.
A5 Limitations and Future Work
Our approach can be divided into three main modules: 1) pixel trajectory tracking, 2) dynamics- based trajectory optimization, and 3) trajectory inversion and image registration. For each module, we conducted a simple exploration in the ablation study (see See B.Il evaluate its effect on registration performance, including the influence of complex scenarios such as high-noise artifacts and wrinkles that may affect snrecctor-ttrccking accuracy. Future work may investigate 3D registration methods that are robust to artifacts and wrinkles, well as explore the use of more advanced architectures architectures for improving registration accuracy and generalizability. Furthermore, Table [T] presents the registration accuracy and execution time on the Mus Liver-3 dataset, where the test slice resolution set 1024 1024. Our method achieves the highest SSIM, demonstrating superior accuracy compared to other approaches. However, due to the computational demands of the ODE solver, it does not achieve the fastest runtime.
To further investigate runtime efficiency, we conducted additional experiments focusing on two optimization strategies: (i) sparse trajectory sampling (every 2/3/4 pixels); and (ii) resolution reduction by downsampling followed by upsampling. Table T shows that sparse sampling substantially reduces runtime (from 2.46s 0.95s) while maintaining ofate-of-the-ate accuracy, with SSIM only slightly decreasing from 0.874 to 0.831. Table I0 demonstrates that reducing the resolution also improves efficiency (from 2.46s to 0.91s), though at the cost of accuracy (SSIM drops from 0.874 to 0.827). Nevertheless, the performance remains superior to the GaussReg baseline. We attribute the observed degradation primarily to the use of simple bilinear upsampling strategy. We argue that could achieve more favorable balance between runtime and accuracy.
28

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Table Execution time and performance.
TrakEM2 [56
EFSR 76
EMReg 42
SEMLeSS [53] GaussReg 82 Ours
Time(s) SSIM
3,92 0.66
3.44
0.28 0.65
27.11
0.92
2.46
0.52
0.61
0.82
0.87
Table 9: Ablation study of sparse sampling.
Table 10: Ablation study of resolution reduction.
All pixels 1/2 pixel 1/3 pixel 1/4 pixel
Full resolution 1/2 resolution 1/4 resolution
SSIM 0.874
0.866
0.842
0.831
SSIM Time(s)
0.874
0.845
0.827
Time(s) 2.46
1.35
1.12
0.95
2.46
1.12
0.91
A6 Broader Impacts and Discussion
Our work focuses on 3D registration for Series Section Electron Microscopy (ssEM). One of the key advantages of our approach is its fully unsupervised training, which eliminates the reliance on ground truth and makes it applicable to real-world ssEM scenarios. For ssEM applications 79 13] [5]. this is particularly important for the increasingly complex and diverse electron microscopy image datasets [15 77), as acquiring comprehensive ground truth across various tissue and cell types is challenging. Our method can be applied to the modeling and analysis of biological cell tissues, assisting researchers in exploring the structure and function of cellular tissues.
47). our method provides well-structured image data that aligns with natural biological morphology. By improving the axial continuity of the raw image data while preserving the structural integrity of superstructural details, our approach can enhance the performance of downstream tasks.
For other downstream tasks in ssEM, such as 3D segmentation 40
For the design of 3D registration algorithms, we propose a novel paradigm from the perspective of trajectory optimization. Our approach be divided into three main components: 1) pixel trajectory tracking, 2) dynamics-based trajectory optimization, and 3) trajectory inversion and image registration. These three modules are decoupled and interchangeable. In theory. each module can be replaced with more efficient algorithm. For example, more efficient feature point tracking algorithms [1]] 10] or optical flow estimation algorithms [62 78 can be used for trajectory tracking. Low-pass filtering 22 and convex quadratic programming (QP) [2 17 can be used for trajectory smoothing. Radial basis function interpolation 73 and other splatting techniques
can used for trajectory inversion. This helps researchers explore more efficient 3D registration methods based on this paradigm.
Due to time and resource constraints, we were unable validate the performance of our method on larger-scale scenarios, particularly on ultra-high-resolutign image data. The challenges posed by computational complexity in larger environments and the variability of data noise remain. In future work, we will refine our approach and adapt it into microscopy image processing tool suitable for large-scale real-world datasets.
29

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Table 11: Performance of different registration methods on downstream segmentation tasks across six datasets. We evaluate segmentation accuracy using the Dice score (Avg %%..
EMReg [42
EFSR [76 SEMLeSS [53
TrakEM2 [56]
GaussReg
82 Ours
Mus Heart
0.35 0.06
0.82 1
0.65
0.68
0.89
0.0
0.88
Mus Kidney
0,41
+ o.n
0.75 1: uoz
0.67 tuz2
0.75 4: 0.22
0.82 :: 0.01
0.84 4: 0.01
Mus Liver-3
0.41
= 0.08
0.90 = nm
0.86 no2
0,90 = 0.00
0.95
= 0.01
0.94 : 001
Mus Liver
0.57 = 0.08
0.89 + 004
0.86 oos
0.92
2 0.04
0.94 0.09
0.94 0.0
Mus Pancreas
0,46 0.04
0.87
0.79 no2
0.90
0.94 0.01
0.93
Mus Skin
0.50 0:37
0.87 +
0.76 nus
0.90 0.02
0.94 0.01
0.94
Table 12: Evaluation of the diffeomorphic property of deformation fields using the Folds metric (%
of [jp| < 0).
% of |Jo 0 TrakEM2
56 EFSR 76
EMReg
42
SEMLeSS
53 GaussReg
S2
Ours
Mus Heart
0.188
0.243
0.135
0.099
0.0171
Mus Kidney
0.206
0.379
0.24
0.174
0.0098
Mus Liver
0.193
0.31
0.155
0.106
0.0124
Mus Skin
0.171
0.252
0.102
0.091
0.0095
B Additional Quantitative Results and Visualization
B.1 More Experimental Results
Robustness to Error Accumulation. In practical applications, serial section electron microscopy (ssEM) datasets often contain hundreds even thousands of images. This large number of slices poses challenge for long-sequee meg-sequence registration, as cumulative errors can easily arise, eventually leading to substantial sequence drift and compromising the accurate reconstruction of the biological specimen's 's true 3D structure. To systematically evaluate the ability of our method to suppress such cumulative errors, we conducted comparative experiments on six long-sequence datasets listed in Table 7 focusing on the robustness of our method versus GaussReg [82 and SEMLeSS [53]
Specifically, Figures T2 illustrate how the registration accuracy changes as the number of slices increases. The results show that our method consistently achieves higher average registration accuracy across the entire sequence, with smaller fluctuations in the accuracy curve. This indicates superior stability and robustness when handling long sequences. Moreover, Figures 7 and g present the 3D reconstruction results on the remaining four datasets, is evident that our method accurately recovers the spatial structures of various biological tissues, further demonstrating its generalizability and robustness across different types of data.
Performance on 3D Segmentation Tasks. 3D segmentation [40 41 is a important task in serial section electron microscopy (ssEM) and has been widely applied in various biological image analysis domains 119 43 High-quality 3D registration plays a crucial role in segmentation tasks, as it can significantly mitigate artifacts caused by structural deformation, scale variations, and differences in imaging modalities, thereby improving segmentation accuracy. To comprehensively evaluate the applicability of our method in 3D segmentation, we conducted experiments on six datasets involving various organelles, including nuclei and mitochondria. Specifically, we employed segmentation network (67) to perform segmentation on the images registered by each method. The segmentation performance was quantified by computing the Dice similarity coefficient between the predicted results and the ground-trume -trth (GT) label stacks.
Table I1 summarizes the Dice scores for 3D segmentation results across the six datasets. As shown, our method achieves the best performance on three datasets and the second-best on the remaining three, with overall results slightly below those of the supervised method GaussReg (with differences no greater than 0.01). This discrepancy may be attributed the segmentation model's sensitivity to subtle misalignments, while our method focuses on global trajectory optimization and may have limitations in handling local details. For instance, as illustrated Figure g for the Mus Liver-3 dataset, minor local misalignments can still be observed in our registration results. Nevertheless, our method attains accuracy comparable to that of the supervised GaussReg. Furthermore, Figs. 9 10 and II present 3D visualizations of the segmentation results on multiple datasets. These visual results
30

