4.1.1. Datasets

The *MVSEC* dataset [34] is a widely used dataset for various vision tasks, such as optical flow estimation [16,18,19,41,42]. Its sequences are recorded on a drone (indoors) or on a car (outdoors), and comprise events, grayscale frames and IMU data from an mDAVIS346 [43] (346 × 260 pixels), as well as camera poses and LiDAR data. Ground truth optical flow is computed as the motion field [44], given the camera velocity and the depth of the scene (from the LiDAR). We select several excerpts from the *outdoor\_day1* sequence with a forward motion. This motion is reasonably well approximated by collapse-enabled warps such as (6). In total, we evaluate 3.2 million events spanning 10 s.

The *DSEC* dataset [39] is a more recent driving dataset with a higher resolution event camera (Prophesee Gen3, 640 × 480 pixels). Ground truth optical flow is also computed as the motion field using the scene depth from a LiDAR [41]. We evaluate on the *zurich\_city\_11* sequence, using in total 380 million events spanning 40 s.

The *ECD* dataset [40] is the de facto standard to assess event camera ego-motion [5,8,28,45–48]. Each sequence provides events, frames, a calibration file, and IMU data (at 1kHz) from a DAVIS240C camera [49] (240 × 180 pixels), as well as ground-truth camera poses from a motion-capture system (at 200Hz). For rotational motion estimation (3DOF), we use the natural-looking *boxes\_rotation* and *dynamic\_rotation* sequences. We evaluate 43 million events (10 s) of the box sequence, and 15 million events (11 s) of the dynamic sequence.

The driving datasets (MVSEC, DSEC) and the selected sequences in the ECD dataset have different type of motions: forward (which enables event collapse) vs. rotational (which does not suffer from event collapse). Each sequence serves a different test purpose, as discussed in the next sections.

## 4.1.2. Metrics

The metrics used to assess optical flow accuracy (MVSEC and DSEC datasets) are the average endpoint error (AEE) and the percentage of pixels with AEE greater than *N* pixels (denoted by " *N*PE", for *N* = {3, 10, 20}). Both are measured over pixels with valid ground-truth values. We also use the FWL metric [50] to assess event alignment by means of the IWE sharpness (the FWL is the IWE variance relative to that of the identity warp).

Following previous works [13,27,28], rotational motion accuracy is assessed as the RMS error of angular velocity estimation. Angular velocity *ω* is assumed to be constant over a window of events, estimated and compared with the ground truth at the midpoint of the window. Additionally, we use the FWL metric to gauge event alignment [50].

The event time windows are as follows: the events in the time spanned by *dt* = 4 frames in MVSEC (standard in [16,18,41]), 500k events for DSEC, and 30k events for ECD [28]. The regularizer weights for divergence (*λ*div) and deformation (*λ*def) are as follows: *λ*div = 2 and *λ*def = 5 for MVSEC, *λ*div = 50 and *λ*def = 100 for DSEC, and *λ*div = 5 and *λ*def = 10 for ECD experiments.

#### *4.2. Effect of the Regularizers on Collapse-Enabled Warps*

Tables 1 and 2 report the results on the MVSEC and DSEC benchmarks, respectively, by using two different loss functions *G*: the IWE variance (4) and the squared magnitude of the IWE gradient, abbreviated "Gradient Magnitude" [13]. For MVSEC, we report the accuracy within the time interval of *dt* = 4 grayscale frame (at ≈45 Hz). The optimization algorithm is the Tree-Structured Parzen Estimator (TPE) sampler [51] for both experiments, with a number of sampling points equal to 300 (1 DOF) and 600 (4 DOF). The tables quantitatively capture the collapse phenomenon suffered by the original CMax framework [12] and the whitening technique [27]. Their high FWL values indicate that contrast is maximized; however, the AEE and *N*PE values are exceedingly high (e.g., >80 pixels, 20PE > 80%), indicating that the estimated flow is unrealistic.





