FLOWING: Implicit Neural Flows for Structure-Preserving Morphing
Authors: Arthur Bizzi
Matias Grynberg Portnoy
Vitor Pereira Matias
Daniel Perazzo
João Paulo Silva do Monte Lima
Luiz Velho
Nuno Gonçalves
João M. Pereira
Guilherme Schardong
Tiago Novello
Venue: NeurIPS 2025 poster ()
Keywords: flows
implicit neural representations
morphing
warping
Primary area: applications

Abstract: Morphing is a long-standing problem in vision and computer graphics, requir-
ing a time-dependent warping for feature alignment and a blending for smooth
interpolation. Recently, multilayer perceptrons (MLPs) have been explored as
implicit neural representations (INRs) for modeling such deformations, due to
their meshlessness and differentiability; however, extracting coherent and accurate
morphings from standard MLPs typically relies on costly regularizations, which
often lead to unstable training and prevent effective feature alignment. To overcome
these limitations, we propose FLOWING (FLOW morphING), a framework that
recasts warping as the construction of a differential vector flow, naturally ensuring
continuity, invertibility, and temporal coherence by encoding structural flow prop-
erties directly into the network architectures. This flow-centric approach yields
principled and stable transformations, enabling accurate and structure-preserving
morphing of both 2D images and 3D shapes. Extensive experiments across a
range of applications—including face and image morphing, as well as Gaussian
Splatting morphing—show that FLOWING achieves state-of-the-art morphing
quality with faster convergence. Code and pretrained models are available in
https://schardong.github.io/flowing.