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Table 13: Sensitivity analysis of the hyperparameter X.
X(SSIM)
0.15
1.5
4.5
7.5
10
Mus Heart 0.855 0.827
0.872 0.851
0.87 0.855
0.864
0.83
Mus Kidney
0.843
0.809
Table 14: Impact of different network architectures on trajectory tracking performance.
SSIM
RAFT [64
GAHlow 45
SEA-RAFT (71
Mus Heart
0.892
0.917
0,909
Kidney
0.894
0,933
0,925
Mus Liver
0.871
0,928
0,918
Mus Skin
0.92
0,943
0.932
Table 15: Effect of large deformations registration performance.
GaussReg (a=1.0) Ours (a=1 GaussReg (a=1 .5) Ours (a=1.5 GaussReg (a=2.0) Ours (a=2.0)
Mus Heart
0.832
0.872
0.811
0.867
0.772
0,856
Mus Kidney
0.781
0.851
0.763
0.843
0.739
0.823
Mus Liver
0.805
0.842
0.782
0,836
0.758
0.817
Mus Skin
0.827
0.864
0.797
0.851
0.763
0.838
Table 16: Effect of noise levels on registration performance,
GaussReg (5%)
Orgin (5%)
GaussReg (10%)
Orgin (10%)
GaussReg (15%)
Orgin (15%)
Mus Heart
0.815
0.851
0.788
0.832
0.748
0.771
Mus Kidney
0.762
0.833
0.734
0.822
0.715
0.759
Mus Liver 0.779
0.814
0.74
0.784
0.72
0.714
Mus Skin 0.804
0,837
0.751
0.803
0.728
0.742
demonstrate that our method can reliably reconstruct the correct 3D structure of biological tissues, leading to clearer and more consistent segmentation boundaries, further validating its practicality in downstream tasks.
In addition, we report the Folds metric Table to further evaluate the diffeomorphic property of the deformation fields. The results demonstrate that our method consistently achieves the lowest Folds values, remaining around 0.01, whereas most baseline methods report values greater than 0.1. This indicates that our approach substantially reduces grid folding in the generated deformation fields, thereby better preserving topological consistency compared to existing baselines.
Results on Real- World Data. To further evaluate the performance of our method on real-world datasets, we present the registration results on three datasets, FemFlyBrain, FAFE and the Mouse Cortical Dataset, as shown Figures 13 17 The results demonstrate that our method achieves accurate and consistent alignment across diverse biological samples, effectively handling complex morphological variations and imaging artifacts.
More Ablation Experiments To gain deeper understanding of our method, we conducted additional ablation studies. Table 13 presents sensitivity analysis of the key hyperparameter X, which primarily controls the smoothness term the loss function of the trajectory tracking module. Table [14 investigates the impact of adopting more advanced network architectures on the performance of the trajectory tracking module. As shown, replacing the baseline with more sophisticated network slightly improves the registration accuracy, demonstrating the modular flexibility of our framework and compatibility with advanced architectures. Tables 15 and 16 further compare registration performance under more challenging conditions, including large deformations and high noise levels. The results indicate a gradual performance degradation as deformation or noise intensity increases, suggesting promising directions for extending approach more complex real-world scenarios.
Due FAFB data acquisition, only registration results shown.
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B.2 Explainable Visualization Study
To better understand the performance and explainability of our new paradigm, we conducted visualization analysis of the intermediate trajectories. Specifically, we performed trajectory tracking on data with nonlinear distortions, ground truth data, and registration results, ensuring consistency between the trajectories across different datas. The visualization results are shown in Figure [8 It is evident from the figure that the trajectories of the nonlinear distorted data exhibit irregular noise jitter, leading to abrupt changes in local curvature, which reflect the impact of distortion on the motion trajectory. contrast, the trajectory of the ground truth data maintains the natural evolutionary pattern of the organism's ' movement, exhibiting smooth and continuous changes. Our method successfully overcame abnormal physical deformations in the registration results, faithfully and precisely restoring the natural motion trajectory.
B.3 Failure Cases
We observe that our method is to some extent dependent on the accuracy of the trajectory tracking module. When the tracking is suboptimal, the subsequent image registration performance may be affected. This issue becomes particularly prominent when handling real-world anisotropic data, where the axial resolution is often significantly lower than the lateral (XY) resolution. Substantial structural and textural differences arise between adjacent slices (as shown in Figure 19 Moreover, real datasets often suffer from high noise levels, imaging artifacts, and missing slices, further complicating reliable trajectory estimation. In Figure 20, we illustrate several representative failure cases, in which the axial resolution of the image stacks approximately one-tenth of the lateral resolution, with noticeable noise and missing slices. Although our method still capable of performing registration under such challenging conditions, the visual quality may degrade due the aforementioned interfering factors. We hope future research will explore more robust and efficient trajectory tracking modules, as well as 3D registration methods that are better suited for anisotropic and noisy real-world data.
32

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GI
Nois
EMRey
ERSR
Mouse
SEAML
Ganssteg
OuE
Neke
EMRay
Mouse
SEAMLeSS
TrakEM2
GanzeRay
Ourr
Figure 7: More 3D visualization of registration results on Mus Pancreas and Mus Skin datasets.
FESR
Mouse Liver
SEAMLESS
TrakEM2
GauzeRey
Ours
GT
Naise
EMRay
EFSR
Mouse Liver-3
SEAMLeSS
TrakEM2
GanseRa
Figure 8: More 3D visualization of registration results Mus Liver and Mus Liver-3 datasets.
33

---

Mouse Heart N
TrakEM2
GaussRay
Notv::
EMiReg
EFSR
Mouse Liver-3
SEAMLESS
TrakEM2
Causseeg
Ours
Figure 9: 3D segmentation visualizations registration results using various methods on the Mus Heart and Mus Liver-3 datasets.
EMRag
Mouse Liver
SEAMLESS
TrakEMē
Gausited
Ours
Noise
EMReg
Mouse
SEAMLeSS
TrakEM2
GauzeRey
Ours
Figure 10: 3D segmentation visualizations of registration results using various methods on the Mus Liver and Mus Pancreas datasets.
34

---

EMReg
EFSR
Mouse Kidney
SEAMLeSS
ThakEM
Gaussteg
Ours
Natse
EMReg
EFSR
Mouse Skin
SEAMLeSS
TrakEM2
GazeRRy
Ours
Figure 11: 3D segmentation visualizations registration results using various methods on the Mus Kidney and Mus Skin datasets.
Mouse Heart
Mouse Kidney
Mouse Liver-3
MI
SSIM
SSIM
SSIM
NCC
NCC
150
010
010
MI
Mouse Liver
Mouse Pancreas
MI
Mouse Skin
S52M
SSIM
SSI
NCC
NCC
Figure 12: Error accumulation comparison between GaussReg [82 and SEMLeSS 53 six dataset.
and ours on
35

---

Y
Orgin Data
GaussReg
Ours
Z
Figure 13: Side views of the original data, the registration results of GaussReg 82 and our method on the FemFlyBrain dataset (63].
36

---

Orgin Data
GaussReg
Ours
Figure 14: Side views of the original data, the registration results of GaussReg 82 and our method
on the FemFlyBrain dataset 63
37

---

Y
Orgin Data
GaussReg
Ours
Z
o
o/lo
Figure 15: Side views of the original data, the registration results of GaussReg 82
and our method
on the Mouse cortical dataset 32
38

---

Orgin Data
Ours
Figure 16: Side views of the original data, the registration results the FAFB dataset
S
39

---

Orgin Data
Ours
Figure 17: Side views of the original data and the registration results on the FAFB dataset 84
40

---

Orgin Data
GT
Ours
Mouse Heart
Mouse Kidney
Mouse Liver-3
Mouse
Mose
Mozse Skin
Figure 18: Trajectory visualization of the original data, ground truth, and registration results.
41

---

previous slice
next
previous slice
next slice
previous slice
next slice
previous slice
next slice
Figure 19: Four examples showing texture structure differences between adjacent slices of anisotropic data.
Orgin
Ours
Orgin
Ours
Orgin
Ours
Orgin
Ours
Figure 20: Four failure cases from the FemFlyBrain dataset. These examples are affected by low axial resolution, missing slices, and high noise.
42