By contrast, our regularizers (Divergence and Deformation rows) work well to mitigate the collapse, as observed in smaller AEE and *N*PE values. Compared with the values of no regularizer or whitening [27], our regularizers achieve more than 90% improvement for AEE on average. The AEE values are high for optical flow standards (4–8 pix in MVSEC vs. 0.5–1 pixel [16], or 10–20 pix in DSEC vs. 2–5 pix [41]); however, this is due to the fact that the warps used have very few DOFs (≤4) compared to the considerably higher DOFs (2*Np*) of optical flow estimation algorithms. The same reason explains the high 3PE values (standard in [52]): using an end-point error threshold of 3 pix to consider that the flow is correctly estimated does not convey the intended goal of inlier/outlier classification for the low-DOF warps used. This is the reason why Tables 1 and 2 also report 10PE, 20PE metrics, and the values for the identity warp (zero flow). As expected, for the range of AEE values in the tables, the 10PE and 20PE figures demonstrate the large difference between methods suffering from collapse (20PE > 80%) and those that do not (20PE < 1.1% for MVSEC and <22.6% for DSEC).

The FWL values of our regularizers are moderately high ( ≥1), indicating that event alignment is better than that of the identity warp. However, because the FWL depends on the number of events [50], it is not easy to establish a global threshold to classify each method as suffering from collapse or not. The AEE, 10PE, and 20PE are better for such a classification.

Tables 1 and 2 also include the results of the use of both regularizers simultaneously ("Div. + Def."). The results improve across all sequences if the data fidelity term is given by the variance loss, whereas they remain approximately the same for the gradient magnitude loss. Regardless of the choice of the proposed regularizer, the results in these tables clearly show the effectiveness of our proposal, i.e., the large improvements compared with prior works (rows "No regularizer" and [27]).

The collapse results are more visible in Figure 6, where we used the variance loss. Without a regularizer, the events collapse in the MVSEC and DSEC sequences. Our regularizers successfully mitigate overfitting, having a remarkable impact on the estimated motion.

#### *4.3. Effect of the Regularizers on Well-Posed Warps*

Table 3 shows the results on the ECD dataset for a well-posed warp (3-DOF rotational motion, in the benchmark). We use the variance loss and the Adam optimizer [53] with 100 iterations. All values in the table (RMS error and FWL, with and without regularization, are very similar, indicating that: (i) our regularizers do not affect the motion estimation algorithm, and (ii) results without regularization are good due to the well-posed warp. This is qualitatively shown in the bottom part of Figure 6. The fluctuations of the divergence and deformation values away from those of the identity warp (0 and 1, respectively) are at least one order of magnitude smaller than the collapse-enabled warps (e.g., 0.2 vs. 2).



#### *4.4. Sensitivity Analysis*

The landscapes of loss functions as well as sensitivity analysis of *λ* are shown in Figure 7, for the MVSEC experiments. Without regularizer ( *λ* = 0), all objective functions tested (variance, gradient magnitude, and average timestamp [16]) suffer from event collapse, which is the undesired global minimum of (20). Reaching the desired local optimum depends on the optimizing algorithm and its initialization (e.g., starting gradient descent close enough to the local optimum). Our regularizers (divergence and deformation) change the landscape: the previously undesired global minimum becomes local, and the desired minimum becomes the new global one as *λ* increases.

Specifically, the larger the weight *λ*, the smaller the effect of the undesired minimum (at *hz* = 1). However, this is true only within some reasonable range: a too large *λ* discards the data-fidelity part *G* in (20), which is unwanted because it would remove the desired local optimum (near *hz* ≈ 0). Minimizing (20) with only the regularizer is not sensible.

Observe that for completeness, we include the average timestamp loss in the last column. However, this loss also suffers from an undesired optimum in the expansion region (*hz* ≈ −1). Our regularizers could be modified to also remove this undesired optimum, but investigating this particular loss, which was proposed as an alternative to the original contrast loss, is outside the scope of this work.

