E-MoFlow: Learning Egomotion and Optical Flow from Event Data via Implicit Regularization
Authors: Wenpu Li
Bangyan Liao
Yi Zhou
Qi Xu
Pian Wan
Peidong Liu
Venue: NeurIPS 2025 poster ()
Keywords: Self-Supervised Learning
3D Vision
Optical flow Estimation
Motion Estimation
Event Camera
Primary area: applications
TL;DR: E-MoFlow unifies egomotion and optical flow estimation through implicit regularizations under a self-supervised paradigm.

Abstract: The estimation of optical flow and 6-DoF ego-motion—two fundamental tasks in 3-D vision—has typically been addressed independently. 
For neuromorphic vision (e.g., event cameras), however, the lack of robust data association makes solving the two problems separately an ill-posed challenge, especially in the absence of supervision via ground truth.

Existing works mitigate this ill-posedness by either enforcing the smoothness of the flow field via an explicit variational regularizer or leveraging explicit structure-and-motion priors in the parametrization to improve event alignment.
The former notably introduces bias in results and computational overhead, while the latter—which parametrizes the optical flow in terms of the scene depth and the camera motion—often converges to suboptimal local minima.

To address these issues, we propose an unsupervised pipeline that jointly optimizes egomotion and flow via implicit spatial-temporal and geometric regularization. First, by modeling camera's egomotion as a continuous spline and optical flow as an implicit neural representation, our method inherently embeds spatial-temporal coherence through inductive biases. Second, we incorporate structure-and-motion priors through differential geometric constraints, bypassing explicit depth estimation while maintaining rigorous geometric consistency.

As a result, our framework (called \textbf{E-MoFlow}) unifies egomotion and optical flow estimation via implicit regularization under a fully unsupervised paradigm. Experiments demonstrate its versatility to general 6-DoF motion scenarios, achieving state-of-the-art performance among unsupervised methods and competitive even with supervised approaches. 
Code will be released upon acceptance.

E-MoFlow: Learning Egomotion and Optical Flow from Event Data via Implicit Regularization
Wenpu Li
Bangyan Liao ,2, Yi Zhou Qi Xu!d, Pian Wan
Peidong Liu't,
Westlake University 2Zhejiang University
Hunan University
*Wuhan University
Georgia Institute of Technology
Project Page: https://akawincent www.wwwwwwwweeiiiiibbb.b.. git.ub.itoiioo iombiiio/Eee /ttthub.oo/MMoooww/ /MooMoooww
Abstract
The estimation of optical flow and 6-DoF ego-motion, two fundamental tasks 3D vision, has typically been addressed independently. For neuromorphic vision (e.g., event cameras), however, the lack robust data association makes solving the two problems separately ill- plseed challenge, especially in the absence of supervision via ground truth. Existing works mitigate this ill posedness by either enforeing the smoothness of the flow field via explicit variational regularizer leveraging explicit structure-and-motion priors in the parametrization to improve alignment. The former notably introduces bias in results and computational overhead, while the latter, which parametrizes the optical flow in terms of the scene depth and the camera motion, often converges to suboptimal local minima. To address these issues, we propose an unsupervised framework that jointly op- timizes egomotion and optical flow via implicit spatial-temporal and geometric regularization. First, by modeling camera's 's egomotion as continuous spline and optical flow as an implicit neural representation, our method inherently em- beds
coherence through inductive biases. Second, we incorporate srrucure-anddmmttion priors through differential geometric constraints, bypassing explicit depth estimation while maintaining rigorous geometric consistency. As result, our framework (called E-MoFlow) unifies egomotion and optical flow estimation via implicit regularization under fully unsupervised paradigm. Exper- iments demonstrate its versatility to general 6-DoF motion scenarios, achieving state-of-thee thrt performance among unsupervised methods and competitive even with supervised approaches.
Introduction
Optical flow estimation [ and 6-DoF camera motion recovery [2] are two core building blocks in many 3D vision tasks, playing crucial role in providing motion and structural information for various downstream applications such as object tracking S 4) scene reconstruction [ 6. and Simultaneous
Localization and Mapping (SLAM) (7.9). In classical computer vision, these two problems have been extensively studied and can be successfully solved independently. thanks to well-established feature extraction techniques [10] and data association [[]] algorithms. However, when applied to the emerging field of neuromorphic vision 12]. these traditional methods face significant challenges.
The unique nature of event cameras 112, which reports asynchronous and sparse events instead of capturing frames, brings challenges estimating optical flow and 6-DoF camera motion reliably. For example, optical flow estimation from event data faces the well-known aperture problem (13), where the learned flow is essentially the normal flow [4]. Furthermore, depth-free 6-DoF motion
Equaal Contribution: (liwenpu, Seweobaannyaanneeest.ese edu. Corresponding author: liupei donpeidongewettaaee edu. en
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Neural Implicit Optical Field
Input Events
E-MoFlow
EgoMation Spline
(a) Our method takes imput of events, and predicts the dense and (b) Recovered dense optical flow fields cor- responding images of warped events (IWE).
continuous optical flow and ego-motion.
Figure I: Illustration of E-MoFlow.
unless a locally constant depth assumption imposed on the event data [16]. These challenges stem primarily from the absence of reliable long-term association in event data, rendering independent estimation of these two problems ill-posed and error- proore
estimation in general scenes has been proved to be theoretically intractable [15
To overcome these challenges, existing approaches have primarily focused on regularization tech- niques constrain the inherently ill ill-posed nature of these problems. One common strategy leverages seatial-tempora regularization to explicitly enforce optical flow continuity over both time and space [17-19) These methods typically incorporate additional loss terms during optimization, which helps stabilize solutions but introduces trade-offs: the regularization constraints may bias flow estimation while simultaneously increasing computational complexity. embeds these regularization priors implicitly through learned representations. We formulate camera egomotion as a spline in the space of first-order kinematics (20) and optical flow as an implicit neural representation [21). which intrinsically encode spatial-temporal continuity. This representation-driven regularization climinates the need for explicit constraint terms while maintaining solution stability.
contrast, our approach
Another line of works 16 19 introduce geometric regularization by jointly estimating motion and depth. These approaches have shown improvements in optical flow accuracy, as depth information provides valuable constraints for motion estimation. However, these methods typically rely on motion fields [22] or re-projection equations 6 to relate optical flow, depth, and camera motion. This explicit depth estimation increases the degrees of freedom in the model, leading to higher risk of local minima and instability during the optimization process. To address this issue, we adopt differential geometric constraints 23 to jointly estimate egomotion and optical flow without requiring explicit depth estimation. This approach implicitly incorporates geometric regularization, stabilizing the solution while retaining the ability to accurately estimate egomotion and flow.
In summary, E-MoFlow as shown in Fig. | unifies egomotion and optical flow estimation through implicit regularization under a fully unsupervised learning paradigm. We conduct extensive experi- ments across variety of 6-DoF motion scenarios, demonstrating the applicability and robustness of our method. Experimental results demonstrate that E-MoFlow outperforms existing unsupervised methods and achieves comparable performance supervised approaches.
2 Related Work
Event-based Optical Flow Learning Event cameras asynchronously measure per-pixel intensity changes, not absolute values at fixed intervals [12.. This enables high dynamic range and low-latency sensing. However, estimating optical flow from local event patches inherently suffers from the aperture problem [13] [4]. Because traditional optimization--asse methods struggle to recover full motion fields from this ambiguous local data, learning-basee approaches have become dominant. These learning-based optical flow methods are broadly categorized as supervised, semi-supervised, and unsupervised.
Supervised methods (24 rely on dense ground-truth optical flow for training, typically requiring large-scale synthetic or real-wor real-world datasets. However, acquiring accurate flow annotations at scale is prohibitively expensive, often leading to significant sim-to-real gap that restricts their practical deployment in real-world environments. Semi-supervised methods, such as [8 33 mitigate this limitation by incorporating grayscale images and enforcing photometric consistency as

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pervisory signal. While these approaches reduce dependency on labeled data, their performance highly sensitive to the quality of reconstructed intensity images, making them unreliable in ex- treme conditions (e.g., high-speed motion or low-light scenarios). In contrast, unsupervised learning methods [7 19 27 32 [5/37] operate solely on event data, eliminating the need for external super- vision. These methods typically optimize optical flow using contrast maximization (CMax) objectives [5]. which align event warping with estimated motion. Depending on their learning paradigm, unsupervised approaches can be further divided into online and offline fashion. The former (e.g.. B2 (7) employs neural networks to predict optical flow directly from event representations (e.g., event images or voxel grids). These approaches enable efficient, real-time inference but may suffer from error accumulation due to their feedforward nature. The latter (e.g.. [17]) iteratively refines flow estimates from scratch for each new event batch. While more computationally intensive, this kind of method avoids the limitations of learned feature extraction, though often at the cost of reduced efficiency compared online techniques.
In this work, we adopt an unsupervised learning paradigm that combines the efficiency of neural networks with high estimation accuracy. Our approach eliminates reliance on labeled data or auxiliary intensity images, ensuring robust performance across diverse and challenging scenarios.
Event Camera Egomotion Estimation Direct egomotion estimation from event streams represents fundamental yet highly challenging problem in event-based computer vision. Existing approaches primarily follow either 1) linear solver or 2) nonlinear optimization paradigms, each with distinct advantages and limitations.
Linear solver methods employ well-designed geometric constraints to derive closed-form motion solutions. For instance, Ev EvLinearSolver [1] achieves 3-DoF rotation estimation and 6-DoF motion (by incorporating depth priors) through event-based normal flow constraints. Related work by [38 39 utilizes event- based line features for translation estimation when angular velocity is known. As a result, these methods fundamentally require some form of prior knowledge, preventing complete egomotion recovery from event data alone. On the nonlinear optimization front, -based approaches [1
warp events to a reference timestamp and optimize motion parameters by maximizing the contrast in the resulting Image of Warped Events (IWE). While effective for certain motion patterns, these methods face critical challenges including degenerate solutions like event collapse [41 and inherent limitations in recovering full 6-DoF motion without depth priors (or necessarily assuming a constant depth shared by local events). Alternative spatial-temporal registration methods 46) leverage time surfaces for motion estimation, offering computational efficiency through sparse event processing but demonstrating increased sensitivity to noise compared to contrast-based techniques. More recently, [47] et al. propose iterative optimization pipeline for 6-DoF motion estimation using event- based line features.
40 42
In conclusion, current solutions whether linear or nonlinear- still cannot achieve reliable 6-DoF motion estimation in general scenarios without restrictive assumptions. This fundamental limitation highlights the need for more robust approaches capable of handling the full complexity of egomotion estimation from event data.
Flow and Motion Joint Estimation Joint estimation of optical flow and camera motion typically builds upon the motion field theory [22] or re-projection constraints (6) which inherently require depth estimation. [99 pioneers an event-based framework using discretized apatial-temporal volumes to jointly predict optical flow and ego-motion, employing motion compensation through motion blur based and temporal re-projection losses for unsupervised learning. Similarly, [48) utilizes the Evenly- Cascaded Convolutional Network to parameterize depth and pose, employing the re-projection error between warped event images and the current frame as a supervisory signal to jointly optimize depth, motion, and optical flow.
distinct approach is presented by [16), which introduces a contrast maximization framework for simultaneous estimation of optical flow, depth, and egomotion through motion field parameterization. However, these joint estimation methods face critical challenge: the inclusion of depth estimation expands the parameterization space, introducing additional degrees of freedom that require stronger regularization to avoid convergence to local minima.
Neural Flow Fields Recent studies 21 49 have demonstrated the significant potential of using deep neural network as continuous and memory-effiiiet implicit representations for tasks such as view synthesis (50) (51), scene flow estimation 552 (53), and Non-Rigia Structure from Motion (54).
3