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Reviews:
[{"review_id": "knRtKlw6nr", "reviewer": "Reviewer_L5DT", "summary": "1. This paper first analyses the intrinsic features of ssEM registration problem, and defines 3D registration as a continuous manifold trajectory optimization problem.\n2. This paper proposes a new unsupervised learning method by integrating biophysical priors with data-driven deformation compensation.\n3. This paper conducts the synthetic and realistic experiments to verify the effectiveness of the proposed method.", "questions": "Please refer to \"cons\" in Strengths And Weaknesses\n\n1. Does the proposed method perform better than other methods, especially for the traditional methods, on the unseen ssEM datasets? In the experiment, the main results focus on the synthetic datasets, and EFPL dataset is used for the ablation studies. And the real-world experiments in the supplementary materials also show the training and testing results? I wonder about the generalizable performance of the proposed method with unsupervised learning schemes. \n2. The ablation experiments on the trajectory tracking module are required. Do the hyperparameters in Eq.15 influence the final performance obviously? And the powerful tracking network, such as more parameters or advanced networks, can give better performance?\n3. The efficiency problem is important. And how do you use the proposed method for the large-scale long-slices ssEM dataset? Can the method save the inference burden by selecting several trajectories rather than considering all pixels?\n4, Dice score and Folds are demanded for the performance comparison. Can the author evaluate the neuron segmentation smoothness and deformation smoothness?", "limitations": "Please refer to \"Questions\" and \"Weaknesses\".", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. The motivation is straightforward and the idea is reasonable. The global optimization along the slice dimension is a solution to the ssEM registration problem, which shares a similar core idea with some traditional methods \n2. The detailed operations in the method make sense and are easy to follow.\n3. The proposed method shows superior performance compared with other comparison methods.\n\n\nCons:\n1. The main experiments focus on the synthetic dataset and lack comprehensive experiments on the real-world/large-scale ssEM datasets.\n2. The effect of the trajectory tracking module is clear. Accurate tracking is an important starting point for the following optimization, but the current discussion is not enough.\n3. The efficiency problem should be considered. Compared with GaussReg, the SSIM improves from 0.82 to 0.87, but the utilized time also improves from 0.92 to 2.46. Though authors claim the better performance compared with GaussReg, more situations without labels should be conducted and analyzed.\n4. More metrics should be utilized, such as Dice score in some labelled ssEM dataset, which can better reflect the instance coherence.", "was_revised": true, "final_justification": "In the rebuttal period, authors give detailed explanations and results to address my concerns.\n\nFor the large-scale EM registration results, the authors emphasise the results in the appendix and promise to supplement more results in the final version.\n\nMeanwhile, other concerns, such as more metrics (Dice and Folds) and better tracking backbones, are discussed and solved to some extent.\n\nAs a result, I will change my score from \"Borderline reject\" to \"Borderline accept\".", "rebuttal": "#### **Cons 1**\nThank you for the valuable suggestion. We already showed registration results on four $1024 \\times 1024 \\times 1299$ subvolumes from different regions of the real FemFlyBrain dataset in Figures 12–13 of our Supplementary Materials. As you can see, our method achieves improved structural continuity and better restoration of correct axial structures compared to baselines. \n\nFollowing your suggestion, we have now additionally evaluated our method on two $1024 \\times 1024 \\times 700$ blocks from the FAFB dataset [1] and four $1024 \\times 1024 \\times 500$ blocks from the mouse cortical dataset [2]. Due to the lack of established quantitative metrics for this task on real data and the restriction of not being able to include figures in rebuttal, we will include corresponding results and visual comparisons in the revised version.\n\n[1] A complete electron microscopy volume of the brain of adult *Drosophila melanogaster*, *Cell*, 2018.  \n[2] Saturated reconstruction of a volume of neocortex, *Cell*, 2015.\n\n#### **Cons 2**\n\nWe agree with the reviewer that accurate tracking is a critical foundation for the subsequent optimization stages. In response, we have now added ablation studies on the trajectory module, including:  \n(i) replacing it with a stronger dense tracking network (Table 1), and  \n(ii) a sensitivity analysis of the key hyperparameter $\\lambda$ (Table 2).  \n\nTo ensure generality, these experiments are also conducted on datasets beyond EFPL. Table 1 shows that incorporating more advanced optical flow models can further improve the performance, yielding 0.3--0.5 higher SSIM scores. Table 2 demonstrates that our method remains robust and maintains SOTA-level accuracy across a wide range of $\\lambda$ values (between 1.5 and 7.5).  \n\nWe would like to emphasize that our work introduces a novel paradigm rather than merely an incremental improvement (as aptly noted by Reviewer TWfx). To highlight this, we intentionally adopted a simple baseline network architecture in the original manuscript, which leaves significant room for further performance gains by integrating more sophisticated tracking components.  All the additional results will be included in the revised version.\n\n**Table 1.** \n| SSIM| RAFT (ECCV 2020) | GAFlow (ICCV 2023) | SEA-RAFT (ECCV 2024) |\n|-|-|-|-|\n| Mus Heart|0.892| 0.917| 0.909|\n| Mus Kidney|0.894| 0.933| 0.925|\n| Mus Liver| 0.871| 0.928|0.918 |\n| Mus Skin| 0.920| 0.943|0.932|\n\nRAFT: Raft: Recurrent all-pairs field transforms for optical flow. ECCV, 2020\n\nGAFlow: GAFlow: Incorporating Gaussian Attention into Optical Flow, ICCV, 2023\n\nSEA-RAFT: SEA-RAFT: Simple, Efficient, Accurate RAFT for Optical Flow, ECCV 2024\n\n**Table 2.** \n| SSIM \\| $\\lambda$ |0.15|1.5|4.5|7.5|10|\n|-|-|-|-|-|-|\n|Mus Heart|0.855|0.872|0.870|0.864| 0.830|\n|Mus Kidney|0.827|0.851|0.855|0.843| 0.809|\n\n#### **Cons 3**\n\nThank you for the insightful suggestion regarding runtime efficiency. In response, we have added new experiments focused on efficiency optimization through:  \n(i) sparse trajectory sampling (every 2/3/4 pixels) (**Table 3**), and  \n(ii) resolution reduction via down-sampling followed by up-sampling (**Table 4**).  \n\n**Table 3** shows that sparse sampling significantly reduces runtime (from 2.46s to 0.95s), while preserving SOTA-level accuracy, with SSIM only decreasing slightly from 0.874 to 0.831.  \n**Table 4** demonstrates that reducing resolution also improves speed (from 2.46s to 0.91s), with a drop in accuracy (SSIM from 0.874 to 0.827), which still outperforms the GaussReg baseline. The observed performance drop may be attributed to the use of a simple bilinear up-sampling strategy. We believe that combining sparse trajectory sampling with resolution reduction via down-sampling followed by up-sampling can achieve a better trade-off between runtime and accuracy.We will include the above results and further discussion in the revised version.\n\nFor more situations without labels, we also include more experiments on real datasets and compare with GaussReg (see our response to Cons 1 above).  We will include the complete results and extended discussion in the revised version.\n\n**Table 3.** \n|               | All pixels | Every 2nd pixel | Every 3rd pixel | Every 4th pixel |\n|---------------|------------|-----------------|-----------------|-----------------|\n| **SSIM**      | 0.874      | 0.866           | 0.842           | 0.831           |\n| **Time (s)**  | 2.46       | 1.35            | 1.12            | 0.95            |\n\n**Table 4.** \n|               | Full resolution | 1/2 resolution | 1/4 resolution |\n|---------------|-----------------|----------------|----------------|\n| **SSIM**      | 0.874           | 0.845          | 0.827          |\n| **Time (s)**  | 2.46            | 1.12           | 0.91           |\n\n\n#### **Cons 4**\nWe appreciate your valuable suggestion. As reported in Table 3 of our Supplementary Materials, we have already provided Dice scores for quantitative evaluation. In accordance with your request, we additionally report the Folds metric in Table 5 to assess the diffeomorphic property of the deformation field. The results show that our method achieves the lowest Folds values, consistently around 0.01, whereas most baseline methods report values above 0.1. This indicates significantly fewer grid foldings and better preservation of topological consistency in the deformation fields produced by our approach.\n\n**Table 5.