**Figure 6.** *Proposed regularizers and collapse analysis*. The scene motion is approximated by 1-DOF warp (zoom in/out) for MVSEC [34] and DSEC [39] sequences, and 3-DOF warp (rotation) for boxes and dynamic ECD sequences [40]. (**a**) Original events. (**b**) Best warp without regularization. Event collapse happens for 1-DOF warp. (**c**) Best warp with regularization. (**d**) Divergence map ((10) is zero-based). (**e**) Deformation map ((15), centered at 1). Our regularizers successfully penalize event collapse and do not damage non-collapsing scenarios.

**Figure 7.** *Cost function landscapes over the warp parameter hz for*: (**a**) Image variance [12], (**b**) gradient magnitude [13], and (**c**) mean square of average timestamp [16]. Data from MVSEC [34] with dominant forward motion. The legend weights denote *λ* in (20).

#### *4.5. Computational Complexity*

Computing the regularizer(s) requires more computation than the non-regularized objective. However, complexity is linear with the number of events and the number of pixels, which is an advantage, and the warped events are reutilized to compute the DIWE or IWA. Hence, the runtime is less than doubled (warping is the dominant runtime term [13] and is computed only once). The computational complexity of our regularized CMax framework is *O*(*Ne* + *Np*), the same as that of the non-regularized one.

#### *4.6. Application to Motion Segmentation*

Although most of the results on standard datasets comprise stationary scenes, we have also provided results on a dynamic scene (from dataset [40]). Because the time spanned by each set of events processed is small, the scene motion is also small (even for complicated objects like the person in the bottom row of Figure 6), hence often a single warp fits the scene reasonably well. In some scenarios, a single warp may not be enough to fit the event data because there are distinctive motions in the scene of equal importance. Our proposed regularizers can be extended to such more complex scene motions. To this end, we demonstrate it with an example in Figure 8.

**Figure 8.** *Application to Motion Segmentation*. (**a**) Output IWE, whose colors (red and blue) represent different clusters of events (segmented according to motion). (**b**) Divergence map. The range of divergence values is larger in the presence of event collapse than in its absence. Our regularizer (divergence in this example) mitigates the event collapse for this complex motion, even with an independently moving object (IMO) in the scene.

Specifically, we use the MVSEC dataset, in a clip where the scene consists of two motions: the ego-motion (forward motion of the recording vehicle) and the motion of a car driving in the opposite direction in a nearby lane (an independently moving object—IMO). We model the scene by using the combination of two warps. Intuitively, the 1-DOF warp (6) describes the ego-motion, while the feature flow (2 DOF) describes the IMO. Then, we apply the contrast maximization approach (augmented with our regularizing terms) and the expectation-maximization scheme in [21] to segmen<sup>t</sup> the scene, to determine which events belong to each motion. The results in Figure 8 clearly show the effectiveness of our regularizer, even for such a commonplace and complex scene. Without regularizers, (i) event collapse appears in the ego-motion cluster of events and (ii) a considerable portion of the events that correspond to ego-motion are assigned to the second cluster (2-DOF warp), thus causing a segmentation failure. Our regularization approach mitigates event collapse (bottom row of Figure 8) and provides the correct segmentation: the 1-DOF warp fits the ego-motion and the feature flow (2-DOF warp) fits the IMO.

## **5. Conclusions**

We have analyzed the event collapse phenomenon of the CMax framework and proposed collapse metrics using first principles of space-time deformation, inspired by differential geometry and physics. Our experimental results on publicly available datasets demonstrate that the proposed divergence and area-based metrics mitigate the phenomenon for collapse-enabled warps and do not harm well-posed warps. To the best of our knowledge, our regularizers are the only effective solution compared to the unregularized CMax framework and whitening. Our regularizers achieve, on average, more than 90% improvement on optical flow endpoint error calculation (AEE) on collapse-enabled warps.

This is the first work that focuses on the paramount phenomenon of event collapse. No prior work has analyzed this phenomenon in such detail or proposed new regularizers without additional data or reparameterizing the search space [14,16,27]. As we analyzed various warps from 1 DOF to 4 DOFs, we hope that the ideas presented here inspire further research to tackle more complex warp models. Our work shows how the divergence and

area-based deformation can be computed for warps given by analytical formulas. For more complex warps, like those used in dense optical flow estimation [16,18], the divergence or area-based deformation could be approximated by using finite difference formulas.