FLOWING Implicit Neural Flows
for Scructure-Preservinn Morphing
Arthur Bizzi'
Matias Grynberg²
Vitor Matias
Daniel Perazzo 4.5
João Paulo Lima" 4.6
Luiz Velho
Nuno Gonçalves
João Pereira
Guilherme Schardong"
Tiago Novello
EPFL
University of Buenos Aires
2 University of São Paulo
'IMPA CSMLLIII
6 Universidade Federal Rural Pernambuco
ISR-UC
*INCM
University of Georgia
Abstract
Morphing is long-standing problem in vision and computer graphics, requir- ing a time-dependent warping for feature alignment and a blending for smooth interpolation. Recently, multilayer perceptrons (MLPs) have been explored implicit neural representations (INRs) for modeling such deformations, due to their meshlessness and differentiabillty; however. extracting coherent and accurate morphings from standard MLPs typically relies on costly regularizations, which often lead unstable training and prevent effective feature alignment. To overcome these limitations, we propose FLOWING (FLOW morphING), framework that recasts warping as the construction of differential vector flow, naturally ensuring continuity, invertibility, and temporal coherence by encoding structural flow prop- erties directly into the network architectures. This flow-centric approach yields principled and stable transformations, enabling accurate and structure-preseviine morphing both 2D images and 3D shapes. Extensive experiments across a range of applications- innicauionnnnnlluiiiin face and image morphing, well Gaussian Splatting morphing- show that FLOWING achieves state-of-the-art morphing quality with faster convergence. Code and pretrained models are available in https:// github. io/flowing
Introduction
Morphing is a long-standing problem computer vision and graphics 12 56] with applications in
image editing [48] 4. biometrics [46] [3). and 3D shape interpolation (27 42 The task consists of continuously interpolating between two signals while ensuring that the intermediate representations remain structurally consistent. Traditionally, morphing is decomposed into two stages: warping stage, which aligns source and target features over time (ideally through a one-parameter family of diffeomorphisms), and blending stage, which interpolates between the aligned signals.
Recently, ifmorph 44 introduced face morphing approach that employs multilayer perceptrons (MLPs) as implicit neural representations (INRs) to model the warping transformation. INRs provide a differentiable and memory-efficient solution for representing low-dimensional signals and have
been successfully applied to surface reconstruction 47
58
30 45 surface evolution 27 31
radiance fields [28 3), and image modeling 34 35). However, applying generic MLPs to represent warping transformations presents important limitations. For instance, enforcing temporal coherence requires explicit regularization the loss function, which greatly increases training time and may lead to convergence issues. In addition, unconstrained MLPs lack structural priors, making the learned deformation prone to undesirable behaviors such as singularities regions where the mapping becomes non-invertible- non-invertille-resutnnn in artifacts that degrade the morphing quality (see the first row in Figure bottom-left).
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Face morphing using FLOWING
3DGS morphing using FLOWING
source
target
target
FLOWING offers instant landmarkk alignments
Warping prior to generative models improves morphing
algrment
ilmorph learn flow properties
Emorph FL FLOWING
TD
12
FLOWING is flow by construction
Steps
100 vaaps
1K staps
LOK staps
DrrMtorherr
curs
target
Figure 1: We present FLOWING, robust and theoretically grounded framework for fast, accurate, and structure- structure-prexeriing morphing. It enables smooth and temporally consistent morphs by learning structure- preseervngg flow, applicable both 2D (top-left) and 3D (top-right) data; the latter uses 3DGS for representing 3D faces. Bottom-left: FLOWING instantly aligns landmarks by construction, outperforming So'T methods in both visual quality and convergence speed. Bottom-right: Integrating FLOWING with generative models improves morphing fidelity and semantic coherence.
To address these limitations, we propose FLOWING, a framework that employs specialized flow- based INRs that perform structurr-preservinng warpings by construction. We reframe morphing as the problem of constructing an interpolating flow, allowing us to inherit the structural properties of flow operators through flow-based neural architectures, such as neural ODEs (NODEs) p and neural conjugate flows (NCFs) [6] FLOWING adapts these architectures to the 2D/3D warping contexts and leverages their architectural priors produce morphings that valid by construction. As result, it enables wear-instant training, significantly faster than generic MLPs that depend on soft regularization to enforce coherence. |Figure showcases FLOWING's ability to generate smooth, consistent, and structure- structure-preserving morphs in both 2D images and 3D Gaussian splatting (3DGS) 20), with clear improvements in visual quality, alignment, speed, and semantic coherence over prior methods. In summary, our contributions are:
A principled, flow-based morphing approach that leverages specialized INRs to enforce continuity and temporal coherence by construction. This enables learning highly detailed, structure-preserving morphings near instantly, using only sparse landmark correspondences.
An adaptation of flow-based architectures to the morphing setting, combined with SIRENs B2), allowing the representation of complex, high-frequency deformations and accurate keypoint matching. To minimize spurious deformations, incorporate thin-plate energy regularization 7 449, enabled by a novel forward-mode differentiation scheme.
Demonstration of our approach across diverse morphing tasks-including f'ac morph- ing, general image morphing, and 3DGS morphing-achine strong results in 3D splat morphing via a novel blending scheme.
2 Related works
Warping and morphing techniques have been widely studied in computer graphics and vision 12 556 [55]. Classical approaches rely on geometric transformations such as thin-plate splines (26), radial basis functions [7]. and triangle meshes to morph between images 4 49 While effective, these methods often struggle to handle smooth and complex deformations. Moreover, standard OpenCV- style morphing combines simple linear interpolation between features with shared triangulated domain, which not always available. In contrast, by introducing flow concepts into the model architecture, our method (FLOWING) achieves high-quality warping for both images and 3D data. It