** \n| Folds       | TrakEM2 | EFSR  | EMReg | SEMLeSS | GaussReg | Ours   |\n|-------------|---------|-------|-------|---------|----------|--------|\n| Mus Heart   | /       | 0.188 | 0.243 | 0.135   | 0.099    | 0.0171 |\n| Mus Kidney  | /       | 0.206 | 0.379 | 0.24    | 0.174    | 0.0098 |\n| Mus Liver   | /       | 0.193 | 0.31  | 0.155   | 0.106    | 0.0124 |\n| Mus Skin    | /       | 0.171 | 0.252 | 0.102   | 0.091    | 0.0095 |\n\n#### **Q.1**\n\nThank you for your valuable suggestion. As stated in the main text (Table 1), all six test datasets are unseen ssEM datasets. We train a single model on training data and evaluate it across these six unseen datasets to demonstrate the generalization ability of our method. Compared to traditional methods TrakEM2, our approach achieves significantly improved performance, for instance, on the mouse heart dataset, the SSIM improves from 0.57 to 0.87. \n\nFor real-world datasets, we also adopt a similar protocol: training on a small region and testing on separate, unseen regions (as described in Line 110, Section A2.3 of our Supplementary Materials). The visualizations in Figures 12–13 of Supplementary Materials demonstrate that our method better restores the correct axial structure and generalizes well to new regions that were not observed during training. \n\nFollowing your suggestion, we additionally include experiments on 6 blocks from two more real-world ssEM datasets (see our response to Cons 1 above). We also provide ablation studies on datasets beyond EFPL (see our response to Weaknesses 2 from Reviewer zfxm), further validating the robustness and generalizability of our framework.\n\n#### **Q.2**\n\nThank you for your helpful suggestion. We now provide a sensitivity analysis of the key hyperparameter λ in Table 2. The results show that our method remains robust and maintains SOTA-level accuracy across a wide range of λ values (from 1.5 to 7.5). \n\nIn addition, regarding the tracking module, we replace the original 2D Unet network with more advanced optical flow models: RAFT (5.3M parameters), GAFlow (10.1M), and SEA-RAFT (4.9M). Table 1 shows that incorporating these more powerful tracking networks can further boost performance, achieving 0.3–0.5 improvements in SSIM scores. Please also refer to our response to Cons 2 above for additional detailed results and discussions.\n\n#### **Q.3**\n\nThank you for your question. For large-scale long-slice ssEM datasets, we adopt a divide-and-conquer strategy by splitting large volumes into manageable blocks. Our method estimates deformation fields in parallel for each block and then fuses them to reconstruct the full field. Please refer to our response to Q.4 from Reviewer zfxm for more details. \n\nFor handling long slices, we process them in blocks (e.g., 100 slices per batch), as described in Line 113, Section A2.3 of our Supplementary Materials. \n\nTo address the inference burden, we introduce two efficient strategies as discussed in our response to Cons 3 above: 1) sparse trajectory sampling, and 2) downsample-then-upsample resolution. Experiments show that these strategies significantly reduce inference time from 2.46s to 0.95s, while preserving SOTA accuracy.\n\n#### **Q.4**\n\nWe appreciate your valuable suggestion. In Table 3 of our Supplementary Materials, we have already provided Dice scores for quantitative evaluation. \n\nIn accordance with your request, we additionally report the Folds metric in Table 5 above to assess the diffeomorphic property of the deformation field. The results show that our method achieves the lowest Folds values, consistently around 0.01, whereas most baseline methods report values above 0.1. This indicates significantly fewer grid foldings and better preservation of topological consistency in the deformation fields produced by our approach."}, {"review_id": "B3udItOE1R", "reviewer": "Reviewer_Luv5", "summary": "The authors provide motivation from their application (3D registration of series section electron microscopy) for a 3D registration method that can handle non-linearities. Neural ODEs are a reasonable approach for this. They compare their method to existing methods in the field and demonstrate superior performance under certain metrics.", "questions": "* You refer to \"biological priors\" throughout the paper (including the abstract) but never define exactly what you mean by it or how they are integrated into the model\n\n* How does your method handle noise and imaging artefacts? Pixel tracking methods are sensitive in particular to the former\n\n* Suggested ablation study - what happens if you remove the ODESolver / neural compensator component in Equation 10? Capturing non-linearity is your central argument, it would be instructive to test how important this is to your methodology\n\nL249 - any deeper insight into why your method outperforms GaussReg?\n\n* The results are a bit underwhelming - only in synthetic data, and no comment on the key requirement of handling non-linearities. How does it perform with real data, with noise and non-linearities?\n\n* Gauss-Seidel operator - this is introduced briefly but needs defining better\n\n* Trajectory inversion - why was Gaussian splatting chosen over other interpolation methods?", "limitations": "Please see my above comments regarding limitations.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n- Neural ODEs are a reasonable approach for the task\n- There are some interesting ideas in the paper\n\nWeaknesses\n- The use of ODE based methods for this task is not novel in itself\n- For an application-driven paper, the Results section is a bit underwhelming - only shows synthetic data, and no comment on the key requirement of handling non-linearities. How does it perform with real data, with noise and non-linearities?", "was_revised": true, "final_justification": "The authors satisfactorily answered my questions, particularly regarding the novelty of the work over established registration methods.", "rebuttal": "#### **Weaknesses 1**\nThank you for raising concerns on novelty. While Neural ODEs have been applied in many other domains, to the best of our knowledge, we are the first to introduce Neural ODEs for 3D registration in ssEM data. \n\nMore importantly, our key contribution lies not merely in the use of Neural ODEs, but in the conceptual reformulation of registration as an unsupervised global trajectory optimization problem—an innovation that directly addresses the long-standing issue of error accumulation in traditional pairwise methods (as succinctly summarized by Reviewer TWfx). Neural ODEs are simply a natural and effective mechanism for realizing this paradigm.\n\n#### **Weaknesses 2**\nThank you for the valuable suggestion. We already showed registration results on four 1024 × 1024 × 1299 subvolumes from different regions of the real FemFlyBrain dataset in Figures 12–13 of our Supplementary Materials. As you can see, our method achieves improved structural continuity and better restoration of correct axial structures compared to baselines. Following your suggestion, we have now additionally evaluated our method on two blocks from the FAFB dataset and four blocks from the mouse cortical dataset (see our responses to Cons 1 from Reviewer L5DT). We will include the corresponding results in the revision.\n\n#### **Q.1**\nThank you for the valuable question. The biological prior refers to the smooth manifold nature of biological structures, where nonlinear deformation primarily arises from gradual local curvature changes (as mentioned at the end of the third paragraph in Section 1 of the main text). \n\nBased on this prior, we introduce two regularization terms in our trajectory optimization module: a temporal smoothness loss and a spatial consistency loss (see Line 140 in the main text). These are designed to promote biologically plausible deformations. \n\nTo validate the correctness of this biological prior modeling, we conduct ablation studies in Table 2 of the main text. In addition, we now include the same ablation experiments on two more datasets beyond EFPL (see our response to Weaknesses 2 from Reviewer zfxm). The results show consistent patterns in peak performance improvements, further supporting the robustness of our prior-based design. We will clarify this point more explicitly in the revised version.\n\n#### **Q.2**\nThank you for the suggestion. We agree with you that noise and imaging artifacts would affect the trajectory tracking stage. To address this issue, we adopt two strategies: 1) introducing data augmentation during training and 2) replacing the baseline tracker with more advanced optical flow networks. To validate these approaches, we conducted experiments on four datasets with varying noise levels, comparing our method with GaussReg (Table 6). \n\nThe results (e.g., on the mouse heart dataset with 10% noise) demonstrate that both strategies significantly improve performance: data augmentation improves SSIM from 0.832 to 0.852, and upgrading the tracking network further boosts it to 0.883. In contrast, GaussReg achieves only 0.788. These results highlight that our method remains robust under noisy conditions. Importantly, this indicates that our work represents a novel paradigm in ssEM registration rather than a mere incremental improvement, with significant room for further gains even using more powerful networks and training tricks.\n\n**Table6**\n||5%GaussReg|5%Orgin|5%Orgin+DataAug|5%SEARAFT+DataAug|10%GaussReg|10%Orgin|10%Orgin+DataAug|10%SEARAFT+DataAug|15%GaussReg|15%Orgin|15%Orgin+DataAug|15%SEARAFT+DataAug|\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|MusHeart|0.815|0.851|0.863|0.896|0.788|0.832|0.852|0.883|0.748|0.771|0.853|0.87|\n|MusKidney|0.762|0.833|0.854|0.915|0.734|0.822|0.847|0.893|0.715|0.759|0.834|0.882|\n|MusLiver|0.779|0.814|0.842|0.895|0.74|0.784|0.845|0.877|0.72|0.714|0.83|0.863|\n|MusSkin|0.804|0.837|0.857|0.922|0.751|0.803|0.824|0.905|0.728|0.742|0.827|0.88|\n\nSEA-RAFT: SEA-RAFT: Simple, Efficient, Accurate RAFT for Optical Flow, ECCV 2024\n\n#### **Q.3**\nThank you for your suggestion. We provided ablation studies in Table 5 of our Supplementary Materials by removing Neural ODEs. The results show that removing the Neural ODE component leads to performance drop, with SSIM decreasing from 0.746 (full model) to 0.721. \n\nTo further validate the importance of this module, we additionally performed the same ablation experiments on multiple datasets beyond EFPL (see our response to Weaknesses 2 from Reviewer zfxm). Consistent patterns were observed across these datasets, confirming that the Neural ODEs are critical to the effectiveness of our method.