**Author Contributions:** Conceptualization, S.S. and G.G.; methodology, G.G.; software, S.S.; validation, S.S.; formal analysis, S.S. and G.G.; investigation, S.S. and G.G.; resources, Y.A.; data curation, S.S.; writing—original draft preparation, S.S. and G.G.; writing—review and editing, S.S., Y.A. and G.G.; visualization, S.S. and G.G.; supervision, Y.A. and G.G.; project administration, S.S.; funding acquisition, S.S., Y.A. and G.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the German Academic Exchange Service (DAAD), Research Grant-Bi-nationally Supervised Doctoral Degrees/Cotutelle, 2021/22 (57552338). Ref. no.: 91803781.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available in publicly accessible repositories. The data presented in this study are openly available in reference number [34,39,40].

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Warp Models, Jacobians and Flow Divergence**

*Appendix A.1. Planar Motion — Euclidean Transformation on the Image Plane, SE*(2)

If the point trajectories of an isometry are **<sup>x</sup>**(*t*), the warp is given by [27]

$$
\begin{pmatrix} \mathbf{x}'\_k \\ \mathbf{1} \end{pmatrix} \sim \begin{pmatrix} \mathbf{R}(t\_k \omega\_\mathbf{Z}) & t\_k \mathbf{v} \\ \mathbf{0}^\top & 1 \end{pmatrix}^{-1} \begin{pmatrix} \mathbf{x}\_k \\ 1 \end{pmatrix},\tag{A1}
$$

where **v**, *ωZ* comprise the 3 DOFs of a translation and an in-plane rotation. The in-plane rotation is

$$\mathbf{R}(\phi) = \begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{pmatrix} . \tag{A2}$$

Since

$$
\begin{pmatrix} \mathbf{A} & \mathbf{b} \\ \mathbf{0}^{\top} & 1 \end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}^{-1} & -\mathbf{A}^{-1}\mathbf{b} \\ \mathbf{0}^{\top} & 1 \end{pmatrix} \tag{A3}
$$

and <sup>R</sup>−<sup>1</sup>(*φ*) = <sup>R</sup>(−*φ*), we have

$$
\begin{pmatrix} \mathbf{x}'\_k \\ 1 \end{pmatrix} \sim \begin{pmatrix} \mathbf{R}(-t\_k \omega\_\mathbf{Z}) & -\mathbf{R}(-t\_k \omega\_\mathbf{Z})(t\_k \mathbf{v}) \\ \mathbf{0}^\top & 1 \end{pmatrix} \begin{pmatrix} \mathbf{x}\_k \\ 1 \end{pmatrix}. \tag{A4}$$

Hence, in Euclidean coordinates the warp is

$$\mathbf{x}'\_{k} = \mathbf{R}(-t\_{k}\omega\_{\mathbb{Z}})(\mathbf{x}\_{k} - t\_{k}\mathbf{v}).\tag{A5}$$

The Jacobian and its determinant are:

$$\mathbf{J}\_k = \frac{\partial \mathbf{x}'\_k}{\partial \mathbf{x}\_k} = \mathbf{R}(-t\_k \omega\_\mathbb{Z}),\tag{A6}$$

$$\det(\mathbf{J}\_k) = 1.\tag{A7}$$

The flow corresponding to (A5) is:

$$\mathbf{f} = \frac{\partial \mathbf{x}^{\prime}}{\partial t} = \mathbf{R}^{\top} \left( \frac{\pi}{2} + t\omega\_{\mathbb{Z}} \right) (\mathbf{x} - t\mathbf{v}) \omega\_{\mathbb{Z}} - \mathbb{R}(-t\omega\_{\mathbb{Z}}) \mathbf{v},\tag{A8}$$

whose divergence is

$$\nabla \cdot \mathbf{f} = -2\omega\_Z \sin(t\omega\_Z). \tag{A9}$$

Hence, for small angles |*<sup>t</sup>ωZ*| 1, the divergence of the flow vanishes.