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Differential
Neural
Optical Flow
plan Courdinate
Network
Spline
EgoMotion
Differential Geometric Loss
Input Events
Figure 2: Training Pipeline of E-MoFlow. Given the input event data, we use differential flow
loss Eq. ( and differential geometric loss Eq. (9) to train the neural network Eq. (2) and spline
parameters Eq. (6). These two losses are optimized together until convergence. Solid arrows denote
the forward process; dashed arrows denote gradient backpropagation.
This class of methods leverages coordinate-basse networks to store the mapping from spatio-temporal coordinates target vectors, such as radiance values or motion fields. The inherent smoothness bias of the neural network architecture itself serves as powerful "neural prior," providing implicit regularization for downstream problems and replacing traditional handcrafted regularizers.
By modeling the differential properties of the scene [22 54], this paradigm can be extended to represent entire temporal dynamics Ordinary Differential Equations (ODEs) and to estimate complex, long-term 3D trajectories. This provides solid foundation for our proposed method.
Methodology
In this section, we will first introduce the continuous representations of optical flow and camera motion (Sec.3.1) 3.1). Next, we will introduce the differential losses used in our work (Sec.3.22), including differential flow loss Eq. (8) and differential geometric loss Eq. (9). Finally, summarize
training pipeline in Sec. 3.3
3.1 Continuous Representations
To include the spatial- temporal consistency prior through continuous representations implicitly, we model ego-motion and optical flow as spline (20] and implicit neural representations (2]] respectively.
Continuous Flow. Implicit neural representations 21 model the continuous signal through a spatial-temporal coordinate based neural network. Specifically, given time and normalized pixel coordinate x, the neural network will output a normalized optical flow vector (t x). We can write it
(t x) (t, x),
(1)
where we denote the parameters of this neural network as The detailed implementation can be found in Sec. [.1 Besides, following the Neural Ordinary Differential Equation (Neural ODE) [55], we can reformulate the warping trajectory of an event ex (xx, 4x neural ODE solution. In particular, given the initial condition, the warping terminal point at time & can be denoted as (t) + (t), t} and it satisfies the following equation:
dxx(1) NN. X (t)),
(x(t)) = Xx.
(2)
dt
This ODE can be integrated by any off-the-shelf ODE solvers (e.g., euler [56]. rk4 57 dopri5 [58] with the solution defining the event warping trajectory:
/
x (t) =x
(No(s,x((s) ds J
For the backward gradient, this can be solved by a reverse adjoint ODE. Given a vcaler-valued loss function at the reference time point L(x; (trer)), the adjoint state a (t) = dL /dxx (t) namely the gradient at time can be represented by
da,(1) a (t) dt
ONN O (1. (x) Ox (t)
(ther) = dI (T((xx((Meet)
(4)
dx (Arot)

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Fls
our approach directly FLow Goomebr (um))
(a) Parameter Landscape
(b) Original IWE at A
(c) IWE at minimal B
(d) IWE minimal C
Figure 3: Landscape of different loss functions. By jointly estimating these two losses, we can avoid getting stuck in local minima, making it possible to solve the ill- ill-posed problem.
Finally, to get the gradient to the parameters, we can perform simple integral along time as:
dL
A (t)
ONN xx))))
de
00
Remark. Unlike previous work, which directly models displacement [7
models optical flow (velocity field). This enables our method to be applicable to scenes with more aggressive motion. Additionally, direct backpropagation of gradients can lead to gradient explosion and excessive memory usage [55]. The Neural ODE (55] approach mitigates this by modeling the backpropagation of gradients as an adjoint ODE, significantly reducing memory consumption.
Continuous Motion. Unlike optical flow, camera ego-motion diwdiimenionna,, and we represent it using cubic B-spline 22)] Specifically, given control points 3r E IR', the angular and linear velocities wa (L), (1) at time can derived as follows:
We (t) v2()) = Σ Bts(t)8s
(6)
1=0
where B,,3 denotes the cubic basis function of B-spline. For more details, please refer to the supplementary material.
3.2 Loss Functions
Building the continuous representations introduced earlier, in this section, continue by present- ing the differential losses used in our work. Specifically, the differential flow loss Eq. ( employs the CMax loss [15] to learn optical flow. Additionally, differential geometric loss is proposed in Eq. (9) to handle the 6-dof motion scenario. By combining these two losses, we are able to overcome the ill-posed problem while simultaneously avoiding the need for depth estimation.
Differential Flow Loss For the differential flow loss, follow the standard contrast maximization pipeline (15) to learn the continuous optical flow field. Given a set of events input {ex) eb = {x2, 4x with size N, following the definition warping trajectory defined Eq, (3). we can warp each of them to a reference time tyets denoted as Ettrr = (ex(trr)) Then, all the event are accumulated into an image of warped events (IWE),
N
N
1(x, E(tret)
Σ (((xxxx(tre)) o²). km
where defines the gaussian smoothing kernel size. Then, by the following differential flow loss, can measure the concentration of events.
w
Lara (E (E(tre)),
Σ Σ -1)2
(8)
HW
W where H, W and μu Σ"1 )= Lg denotes the hight, width and the mean of IWE, respectively.
5

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Differential Geometric Loss In the multi-view geometry literature, there are two equations connect
optical flow and ego-motion: the motion field equation [2 and the epipolar equation 23]. Although not widely known, the differential epipolar equation (3 can theoretically be viewed as the differential
forms of the epipolar constraint. Our differential geometric loss primarily based on this equation.
Specifically, the 6-DOF motion differential geometric loss in homogeneous coordinate can be defined to be:
where st(t) Lgeomenry (t, 8) = (t x) " [tx(t)) - x (t())
hat denotes the homogeneous coordi-
nate and [: denotes the skew-symmetric operation.
Remark. As shown in Fig. 3 by jointly estimating these two losses, we can prevent optical flow estimation from getting trapped in local minima, making it possible to solve the inherently ill-posed problem involving translational motion.
3.3 Training Pipeline
After collecting the representations and losses, pipeline. As shown in Fig. 2 given a sequence of event data E, we use differential flow and differential geometric losses to train the neural network and spline parameters.
ready to build the unsupervised training
min Ees [L (Eneigh(( 0)] + B(x, (egeomerry (t, x, 0,3) 0,3
(10)
For the differential flow loss, we randomly select a reference time heer and a set of surrounding events Eneigh ((ee) (fzer)
around it, then evaluate the differential flow loss T- (Eneigh (her), 0) and back propagate the gradient using Neural ODE to train the network parameters 0. For the differential geometric loss, we randomly sample some events e = fxx 1} E, collect the corresponding optical flow for each event (t, x) and camera velocity (
derived from the spline 3 at time t, compute and accumulate the differential geometric loss, and then back propagate the gradient to train both the neural network 0 and the spline B. These two losses are optimized simultaneously until convergence.
4 Experiments
4.1 Experimental Setup
Datasets and Metrics. We conduct comprehensive evaluations the MVSEC dataset (64), which is the de facto standard dataset used in prior work to benchmark optical flow and 6-DoF ego-motion estimations. This dataset contains both indoor sequences recorded by drones and outdoor sequences recorded by vehicles. Additionally, we benchmarked optical flow estimation on the more challenging DSEC 65 dataset, which features complex textures, diverse motion patterns, and varying lighting conditions. For optical flow evaluation, we compute three standard metrics: endpoint error (EPE), angular error (AE) and the percentage of pixels with EPE 3 pixels (denoted by "% Out"), exclusively over valid ground truth regions with event activity in the evaluation interval. For motion estimation accuracy, we adopt the root mean square error (RMSE) metric proposed in [14 to measure angular velocity (%/9 (*/s) and linear velocity (m/s) errors.
Baselines. Optical flow estimation using event camera is a widely explored task with various methodological paradigms. We categorize existing approaches into four primary classes: Supervised Learnin(SL.), which requires ground truch optical flow for training (e.g., E-RAFT 26 EV-FlowNet-
EST 25 EV-FlowNet+ (29]. DCFIFlow [59]. TMA 24, ADM-Flow (31) Semi-Supervised
Learning (SSL), leveraging grayscale images as supervision through photometric loss construction (e.g., EV-FlowNet [18), STE-FlowNet [33)); Unsupervised Learning (USL), relying solely on event data by warping events reference time and maximizing accumulated image contrast (c.g.. MotionPriorCMax [36.. ConvGRU-EV-FIOWNY
[32. FireFlowNet (27). USL-EE-FlowWNr [19]. ET-FlowNeT 35), EV-MGRFFloNNet (63. Paredes et al. (37]); and Model-Based (MB) methods, which also adopt contrast maximization objectives but employ traditional nonlinear optimization instead of neural networks (e.g., M22 MultiCM [17], Akolkar el al. [3], Nagata el al. (60, Brebion el al. [61, Cuadrado el al. (28). Shiba et al. (62).. From these paradigms, we select representative prior works as strong baselines for comprehensive quantitative benchmarking of optical flow estimation performance.