\n\n#### **Q.4**\nThank you for the valuable question. GaussReg performs optimization within local slice windows, which inherently limits its ability to maintain global consistency across long-range sections. In contrast, our method adopts a global trajectory-based formulation that enables coherent alignment over large spatial extents. This key advantage is quantitatively supported by the error accumulation scatter plots shown in Figure 4 of the main text and Figures 6–11 of the Supplementary materials, where our method demonstrates reduced error accumulation compared to GaussReg. We will further elaborate on this important distinction in the revision.\n\n#### **Q.5**\nThank you for your suggestion. Regarding the effectiveness of our method on more real-world data, we already presented registration results on four 1024 × 1024 × 1299 subvolumes from different regions of the real FemFlyBrain dataset in Figures 12–13 of the Supplementary Materials. As you can see, our method demonstrates improved structural continuity and more accurate restoration of axial anatomical structures compared to the baselines. Following your suggestion, in addition to the four real FemFlyBrain blocks included in the supplementary material, we further conduct experiments on six blocks from two widely used real datasets (see our responses to Cons 1 from Reviewer L5DT), we will report the corresponding results in the revised version.\n\n#### **Q.6**\nThank you for the valuable suggestion. We employ the Gauss-Seidel method, a classical iterative solver for linear systems $A\\mathbf{x} = \\mathbf{b}$ that leverages the most recent updates to accelerate convergence, particularly in sparse, tridiagonal systems.\n\nOur smoothing objective (Eq. (2) in Section A.3 of Supplementary Materials) is:\n\n$$\nE(\\mathcal{P}) = \\sum_{z=1}^{N-1} \\left\\| \\mathcal{P}(z{-}1) - 2\\mathcal{P}(z) + \\mathcal{P}(z{+}1) \\right\\|^2 + \\lambda \\sum_{z=1}^{N-1} \\left\\| \\mathcal{P}(z) - \\mathcal{P}^{(0)}(z) \\right\\|^2.\n$$\n\nAs derived in Section A.3 of Supplementary Materials, minimizing this energy leads to the following linear system:\n\n$$\n(4 + \\lambda)\\mathcal{P}(z) - 2\\mathcal{P}(z{-}1) - 2\\mathcal{P}(z{+}1) = \\lambda \\mathcal{P}^{(0)}(z), \\quad z = 1, \\ldots, N{-}1.\n$$\n\nThis system can be equivalently written as a symmetric tridiagonal linear system:\n\n$$\nA \\mathbf{p} = \\mathbf{b},\n$$\n\nwhere:\n\n* $\\mathbf{p} = [\\mathcal{P}(1), \\ldots, \\mathcal{P}(N{-}1)]^\\top$ is the unknown trajectory,\n* $\\mathbf{b} = [\\lambda \\mathcal{P}^{(0)}(1), \\ldots, \\lambda \\mathcal{P}^{(0)}(N{-}1)]^\\top$,\n* and $A \\in \\mathbb{R}^{(N{-}1) \\times (N{-}1)}$ is a tridiagonal matrix with structure:\n\n$$\nA =\n\\begin{bmatrix}\n4+\\lambda & -2         &           &        &  \\\\\n-2        & 4+\\lambda & -2         &        &  \\\\\n          & \\ddots    & \\ddots     & \\ddots &  \\\\\n          &           & -2         & 4+\\lambda & -2 \\\\\n          &           &            & -2     & 4+\\lambda\n\\end{bmatrix}.\n$$\n\nApplying the Gauss-Seidel method to this system, the general update rule for the $i$-th variable is:\n\n$$\nx_i^{(k+1)} = \\frac{1}{A_{ii}} \\left( b_i - \\sum_{j<i} A_{ij} x_j^{(k+1)} - \\sum_{j>i} A_{ij} x_j^{(k)} \\right).\n$$\n\nFor our tridiagonal matrix, each row $z$ only has nonzero entries at $z-1$, $z$, and $z+1$. Substituting these into the general Gauss-Seidel formula, the update rule for $\\mathcal{P}(z)$ becomes:\n\n$$\n\\mathcal{P}^{(k+1)}(z) = \\frac{1}{A_{zz}} \\left( \\lambda \\mathcal{P}^{(0)}(z) - A_{z,z-1} \\mathcal{P}^{(k+1)}(z{-}1) - A_{z,z+1} \\mathcal{P}^{(k)}(z{+}1) \\right),\n$$\n\nwhere:\n\n* $A_{zz} = 4 + \\lambda$,\n* $A_{z,z-1} = A_{z,z+1} = -2$,\n* and we use the latest available updates for $\\mathcal{P}^{(k+1)}(z{-}1)$, along with the previous values for $\\mathcal{P}^{(k)}(z{+}1)$.\n\nSubstituting these in yields:\n\n$$\n\\mathcal{P}^{(k+1)}(z) = \\frac{1}{4 + \\lambda} \\left( 2 \\mathcal{P}^{(k+1)}(z{-}1) + 2 \\mathcal{P}^{(k)}(z{+}1) + \\lambda \\mathcal{P}^{(0)}(z) \\right).\n$$\n\nFor simplicity, we divide both numerator and denominator by 2, resulting in the final explicit Gauss-Seidel update:\n\n$$\n\\mathcal{P}^{(k+1)}(z) = \\frac{\\mathcal{P}^{(k+1)}(z{-}1) + \\mathcal{P}^{(k)}(z{+}1) + \\lambda \\mathcal{P}^{(0)}(z)}{2 + \\lambda}, \\quad z = 1, \\ldots, N{-}1.\n$$\n\nThis completes the derivation of the iterative update rule used for trajectory smoothing, as presented in Eq.8 of the main text.\n\n#### **Q.7**\nThank you for the valuable question. We adopt bilinear splatting (not Gaussian splatting) in the inversion step due to its favorable speed–accuracy trade-off. To validate this choice, we now provide a comparison among bilinear, trilinear, and classical radial basis function (RBF) interpolation in Table 7. \n\nThe results show that bilinear interpolation achieves an SSIM of 0.87 within only 0.008 seconds. In contrast, classical RBF interpolation, even when accelerated by GPU, takes 1.2 seconds but yields only marginal improvement (SSIM = 0.89). While learning-based interpolation methods may potentially yield further improvements, we leave this for future work due to time constraints for rebuttal.\n\n**Table7**\n|Method|Time(s)|SSIM|\n|-|-|-|\n|Bilinear splatting|0.008|0.87|\n|Trilinear splatting|0.0056|0.87|\n|RBF(CPU)|21.4|0.89|\n|RBF(GPU)|1.2|0.89|"}, {"review_id": "6gkYYRy7mG", "reviewer": "Reviewer_zfxm", "summary": "This paper proposes a novel unsupervised framework for 3D registration in ssEM by formulating it as a continuous trajectory optimization problem. Combining Gauss-Seidel iteration with Neural ODEs, the method balances global smoothness with local structure preservation without requiring ground truth. Experiments across diverse tissue types show improved accuracy and robustness over existing approaches.", "questions": "1. The benchmark dataset generation should be further clarified: Please clarify how the synthetic benchmarks are constructed, particularly with respect to the OpenOrganelle datasets. Are the original volumes considered as ground truth, with artificial z-axis distortions added to simulate noise? If so, what procedure is used to introduce these deformations, and how is the realism of such synthetic noise validated?\n\n\n2 The paper benchmarks against models such as EMReg and TrakEM2. However, EMReg, while primarily pairwise, can be adapted to optimize over larger blocks with global regularization. This appears to contradict the claim that prior methods are strictly local. Could the authors clarify how EMReg and other baselines were implemented or adapted for the comparison, particularly in terms of their global vs. local optimization capabilities? Also, note that the EMReg citation in Table 1 (ref [29]) is inconsistent with its mention in the main text (ref [24]); please ensure citation accuracy.\n\n\n3. The comparison with biophysically-informed alternatives can be made. A key contribution of this work is the integration of biophysical priors into the registration process. However, other methods such as SOFIMA (https://github.com/google-research/sofima)(based on finite element modeling) similarly aim to preserve biological morphology through elastic deformation modeling. It would strengthen the paper to compare against such methods that also explicitly model biomechanical plausibility. If models like this are out of scope for direct benchmarking, a discussion of the similarities and differences in modeling philosophy would still be helpful.\n\n\n4. The paper would benefit from additional detail on how the method scales to TB-scale ssEM datasets. Specifically, how is memory usage managed for pixel-wise trajectory tracking across thousands of slices, and are any tiling, chunking, or multi-CPU strategies employed? Can users apply the model with a 40 GB memory GPU? Including this information would clarify the practical feasibility of the method for real-world datasets.", "limitations": "1. The proposed method optimizes global pixel trajectories, which may introduce higher computational overhead compared to locally constrained models. Given the large size of typical EM volumes, runtime is a critical factor in practical deployment. It would be valuable for the authors to report time consumption and discuss the trade-off between registration accuracy and computational cost relative to existing methods.\n\n2. Many real-world ssEM pipelines include stitching as a pre-alignment step. Some baseline methods such as TrakEM2 are capable of both stitching and registration. It would strengthen the paper to clarify whether the proposed method could be extended to incorporate stitching, or to discuss its potential limitations in scenarios where stitching errors are significant.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- The paper presents a well-motivated and clearly explained workflow for global trajectory optimization in EM 3D registration. The use of Neural ODEs and the dual-objective formulation are conceptually sound and well-supported by the mathematical derivation. \n- Experimental results across multiple datasets show strong performance compared to existing methods, and the theoretical exposition aids in understanding the model's underlying mechanisms.\n\n## Weaknesses\n\n- Several figures lack sufficient clarity. For example, in Figure 1(b), it is unclear what the “previous method” refers to, what the color-coded outputs represent, and which dataset is used. Figure 3(a) is not referenced or discussed in the main text, leaving the role of grid sampling and bilinear splatting unexplained. \n- The ablation study is limited to a single dataset (EFPL), which restricts the generalizability of the analysis. \n- Runtime comparisons are missing—an important concern for large-scale EM users.