---

requires only feature correspondences as input and supports nonlinear interpolation between features, enabling more flexible and robust morphing.
Implicit neural representations (INRs) have gained significant attention for encoding continu- ous low-dimensional signals directly in the parameters of neural networks. Early works such as SIREN 47 and Fourier feature networks [52] enhanced INR expressiveness by mapping input coordinates to sinusoidal functions, enabling detailed reconstructions of images, 3D shapes, and physical fields. More recently, INRs have been applied to 2D face morphing 44). where SIRENs are used to parameterize the warping and additional costly constraints are introduced to enforce flow properties. In contrast, our method leverages flow-based architectures that enable faster, more efficient, and robust morphing of both 2D and 3D data, with deformations learned as continuous, Seuuctrre-presenvee flows.
Flow-based networks have been explored for modeling continuous-time dynamics and transfor- mations. NODES (9) provide a powerful yet computationally intensive framework for learning continuous-time dynamical systems, where inference through discrete layers replaced by numerical integration. Originally developed for continuous normalizing flows, NODEs have since been applied to medical image registration 57 50 [T, solving PDEs [5). More recently, NCFs 6 were introduced as an alternative approach to model dynamics by topologically deforming affine flows, enabling greater efficiency through parallelism. In this work, we adapt both NODEs and NCFs to the morphing setting for images and 3DGS, further enhancing their representational power with sinusoidal activations.
Generative methods have also emerged as powerful and flexible approach for morphing, with DiffMorpher [59] presenting general image morphing without reference landmarks. However, they face two key limitations: their outputs are constrained to a fixed resolution, and they require pre- aligned target images, adding significant preprocessing overhead. Our method addresses these challenges by introducing flow-based warping, which acts as a sophisticated, non-linear alignment mechanism. This enables seamless integration with generative blending techniques, improving both alignment quality and applicability.
3 FLOWING
3.1 Problem setup: morphing flows
Given source and target media objects f : Ω C IR" + R", where S2 denotes their supports, annotated with K feature correspondences {p" p C Ω. our goal is to construct continuous, time-dependent family of warpings between the source features at t 0 and the corresponding target features at 1, such that @ (p) = p!. For morphing, Tb should have additional properties:
Ω2 x [0, - R" that allows us to smoothly interpolate
Uniqueness. The path traced by each feature must be unique, otherwise distinct features may overlap, leading to unstructured, incoherent interpolations.
Invertibility. The forward and backward mappings of source and target features should match at intermediate times and remain symmetric under time reversal avoid artifacts during blending.
Feature paths should be minimal and smooth for coherent interpolation; otherwise, overfitting may introduce artifacts singularities that degrade morphing quality.
Energy-minimizatte
The key idea behind FLOWING is that the first two properties arise from the definition of flows:
by constraining features to move as an integral over an underlying vector field, we may leverage the uniqueness and reversibility of its orbits. This allows us to recast the morphing problem as the construction of a flow operator [54], enforcing corresponding features to remain consistent over time:
(pl, () = (pp.. 1) for i E (1,..., K} and I. E 1].
(1)
To enforce these structural propertics on the warping map. we employ a 9 p-paraeerrized flow repre- sentation &g (see Sec. 3.3) The third property is imposed by minimizing the curvature of the orbits of &g across the domain Q2. This is achieved with the following thin-plate-lie optimization problem:
arg min, [| (x, 0)112 subject to @ t) &e 1),
(2)
where (x, 0) denotes the time derivative of (x, at t 0 and J its Jacobian. Once the features {PL p are aligned, the resulting flow can be used to warp the source and target signals to
3

---

an intermediate time, after which the resulting warpings are blended together. Figure overview the morphing process.
provides an
feature alignment
p
forward warping
p t)
backward
warping
feature extraction
feature extraction
blending
source
linear
generative
target
Figure 2: Overview of FLOWING. Given source and target images 1 I we extract landmark pairs (p, pt) with feature extractor (dlib, Xfeat, etc). We train flow e such that <(p", t) = "(p' , 1), effectively mapping p to p². At inference, we warp I forward by units and / and backward by t- units, then blend them together with methods such as linear blending or generative models.
3.2 Training
Training flow-based architecture bg requires defining loss function to solve the optimization problem in (2). Note that the uniqueness and invertibility properties are guaranteed by construction, since t is flow. Therefore, the loss function L consists of a data constraint to enforce feature matching and regularization term penalize path distortion:
[(1) = Σ / [o.1] I 2 ( 1)11 di + (0,%(x,)113 dx,
-(0==E
Thin-plate constraint
Data constraint
where À is a parameter. In practice, the data term enforced on only few time steps, since uniqueness ensures that feature correspondences hold across the interval. We find out that using t = 0.5 for NODE and {0, 0.25, 0.5, 0.75, 1} for NCF is sufficient.
For thin-plate term, we minimize the Jacobian of & (x, 0) over the domain, which is equivalent to minimizing the second derivative of the integral path o f) for each Since T is flow, explicit time sampling is unnecessary-ome reason flow-based architectures train faster than generic MLPs. The integral is approximated via Monte Carlo methods, with À the regularization weight. To further accelerate computation, we implement a forward differentiation (FD) scheme based on generalized dual-number arithmetic, enabling derivatives be computed in parallel with inference, significantly reducing overhead. Our ablation study shows that FD makes the thin-plate loss calculation 23 times faster than standard autograd. Full results and details are given in Appendix A
3.3 -lowd architectures
Flow-based networks provide principled representation for the time-dependent warping & as they hold desirable properties such as continuity, invertibility, and temporal coherence. Compared to using MLPs to parametrize the flow operator &g. which require expensive additional terms to approximate uniqueness and invertibility in addition (2). flow-based architectures enforce these properties by construction. As a result, they rely on simpler losses, may achieve near-instant training, and provide accurate alignment while avoiding the catastrophic artifacts that arise when MLPs fail capture proper flow properties.
with our approach (FLOWING). For this experiment, we employed dlib to extract features centered on the faces, which explains why the ears are not aligned. In this work, we focus on two flow-based networks: Neural ODEs and Neural Conjugate Flows.
Figure 3
compares an MLP-based flow (ifmorph 44