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Table Quantitative comparison of optical flow estimation task on MVSEC dataset. Bold the best among all methods; underlined is second best. Pink represents the best in the 'USL'; Orange represents the second best in the 'USL.
indoor_ indoor_fyingl indoor_ indoor_fying2
indoor_ indoor_fyyng3
outdoor_day
average
dt
EPE %0ut +
EPE + %0ut
EPE J %0ut L
EPE %0ut 4 EPE + %0ut.
FV-FFowwNet-FST
0.97
0.91
1,38
8.20
1.43
6.47
NV-FlowNet+
0.56
1.00
0.66
1.00
0.59
1.00
0.68
0.623
0.998
E-RAFT 26
1.10
5.72
1.94
30.79
1.66
25.20
0.24
1.70
1.235
15.853
DCEIFlow 59
0.75
1.55
0.90
2.10
0.80
1.77
0.22
0.00
0.668
1.355
TMA 224
1.06
3.63
1.81
27.29
1.58
23.26
0.25
0.07
1.175
13.563
ADM-Flow
31
0.52
0.14
0.68
1.18
0.52
0.04
0.41
0.00
0.533
0.340
EV-FlowNet 18
1.03
2.20
1.72
15.10
1.53
11.90
0.49
0.20
1.193
7.350
STE-FlowNet (33
0.57
0.10
0.79
1.60
0.72
1,30
0.42
0.00
0.625
0,750
Akolkar al.
13
1.52
1.59
1.89
2.75
1.938
Nagata et al.
60
0.62
0.93
0.84
0.77
0.790
Brebion et al.
6
0.52
0.10
0.98
5.50
0.71
2.10
0.53
0.20
0.685
1.975
Cuadrado
28
0.58
0.72
0.67
0.85
0.705
Shiba al.
62
1.05
2.90
1.68
13.44
1.43
8.97
0.94
3.08
1.275
7.098
MultiCM 17
0.42
0.09
0.60
0.59
0.50
0.29
0.30
0.11
0.455
0.270
MultiCM- V2
[6
0.30
0.00
0.47
0.01
0.34
0.00
0.28
0.21
0.348
0.055
USL-EV-FlowNNG
19
0.58
0.00
4.00
0.87
3.00
0.32
0.698
1,750
FireFlow.Net 27
0.97
2.60
1.67
15.30
1.43
11.00
1.06
1.283
8.875
USL
ConGGRU-EV-FloNNee
0.60
0.51
1.17
8.06
0.93
5.64
0.47
0.25
0.793
3.615
ET-FlowNeT 35
0.57
0.53
1.20
8.48
0.95
5.73
0.39
0.778
3.715
EV-MGRFlowNNct
03
0.41
0.17
0.70
2.35
0.59
1.29
0.28
0.02
0.495
0.958
Paredes
77
0.44
0.00
0.88
4.51
0.70
2.41
0.27
0.05
0.573
1.743
MotionPriorCMuxx
36
0.45
0.09
0.71
2.40
0.60
0.93
Ours
0.40
0.30
0.52
0.18
0.46
0.29
0.42
0.54
0.450
0.328
di
E-RAFT
26
2.81
40.25
5.09
64.19
4.46
57.11
0.72
1.12
3,270
40.668
DCE1Flow
59
2.08
21.47
3.48
42.05
2.51
29.73
0.89
3.19
2.240
24.110
TMA 24
2.43
29.91
4.32
52.74
3.60
42.02
0.70
1.08
2.762
31.438
ADM-Flow
31
1.42
7.78
1.88
16.70
1.61
11.40
1.51
10.20
1.605
11.520
EV-FlowNet
18
2.25
24.70
4.05
45.30
3.45
39.70
1.23
7.30
2.745
29.250
STF--FloWNet 33
1.77
14.70
2.52
26.10
2.23
22.10
0.99
3.90
1.878
16.700
Shiba al.
62
4.06
53.88
6.39
71.82
5.36
65.57
3.60
49.04
4.853
60.077
MultiCM 11
1.68
12.79
2.49
26.31
2.06
18.93
1.25
9.19
1.870
16.805
MultiCM-V V2
16
1.18
4.77
1.87
15.51
1.38
7.62
1.05
5.68
1.108
8.305
USL-EV-Flow.Ndt
19
2.18
24.20
3.85
46.80
3.18
47.80
1.30
9.70
2.628
32.125
ConvGRU-EY--lowNee
32
2.16
21.51
3,90
40.72
3.00
29.60
1.69
12.50
2.688
26.083
ET-FlowNeT
2.08
20.02
41.33
3.13
31.70
1.47
2.667
25.555
EV-MGRFlowNel
63
1.50
23.70
2.06
18.00
1.10
1.763
14.147
Ours
1.58
2.04
18.54
1.84
13.57
1.63
14.42
1.773
13.933
For 6-DoF motion estimation, existing methods can be categorized into two classes based on whether depth prior knowledge is required. Approaches such as AEmmi [42), Inemin (66), and PEME [43] require depth-augmented event data to achieve 6-DoF estimation, while ECN (88 and MultiCM-V2 [16] can perform a 6-DoF estimation relying solely on raw event streams without depth priors.
Implementation Details. The neural implicit flow field adopts the MLP architecture. The detailed network architecture can be found in appendix. Following prior works [17 43 we partition the entire event sequence into multiple segments during training, with each segment containing 30k events for MVSEC [64] and 300k events for DSEC [65] Because the time interval of each segment is short, the cubic spline modeling continuous camera motion employs only 4 control knots, whose 6-dimensional vectors are initialized to constant value of 0.2. To solve the event warping trajectory Eq. (2), we employ the euler solver for its computational efficiency in numerical integration. The weight of the differential geometric loss set to 0.25 and the differential flow loss to We employ two separate AdamW optimizers
[67] for the neural implicit flow field NN, and the camera motion parameters 3. For the MVSEC dataset, the learning rate for the flow field exponentially decayed from x 10 to 6.3 x 10 whereas for the DSEC dataset, is cosine annealed from 2 x 10-3
7

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Table 2: Quantitative comparison of optical flow estimation task DSEC dataset. Bold is the best among all methods; underlined is second best. Pink represents the best in the 'USL'; Orange represents the second best in the 'USL'.
All
interlaken 00_b
interlaken 01
thun 01
Method
EPE 1
AE L 90Out
EPE 4 AE 1 %0Ott
EPE 4
AE ] %0Ou
EPE L
AE J
S/Out
(SL) E-RAFT 26
0.79
2.85
2.68
1.39
2.36
6.19
0.90
2.54
3.91
0.65
2.94
1.87
(SL) TMA
0.74
2.68
2.30
1.39
2.16
5.79
0.81
2.23
3.11
0.62
2.88
1.61
(MB) MultiCM-V2
16
3.47
13.98
30.86
5.71
9.19
38.93
3.71
9.77
31.37
2.12
11.06
17.68
(USL) Paredes al.
37
2.33
10.56
17.77
3.34
6.22
25.72
2.49
6.88
19.15
1.73
9.75
10.39
(USL.) MotionPrrar( Max
16
3.20
8.53
15.21
3.21
4.89
20.45
2.38
5.46
17.40
1.39
6.99
7.36
(USL) EV-FlowNNet
19
3.86
31.45
6.32
47.95
36.07
2.33
20.92
(USL) Ours
3.14
10.87
19.43
7.21
14:13
35.53
3.18
7.52
19:21
1.83
6.89
12:65
thun_ _0____1bb
zurich city 12_
zurich_eitt
14_c
zurich city 15_a
Method
KPH 4
AE J.
98OOt
KPE 4.
AE J.
(5Out
KPH 4.
AE 4..
18/Ot
KPH 4.
AE J.
95/Out
(SL) E E-RAFT 26
0.58
2.20
1.52
0.61
4.50
1.06
0.71
3.43
1.91
0.59
2.55
1.30
(SL) TMA 24
0.55
2.10
1.31
0.57
4.38
0.87
0.66
3,09
1,99
0.55
2.51
1.08
(MB) Mu
17
2.48
12.05
23.56
3.86
28.61
43,96
2.72
12.62
30.53
2.35
11.82
20.99
(USL) Paredes al
37
1.66
8.41
9.34
2.72
23.16
26.65
2.61
10.23
23.01
1.09
8.88
9.98
(USL) MotionPriorCMM Max
36
1.54
6.55
9.69
8 8.33
20,16
22.39
1.78
8.79
12.99
1.45
6.27
8.34
(USL.) HV-Hoow Net
3.04
25.41
2.62
25.80
3.36
36.34
2.97
25.53
(USL) Ours
1.66
5.62
10.55
3.52
24.92
28.51
1.89
6.13
15.44
1.51
5.66
9.08
Table 3: Quantitative comparison of 6-DoF egomotion estimation task on MVSEC dataset. The notation "w/ D" denotes requiring depth prior information, whereas "w/o D" indicates no need for depth information. Bold is the best among all methods; underlined is second best.
indoor_ indoor_flying!
indoor_ flying2
indoor_flyngg
outdoor_day _d_y
average
R
RMS.
RMS_ RMS
RMS RMS.
RMS RMS.
RMS RMS.
AEmin 42
1.38
0.069
4,00
0.677
IncEmin 66
1.65
0.085
4.05
1.035
PEME +
1.05
0.039
2.72
0.598
ECN 48
0.70
w/o
MultiCM-Vi V2 16
7.72
0.24
11.50
0.27
9.53
0.31
6.85
5.90
8.900
1.680
Ours
3.44
0.11
5.31
0.12
4.12
0.15
3.38
0.76
4.062
0.285
to 10-7 The learning rate for the camera motion is kept constant at 10-3 for both datasets. Each short event segment is trained for Ik iterations. All experiments were conducted on a NVIDIA RTX 4090.
4.2 Results on Optical Flow Estimation
Quantitative Results. We conduct comprehensive quantitative benchmarking for optical flow
estimation on the MVSEC dataset [644] and DSEC dataset 168] detailed in Tab. [nd Tab. 2
The MVSEC benchmarks optical flow estimation over one-frame (dt = 1) and four-frame (dt- = 4) intervals, respectively. Our method outperforms all existing unsupervised learning methods on the MVSEC dataset. Notably, under the dt = setting, our method ranks second among all methods, only behind the traditional optimization- method MultiCM-V2 [16], and outperforms all supervised learning methods. On the DSEC benchmark, our method performs comparably to existing unsupervised learning approaches. However, there remains performance gap compared to supervised methods. This is primarily because driving scenes involve more complex motion patterns, abrupt illumination changes, and unstructured scene depth, all of which make it more challenging for unsupervised methods to learn accurate optical flow. The experimental results demonstrate that by introducing implicit spatial-temporal continuity constraints Eq. (2) and differential geometric constraints Eq. ( our method can unlock the network's 's representation capacity in an unsupervised learning manner, thereby predicting more accurate optical flow.
Qualitative Results. We performed a qualitative comparison with strong baselines on the MVSEC dataset and DSEC datasets. As shown in Fig the results on the MVSEC datsets demonstrate that our method predicts optical flow closer to the ground truth. Fig. o dillustrates that our method also
8

---

Indoor Rying!
indoor flying2
indoor Rying3
Figure 4: Qualitative results of 6-DoF motion estimation on the MVSEC dataset. Due to space constraints, this figure only presents the linear velocity estimation results. The angular velocity results are provided in the supplementary material. The top, middle, and bottom rows in each subfigure correspond to the z-axis, y-axis, and z-axis results, respectively.
GT
Oura
MuliCM
ComACRU--VV-FIWWN.
E.V. Flowwee
Figure Qualitative results of optical flow estimation on the MVSEC dataset. The results in the first, second, and third rows correspond to sequences indoor flying,, andoor_fying. and indoor flying3, respectively.
achieved good performance on the DSEC dataset and features smooth optical flow visualizations. Note that we visualize optical flow only at pixels where events are triggered.
4.3 Results on 6-DoF EgoMotion Estimation
Quantitative Results. To evaluate the performance of 6-DoF motion estimation, we compared our method with existing approaches on the MVSEC dataset [64], provides ground truth camera motion including angular and linear velocities. As shown in Tab. 3 our method achieves state-of- the-art 6-DoF motion estimation performance in both indoor and outdoor scenes. This primarily attributed our method's use of splines to represent camera motion Eq. (6). which provides a strong
continuity prior, combined with our designed differential geometric loss Eq. (9). This loss enables optical flow information to offer geometrically meaningful guidance for motion learning without requiring depth priors.
Qualitative Results. We also conducted qualitative evaluation on 6-DoF motion estimation. As illustrated in Fig. 4 the 6-DoF motion estimated by our method closely matches the ground truth, demonstrating that our approach can achieve outstanding motion estimation performance without requiring depth information in unsupervised paradigm.