\n- The paper lacks discussion of failure cases or scenarios where the method may struggle, such as with highly noisy data. While the framework is said to incorporate biophysical priors, the modeling and validation of these priors remain vague. Recent deep learning-based methods for dense optical flow or registration are not mentioned in the comparisons. Finally, the scalability and memory demands of pixel-wise trajectory tracking across large volumes are not addressed, which raises concerns for TB-scale ssEM datasets.", "was_revised": true, "final_justification": "The authors have addressed my concerns in the response.", "rebuttal": "#### **Weaknesses 1**\nThank you for the question. The “previous works” in Figure 1 refer to pairwise registration methods. The demo in Figure 1 uses the EFPL dataset, and the color-coded represents deformation field. We apologize for the missing reference to Figure 3(a), which is intended to contrast grid interpolation and splatting. We will clarify these figures with more explicit explanations in the revision.\n\n#### **Weaknesses 2**\nThank you for the suggestion. We have extended our ablation studies to four additional datasets, as presented in Tables 8–11 below. The consistent performance patterns observed across all datasets demonstrate the generalizability and robustness of our method beyond a single test set. In addition, we have included several other ablation experiments in response to suggestions from other reviewers (see our responses to Cons 2 from Reviewer L5DT). We will include all results and discussion in the revised version.\n\n**Table 8**\n||TS Loss term|DF Loss term|SC Loss term|SSIM|\n|-|-|-|-|-|\n|Mus Heart|✓|✗|✗|0.721|\n||✓|✓|✗|0.855|\n||✓|✗|✓|0.793|\n||✓|✓|✓|0.872|\n|Mus Kidney|✓|✗|✗|0.698|\n||✓|✓|✗|0.822|\n||✓|✗|✓|0.774|\n||✓|✓|✓|0.851|\n\n**Table9**\n||SSIM|0.0001|0.001|0.01|0.1|\n|-|-|-|-|-|-|\n|MusHeart|0.001|0.775|0.782|0.771|0.762|\n||0.01|0.795|0.806|0.783|0.779|\n||0.1|0.852|0.872|0.841|0.822|\n||1|0.811|0.819|0.805|0.787|\n|MusKidney|0.001|0.754|0.762|0.752|0.741|\n||0.01|0.781|0.794|0.772|0.766|\n||0.1|0.833|0.851|0.821|0.815|\n||1|0.768|0.773|0.758|0.743|\n\n**Table10**\n|SSIM\\Iterations|50|100|150|200|250|300|\n|-|-|-|-|-|-|-|\n|MusHeart|0.719|0.802|0.855|0.872|0.874|0.874|\n|MusKidney|0.713|0.796|0.828|0.851|0.851|0.851|\n|MusLiver|0.695|0.774|0.811|0.842|0.844|0.844|\n|MusSkin|0.682|0.788|0.842|0.864|0.868|0.866|\n\n**Table 11**\n||Gauss-Seidel|NeuralODEs|SSIM|\n|-|-|-|-|\n|Mus Heart|✔|✘|0.847|\n||✘|✔|0.835|\n||✔|✔|0.872|\n|Mus Kidney|✔|✘|0.833|\n||✘|✔|0.827|\n||✔|✔|0.851|\n|Mus Liver|✔|✘|0.816|\n||✘|✔|0.803|\n||✔|✔|0.842|\n|Mus Skin|✔|✘|0.843|\n||✘|✔|0.825|\n||✔|✔|0.864|\n\n#### **Weaknesses 3**\nThank you for your question. As reported in Table 2 of the Supplementary Materials, our method achieves the best accuracy with an SSIM of 0.87, though the runtime is 2.46s. To further improve efficiency, we have now added new experiments focused on optimization strategies. We significantly reduce the runtime (from 2.46s to 0.95s) while maintaining SOTA-level accuracy. This makes our approach more applicable for large-scale EM use cases. Please refer to our response to Cons3 from Reviewer L5DT for detailed experimental results and analysis. \n\n#### **Weaknesses 4**\nThank you for your comments. We addressed challenging scenarios like high noise and nonlinear deformations by: 1) data augmentation by adding noise during training and 2) upgrading the baseline tracker with advanced optical flow networks (see our response to Q.2 from Reviewer Luv5). The results, e.g., on the mouse heart dataset with 10% noise, demonstrate that both strategies significantly improve performance: data augmentation improves SSIM from 0.832 to 0.852, and upgrading the tracking network further boosts it to 0.883. \n\nFor varying nonlinear deformations, Table 12 shows our method consistently outperforms GaussReg (e.g., SSIM on mouse heart drops from 0.832 to 0.742 for GaussReg, but only from 0.872 to 0.803 for ours). \n\nOur biological prior—based on the smooth manifold nature of structures—introduces temporal smoothness and spatial consistency losses (see Line 140 of the main text) to ensure plausible deformations. Ablation studies in Table 2  of the main text and additional datasets beyond EFPL (see our above response to Weaknesses 2) confirm the effectiveness and robustness of this design. \n\nWe agree that using a stronger tracking network improves results, with advanced optical flow models boosting SSIM by 0.3–0.5 (see our response to Cons 2 from  Reviewer L5DT). However, our work presents a novel paradigm, not just an incremental upgrade (as noted by Reviewer TWfx). A simple network already demonstrates great potential of our framework, though it leaves room for further improvements with better tracking components. \n\nWe optimized the speed–accuracy trade-off for large-scale EM data, reducing runtime from 2.46s to 0.95s while maintaining SOTA SSIM accuracy (see our response to Cons 3 from Reviewer L5DT). Scalability and memory for TB-scale datasets are also addressed (see our below response to Q.4). All results will be included in the revised version.\n\n**Table 12**\n|| GaussReg (α=1.0) | Ours (α=1.0) | GaussReg (α=1.5) | Ours (α=1.5) | GaussReg (α=2.0) | Ours (α=2.0) | GaussReg (α=2.5) | Ours (α=2.5) | GaussReg (α=3.0) | Ours (α=3.0) |\n|-|-|-|-|-|-|-|-|-|-|-|\n|Mus Heart|0.832|0.872|0.811|0.867|0.772|0.856|0.753|0.834|0.742|0.803|\n|Mus Kidney|0.781|0.851|0.763|0.843|0.739|0.823|0.725|0.809|0.702|0.776|\n|Mus Liver|0.805|0.842|0.782|0.836|0.758|0.817|0.733|0.782|0.721|0.762|\n|Mus Skin|0.827|0.864|0.797|0.851|0.763|0.838|0.746|0.821|0.737|0.79|\n\n#### **Q.1**\nThank you for raising this important question. For the synthetic experiments, we adopt the high-quality volumes from the OpenOrganelle dataset as the ground-truth references. These volumes are acquired with isotropic resolution and minimal distortion, making them suitable for constructing controlled benchmarks. \n\nTo simulate realistic z-axis deformations, we synthetically warp the volume along the axial direction using smooth deformation fields generated by randomly sampling Gaussian noise fields (see Eq. (1) in the Supplementary Materials). Independent perturbations are applied to each slice (except for the first) to introduce realistic inter-slice misalignments (see Section A2.1 for details in the Supplementary Materials). \n\nTo ensure the realism of the synthetic distortions, we strictly follow the deformation protocols established in widely used baselines such as GaussReg, EFSR, and EMReg, which introduce slice-wise perturbations that closely mimic real-world misalignments observed in ssEM data. This benchmark design enables controlled evaluation while preserving biological plausibility.\n\n#### **Q.2**\nThank you for the question. To the best of our knowledge, EMReg, as presented in the original paper, focuses solely on pairwise registration, and neither the method nor the reported results extend to full-stack or global 3D registration. Therefore, EMReg has not been adapted to optimize over larger blocks with global regularization. If the reviewer is aware of subsequent work where EMReg has been applied for global regularization across larger volumes, we would be happy to analyze and compare accordingly. \n\nRegarding baseline implementations, we used the official EMReg implementation and extended it to 3D registration via sequential pairwise registration. For TrakEM2, we employed the FIJI “Elastically Align Stack” plugin. For SEAMLeSS, we used the official Corgie package for 3D registration. For EFSR and GaussReg, we followed the settings of the original papers. All baselines were trained on the same data as our method, with hyperparameters set according to their respective papers. We will clarify all baseline configurations and resolve citation inconsistencies in the revised version.\n\n#### **Q.3**\nWe appreciate your mention of SOFIMA, whose distributed block processing strategy has been inspiring to us. After carefully reviewing its source code, we found that it employs an \"elastic mesh optimizer\" for registration but acknowledges that the optimization proceeds sequentially, section by section. This approach differs fundamentally from our design of global trajectory optimization.As Reviewer TWfx noted, we conceptually redefine registration as a global trajectory optimization problem—an elegant and significant contribution that directly addresses the fundamental issue of error accumulation inherent in pairwise methods. Due to the time constraint for rebuttal, we will discuss this in the revised version.\n\n#### **Q.4**\nThank you for the suggestion. We optimized the speed–accuracy trade-off, reducing runtime from 2.46s to 0.95s while maintaining SOTA SSIM accuracy (see our response to Cons 3 from Reviewer L5DT). Our framework can scale to TB-scale ssEM datasets using a parallel, block-wise strategy like SOFIMA: 1) split volumes into chunks; 2) run parallel trajectory tracking and inversion on multi-GPUs; 3) stitch local deformation fields; 4) apply the global deformation. \n\nAs noted in Section A2.3 of Supplementary Materials, for datasets containing thousands of slices, we split them into blocks of 100 slices, which can be processed comfortably on a 40GB GPU. We will detail this scalable design in the revision.\n\n#### **Limitation 1**\nThank you for your suggestion. We have explored the speed–accuracy trade-off for large-scale EM data. Our optimization significantly reduces runtime (from 2.46s to 0.95s), while preserving SOTA-level SSIM accuracy (see our responses to Cons 3 from Reviewer L5DT).