---

Neural ODEs. First, we propose us- ing NODEs ( to model the vector field V' the time derivative of the warping deformation 4 with a neural network Fg. Thus, inference for these models in- volves performing numerical integration over the vector field Fe; analogously, it may be interpreted as deep ResNets [[5] with "continuous depth". Specifically, NODEs take the following formulation:
d x = Fo(x)
(4)
source
ifmorph
FLOWING (ours)
target
(xxo, =x())=+++++)))))) x(0)+ / Fo dx
Figure 3: Comparison between ifmorph and FLOWING. While ifmorph fails to preserve structure, FLOWING pro- duces clean and realistic interpolations by enforcing flow properties such invertibility and trajectory uniqueness.
To achieve high-quality matching, the vector field Fo is modeled SIREN (47), allowing it to capture highly detailed spatial deformations necessary to handle the diverse range of distances and orbits traversed by each pair of correspondences. Moreover, to ensure that orbit deformations remain minimal, we penalize the norm of the Jacobian matrix of Fo over the domain Ω.
During training. we integrate Fe using a fourth-order Runge- Kutta method. We integrate the source and target points t 0.5 using the same number of integration steps for the forward and backward dynamics. Since NODE- based approaches are often prone to numerical errors during integration, include in Appendix B an ablation study the total number of integration steps involved in this process, showing that we can use very few integration steps while both preserving an accurate approximation of the vector field Fe and keeping the training efficient. Moreover, inference time, the number of integration steps can be increased arbitrarily, without additional training cost, to more accurately capture the dynamics of Fg.
Neural conjugate flows. While NODES explicitly integrate neural vector fields, they can become computationally expensive when many steps are required during inference. Neural conjugate flows (NCFs) 66 offer an alternative formulation based on topological conjugation. Instead of sequential integration, NCFs employ invertible neural networks to deform the orbits of simple affine flows, enforcing simplified topology. Precisely, the flow-based warping is given by:
To (x, t) Hg v (H)()),t), (x,t) R R.
We parameterize He and 11 using invertible architectures known as coupling layers [1]. These apply alternating, memory-cquipped transformations in sequence, guaranteeing invertibility by construction. Importantly, the resulting conjugated flow Φ remains valid flow, inheriting key properties such as continuity, invertibility, and associativity from the original affine system.
3.4 Blending
Having established how FLOWING constructs flow-based warpings that guarantee structural proper- ties and accurate feature alignment, we now turn to the second stage of morphing: blending.
Image morphing. Let 70, 11 IR. - C denote two input images and &g IR22 flow aligning their features over time. We define the warped images as Jx t) = I ( t)). ensuring spatial alignment at each intermediate timestep E [0, 1). straightforward morphing then obtained by linearly blending the aligned images:
R be
I [ t) (1 tx t) + (x,t)
(6)
yielding smooth function J
2 I C that interpolates between 7 and 11_
While generative models can interpolate between images, they typically lack explicit feature alignment. To address this limitation, first apply our blow- warping 4 to align features over time, and
5