---

Ours
MotionPriorCMaax
Paredes et al.
E-RAFT
Figure 6: Qualitative results of optical flow estimation the DSEC dataset. The results in first, second, and third rows correspond to sequences jurich_city_1aa surich_city__ _4c and interlaken respectively.
Table 4: Ablation studies on differential geometrie constraints.
indoor ffying]
indoor indoor_ftying2
indoor flying3
outdoor outdoor_dayl
dt =1=4
EPE
%Out |
EPE J
%0ut
EPE 4
%Out 4
EPE 4
%0ut +
w/o geometric constrain
1.84
9.68
2.27
18.76
2.27
13.72
1.67
15.13
w/ geometric constrain
1.58
9.2
2,04
18.54
1.84
13.57
1.63
14.42
ground truth motion
1.56
9.13
2.04
18.43
1.84
13.57
1.61
14.25
4.4 Ablation Study
To validate the efficacy of differential geometric constraints Eq. conducted ablation studies the MVSEC dataset [64] under dt 4 with three configurations: a) no geometric constraints, b) with differential geometric constraints, and e) direct supervised by ground truth motion. As demonstrated in Tab. 4 the differential geometric constraints yield improvements in optical flow estimation, even achieving performance comparable those supervised with ground truth motion on undoor_fying and indoor_fyings? This indicates that our design enhances the geometric plausibility of the estimated optical flow while effectively avoiding convergence local minima, as visualized in Fig.
5 Conclusion
This work presents E-MoFlow, novel framework that unifies 6-DoF egomotion and optical flow estimation using implicit spatial-temporal and geometric regularization within an unsupervised learning paradigm. By incorporating implicit neural representations with differential geometry constraints, our approach effectively tackles the ill-posed challenges separate estimations of flow and egomotion from event data. Extensive experiments demonstrate that E-MoFlow achieves state- of-the-art performance across diverse motion scenarios, matching or surpassing many supervised approaches.
This work was supported in part by NSFC under Grant 62202389, in part by a grant from the Westlake University-Muyuan Joint Research Institute, and in part by the Westlake Education Foundation.
Acknowledgements.
10

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Appendix
A
Network Architecture
Our network adopts a simple MLP architecture that takes spatial-temporal coordinates (x, 1) as input and outputs optical flow signal u (u, v). Compared to 19 based MLP implicitly represents optical flow at spatial-temporal coordinates, essentially velocity field, without relying on explicit discrete structures (c.g.. voxel grid, event count image), enabling temporally continuous and dense flow estimation.
27
32
36
37 63 this coordinate-
Figure 7: Schematic Diagram the Neural Implicit Optical Flow Field Network Architecture. The input of the network is three-channel spatial-temporal coordinates (x, l), and the output is optical flow = (u, v). The yellow rectangles represent 256-dimensional hidden units. The orange diamonds denote ReLU activation functions. The blue arrows indicate residual connections, The green arrows represent concatenating the original input to the output the fifth layer.
employs 9 fully-connected layers with 256 dimensional hidden units. The first eight layers utilize ReLU activations to enforce low Lipschitz constant, ensuring smoother responses to input variations [70]. 77 This design suppresses high-frequency features while favoring learning of low- cow-frequency features, aligning with the prior that optical flow exhibits spatial-temporal smoothness [17-1]. Notably, no activation function (e.g., ReLU or sigmoid) is applied to the output layer, optical flow inherently spans both positive and negative values. To further stabilize network training, we introduce residual connections between the second layer to the eighth layer and implement skip connections that concatenate the raw input with the activation outputs of fifth layer. The complete architecture is illustrated in 7
Specifically, our network architecture, inspired by NeRF 69
Although the original NeRF architecture employs positional encoding that enhances high-frequency feature learning [69], our framework deliberately omits such encoding. This design aligns with our goal to model optical flow field which is inherently low-frequency spatial-temporal signals, while avoiding spectral bias toward high-freet tighffreuunncy feature 722
B Continuous Motion Representation
In this section, we discuss how to select an appropriate motion parameterization based on the characteristics of camera egomotion. Given a time t, F maps it to the camera's 's angular velocity w and linear velocity v that moment.
(w,D(, R+E 3
(11)
In scenarios such drones, handheld devices, and rehicle-mounter systems, camera ego-motion is constrained by strong prior assumptions. Specifically, camera motion exhibits temporal continuity and smoothness, meaning abrupt changes oceur within infinitesimal time intervals A/. This prior is formalized as:
dh
& 20...
(12)
d
denotes the order of the derivative and specifies the upper bounds for their respective derivatives. The equation indicates that the k-th order motion derivatives exist and continuous. This can be
22

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simplified as:
F E EC
(13)
C denotes the set of functions that have continuous derivatives up to the k-th order. Additionally, the motion of the camera low-dimensional (2) Thus, there is no need to over-parameterize the camera motion (e.g., using neural networks).
In summary, employ cubic B-spline as F to parameterize the camera motion, its basis functions exhibit C² continuity and compact representation via sparse control knots [20]. Specifically, we four control knots 3 ,,,
R over a time interval [0. 1]. Therefore, the motion parameterization F can be formally defined as:
, $2, fa] /
F(t) (wo (t), (t)) ERR x R
wt(() B(t) / lo:2
(14)
(t))) [B() $ 13:5
This definition allows us to derive the camera's 's angular velocity ()(() and linear velocity (t(() at time t, where (t) E R denotes the cubic B-spline basis functions, defined as follows:
1
3
-3
1 3 2
3
B(t)
1]
(15)
3
0
0
1
0j
This design choice inherently satisfies the prior assumptions: 1) Cubic B-spline intrinsically enforces C² smoothness priors, ensuring natural continuity in velocity, acceleration and jerk without requiring explicit smoothness constraints. 2) By utilizing sparse control knots, this approach model continuous camera motion while maintaining low-dimensional parameterization of the 6-DoF egomotion.
C Differential Geometric Loss
In 3D vision, motion of the camera (w, v) induces a motion field m projected points on the normalized image plane x. Assuming the camera is rigid body, the relationship between the motion field and the camera motion can be expressed by the following equation, which we formulate in homogeneous coordinates:
m(x)
A(x)v + B(x)w, x [xr, 1J
(16)
Z(x)
The matrices A(x) and (x) are functions of homogencous image coordinates defined as follows:
0 x7
ay
(1 x2)
A(x) =
0
-1 y/
B(x) = (1 + v²)
(17)
In practice, m(x) is approximated by optical flow field u(x) [u, u, 0 under brightness constancy
assumption.
u(x)
Z(x) A(x)v + B(x)w
(18)
[q.(18) is a critically important motion field equation, which establishes the relationship between
optical flow and camera egomotion (22)
73
However, the presence of (x) in this equation implies that recovering camera motion from optical flow or deriving optical flow from camera motion requires knowledge of depth values at each image coordinate. Prior works such 14 15 42) rely on depth priors or assume locally shared depth values when performing 6-DOF motion estimation, while methods like [6
19 48 jointly estimate
depth alongside optical flow and 6-DOF motion. However, this expands the parameterization space of the optimization problem, introducing additional degrees of freedom that may lead to convergence to local minima. Therefore, to enable the formulation of an unsupervised loss function that can simultaneously estimate optical flow and 6-DOF motion with high accuracy, we need to eliminate the dependence on (xx)
23

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Table 5: Ablation studies on early stopping strategy.
indoor_ftynng andoor_flingg andoor_flinng outdoor_de _
Time
w/o early stopping
9.30s
9.41s
9.50s
10.77s
w/ early stopping
4.21s
4.30s
4.87s
4.56s
efficiency improvement
2.21
2.19x 1
1.99x 1
2.36x 1
EPE
w/o early stopping
1.58
2.04
1.84
1.63
w/ early stopping
1.61
2.09
1.90
1.68
performance drop
1.90%
2.45% 4
3.26% 4
3.07% 4
We transpose [q and then left-multiply by form the inner product of u(x) and v x x, yieding scalar equation to isolate (xx as follows. x denotes the cross product operation.
u(x)² (v x)
1 2(x)
A(x)v +B(x)w) (w xxx
(19)
Simplify the above equation to obtain:
"xx) p
UT A(x) [xx) x Z(x)
+
(20)
where Hxx denotes the skew-symmetric operation. Interestingly, it can be proven that the coefficient of the term that involves (xx in Eq. is identically zero.
of A(x) 1(x)7(u)x* x
(21)
Therefore, (0) can be further simplified as follows:
u(x)² [ - B(x) M=
(22)
By expanding " [ xx, Eq. 22) can rewritten the following form:
u(x)² [ x sx 0, s ([W) [m]x + [ [w]x)
(23)
Finally, we obtained an equation that connects the optical flow field and camera motion without relying on depth values. (33) can theoretically regarded as differential form of the epipolar constraint. We use this our differential geometric loss to jointly learn optical flow and 6-DoF motion, shown in the following equation.
Ligoomerryy (t, x, 0, () = l|u x) (Ms())) XT s
(24)
(t()) (Iws(t))[[ss(t)) + ( (t)))))
Here, uo(t, x) represents the optical flow obtained from our neural implicit representation, while wt(()) and (t) denote the angular velocity and linear velocity of the camera, derived from the cubic B-spline continuous motion representation.
D More Ablation Studies
Early Stopping Strategy. To enhance computational efficiency, we employed the early-stopping strategy from [52 on the MVSEC dataset (64) under dt = 4. Specifically, we set the patience to 45 and the minimum improvement threshold to 10-3 applying the early stopping strategy after 300 iterations. The results in Tab. indiiaaee that the training speed was boosted by 2.2x, while the optical flow estimation accuracy showed only a slight 2.67% drop the EPE. This demonstrates that the strategy significantly reduces training time at the cost of only minor loss in accuracy, achieving an excellent trade-off.
E More Qualitative Results
We further provide additional qualitative results. As shown in 8 our method achieves comprehensive 6-DoF motion estimation on the MVSEC dataset (64]. The angular velocity and linear velocity estimated by our approach closely match the ground-truth motion.

---

indoor flying!
indoor fying?
indoor flying
Figure 8: Complete qualitative results of 6-DoF motion estimation on the MVSEC dataset. The top section displays the linear velocity estimation results (in m/s), while the bottom section shows the angular velocity estimation results (in deg/s), s). The top, middle, and bottom rows in each subfigure correspond to the z-axis, y-axis, and a-axis results, respectively.
Figure 9: Color wheel for visualizing optical flow. A green color in the optical flow visualization corresponds to motion directed toward the lower-left corner of the image, while the saturation the color encodes the flow magnitude more vivid hues indicate larger displacement values.
For MVSEC datasets [64]. We provide additional qualitative comparisons of optical flow estimation between our method and MultiCM [17]. the second-best performing baseline. As shown in I0 our approach predicts optical flow with superior continuity and smoothness, validating the effectiveness of our neural implicit optical flow field representation. The color wheel used visualize optical flow is shown in g
where different colors encode the magnitude and direction of the optical flow.
For DSEC datasets [68]. We provide additional visualization comparisons of optical flow estimation between our method, state-of-the-art unsupervised learning methods, and supervised learning methods on more sequences. The results in Fig. 11 demonstrate that in scenarios, our method yields visually superior results with smoother optical flow, while in scenes with complex textures and drastic depth changes, it may produce errors at the edges.
Furthermore, we also provide visualization results of the flow field in - y - 3D space on the MVSEC dataset (64). The results in Fig. 12 indicate that method exhibits emergent capability for point tracking, which is the ability to track the movement of points in pixel space.
25

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GT
Ours
MultiCM
GT
Ours
MultiCM
Figure 10: More qualitative results of optical flow estimation on the MVSEC dataset. It can be clearly observed that method estimates smoother optical flow, free from abrupt variations, and demonstrates closer alignment with the ground truth optical flow. This indicates that our approach more effectively models the intrinsically ppatial-temporally continuous optical flow field.
Ours
MotionPriorCMMax
Paredes al.
E-RAFT
Figure 11: More gualitative results of optical flow estimation on the DSEC dataset. Qualitative results of optical flow estimation on the DSEC dataset. The results in the first, second. third, and fourth rows correspond to sequences interlaken 00b, thin Ola, thun 0la, and swrich_ctt__44 respectively.
26