\n\n#### **Limitation 2**\nThank you for your suggestion. TrakEM2 is a widely used EM image processing toolkit that supports both stitching and registration. However, existing frameworks—including TrakEM2—typically adopt a sequential pipeline that performs stitching first, followed by registration. Our method is also compatible with such pipelines. \n\nBut we agree that integrating stitching and registration into a unified framework is a promising direction. A potential solution is to leverage shared encoder features between the stitching and trajectory tracking modules, enabling joint optimization. \n\nMoreover, we agree that stitching errors can affect the trajectory tracking. To assess the impact, we added evaluations under more challenging conditions (see our responses to Q.2 from Reviewer Luv5), and further replaced the tracking module with more advanced networks to improve robustness against artifacts (see our responses to Cons 2 from Reviewer L5DT). We will incorporate all these clarifications in the revised version."}, {"review_id": "mTzio1zutr", "reviewer": "Reviewer_TWfx", "summary": "This paper addresses the challenge of 3D registration for Series Section Electron Microscopy (ssEM), a critical step for reconstructing biological structures that is often hampered by nonlinear distortions and the accumulation of errors during physical sectioning. The paper proposes a fundamental shift, moving from traditional local, pairwise registration to a global optimization framework. They redefine 3D registration as a continuous manifold trajectory optimization problem. This formulation is motivated by the physical insight that biological tissue deforms smoothly, whereas sectioning artifacts introduce abrupt, non-physical curvature mutations into the motion paths of cellular structures as they move through the image stack.\nThe method consists of three main stages:\n1. Pixel Trajectory Tracking: An initial, noisy trajectory for each pixel is established by using a 2D U-Net to estimate displacement fields between adjacent slices.\n2. Dynamics-based Trajectory Optimization: In the core contribution, a hybrid solver smooths the noisy trajectories. It combines the Gauss-Seidel method, a classical, iterative solver for rapid, stable convergence based on a physical model with a Neural Ordinary Differential Equation, which serves as a data-driven compensator to learn and correct for complex, nonlinear deviations. An adaptive weighting scheme dynamically balances these two components.\n3. Trajectory Inversion and Registration: The final, smoothed trajectories are used to compute a displacement field, which is mapped back to the discrete pixel grid via an efficient bilinear splatting technique to produce the final deformation fields.\nA key advantage of the pipeline is its fully unsupervised training, making it highly applicable to real-world ssEM data where ground-truth alignments are rarely available.", "questions": "1. The ablation study justifies using the Euler solver for the Neural ODE to balance performance and cost. To better understand this trade-off, could you please:\n\n   * Quantify the concrete runtime differences between the Euler, Dopri5, and RK4 solvers on your test hardware?\n   * Clarify if you used the adjoint sensitivity method for more efficient gradient computation during training? If not, have you considered it or other continuous-depth models to improve the speed/accuracy profile, given that runtime is a key limitation?\n\n      2. The framework's modularity is presented as a strength, suggesting that more efficient algorithms could be swapped in for each component. Given the pipeline's acknowledged dependence on the initial tracking module, could you clarify the practicalities of this modularity? For example, if the 2D U-Net were replaced by a state-of-the-art sparse feature tracker like TAPIR, would the trajectory smoothing module require complete retraining to adapt to the different statistical properties of sparse vs. dense trajectories? How does the module generalize?\n\n      3. Could you provide any theoretical or empirical analysis regarding the convergence of the combined Gauss-Seidel + Neural ODE solver, particularly in relation to the behavior of α?\n\n      4. The loss hyperparameters (λ, μ) show a clear performance peak. How were the search ranges determined, and how well do you expect these values to generalize to new datasets without another extensive search?\n\n      5. Have you tested on synthetic data with known artifacts (folds, missing sections) beyond those shown in failure cases?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n* The paper's primary strength is its conceptual reframing of registration as a global trajectory optimization problem. This is an elegant and significant contribution that directly addresses the fundamental issue of error accumulation that plagues traditional pairwise methods.\n* The fully unsupervised nature of the framework is a major practical advantage, making it applicable to diverse ssEM datasets where obtaining ground-truth annotations is prohibitive.\n* The methodology is of high quality, featuring a sophisticated hybrid solver that combines a physics-based prior with data-driven learning in a theory-guided, data-corrected approach. \n* The experimental validation is extensive and robust. The method is tested across six diverse tissue types and compared against multiple state-of-the-art baselines. \n* The authors provide derivations, algorithmic details, and a thorough discussion of limitations and failure cases in the supplementary material, which greatly enhances the paper's quality and reproducibility.\n\nWeaknesses:\n* A significant weakness is the method's computational expense, which the authors acknowledge is due to the ODE solver. While presented as a trade-off for accuracy, this could be a barrier to adoption for labs needing to process the very large-scale datasets that motivate the work. The reporting is also incomplete, with detailed runtime analysis missing for most datasets.\n* The entire pipeline's performance is critically dependent on the quality of the initial trajectory tracking module. This module's architectural design uses a 2D U-Net that operates on strictly local, adjacent slice pairs, giving it no wider temporal context. This lack of context makes the tracker prone to errors in regions of low texture or ambiguity. As acknowledged by the authors and demonstrated in the paper's failure cases, these initial tracking errors can propagate through the pipeline, leading to suboptimal registration results that the global smoothing module cannot fully correct. This fundamentally restricts the method's robustness, especially when applied to challenging real-world data with significant noise or anisotropy.\n* While the formulation is innovative, the paper lacks a theoretical convergence analysis for the novel combined Gauss-Seidel + Neural ODE framework. The conditions under which the adaptive blending mechanism (Eq. 11) is guaranteed to converge are not discussed.\n* There is limited analysis of key parameters like β in the adaptive weighting. While Table 4 shows ablation for Gauss-Seidel iterations (50-300) and Table 3 provides hyperparameter grid search for λ and μ, the sensitivity analysis for β is missing and the hyperparameter studies were conducted only on the EFPL dataset, not across all datasets.\n* The method's quantitative evaluation relies on synthetic distortions created by applying a Gaussian-smoothed random displacement field. This model may not fully represent the diverse range of real-world artifacts from physical sectioning, such as compressive shearing, chatter, or large tissue folds. Consequently, the method's strong performance on this specific type of simulated noise may not guarantee generalization to all distortion types encountered in practice.\n* In the final stage, continuous trajectories are resampled onto a discrete grid by bilinear splatting which introduces an unevaluated risk to the preservation of fine biological structures. This interpolation method inherently smooths information as it distributes a single point's value across multiple pixels, which could be problematic to the single-voxel precision required for connectomics. Hence, the potential for this operator to blur these high-frequency details is a significant concern.", "was_revised": true, "final_justification": "I appreciate the authors’ detailed rebuttal. The authors have addressed key weaknesses, such as adding more data validation and improving runtime trade-offs.", "rebuttal": "#### **Weaknesses 1**  \nThank you for your suggestion. We have explored the speed–accuracy trade-off for large-scale EM data. Our optimization significantly reduces runtime (from 2.46s to 0.95s), while preserving SOTA SSIM accuracy (see our responses to Cons 3 from Reviewer L5DT).\n\n\n#### **Weaknesses 2**\nWe agree that trajectory tracking plays an important role in the overall pipeline performance. Our current design adopts a lightweight 2D U-Net to enable fast and memory-efficient tracking between adjacent slices, but we acknowledge that its limited temporal context may cause errors in ambiguous regions. To address this, we explored two strategies to improve performance under more challenging scenarios (see our responses to Q.2 from Reviewer Luv5).  But we acknowledge the importance of modeling longer temporal context, and we plan to explore trajectory models with extended time horizons to handle challenging scenarios with low texture or strong noise.\n\nMoreover, to evaluate robustness, we tested our method on two additional real datasets (see responses to Cons 1 from Reviewer L5DT). All corresponding results will be included in the revision.\n\n#### **Weaknesses 3**\nWe apologize for the typo in Eq. (10) of the main text (implementation and experiments were correct). The original equation:\n$$\n\\mathcal{P}^{(t_N)} = \\alpha \\cdot \\mathcal{G}(\\mathcal{P}^{(t_0)}) + (1-\\alpha) \\cdot \\mathcal{F}_\\theta(\\mathcal{P}^{(t_0)}),\n$$\n\nwhere $\\mathcal{F}_\\theta(\\mathcal{P}^{(t_0)})$ denotes a shorthand for $\\text{ODESolver}$\n\nshould be corrected to:\n\n$$\n\\mathcal{P}^{(t_N)} = (1-\\alpha) \\cdot \\mathcal{G}(\\mathcal{P}^{(t_0)}) + \\alpha \\cdot \\mathcal{F}_\\theta(\\mathcal{P}^{(t_0)}).