---

then perform blending in the latent space of a pretrained generative model. Let 8 and D denote the encoder and decoder of a generative model. We define generative morphing as:
I (-,t) 2 ((1 - (1-t)c (t) +
(t)), where c'(t) = (()))
(7)
This strategy combines explicit spatial alignment with perceptual quality, producing temporally coherent transitions that significantly outperform naive latent-space blending.
3D Gaussian Splatting morphing, Beyond images, we show that morphing with FLOWING can be extended to 3DGS (20) 36) which has recently emerged as a popular format for 3D repre- sentation. 3DGS enables photorealistic rendering from a set of Gaussians ' with each Gaussian g. {pE, o Ck,y ] consisting of center pv E RR, opacity arg, view-dependent color O E R and covariance matrix Σn E R3x3 that encodes orientation and spread.
Our method morphs between two Gaussian sets, y and E producing an intermediate set B(t) with 8()) = "g" and (((1= = 8¹ We employ a FLOWING network Dg to smoothly align Gaussian centers over time, while linearly blending the opacity parameters aj to ensure seamless transitions. Specifically, for each Gaussian g E 'S', we compute:
21(1) (vo t- i), - 11-1/oo o &e E
(8)
which jointly applies warping and blending. The final morphed set is then defined by 9((1 ()) U (t). This 3DGS morphing procedure preserves structural coherence through flow-based alignment and achieves photorealistic 3D transitions via linear blending strategy.
4
Experiments
We evaluate FLOWING on face and image morphing, as well as on 3DGS morphing, using four diverse datasets. These experiments demonstrate the effectiveness of our approach in both 2D and 3D settings. Additionally, we provide ablation studies in Appendix B to validate our architectural choices and regularization strategies.
Evaluation datasets. For face images, we use the FRLL dataset 10]. which contains 102 identities with 5 images captured at fixed angles and two expressions (neutral and smiling). We use the neutral, front-facing images. Following the protocol in [43,, we construct 1220 pairs, with landmarks extracted using dlib 19 As a post-processing step, we apply FFHQ alignment and cropping, producing images at resolution of 1350 The second dataset is subset of MegaDepth (25), which provides multi-view scene landmarks. From this subset, we extract 275 image pairs based on the cosine similarity between their ResNet (15) embeddings, with feature correspondences obtained using Xfeat [38]. Third, we use eight subjects from NeRSemble 22 a multi-view collection of human heads. Following (39], we fit an FLAME model to each subject to extract 254 landmarks per head. Finally, we include in-the-wild face images from FFHQ 18 for qualitative evaluation.
FLOWING is implemented in PyTorch [33] and trained with the Adam optimizer [21]. At each training step, we sample 20,000 points from the spatial domain [-1, for the selected values of
4.1 Quantitative comparisons
To evaluate the performance of FLOWING on the morphing task, we consider two key components: warping, which aligns landmarks across signals over time, and blending, which generates smooth intermediate transitions. Accordingly, we organize our comparisons into two aspects: landmark alignment 4.1.I) and blending quality (..1.).
4.1.1 Landmark alignment
We compare FLOWING with ifmorph 44 and SIREN activations. For fair comparison, all methods are supervised under the same set of landmark points.
testing both NCF and NODE backbones with sigmoid
Metries. We report the mean squared error (MSE) between warped target landmark positions across intermediate timesteps. For each landmark pair, we sample 10 evenly spaced values of E [0, 1), warp the landmarks accordingly. and compute their MSE, averaging the results across all timesteps. A