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Figure 12: Point tracking results of flow field on the MVSEC dataset.
27


Reviews:
[{"review_id": "sKDCb3BEno", "reviewer": "Reviewer_j1WT", "summary": "E-MoFlow presents a novel approach to the problem of jointly predicting optical flow and ego-motion from sequential data obtained from an event camera. It's main contribution is an implicit neural flow field formulation which can be optimized at test time and solved integrated like an ODE to produce a continuous flow fields. In addition to optimizing the neural flow field, a cubic spline is also fit for the camera's motion. Both the spline and the flow field are optimized together in order to produce the final output, and they are trained using a combination of two losses. One is a flow specific loss which uses contrast maximization (as in prior work) and the other is a differentiable geometric loss which connects the flow and ego motion together as is commonly done in multi-view geometry.", "questions": "1. In table 3, why are some of the methods missing for certain scenes? Is this because they were using those scenes for training?\n2. Since your method does not crop inputs, is this an advantage in the qualitative results? Why do other methods use a crop? Runtime issues?\n3. Have you tried early-stopping instead of training every event for 1000 steps? In related works (see below) early stopping can improve runtime performance quite significantly.\n\nI think this paper would be well served by making more parallels to other related literature that also does neural representations and optimization. In particular the NeRF literature [1], as well as the related work in scene flow such as Neural Scene Flow Prior [2], Neural Trajectory Prior [3], and Neural Eulerian Scene Flow Fields [4].\n\nOn the topic of the related works, in Neural Eulerian Scene Flow Fields it was shown that by performing Euler integration over the neural ODE representation, continuous point tracking was an emergent property. I imagine that E-MoFlow could exhibit similar properties and this would be a compelling visual to add (even if no metrics for point tracking can be provided).\n\n1. https://arxiv.org/abs/2003.08934\n2. https://arxiv.org/abs/2111.01253\n3. https://openaccess.thecvf.com/content/CVPR2022/papers/Wang_Neural_Prior_for_Trajectory_Estimation_CVPR_2022_paper.pdf\n4. https://arxiv.org/abs/2410.02031", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\nThe main strengths of this method is that it is fully unsupervised and does not require depth priors, while still producing optical flow estimates that are quite good. In certain circumstances, other methods may perform better than E-MoFlow through the use of depth or supervised learning. The overall method is relatively well explained, though a pseudocode section to explain the optimization loop would probably be helpful.\n\n## Weaknesses\nThis paper's main weaknesses are in its presentation. The method could use some extra explanation (it's never explicitly stated in the paper that it performs test time optimization). Similarly, the runtime is not stated which is a notable con of test time optimization approaches.", "was_revised": true, "final_justification": "The authors have satisfactorily answered my questions, and will include the relevant parallels to related work in their camera ready. Their method benefits from the same tricks as many other neural ODE solvers for flow methods such as early stopping, and continuous point tracking behavior, and they included these results in the rebuttal. My score is now accept.", "rebuttal": "## Response to Reviewer j1WT\n\nWe sincerely thank the reviewer for recognizing the key strengths of our method, including its fully unsupervised nature and its ability to produce competitive optical flow estimates without requiring depth priors. We also appreciate the valuable suggestions, which we will incorporate to improve our method. \n\nWe address the main concerns below.\n\n---\n\n> **[W1] Lack of explicit explanation that the paper performs test time optimization.**\n\nIn the camera-ready version, we'll explicitly explain that our method is a test-time optimization approach.\n\n---\n\n> **[W2] Lack of runtime analysis.**\n\nWe report the training time and the inference time of our method in the following table. The analysis has been tested across all sequences in dataset MVSEC. Each sequence is divided into multiple clips with a time interval of approximately 80-100 ms, corresponding to $dt=4$, and each clip contains approximately 30,000 event data. In the table below, we report the average training time and inference time per clip.\n| **Sequence**| &emsp;&emsp;**Training Time**&emsp;&emsp; | &emsp;&emsp;**Inference Time**&emsp;&emsp; |\n|-|:-:|:-:|\n| *indoor_flying1*  | &emsp;&emsp; $9.30s$ &emsp;&emsp; | &emsp;&emsp; $0.86s$ &emsp;&emsp;|\n| *indoor_flying2*  | &emsp;&emsp; $9.41s$ &emsp;&emsp; | &emsp;&emsp; $0.87s$ &emsp;&emsp;|\n| *indoor_flying3*  | &emsp;&emsp; $9.50s$ &emsp;&emsp; | &emsp;&emsp; $0.88s$ &emsp;&emsp;|\n| *outdoor_day1*    | &emsp;&emsp; $10.77s$ &emsp;&emsp; | &emsp;&emsp; $0.86s$ &emsp;&emsp;|\n\nIt's important to note that these runtime results are preliminary, as our method doesn't employ any acceleration techniques, and a simple acceleration strategy (such as the early-stopping evaluated in Q3) could significantly accelerate the method. We will include the detailed runtime analysis in the camera-ready version. \n\n---\n\n> **[W3] Runtime is a notable con of test time optimization approaches.**\n\nWe acknowledge that runtime is a drawback of test-time optimization approaches. However, we'd like to point out that in future work, we'll try using the proposed implicit regularization to unsupervisedly train a feed-forward optical flow prediction network. At that point, its runtime will no longer be a limitation.\n\n---\n\n>  **[Q1] Explanation for some missing scenes in table 3.**\n\nAEmin [1], IncEmin [2], and PEME [3] methods fail on these scenes and ECN [4] struggles to generalize well in complex indoor scene. The detailed analysis is as follows.\n\n**AEmin [1], IncEmin [2], and PEME [3]** methods for estimating 6-DOF motion all require ground-truth depth measurements to augment the event data. However, the depth values obtained from LiDAR are measured at a frequency of only 20 fps, which is much lower than the temporal resolution of the event data. This discrepancy introduces significant noise into the depth-augmented event data. The *indoor_flying2* and *indoor_flying3* sequences feature complex motion patterns and rich scene texture information, which make these methods less robust to noise, leading to their failure on these sequences.\n\n**ECN [4]** requires training on a large amount of event data, but the indoor scene sequences in MVSEC are too short to generate sufficient training data and contain more complex motion patterns. As mentioned in the ECN paper [4], due to the limited training data, ECN struggles to generalize well in complex indoor scenes. Furthermore, its strong performance in outdoor scenes can be attributed to the fact that only two motion parameters play a critical role (forward translation and yaw axis rotation in the vehicle coordinate frame), leading the network to tend to overfit.\n\n---\n\n>  **[Q2] Explanation for no image crop.**\n\nOur method does not require cropping the input, which indeed provides an advantage in the qualitative results. Other methods use cropping primarily due to limitations of their approach. The detailed analysis is as follows.\n\n**a)**  Our model uses a coordinate-MLP to track the motion of events in the $(x, y, t)$ space. We aim for the coordinate-MLP to fully utilize the information from all event data to learn continuous-time motion in the spatio-temporal domain, while leveraging the implicit regularization of neural ODEs to constrain the motion smoothness of adjacent events. Therefore, it is crucial to avoid cropping, which could result in the loss of local spatio-temporal information and disrupt the implicit regularization of motion patterns. This allows us to recover optical flow at the original resolution using the raw event data.\n\n**b)** Other learning-based methods, such as EV-FlowNet [5] and ConvGRU-EV-FlowNet [6], require preprocessing the raw event data into event images as inputs to their networks. Due to the spatial sparsity of event data, event images may contain a large number of regions where no events occur, which contribute little to optical flow estimation. As a result, these methods often use cropping to focus primarily on regions with dense events. In contrast, our method does not require preprocessing the raw events into event images. Instead, it directly uses the $(x, y, t)$ coordinates of the raw events obtained from actual measurements to train the coordinate-MLP. Our method does not introduce a large number of coordinates from regions without events, which lack effective information, for training. Instead, it naturally supports focusing only on regions where real events are triggered. Additionally, the CNN employed by these methods require input data with a fixed resolution, and cropping the event images to a fixed size simplifies the design and training of the model.\n\n**c)** Cropping the input primarily affects the size of data ultimately fed into the network for training, which significantly impacts memory usage, especially with high-resolution inputs. However, for runtime performance, since neural network training and inference leverage the parallel computing power of GPUs, cropping does not have a significant impact on runtime.\n\n---\n\n>  **[Q3] Results of early stop training strategy.**\n\nWe adopt the early-stopping strategy and report the detailed results in the following tables. We find that it significantly reduces the training time with only a slight loss in accuracy, achieving an excellent trade-off. Specifically, we set the patience to 45 and the minimum improvement threshold to 1.0e-3, applying the early stopping strategy after 300 iterations. We find that the training speed was boosted by 2.2x, while the optical flow estimation accuracy showed only a slight 3.41% drop in the EPE. The motion estimation accuracy still maintain SOTA performance. The detailed results are shown in the table below.\n\n**a)** `Training time.` The number in brackets indicates the average training iterations when using early-stopping strategy.\n| **Sequence**        | &emsp;**w/ early stopping**&emsp; | &emsp;**w/o early stopping**&emsp; | &emsp;**Efficiency Improvement**&emsp; |\n|-|:-:|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $4.21s$ $(452.93)$ &emsp;&emsp; | &emsp;&emsp; $9.30s$ &emsp;&emsp; | &emsp;&emsp; $2.21 \\times$ $\\uparrow$ &emsp;&emsp; |\n| *indoor_flying2* | $4.30s$ $(461.84)$ | $9.41s$ | $2.19 \\times$ $\\uparrow$ |\n| *indoor_flying3* | $4.78s$ $(496.77)$ | $9.50s$ | $1.99 \\times$ $\\uparrow$ |\n| *outdoor_day1* | $4.56s$ $(487.51)$ | $10.77s$ | $2.36 \\times$ $\\uparrow$ |\n| *average* | $4.46s$ $(474.76)$ | $9.75s$ | $2.19 \\times$ $\\uparrow$ |\n\n**b)** `Optical flow estimation accuracy(EPE).` \n| **Sequence**        | &emsp;**w/ early stopping**&emsp; | &emsp;**w/o early stopping**&emsp; | &emsp;**Performance Change**&emsp; |\n|-|:-:|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $1.61$ &emsp;&emsp; | &emsp;&emsp; $1.56$ &emsp;&emsp; | &emsp;&emsp; $3.21$ % $\\downarrow$ &emsp;&emsp; |\n| *indoor_flying2* | $2.09$ | $2.04$ | $2.45$ % $\\downarrow$ |\n| *indoor_flying3* | $1.90$ | $1.84$ | $3.26$ % $\\downarrow$ |\n| *outdoor_day1* | $1.68$ | $1.61$ | $4.35$ % $\\downarrow$ |\n| *average* | $1.82$ | $1.76$ | $3.41$ % $\\downarrow$ |\n\n**c)** `6-DoF egomotion estimation.` Each column reports RMSw on the left and RMSv on the right.