In short, this warp has the same determinant and approximate zero divergence as the 2-DOF feature flow warp (Section 3.5.1), which is well-behaved. Note, however, that the trajectories are not straight in space-time.

#### *Appendix A.2. 3-DOF Camera Rotation, SO*(3)

Using calibrated and homogeneous coordinates, the warp is given by [5,12]

$$\mathbf{x}\_{k}^{h\prime} \sim \mathbb{R}(t\_{k}\omega) \,\,\mathbf{x}\_{k\prime}^{h} \tag{A10}$$

where *θ* = *ω* = (*<sup>ω</sup>*1, *ω*2, *<sup>ω</sup>*3) is the angular velocity, and R (3 × 3 rotation matrix in space) is parametrized using exponential coordinates (Rodrigues rotation formula [35,36]).

By the chain rule, the Jacobian is:

$$\mathbf{J}\_{k} = \frac{\partial \mathbf{x}\_{k}^{\prime}}{\partial \mathbf{x}\_{k}} = \frac{\partial \mathbf{x}\_{k}^{\prime}}{\partial \mathbf{x}\_{k}^{h\prime}} \frac{\partial \mathbf{x}\_{k}^{h\prime}}{\partial \mathbf{x}\_{k}^{h}} \frac{\partial \mathbf{x}\_{k}^{h}}{\partial \mathbf{x}\_{k}} = \frac{1}{(\mathbf{x}\_{k}^{h\prime})\_{3}} \begin{pmatrix} 1 & 0 & -\mathbf{x}\_{k}^{\prime} \\ 0 & 1 & -\mathbf{y}\_{k}^{\prime} \end{pmatrix} \mathbb{R}(t\_{k}\omega) \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}. \tag{A11}$$

Letting **r**3,*k*be the third row of <sup>R</sup>(*tkω*), and using (32)–(34) in [38], gives

$$\det(\mathbf{J}\_k) = (\mathbf{r}\_{3,k}^\top \mathbf{x}\_k^h)^{-3}.\tag{A12}$$

Connection between Divergence and Deformation Maps

If the rotation angle *tkω* is small, using the first two terms of the exponential map we approximate <sup>R</sup>(*tkω*) ≈ Id + (*tkω*)<sup>∧</sup>, where the hat operator ∧ in *SO*(3) represents the cross product matrix [54]. Then, **<sup>r</sup>**3,*k***<sup>x</sup>***hk* ≈ (−*tk<sup>ω</sup>*2, *tk<sup>ω</sup>*1, <sup>1</sup>)(*xk*, *yk*, 1) = 1 + (*yk<sup>ω</sup>*1 − *xkω*2)*tk*. Substituting this expression into (A12) and using the first two terms in Taylor's expansion around *z* = 0 of (1 + *z*)−<sup>3</sup> ≈ 1 − 3*z* + 6*z*<sup>2</sup> (convergent for |*z*| < 1) gives det(<sup>J</sup>*k*) ≈ 1 + <sup>3</sup>(*xkω*2 − *yk<sup>ω</sup>*1)*tk*. Notably, the divergence (18) and the approximate amplification factor depend linearly on <sup>3</sup>(*xkω*2 − *yk<sup>ω</sup>*1). This resemblance is seen in the divergence and deformation maps of the bottom rows in Figure 6 (ECD dataset).

#### *Appendix A.3. 4-DOF In-Plane Camera Motion Approximation*

The warp presented in [20],

$$\mathbf{x}\_{k}^{\prime} = \mathbf{x}\_{k} - t\_{k} \left( \mathbf{v} + (h\_{z} + 1)\mathbf{R}(\boldsymbol{\phi})\mathbf{x}\_{k} - \mathbf{x}\_{k} \right) \tag{A13}$$

has 4 DOFs: *θ* = (**<sup>v</sup>**, *φ*, *hz*). The Jacobian and its determinant are:

$$\mathbf{J}\_{k} = \frac{\partial \mathbf{x}\_{k}^{\prime}}{\partial \mathbf{x}\_{k}} = (1 + t\_{k})\mathbf{I}\mathbf{d} - (h\_{z} + 1)t\_{k}\mathbf{R}(\boldsymbol{\phi}),\tag{A14}$$

$$\det(\mathbf{J}\_k) = (1 + t\_k)^2 - 2(1 + t\_k)t\_k(h\_z + 1)\cos\phi + t\_k^2(h\_z + 1)^2. \tag{A15}$$