\n$$\n\nDefine\n$$\n\\mathcal{P}^* := \\mathcal{G}(\\mathcal{P}^{(t_0)}), \\quad \\mathcal{P}^\\dagger := \\mathcal{F}_\\theta(\\mathcal{P}^{(t_0)}).\n$$\n\n1. **Gauss-Seidel:**  \nConverges under standard conditions for SPD linear systems, solving the quadratic objective  \n$$\n\\mathcal{L}(\\mathcal{P}) = \\sum_{z=1}^{N-1} \\|\\mathcal{P}(z-1) - 2\\mathcal{P}(z) + \\mathcal{P}(z+1)\\|^2 + \\lambda \\sum_{z=1}^{N-1} \\|\\mathcal{P}(z) - \\mathcal{P}^{(0)}(z)\\|^2,\n$$  \nvia iterations  \n$$\n\\mathcal{P}^{(k+1)} = T \\mathcal{P}^{(k)} + c,\n$$  \nwith spectral radius $\\rho(T) < 1$.\n\n2. **Neural ODE:**  \nThen the ODE has a unique solution $\\mathcal{P}(t)$ on $[t_0, t_N]$, and numerical solvers yield approximations with bounded error:\n$$\n\\|\\mathcal{P}_{\\text{num}}^{(t_N)} - \\mathcal{P}^{(t_N)}\\| \\leq \\epsilon.\n$$\n\n3. **Adaptive Blending:**  \n- Both modules converge individually.  \n- Blending weight $\\alpha$ increases monotonically as error decreases via a sigmoid function.  \n- If $\\|\\mathcal{P}^* - \\mathcal{P}^\\dagger\\|$ is sufficiently small, the combined update is stable and converges.\n\nUnder these conditions, the adaptive blending update $\\mathcal{P}^{(t_N)}$ converges reliably.\n\n\n#### **Weaknesses 4**  \nWe thank the reviewer for emphasizing the importance of parameter \\$\\beta\\$ in Eq. (11). It controls the sigmoid steepness and thus the transition speed between Gauss-Seidel and Neural ODE modules. A larger \\$\\beta\\$ makes \\$\\alpha\\$ shift faster from 0 to 1, enabling a quicker switch, while a smaller \\$\\beta\\$ yields a smoother transition. We will add convergence curves and performance comparisons for different \\$\\beta\\$ values in the revised version. \n\nAblation studies were also repeated on four additional datasets (see responses to Weaknesses 2 from Reviewer zfxm).\n\n#### **Weaknesses 5**  \nWe employ Gaussian-smoothed random displacement fields to ensure fair comparison. Specifically, we strictly follow the deformation protocols established in widely used baselines such as GaussReg, EFSR, and EMReg, which introduce slice-wise perturbations that closely mimic real-world misalignments observed in ssEM data. \n\nWe agree with the reviewer that this model may not fully represent the diverse range of real-world artifacts resulting from physical sectioning. Therefore, in addition to the four real FemFlyBrain blocks included in the supplementary material, we further conduct experiments on six blocks from two widely used real datasets (see our responses to Cons 1 from Reviewer L5DT), providing stronger validation under real-world scenarios. Additional results will be included in the revision.\n\n\n#### **Weaknesses 6**  \nThank you for the thoughtful question. Our trajectory inversion step aims to obtain displacement vectors defined on a pixel-aligned grid, which are then used to construct the full deformation field. \n\nIt is important to note that this inversion operates on displacements rather than pixel intensities. But it does not distribute a single point’s value across multiple pixels, and thus does not introduce the kind of smoothing effect typically associated with image-based interpolation. The resulting deformation field is applied once to warp the input stack using STN [1]. Therefore, bilinear splatting is only used as a technical means to discretize continuous displacements, and it does not additionally degrade spatial precision.\n\n[1] Spatial transformer networks. NeurIPS 2015\n\n#### **Q.1**  \nThank you for the suggestion. We have added runtime comparisons of Euler, Dopri5, and RK4 solvers on datasets beyond EFPL (see Table 13). The results show that while Dopri5 and RK4 provide marginal improvements in performance compared to Euler (e.g., SSIM increases from 0.872 to 0.877 and 0.876 on the mouse heart dataset), they come at a significant computational cost (runtime increases from 2.45s to 8.33s and 7.68s, respectively). Given this trade-off, the Euler solver offers a better balance between accuracy and efficiency in our pipeline. For the training of Neural ODEs, we already employ the adjoint sensitivity method to enable memory-efficient and scalable gradient computation.\n\n**Table13**\n|Dataset|Solver|MusHeartTime(s)|MusHeartSSIM|MusKidneyTime(s)|MusKidneySSIM|MusLiverTime(s)|MusLiverSSIM|MusSkinTime(s)|MusSkinSSIM|\n|-|-|-|-|-|-|-|-|-|-|\n||Euler|2.45|0.872|2.41|0.851|2.50|0.842|2.46|0.864|\n||Dopri5|8.33|0.877|8.52|0.855|8.71|0.848|8.39|0.869|\n||RK4|7.68|0.876|7.66|0.854|7.72|0.847|7.25|0.868|\n\n#### **Q.2**\nThank you for your suggestion. We explored replacing the original U-Net with more advanced architectures while keeping the same training pipeline (see our responses to Cons 2 from Reviewer L5DT). The results show that incorporating more advanced models can further improve performance, yielding 0.3–0.5 higher SSIM scores. \n\nDifferent dense prediction modules don’t require retraining the trajectory smoothing module. However, sparse prediction are unsuitable because our trajectory inversion uses bilinear splatting to get displacement vectors on the pixel grid. Sparse trajectories can cause missing pixel values and reduce registration quality. Since accurate registration needs a dense, continuous deformation field, dense prediction modules are preferable. \n\nWe have added ablation studies on trajectory sampling intervals (see our responses to Cons 3 from Reviewer L5DT). The results demonstrate that overly sparse trajectory sampling leads to performance degradation (e.g., SSIM drops from 0.874 to 0.831), especially for EM images where subpixel-level alignment is critical.\n\n\n#### **Q.3**\nThank you for your insightful question. We have provided a theoretical analysis regarding the convergence of the combined Gauss-Seidel + Neural ODE solver in our above response to Weaknesses.3. As for the behavior of the weighting coefficient $\\alpha$, it is dynamically modulated by the relative residual through a sigmoid-shaped function controlled by the parameter $\\beta$. Specifically, a larger $\\beta$ causes $\\alpha$ to transition more rapidly from 0 to 1 once the relative error has sufficiently decreased, thereby accelerating the switch from the linear Gauss-Seidel module to the nonlinear Neural ODE module. In contrast, a smaller $\\beta$ leads to a more gradual and smoother transition (see our above response to Weaknesses 4).\n\n#### **Q.4**\nWe thank the reviewer for raising this important question. The $(\\lambda, \\mu)$ search ranges were initially determined through a grid search on a small subset of the EFPL dataset during preliminary experiments. To assess generalization, we further conducted ablation studies on additional datasets (see responses to Weaknesses 2 from Reviewer zfxm) and consistently observed clear performance peaks around the same parameter values. These results suggest that our choice of $(\\lambda, \\mu)$ offers good generalization across datasets, reducing the need for extensive re-tuning when applying our method to new domains.\n\n#### **Q.5**\nWe thank the reviewer for the suggestion. We have added new results on four datasets with varied inter-slice intervals to simulate missing sections (see Table 14). Due to time constraints, we were unable to retrain the trajectory tracking module, resulting in reduced performance as the slice interval increases. However, based on our ablation studies on the tracking module (see our response to Cons 2 from Reviewer L5DT), we believe that training on anisotropic data and replacing the tracker with more advanced networks can effectively mitigate this issue. \n\nAs for folded sections, handling them is particularly challenging because it requires not only detecting the folds but also unfolding the distorted tissue, which is a complex nonlinear operation [2]. In practice, it is common to treat folded slices as missing sections and exclude them from the registration process to avoid introducing errors [3]. We will include these clarifications and relevant discussions in the revision.\n\n[2] A unified deep learning framework for ssTEM image restoration. IEEE TMI, 2022.\n\n[3] A complete electron microscopy volume of the brain of adult Drosophila melanogaster. Cell, 2018.\n\n**Table14**\n|SSIM\\Intervals|1|2|4|6|\n|-|-|-|-|-|\n|MusHeart|0.872|0.851|0.832|0.804|\n|MusKidney|0.851|0.832|0.811|0.796|\n|MusLiver|0.842|0.828|0.801|0.795|\n|MusSkin|0.864|0.850|0.835|0.807|"}]

Author rebuttal:

Meta-review: This is an application-driven paper that addresses registration in 3D ssEM. To mitigate error accumulation, the authors frame registration as a trajectory optimization problem and design an unsupervised approach to solve it. Reviewers initially criticized the lack of realistic large-scale evidence beyond synthetic data; the clarity of definitions (e.g., “biological priors”), figures, and formatting; and the depth of analysis (scalability, hyperparameters, convergence, …). In rebuttal, the authors effectively addressed many of these points, including adding (or pointing to) results on FemFlyBrain, FAFB, and mouse cortex, adding ablations, reporting additional metrics, and analyzing runtime and various trade-offs. My impression is that this is a solid applied contribution to NeurIPS, of interest in particular to the ssEM community, but also more generally to the ML for microscopy community.

For the camera-ready version, the authors promised to consolidate real-data metrics, runtime and memory tables, sensitivity to the various hyperparameters, and improved figures and definitions. I hope the promised experiments and clarifications are indeed included in the final version.
Decision: Accept (poster)