\n| **Sequence**        | &emsp;**w/ early stopping**&emsp; | &emsp;**w/o early stopping**&emsp; |\n|-|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $5.32$ \\| $0.12$ &emsp;&emsp; | &emsp;&emsp; $3.44$ \\| $0.11$ &emsp;&emsp; |\n| *indoor_flying2* | $5.77$ \\| $0.14$ | $5.31$ \\| $0.12$ |\n| *indoor_flying3* | $4.94$ \\| $0.17$ | $4.12$ \\| $0.15$ |\n| *outdoor_day1* | $5.22$ \\| $1.41$ | $3.38$ \\| $0.76$ |\n\n---\n\n>  **[Q4] More discussions on neural representations and optimization.**\n\nWe will include discussions of these related works in implicit neural flow representation in our camera-ready version.  Our work, which focuses on representing continuous flow in the $(x,y,t)$ spatiotemporal domain, shares a very similar inspiration with these works that model object motion and deformation in 3D space. Both aim to implicitly constrain the smoothness of continuous-time flow through neural representation priors, while modeling continuous trajectories via ODEs or interpolation. \n\n---\n\n>  **[Q5] More continuous point tracking visual results.**\n\nWe will include visualized results of point tracking in the camera-ready version. Due to the rebuttal policy prohibiting image uploads, we're unable to show you these visual results.\n\n---\n\n>  **[MW1] Typo.**\n\nWe will correct this typo in the future version.\n\n---\n\n**Reference:**\n\n[1] Entropy Minimisation Framework for Event-based Vision Model Estimation\n\n[2] Robust Event-based Vision Model Estimation by Dispersion Minimisation\n\n[3] Progressive Spatio-temporal Alignment for Efficient Event-based Motion Estimation\n\n[4] Unsupervised Learning of Dense Optical Flow, Depth and Egomotion from Sparse Event Data\n\n[5] EV-FlowNet: Self-Supervised Optical Flow Estimation for Event-based Cameras\n\n[6] Self-Supervised Learning of Event-Based Optical Flow with Spiking Neural Networks"}, {"review_id": "pN0rJzFr5v", "reviewer": "Reviewer_3SAm", "summary": "This paper introduces E-MoFlow, an unsupervised framework for jointly estimating optical flow and 6-DoF egomotion, which are typically addressed independently in 3D vision. The authors propose to overcome the ill-posed challenge of separate estimation, especially pertinent for neuromorphic vision without ground truth, by implicitly regularizing spatial-temporal and geometric coherence. E-MoFlow models camera egomotion as a continuous spline and optical flow as an implicit neural representation, integrating spatiotemporal coherence through inductive biases. Furthermore, it incorporates structure-and-motion priors via differential geometric constraints, avoiding explicit depth estimation while maintaining geometric consistency. Experiments demonstrate the framework's versatility and state-of-the-art performance in various 6-DoF motion scenarios under an unsupervised paradigm, even competing with supervised approaches.", "questions": "See weakness.", "limitations": "See weakness.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "The author's writing is clear, and the method is also easy to understand, specifically how the author optimizes optical flow and egomotion simultaneously in an unsupervised manner through events. However, I still have some doubts about the design of the loss function.\n\nThe design of the Differential Flow Loss uses the event's IWE from the CM algorithm as ground truth, then aligns it using the optical flow's ODE. Here I have some questions: If the supervisory signal comes from the CM algorithm, is the model's estimation upper limit determined by the CM algorithm (if not, please provide more explanation). What are the differences between this network-based estimation strategy and directly using the CM algorithm to estimate optical flow? Also, when the CM algorithm aligns events, it does so at time t and t_ref. How is it ensured that the network can learn continuous optical flow between these two moments? And how are the gradients for Equation (3) specifically calculated during backpropagation? Does it require multiple gradient accumulations for the flow model, and could this lead to potential OOM issues? Since I am not familiar with the gradient analysis for ODEs in Equations 3-5, I hope the author can provide more explanation in this part.\n\nThe ablation experiment shown in Tab.1 indicates that with the addition of unsupervised loss 'a', its performance is comparable to that with given gt motion. Has the model's robustness been tested here? Or are there any failure cases? Because the unsupervised loss designed by the author requires coupling between two losses, it certainly needs a more refined design compared to supervised methods. I am curious whether this algorithm can still achieve results comparable to supervised effects in cases of very fast motion.\n\nFinally, the idea of jointly optimizing optical flow and egomotion is very interesting. If the authors are interested, they could try adding a multi-view constraint loss provided by 3DGS on top of this. This is quite similar to the geometric loss in the paper, but here it is afforded by 3DGS. Reference: USP-Gaussian: Unifying Spike-based Image Reconstruction, Pose Correction and Gaussian Splatting CVPR 2025", "was_revised": true, "final_justification": "I keep my score.", "rebuttal": "## Response to reviewer 3SAm\n\nWe sincerely appreciate the reviewer’s recognition of our efforts in addressing the challenges of unsupervised optical flow and 6-DoF egomotion estimation through implicit neural representations and spatiotemporal coherence. Also, Thanks for the constructive and thoughtful comments.\n\nWe address the main concerns below.\n\n---\n\n> **[W1] Is CM algorithm the upper limit of E-MoFlow.**\n\nThe performance of E-MoFlow is not determined by the upper limit of the CM algorithm. In contrast, our model uses two types of loss functions, which theoretically leads to better accuracy than relying solely on differential flow loss. This is further substantiated by the ablation studies presented in Table 1 of the main paper.\n\nThe reason for this lies in the inherent limitations of a standalone differential flow loss, which, as illustrated in Figure 3 of our main paper, cannot provide enough constraints (known as the aperture problen or event collapse). Our differential geometric loss, however, offers additional implicit regularization for optical flow estimation, preventing the model from converging to an ill-posed local solution.\n\n---\n\n> **[W2] Differences between the network-based estimation and directly using the CM algorithm.**\n\nCompared to traditional CM methods, our network-based estimation strategy leverages implicit neural representations to model optical flow, achieving implicit regularization of spatio-temporal flow continuity.\n\nSpecifically, traditional non-learning CM algorithms [1] [2] typically model optical flow directly as a vector. This approach has several key limitations:\n\n1. The optimized flow often struggles to effectively fit non-linear motion in complex scenarios.\n\n2. The estimated flow only corresponds to the flow at the starting moment, making it impossible to obtain flow at arbitrary moments between the start and end timestamps. Consequently, it cannot estimate continuous-time flow.\n\n3. Due to the absence of inherent implicit constraints on the flow, traditional CM often necessitates an additional loss function to enforce consistency between neighboring flow vectors.\n\nIn contrast, our method models optical flow using implicit neural representations. This naturally introduces the Multi-Layer Perceptron (MLP) network's inherent bias towards learning low-frequency signals, which effectively constrains the spatio-temporal continuity of the estimated flow.\n\n---\n\n> **[W3] Can E-MoFlow learn continuous optical flow.**\n\nIt is ensured that our method can learn continuous optical flow. Specifically, there are two main reasons for this:\n\n1. Our reference time is randomly selected in each iteration. This effectively prevents the degradation that can occur when only a fixed reference time is chosen.\n\n2. This is also attributed to our use of Neural Ordinary Differential Equations (Neural ODEs) to parameterize the optical flow. This enables us to obtain optical flow at any given moment and to acquire displacement over any time interval through integration.\n\n---\n\n> **[W4] Details of the gradient calculation.**\n\nDuring backpropagation, the gradients for Equation (3) do not require multiple gradient accumulations for the flow model, and do not lead to potential OOM issues. Below are the specific reasons and analysis.\n\nEquation (2) represents a neural ODE system, which parameterizes the optical flow at different times and states using a neural network. Equation (3) illustrates how to calculate the displacement from time $t_k$ to time t by forward-integrating the neural ODE-parameterized flow. Equation (4) is the adjoint ODE system of the neural ODE system. This is a backward system, which can be understood as the sensitivity of the loss with respect to the state x. When Equation (4) is combined with the partial derivatives of the neural network's output with respect to its parameters, we can obtain the partial derivatives of the loss with respect to the neural network parameters at each moment. At this point, Equation (5) shows that to get the derivative of the loss with respect to the neural network parameters along the entire path, we only need to integrate backward to the starting point.\n\nThis process is fundamentally no different from backpropagation; in fact, you can prove they are exactly the same. The only distinction is that, compared to the \"discretize then differentiate\" approach of classical backpropagation, the adjoint ODE method follows a \"differentiate then discretize\" strategy. It transforms backpropagation into an ODE integration, thereby avoiding the storage of large quantities of intermediate variables and complex gradient computation graphs. This characteristic prevents the problems of gradient explosion and excessive memory usage often encountered when applying classical backpropagation to neural ODEs.\n\n---\n\n> **[W5] Robustness to very fast motion.**\n\nOur method remains robust and demonstrates good performance even in very fast motion scenarios. Thanks to our continuously parameterized optical flow and motion, our method can theoretically handle fast, non-linear movements without any performance loss. We conducted experiments on the *outdoor_day1* sequence to further verify this. The *outdoor_day1* sequence in the MVSEC dataset was captured by a vehicle-mounted event camera, with camera motion speeds reaching 12m/s. The results in the table below validate the robustness of our method in very fast motion scenarios.\n| **Sequence**        | &emsp;&emsp;$\\mathrm{EPE}$&emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{\\omega}$ &emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{v}$&emsp;&emsp; |\n|-|:-:|:-:|:-:|\n| *outdoor_day1* | &emsp;&emsp; $1.61$ &emsp;&emsp; | &emsp;&emsp; $3.38$ &emsp;&emsp; | &emsp;&emsp; $0.76$ &emsp;&emsp; |\n\n---\n\n> **[W6] More discussions on related works.**\n\nWe will expand our discussion to include 3D Gaussian Splatting (3DGS). Our method utilizes a differential geometric error, which shares inspiration with the multi-view geometric error used in 3DGS. In future work, we will intend to explore combining these two loss functions.\n\n---\n\n**Reference:**\n\n[1] A Unifying Contrast Maximization Framework for Event Cameras,\nwith Applications to Motion, Depth, and Optical Flow Estimation\n\n[2] Focus Is All You Need: Loss Functions For Event-based Vision"}, {"review_id": "nQE1UTfjeA", "reviewer": "Reviewer_f2hB", "summary": "This paper proposes E-MoFlow, a novel method for egomotion-aware optical flow estimation from event cameras, aimed at improving motion segmentation in dynamic environments. Unlike prior approaches that often ignore egomotion or handle it separately, E-MoFlow introduces a two-stage architecture:\n- A self-supervised egomotion estimator that infers camera motion from a short event window.\n- A rigid flow predictor conditioned on the predicted egomotion.\n- A residual flow module to account for independent object motion.