The flow corresponding to (A13) is given by

$$\mathbf{f} = \frac{\partial \mathbf{x}'}{\partial t} = -\left(\mathbf{v} + (h\_{\bar{z}} + 1)\mathbf{R}(\boldsymbol{\phi})\mathbf{x} - \mathbf{x}\right),\tag{A16}$$

whose divergence is:

$$\nabla \cdot \mathbf{f} = -(h\_z + 1)\nabla \cdot \left(\mathbf{R}(\phi)\mathbf{x}\right) + \nabla \cdot \mathbf{x} \tag{A17}$$

=2 − 2(*hz* + 1) cos(*φ*). (A18)

As particular cases of this warp, one can identify:

	- • 2-DOF translation (*φ* = 0, *hz* = 0). **x** *k*= **<sup>x</sup>***k*− *tk***<sup>v</sup>**.
	- • 1-DOF "rotation" (**v** = **0**, *hz* = 0). **x** *k* = **<sup>x</sup>***k* − *tk* -<sup>R</sup>(*φ*)**<sup>x</sup>***k* − **<sup>x</sup>***k*. Using a couple of approximations of the exponential map in *SO*(2), we obtain

$$\mathbf{x}'\_{k} = \mathbf{x}\_{k} - t\_{k} \left( \mathbf{R}(\boldsymbol{\phi}) - \mathbf{Id} \right) \mathbf{x}\_{k} \tag{A19}$$

$$
\delta \approx \mathbf{x}\_k - t\_k \boldsymbol{\phi}^{\wedge} \mathbf{x}\_k \tag{A20}
$$

$$\mathbf{x} = (\mathbf{1}\mathbf{d} + (-t\_k \boldsymbol{\phi})^\wedge)\mathbf{x}\_k \tag{A21}$$

$$\approx \mathbf{R}(-t\_k \boldsymbol{\phi}) \mathbf{x}\_k \tag{A22}$$

Hence, *φ* plays the role of a small angular velocity *ωZ* around the camera's optical axis *Z*, i.e., in-plane rotation.

•3-DOF planar motion ("isometry") (*hz* = 0). Using the previous result, the warp splits into translational and rotational components:

$$\mathbf{x}'\_{k} = \mathbf{x}\_{k} - t\_{k} \left( \mathbf{v} + \mathbb{R}(\boldsymbol{\phi}) \mathbf{x}\_{k} - \mathbf{x}\_{k} \right) \tag{A23}$$

$$
\stackrel{(A.22)}{\approx} -t\_k \mathbf{v} + \mathbf{R} (-t\_k \phi) \mathbf{x}\_k. \tag{A.24}
$$

#### *Appendix A.4. 4-DOF Similarity Transformation on the Image Plane, Sim(2)*

Another 4-DOF warp is proposed in [27]. Its DOFs are the linear, angular and scaling velocities on the image plane: *θ* = (**<sup>v</sup>**, *<sup>ω</sup>Z*,*<sup>s</sup>*).

Letting *βk* = 1 + *tks*, the warp is:

$$
\begin{pmatrix} \mathbf{x}'\_k \\ 1 \end{pmatrix} \sim \begin{pmatrix} \beta\_k \mathbf{R} (t\_k \omega\_Z) & t\_k \mathbf{v} \\ \mathbf{0}^\top & 1 \end{pmatrix}^{-1} \begin{pmatrix} \mathbf{x}\_k \\ 1 \end{pmatrix}. \tag{A25}
$$