---

perfect alignment would yield an average MSE of O, since source and target landmarks would coincide exactly every value of Experiments are conducted on three benchmarks: face monument 3D Gaussian avatar alignment.
Sinusoidal non-sinusoidal activations. We compare NCF- and NODE-based variants of FLOWING using SIREN activations and their sigmoid counterparts to justify our architectural choices for high-frequency representations in morphing quality and convergence, Table reports the average MSE across the FRLL, MegaDepth, and NeRSemble datasets for face warping, monument alignment, and 3D Gaussian avatar morphing. Results are shown for ifmorph. NCF, and NODE. For NCF and NODE, we evaluate both SIREN and sigmoid activations. Note that NCF (SIREN) and NODE (SIREN) outperform ifmorph by one three orders of magnitude, while SIREN activations consistently yield lower errors than the sigmoid case. In particular, NODEs with sigmoid ("vanilla" configurations) perform significantly worse, confirming the importance of sinusoidal activations for accurate warping alignment.
Table 1: Results for the landmark alignment across the FRLL [10] MegaDepth [25 and NeRSem- ble 22 datasets. The best, second and third best results for each dataset/model combination are shown in green, yellow, and orange, respectively.
Model (Activation)
FRLL ())
MegaDepth (1)
NeRSemble (L)
ifmorph 44
1.5E-3
2.9E-1
1.0E-3
NCF (sigmoid)
8.7E-3
2.0E+1
7.2E-5
NCF (SIREN)
3.9E-5
4.2E-4
7.3E-5
NODE (sigmoid)
6.2E-2
1.9E-1
5.0E-4
NODE (SIREN)
1.4E-4
4.9E-4
5.9E-5
NCF vs. NODE. Figure 4 shows that NCFs and NODEs achieve better results with significantly fewer training steps compared to ifmorph. While NODEs converge quickly, they typically show poorer feature aligmment. In contrast, NCFs require more steps to converge, resulting in better feature alignment overall. However, due minor pixel-level variations between source and target features, this does not necessarily translate to noticeable visual differences.
Morphing at different trainingg steps
Alignment error at t-0.5
10
ifmorph
ifmorph
NCF
Thin- Thin-plate term
FLOWING-NCT tours)
10
FLOWING NOWNNGGNDDE
Jacobian term
0.4
0.2
steps
100
Figure 4: Convergence analysis of ifmorph (green), NCF (orange), and NODE (violet). Left: alignment and deformation metrics over training steps. For NODE, alignment is measured using the Jacobian term, while ifmorph and NCF the bassian-based thin-plate term. Right: qualitative morphing results at / = 0.5 after 100, 1000, 2000, 10000, and 20000 training steps. NCF and NODE achieve accurate alignment by 1000 steps, whereas ifmorph requires least 2000 steps.

---

4.1.2 Blending quality
We evaluate blending quality using FRLL [0] for linear face morphing, following the same experi- mental setup as in the landmark alignment experiment (Sec. 4.I.J). We compare FLOWING with SIREN activations against ifmorph and a standard OpenCV baseline using Delaunay triangulation warping with linear blending" Additional comparisons with the classical thin-plate spline method are given in Appendix F. and further experiments on video interpolation are discussed in Appendix E
Metrics. Perceptual quality is measured using the learned perceptual image patch similarity (LPIPS) metric [60], computed between intermediate frames at t = 0.5 and both source and target images. We also measure the Fréchet inception distance (FID) [6] between generated morphs and original images, using 9048-dimensional feature vectors.
Table 2:
Linear image blending results on the FRLL dataset 10. The best, second- best, and third-best results are highlighted in green, yellow, and orange, respectively.
Morphing type
LPIPS(I", 1) (L)
LPIPS (2) (4)
FID (4)
OpenCV
0.233
0.236
32.426
ifmorph
0.250
0.252
38.427
NCF (Ours)
0.221
0.224
33.300
NODE (Ours)
0.210
0.213
31.952
Table 2 shows that both NCF and NODE outperform ifmorph across all metrics. NCF and NODE also obtained better results than OpenCV in LPIPS. For FID, NODE achieves the best overall performance, while NCF re- mains competitive, slightly below OpenCV. Overall, NODE produces tigher-quality and more consistent morphing results than NCF.
SE
LIVE
4.1.3 Warping and morphing times
Table summarizes training and morphing times obtained on an RTX 4090 GPU for both warp training and morphing (warping infer- ence + blending). We report NODE with both 2000 and 20000 training steps to highlight its fast convergence. As shown, training times for NODE and ifmorph remain stable across different landmark counts, whereas NCF ex- hibits increased training time with additional landmarks. Morphing time depends on im- age resolution: NODE provides faster training but slower morphing, while NCF has slower training and intermediate morphing speed.
Lieer blending
DiflMorpher
DitMMrrpher
target
Figure 5: Applications of FLOWING to view in- terpolation (Rows 1-3), and stylization and ob- ject morphing (Rows 4-6). FLOWING produces smooth transitions even under environmental varia- tions, maintaining geometric and photometric consis- tency. When combined with DiffMorpher, it further enhances structural coherence and visual fidelity, re- covering details missed by DiffMorpher alone, such the missing chimneys and the loss of column de- tail (Rows 2-3).
4.2 Image and face morphing
As shown in the previous sections, FLOWING can be used to warp between two images given a set of landmark correspondences. The warped images can then be blended using var- ious techniques, such as linear blending, Pois- son image editing (37], or generative blend- ing [59]. In this work, we focus primarily on linear blending and generative blending via DiffMorpher [59]. Rows 1-3 of Figure 5lillustrate the view interpolation task, where FLOWING produces smooth transitions while better preserving both
httpp:

---

geometric and photometric consistency across viewpoints. These examples show the benefit of applying FLOWING prior generative blending in contrast the usage of generative blending on its own. It improves both the consistency of the scene structure and reduces the number of missing elements or details. For instance, the building chimneys (Row 2) and architectural details (Row 3) remain intact, unlike in competing approaches. Rows 4-6 demonstrate stylization and object morph- ing, where combining FLOWING with generative models enhances structural coherence and visual fidelity across diverse visual domains, for example, in the photo-to-cartoon Einstein interpolation.
Table 3: Warp training times (68 and 130 landmarks) and morphing times at resolutions 256 and 1350 NODE is reported for both 2k and 20k training steps to highlight its fast convergence.
Warping training
Morphing time
Method
Steps
68 landmarks
130 landmarks
Res. 256
Res. 13502
OpenCV
0.01s
0.06s
ifmorph
44
20k
05m26s
05m22s
0.05s
0.10s
NCF (Ours)
20k
08m48s
15m05s
0.02s
0.31s
NODE (Ours)
2k
00m
00m
0.03s
1,48s
NODE (Ours)
20k
02m39s
02m38s
0.03s
1.48s
Figure 6) shows the application of FLOWING non-aligned, in-the-wild face images. Using the NODE backbone for warping and Difhhorpher for blending, our method produces morphings that both visually coherent and perceptually realistic. Additional examples of generative morphing results are provided in Appendix D
target
DiflMorpher
FLOWING (ours)
source
DiffMorpher
FLOWING (ours)
target
Figure 6: Qualitative results in-the-wild faces from the FFHQ dataset [18]. By applying our flow- based warping prior to generative blending, FLOWING produces morphings that are significantly more coherent and realistic than those generated by DiffMorpher.
4.3
3D face morphing using Gaussian splatting
In this section, we evaluate our 3D face morphing framework using GaussianAvatars 339 an extension of 3DGS for photorealistic human head representation. Each face GaussianAvatars associated with FLAME model aligned with the Gaussian distribution. To morph between two GaussianAvvaarrs we apply the warping and blending formulation defined in (8). Figure 7 provides an overview of FLOWING applied to 3DGS morphing. On the left, we visualize the 3D flow field between two subjects derived from their landmarks and illustrate how the warped Gaussians can be combined to form intermediate representation at = 0.5, blending structural and appearance features from both faces. The middle panel shows additional morphing results for different subjects across multiple time steps. Our method achieves smooth and geometrically consistent transitions between subjects with distinct facial structures and appearance attributes. In particular, it naturally handles complex variations such as hair (second and third rows), thanks to the volumetric nature 3DGS.
[24]. which enables the extraction of 3D facial landmarks spatially
Finally, the right panel compares different blending strategies. The top row corresponds to purely 2D setting, where warping applied to the rendered images followed by linear blending, which results in severe misalignments and ghosting artifacts. In contrast, the middle and bottom rows show our 3D warping results, first with linear blending 3D Gaussians. The fully 3D pipeline produces smoother and more coherent transitions, effectively eliminating ghosting and preserving fine structural details across viewpoints. Additional qualitative examples of 3D morphing results are presented in Appendix C
rendered images, and then with direct blending