\n\nThe method is trained entirely using self-supervised losses without requiring labels, and it is evaluated on the EV-IMO and EED datasets, where it outperforms or matches state-of-the-art methods in flow estimation and motion segmentation.", "questions": "Is the performance gain of MoFlow primarily due to the increased model capacity (e.g., modularization into egomotion + flow), or due to the proposed motion decomposition strategy itself?\nA capacity-controlled baseline (e.g., same parameter count, no egomotion conditioning) would help clarify this.\n\n\nThe paper does not include runtime, latency, or computational load analysis. This is especially important for event-based systems deployed on mobile robots or embedded devices. Can the authors provide FLOPs, FPS, or real-time feasibility benchmarks?\n\n\nPrior methods for event-based flow have been tested on the DSEC benchmark. Has MoFlow been evaluated on DSEC, or is it fundamentally limited to smaller datasets like EV-IMO or EED?\n\nMoFlow performs worse than some baselines on outdoor_day1, possibly due to pose estimation errors in dynamic scenes. Can the authors comment on this failure case?\nFor example, is the egomotion module struggling due to high foreground motion, or is it a scale/generalization issue?", "limitations": "Lack of Generalization to Highly Dynamic Scenes\n\nIn challenging scenarios like outdoor_day1, MoFlow underperforms compared to some baselines. This seems to stem from egomotion prediction being confused by scenes with many independently moving objects, which violate the static-world assumption.\n\nNo Runtime or System Efficiency Analysis\n\nThe paper does not include any analysis of inference speed, resource usage, or real-time viability, which are critical in event-based vision systems intended for robotics or edge deployment.\n\nNo Evaluation on Larger or Standard Benchmarks (e.g., DSEC)\n\nWhile MoFlow performs well on EV-IMO and EED, it lacks evaluation on more comprehensive benchmarks like DSEC, which are widely used in event-based egomotion and optical flow research.\n\nDependency on Static Scene Assumption for Egomotion\n\nThe egomotion module assumes the majority of the scene is static. In highly dynamic or crowded environments, this assumption breaks down, leading to errors in both rigid and residual flow predictions.\n\nLimited Discussion on Failure Modes or Robustness\n\nThe paper does not analyze when or why the system fails—e.g., fast camera motion, lighting changes, or noisy event data—leaving open questions about robustness in the wild.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "# Strengths\nWell-Motivated Problem Setting: Tackles a practically important issue—event-based motion segmentation in dynamic scenes—which is often complicated by egomotion. The paper clearly identifies this gap.\n\nNovel Architecture with Modular Design: The separation between egomotion prediction, rigid flow, and residual flow is intuitive and effectively models dynamic motion components.\n\nSelf-Supervised Learning: The system is trained without requiring ground-truth flow, which increases its applicability to real-world, unlabeled event data.\n\nStrong Experimental Results: Outperforms existing baselines on EV-IMO and EED datasets in both optical flow and segmentation metrics.\n\nAblation Studies & Visualization: The paper includes meaningful ablation studies to validate the contribution of each module, and provides qualitative results that clearly show performance improvements.\n\n\n\n# Weaknesses\nLimited Discussion on Failure Cases / Limitations: The paper does not deeply discuss failure modes, such as when the egomotion estimator might be inaccurate (e.g., under fast or rotational motion), or how residual flow behaves under cluttered scenes.\n\nLack of Runtime or System Overhead Analysis: There is no profiling of computational efficiency, latency, or inference time—critical for real-time robotics or deployment on edge devices.\n\nNo Comparison on More Challenging Benchmarks: Although EV-IMO and EED are good starting points, newer event datasets or higher-resolution scenarios could have been used to better validate scalability.\n\nAssumes a Static Background for Egomotion Estimation: The model assumes that most of the scene is static to compute egomotion, which could fail in highly dynamic environments.", "was_revised": true, "final_justification": "This paper proposes a self-supervised event-based flow estimation method that also considers motion. At the time of the initial submission, I had several concerns, but I gave it a borderline accept, considering that the authors would likely address them in their response. As expected, the authors have resolved most of these concerns. However, certain weaknesses in the approach remain, and I found myself weighing its strengths—being simple and effective—against its novelty. After much consideration, I decided to maintain my current score. I do not oppose the paper’s acceptance, but if other reviewers raise negative opinions, I believe it would be difficult to strongly argue against them. Therefore, I am assigning a final recommendation of borderline accept.", "rebuttal": "## Response to Reviewer f2hB\n\nWe sincerely thank the reviewer for their thoughtful and detailed feedback, as well as for recognizing the key strengths of our work. We are particularly grateful for acknowledging the practical importance of our problem setting, the novelty and modularity of our architecture, and the value of our self-supervised approach for real-world applications.\n\nWe address the main concerns below.\n\n---\n\n> **[W1, Q4, L5] Discussion about failure cases**\n\nOur method can handle fast motion and pure rotational motion, but its performance degrades slightly in dynamic scenarios and fails under rapidly changing lighting conditions. We will include this failure cases discussion in the main text.\n\n**a) Very fast motion**\n\nThanks to our continuously parameterized optical flow and motion, our method can theoretically handle fast, non-linear movements without any performance loss. We conducted experiments on the *outdoor_day1* sequence to further verify this. The *outdoor_day1* sequence was collected outdoors using an event camera mounted on a vehicle, with the camera's motion speed reaching up to 12 m/s, representing very fast motion. The optical flow estimation and motion estimation performance on this sequence are shown in the table below. The results demonstrate that our method maintains strong performance even in scenarios with rapid motion.\n| **Sequence**        | &emsp;&emsp;$\\mathrm{EPE}$&emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{\\omega}$ &emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{v}$&emsp;&emsp; |\n|-|:-:|:-:|:-:|\n| *outdoor_day1* | &emsp;&emsp; $1.61$ &emsp;&emsp; | &emsp;&emsp; $3.38$ &emsp;&emsp; | &emsp;&emsp; $0.76$ &emsp;&emsp; |\n\n\n**b) Rotational motion**\n\nAs discussed in the paper, our differential geometric loss is derived from the motion field equation. According to multi-view geometry theory, the epipolar constraint degenerates under pure rotation, making it impossible to obtain a correct solution. However, we want to emphasize that optical flow estimation for pure rotation was already addressed in prior CMax [1] work. Our focus is on the more challenging 6 Degrees of Freedom (DoF) motion scenarios. In fact, we only need to change our differential geometric loss to a differential rotation loss to handle pure rotational scenes.\n\n$$L_{\\text{R}}(t,\\mathbf{x},\\theta,\\beta)= \\left \\| \\mathbf{u} (t,\\mathbf{x}, \\theta) - \\mathbf{B}(\\mathbf{x})     {\\boldsymbol{\\omega}} (\\beta)  \\right \\|^2_2$$\n\n$$\\mathbf{B}(\\begin{bmatrix}\n       x; \\\\ y\n  \\end{bmatrix})=\\begin{bmatrix}\nxy &-(1+x^2)  &y; \\\\\n(1+y^2) &-xy  &-x\n\\end{bmatrix}$$\n\nwhere $\\mathbf{u} (t,\\mathbf{x}, \\theta) $ is the neural-network parameterized optical flow and ${\\boldsymbol{\\omega}} (\\beta)$ represents the angular velocity.\n\nSince the MVSEC dataset does not include sequences with pure rotational motion, we will additionally synthesized event data under pure rotational motion and tested the performance of our method. We will include these results in the camera-ready version.\n\n**c) Dynamic scene**\n\nThe differential geometric loss used in our method assumes static scenes; any dynamic objects can be treated as outliers and removed using robust optimization techniques. We test the robustness of our method to dynamic scenes on frames 528 to 730 of the *outdoor_day2* sequence in MVSEC dataset. This segment contains a moving vehicle with a large foreground. The moving vehicle indeed violates the static assumption of the differential geometric loss. To enhance the robustness of our method to outliers, we also test the performance of the Huber kernel. The table below shows the performance of our method in optical flow estimation and motion estimation with and without the Huber kernel. The results indicate that our method still achieves good performance even without the Huber kernel, thanks to another key component: the implicit neural flow, which is not constrained by the static assumption. Moreover, the Huber kernel function significantly improves the robustness of our method to dynamic scenes. We will include these results in the paper.\n\n| **Metirc**        | &emsp;&emsp;&emsp;**w/ Huber Kernel**&emsp;&emsp; | &emsp;&emsp;&emsp;**w/o Huber Kernel**&emsp;&emsp; |\n|-|:-:|:-:|\n| $\\text{EPE}$ | &emsp;&emsp; $3.62$ &emsp;&emsp; | &emsp;&nbsp;&nbsp;&nbsp;&nbsp; $3.68$ &emsp;&emsp; | \n| $\\mathrm{RMS}_{v}$ | $1.30$ | $1.56$ | \n| $\\mathrm{RMS}_{\\omega}$ | $4.86$ | $4.94$ | \n\n**d) Rapidly changing lighting changes**\n\nWhen encountering rapidly changing lighting conditions, event cameras respond to global scene brightness shifts. This triggers a large number of outlier events unrelated to optical flow. Like other event-only methods, our approach will fail when the proportion of these outlier events becomes sufficiently large.\n\n---\n\n> **[W2, Q2, L2] Lack of Runtime Analysis.**\n\nWe report the training time and the inference time of our method across all sequences in MVSEC. Each sequence is divided into multiple clips with a time interval of approximately 80-100 ms, corresponding to $dt=4$, and each clip contains approximately 30,000 event data. In the table below, we report the average training time and inference time per clip. All experiments were conducted on a NVIDIA RTX 4090.\n| **Sequence**| **Training Time** | **Inference Time** |\n|-|:-:|:-:|\n| *indoor_flying1*  | $9.30s$ | $0.86s$ |\n| *indoor_flying2*  | $9.41s$ | $0.87s$ |\n| *indoor_flying3*  | $9.50s$ | $0.88s$ |\n| *outdoor_day1*    | $10.77s$ | $0.86s$ |\n\nAccording to Reviewer j1WT's suggestion, if we use an early-stopping strategy, the training speed can be increased by 2.2x, as shown in the table below. The number in brackets indicates the average training iterations when using early-stopping strategy.\n| **Sequence**        | &emsp;**w/ early stopping**&emsp; | &emsp;**w/o early stopping**&emsp; | &emsp;**Efficiency Improvement**&emsp; |\n|-|:-:|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $4.21s$ $(452.93)$ &emsp;&emsp; | &emsp;&emsp; $9.30s$ &emsp;&emsp; | &emsp;&emsp; $2.21 \\times$ $\\uparrow$ &emsp;&emsp; |\n| *indoor_flying2* | $4.30s$ $(461.84)$ | $9.41s$ | $2.19 \\times$ $\\uparrow$ |\n| *indoor_flying3* | $4.78s$ $(496.77)$ | $9.50s$ | $1.99 \\times$ $\\uparrow$ |\n| *outdoor_day1* | $4.56s$ $(487.51)$ | $10.77s$ | $2.36 \\times$ $\\uparrow$ |\n| *average* | $4.46s$ $(474.76)$ | $9.75s$ | $2.19 \\times$ $\\uparrow$ |\n\n---\n\n> **[W3, Q3, L3] More Challenging Datasets.**\n\nWe tested our method on the DSEC dataset. Despite the DSEC dataset capturing outdoor driving scenes with complex textures, motion patterns, and varying lighting conditions, the results in the table demonstrate that our method still exhibits impressive performance in optical flow estimation.