Using (A3) gives

$$
\begin{pmatrix} \mathbf{x}'\_k \\ 1 \end{pmatrix} \sim \begin{pmatrix} \boldsymbol{\beta}\_k^{-1} \mathbb{R}(-\boldsymbol{t}\_k \boldsymbol{\omega}\_{\mathbb{Z}}) & -\boldsymbol{\beta}\_k^{-1} \mathbb{R}(-\boldsymbol{t}\_k \boldsymbol{\omega}\_{\mathbb{Z}}) (\boldsymbol{t}\_k \mathbf{v}) \\ \mathbf{0}^\top & 1 \end{pmatrix} \begin{pmatrix} \mathbf{x}\_k \\ 1 \end{pmatrix}. \tag{A26}$$

Hence, in Euclidean coordinates the warp is

$$\mathbf{x}'\_{k} = \beta\_{k}^{-1} \mathbf{R}(-t\_{k}\omega\_{\mathbb{Z}}) (\mathbf{x}\_{k} - t\_{k}\mathbf{v}).\tag{A27}$$

The Jacobian and its determinant are:

$$\mathbf{J}\_k = \frac{\partial \mathbf{x}\_k'}{\partial \mathbf{x}\_k} = \beta\_k^{-1} \mathbf{R}(-t\_k \omega\_{\overline{Z}}),\tag{A28}$$

$$\det(\mathbb{J}\_k) = \beta\_k^{-2} = \frac{1}{(1 + t\_k s)^2}. \tag{A29}$$

The following result will be useful to simplify equations. For a 2D rotation <sup>R</sup>(*φ*(*t*)), it holds that:

$$\frac{\partial \mathbf{R}(\phi(t))}{\partial t} = -\mathbf{R}^{\top} \left(\frac{\pi}{2} - \phi\right) \frac{\partial \phi}{\partial t}.\tag{A30}$$

To compute the flow of (A27), there are three time-dependent factors. Hence, applying the product rule we obtain three terms, and substituting (A30) (with *φ* = −*tωZ*) gives:

$$\mathbf{f}\_{k} = \left(\frac{\partial \boldsymbol{\beta}\_{k}^{-1}}{\partial t\_{k}} \mathbf{R}(-t\_{k}\boldsymbol{\omega}\_{\mathcal{Z}}) + \boldsymbol{\beta}\_{k}^{-1}\boldsymbol{\omega}\_{\mathcal{Z}}\mathbf{R}^{\top}\left(\frac{\boldsymbol{\pi}}{2} + t\_{k}\boldsymbol{\omega}\_{\mathcal{Z}}\right)\right) (\mathbf{x}\_{k} - t\_{k}\mathbf{v}) - \boldsymbol{\beta}\_{k}^{-1}\mathbf{R}(-t\_{k}\boldsymbol{\omega}\_{\mathcal{Z}})\mathbf{v}, \tag{A31}$$

where, by the chain rule,

$$\frac{\partial \beta\_k^{-1}}{\partial t\_k} = -\beta\_k^{-2} \frac{\partial \beta\_k}{\partial t\_k} = -\beta\_k^{-2} s = -\frac{s}{(1 + t\_k s)^2}.\tag{A32}$$

Hence, the divergence of the flow is:

$$\nabla \cdot \mathbf{f}\_k = \frac{\partial \boldsymbol{\beta}\_k^{-1}}{\partial t\_k} \nabla \cdot \left( \mathbf{R} (-t\_k \boldsymbol{\omega}\_{\mathbb{Z}}) \mathbf{x}\_k \right) + \boldsymbol{\beta}\_k^{-1} \boldsymbol{\omega}\_{\mathbb{Z}} \nabla \cdot \left( \mathbf{R}^\top \left( \frac{\boldsymbol{\pi}}{2} + t\_k \boldsymbol{\omega}\_{\mathbb{Z}} \right) \mathbf{x}\_k \right) \tag{A33}$$

$$=\frac{\partial \beta\_k^{-1}}{\partial t\_k} 2\cos(t\_k \omega\_Z) + \beta\_k^{-1} \omega\_Z 2\sin(-t\_k \omega\_Z) \tag{A34}$$

The formulas for *SE*(2) are obtained from the above ones with *s* = 0 (i.e., *βk* = 1).