\n\n| **Sequence**| &emsp;&emsp;&nbsp;&nbsp;&nbsp;**EPE**&emsp;&emsp; | &emsp;&emsp;&emsp;&nbsp;**Sequence**&emsp;&emsp; | &emsp;&emsp;&nbsp;&nbsp;&nbsp;**EPE**&emsp;&emsp; |\n|-|:-:|:-:|:-:|\n| *average*  | &emsp;&emsp; $6.649$ &emsp;&emsp; | &emsp;&emsp; *thun_01_b* &emsp;&emsp;|&emsp;&emsp; $6.925$ &emsp;&emsp;|\n| *interlaken_00_b*  | &emsp;&emsp; $12.000$ &emsp;&emsp; | &emsp;&emsp; *zurich_city_12_a* &emsp;&emsp;| &emsp;&emsp; $4.272$ &emsp;&emsp;|\n| *interlaken_01_a*  | &emsp;&emsp; $6.784$ &emsp;&emsp; | &emsp;&emsp; *zurich_city_14_c* &emsp;&emsp;| &emsp;&emsp; $7.604$ &emsp;&emsp; |\n| *thun_01_a*    | &emsp;&emsp; $5.209$ &emsp;&emsp; | &emsp;&emsp; *zurich_city_15_a* &emsp;&emsp;| &emsp;&emsp; $4.612$ &emsp;&emsp; |\n\n---\n\n> **[W4, L1, L4] Static Background Assumption.**\n\nWe have already discussed this scenario under **c) Dynamic scene** in **[W1, Q4, L5] Discussion about failure cases**. Please refer to that section.\n\n---\n\n> **[Q1] Analysis of model capacity and motion decomposition strategy on performance.**\n\nThe performance gain of MoFlow is not related to increased model capacity or the proposed motion decomposition strategy.\n\n**increased model capacity:** In our method, the Neural ODE is solely used to preserve the optical flow field; even a shallow MLP is sufficient to represent complex flow fields. Theoretically, increasing model capacity might yield slightly better fitting accuracy, but this improvement becomes negligible given the significant increase in memory and runtime. We tested the impact of model capacity on performance by adjusting the network depth of the MLP. In the original setting, the MLP network depth was 8, and we tested the performance with a network depth of 4. As shown in the results in the table below, the impact of model capacity on performance is minimal, which demonstrates that our method achieves strong performance without relying on simply increasing model capacity. In the table, the left column is $\\text{EPE}$, the middle column is $RMS_{\\omega}$, and the right column is $RMS_{v}$.\n| **Sequence**        | &emsp;&emsp;&emsp;&emsp;&emsp;**NN=4**&emsp;&emsp; | &emsp;&emsp;&emsp;&emsp;&emsp;**NN=8**&emsp;&emsp; |\n|-|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $1.52$ \\| $3.40$ \\| $0.11$ &emsp;&emsp; | &emsp;&nbsp;&nbsp;&nbsp;&nbsp; $1.56$ \\| $3.44$ \\| $0.11$ &emsp;&emsp; | \n| *indoor_flying2* | $2.06$ \\| $5.21$ \\| $0.13$ | $2.04$ \\| $5.31$ \\| $0.12$ | \n| *indoor_flying3* | $1.78$ \\| $4.06$ \\| $0.15$ | $1.84$ \\| $4.12$ \\| $0.15$ | \n| *outdoor_day1* | $1.71$ \\| $3.44$ \\| $0.86$ | $1.61$ \\| $3.38$ \\| $0.76$ | \n| *average* | $1.78$ \\| $4.03$ \\| $0.31$ | $1.76$ \\| $4.06$ \\| $0.29$ |\n\n**motion decomposition strategy:** As we just discussed, Neural ODEs and splines are merely representation tools. As long as their representational capacity is strong enough, adopting a motion decomposition strategy won't affect performance. While we could use a single network to represent both optical flow and motion simultaneously, we believe its performance would not surpass the current approach.\n\n\n[1] A Unifying Contrast Maximization Framework for Event Cameras, with Applications to Motion, Depth, and Optical Flow Estimation"}, {"review_id": "7iWf6plR0J", "reviewer": "Reviewer_tZ5K", "summary": "This paper tackles the problem of jointly estimating both optical flow and camera pose estimation from event camera data. They introduce a novel parameterization by fitting an implicit function (a neural ODE) to the event data to explain both optical flow and camera poses. Camera poses are parameterized as a spline and optical flow as a velocity field. The model is supervised using a self-supervised loss coupling both the estimated optical flow and poses.", "questions": "- How does the method compare to other pose estimation methods with more standard benchmarking (e.g. ATE)?\n- How long is the optimization time in seconds/minutes?", "limitations": "The method requires per-scene optimization for the neural ODE parameterizing the optical flow and pose estimation.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n- Novel unification of optical flow and pose estimation without using geometry information relating the neural ODE and the differential epipolar equation\n- Competitive results on event-image optical flow and 6DOF velocity estimation\n\nWeaknesses:\n\n- Has to be fit per-video, making it potentially impractical. Authors should also describe how long optimization takes per video.\n- Pose estimation results have poor illustrations and quantitative comparisons:\n    - The metric for pose estimation is not well described, and a more standard metric for pose comparison should be used, such as ATE.\n    - The pose trajectories should be visualized, rather than the just the 6DOF velocities: it is currently difficult to interpret the accuracy of the pose estimation results.\n- The method is only evaluated on a limited number of event-data videos (4 videos) — why not on the entire dataset?", "was_revised": true, "final_justification": "As someone who comes from the RGB structure-from-motion landscape it's possible I do not understand the full event-camera landscape fully and its limitations, but I think this paper presents a nice formulation for egomotion estimation on event-camera data. Assuming they add the tables in the rebuttal and emphasize in the camera-ready that it's an optimization-based approach, I move my judgement to borderline accept.", "rebuttal": "## Response to Reviewer tZ5K\n\nWe sincerely appreciate the reviewer’s recognition of our work, particularly the novel unify unsupervised learning of optical flow and 6-DoF motion through the integration of neural ODEs and the differential epipolar equation without relying on explicit geometry.\n\nWe address the main concerns below.\n\n---\n\n> **[W1, Q2] Lack of runtime analysis.**\n\nWe report the training time and the inference time of our method in the following table. The analysis has been tested across all sequences in dataset MVSEC. Specifically, each sequence is divided into multiple clips with a time interval of approximately 80-100 ms, corresponding to $dt=4$, and each clip contains approximately 30,000 event data. Additionally, based on the suggestion of Reviewer j1WT, we tested the impact of the early-stopping strategy on training speed and found that it can significantly boost the training speed to 2.2x. The table below shows the average training time per clip for 1000 training iterations and with the early-stopping strategy. The number in brackets indicates the average training iterations when using early-stopping strategy.\n| **Sequence**        | &emsp;**w/ early stopping**&emsp; | &emsp;**w/o early stopping**&emsp; | &emsp;**Efficiency Improvement**&emsp; |\n|-|:-:|:-:|:-:|\n| *indoor_flying1* | &emsp;&emsp; $4.21s$ $(452.93)$ &emsp;&emsp; | &emsp;&emsp; $9.30s$ &emsp;&emsp; | &emsp;&emsp; $2.21 \\times$ $\\uparrow$ &emsp;&emsp; |\n| *indoor_flying2* | $4.30s$ $(461.84)$ | $9.41s$ | $2.19 \\times$ $\\uparrow$ |\n| *indoor_flying3* | $4.78s$ $(496.77)$ | $9.50s$ | $1.99 \\times$ $\\uparrow$ |\n| *outdoor_day1* | $4.56s$ $(487.51)$ | $10.77s$ | $2.36 \\times$ $\\uparrow$ |\n| *average* | $4.46s$ $(474.76)$ | $9.75s$ | $2.19 \\times$ $\\uparrow$ |\n\nWe will include the detailed runtime analysis in the camera-ready version.\n\n---\n\n> **[W1, L1] Per-video fitting is potentially impractical.**\n\nWe believe per-video fitting doesn't make our method seem impractical; instead, it demonstrates its powerful generalization capability. Thanks to this robust generalization, our approach can handle complex real-world scenarios and accurately estimate high-quality optical flow. This makes it a practical tool for providing ground truth optical flow for real-world event camera datasets, enabling large-scale supervised learning for optical flow neural networks. Furthermore, in future work, we can leverage implicit regularization to unsupervisedly train a feed-forward optical flow prediction network, thus eliminating the need for per-video fitting.\n\n---\n\n> **[W2, Q1]  Poor illustrations and quantitative comparisons of pose estimation.**\n\n**Poor quantitative comparisons**\n\n1. We can provide RPE (Relative Pose Error), but for very short time intervals, it's equivalent to the RMS velocity error. Therefore, providing the RMS velocity error is sufficient to quantify the pose's RPE.\n\n2. Many prior works [1] [2] [3] have chosen to use the RMS velocity error, and we are following this established standard.\n\n3. We cannot provide ATE (Absolute Trajectory Error) because our linear velocity lacks scale information, preventing us from comparing absolute trajectory errors. Furthermore, even with absolute scale information, the short-duration events from event cameras only provide instantaneous motion data. They cannot establish multi-view correspondences over longer time intervals, which means accumulated errors cannot be eliminated. This makes ATE a suboptimal error metric for our context.\n\n\n**Poor illustrations** We are unable to illustrate the camera's pose trajectory. The reason is that we cannot integrate the event camera's linear velocity, which inherently lacks scale information. Nonetheless, we have provided many plots illustrating the event camera's velocity changes.\n\n---\n\n> **[W3] Limited evaluations of datasets.**\n\n**a)** Although the MVSEC dataset provides other sequences, such as *indoor_flying4*, *outdoor_day2*, and *outdoor_night1*, many previous works [][][...] have gradually established a standard for evaluation on the *indoor_flying1*, *indoor_flying2*, *indoor_flying3*, and *outdoor_day1* sequences of the MVSEC dataset, rather than on other sequences. Therefore, our method follows the practice of prior works. Specifically, most of the field of view in the *indoor_flying4* sequence consists of the floor, which lacks sufficient texture, resulting in relatively fewer events being generated. The *outdoor_day2* sequence is used as a training set rather than a test set. The *outdoor_night1* sequence cannot be used as a test set due to the poor quality of the event data, the presence of many flickering events, and the absence of high-quality ground truth optical flow.\n\n**b)** Although the *indoor_flying4* and *outdoor_day2* sequences are not typically used for evaluation, we conducted training and evaluation on these two sequences following the settings reported in the paper to validate the effectiveness of our method. The results show that our method still demonstrated impressive performance on these sequences. The table below shows the accuracy of optical flow estimation and motion estimation.\n\n| **Sequence**        | &emsp;&emsp;$\\mathrm{EPE}$&emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{\\omega}$ &emsp;&emsp; | &emsp;&emsp; $\\mathrm{RMS}_{v}$&emsp;&emsp; |\n|-|:-:|:-:|:-:|\n| *indoor_flying4* | &emsp;&emsp; $2.38$ &emsp;&emsp; | &emsp;&emsp; $5.10$ &emsp;&emsp; | &emsp;&emsp; $0.13$ &emsp;&emsp; |\n| *outdoor_day2* | $3.49$ | $5.76$ | $1.64$ |\n\n---\n\n> **[MW1] Paper Formatting.**\n\nThank you for your suggestion. We will optimize the layout in the camera-ready version to make it more visually appealing and easier to read.\n\n---\n\n**Reference:**\n\n[1] Motion and Structure from Event-based Normal Flow\n\n[2] Secrets of Event-Based Optical Flow, Depth and Ego-Motion Estimation by Contrast Maximization\n\n[3] Entropy Minimisation Framework for Event-based Vision Model Estimation"}]

Author rebuttal:

Meta-review: The paper proposes a method to estimate egomotion and optical flow jointly from event data. The method is unsupervised, an important feature as large event datasets are infeasible to collect. The reviewers appreciated the problem setting and the method. The main weaknesses in the reviews were related to missing evaluations (e.g., highly dynamic scenes and ablations). The rebuttal addressed these concerns and presented several new results. All reviewers were satisfied with the additional experiments and recommended accepting the paper.  I agree--the paper provides useful contributions to perception from event stream data.
Decision: Accept (poster)