---

Blending 3D Gaussian splatting
3D Morphing across poses and timesteps
Comparisons
source
target
Morphing rendered images
& S 0.5)
dulp 0.5)
Warping 3D05 Blending
randered images
warped (0.5)
Blending union (0.5) w (0.5)
warped target (0.5)
source
torget
Warping blanding (Curs)
Figure 7: Overview of FLOWING for morphing between 3D faces using Gaussian splatting. Our approach warps the space to enforce 3D landmark alignment and applies umion- based 3DGS blending, directly combining Gaussians 3D space (left). This yields geometrically consistent morphs and preserves photorealistic appearance across poses and viewpoints (middle). The right panel compares blending strategies, showing that our union-based 3DGS blending produces smoother and more coherent results than morphing rendered images blending after warping.
Conclusion and limitations
This work introduced FLOWING, novel framework for morphing between graphical objects using flow-structured INRs. By leveraging the mathematical properties of flows, our approach enables faster convergence, more stable training, and robust warping behavior that avoids catastrophic singularities. We demonstrated the versatility of FLOWING across a wide range of tasks-including face morphing, view interpolation, and Gaussian- based 3D morphing-achievinng temporally coherent results. Moreover, FLOWING can be seamlessly integrated into generative pipelines as a replacement for traditional alignment procedures, producing morphings comparable to state-of-the-ar generative approaches.
high-quality and
Despite these advantages, FLOWING inherits certain limitations intrinsic flow-based formulations. Since it enforces invertibility by construction, it cannot model transformations involving occlusions topological changes, such as morphing between faces with open and closed mouths. In such cases, generative models can complement our approach during the blending stage to recover missing structures. Additionally, as in other warping-based techniques, FLOWINGG depends the quality and accuracy of landmark correspondences, whether extracted manually or through automated detectors.
As future work, plan extend FLOWING to point cloud alignment and registration, and further improve our Gaussian morphing approach.
Acknowledgments
Guilherme and Nuno would like to thank Fundação de Ciência Tecnologia (FCT) projects UIDB/00048/2220 and UIDP/00048/2220 for partially funding this work. Guilherme would also like to thank FCT project 2024.0768 LACDQ partially funding this work. João Paulo would like thank Fundação Carlos Chagas Filho de Amparo Pesquisa do Estado do Rio de Janeiro (FAPERJ) grant SE1-220003301 2800/20224 for funding this work. Vitor and Daniel gratefully acknowledge support from CAPES, grants 88887.8 88887.842588 842584/2023-00 and 01887.832821/2223.2 supporting this research. João Pereira is thankful for start-up grant from the University of Georgia. We also thank Google for funding this research.
respectively for
PDoI: http:///oooorg////// httcpp//aaaa..rr0000444444/00000
PDOl: attps://ddan http://doo.orr/100// 54499/2024 07681 IACDC
10

---

References
[1] Ivan Anokhin, Kirill Demochkin, Taras Khakhulin, Gleb Sterkin, Victor Lempitsky, and Denis Korzhenkov. Image generators with conditionally-indeeeneen pixel synthesis. CVPR, pages 14278-14287, 2021.
[2] Guha Balakrishnan, Amy Zhao, Mert Sabuncu, John Guttag, and Adrian V Dalca. Voxelmorph: a learning framework for deformable medical image registration. IEEE transactions on medical imaging, 2019.
[3] Jonathan T. Barron, Ben Mildenhall, Dor Verbin, Pratul P. Srinivasan, and Peter Hedman. Mip-nerf 360: Unbounded anti-aliased neural radiance fields. CVPR, 2022.
[4] Thaddeus Beier and Shawn Neely. Feature-based image metamorphosis. ACM SIGGRAPHI computer graphics, 26(2):35-42, 1992.
[5] Arthur Bizzi, Leonardo M Moreira, Márcio Marques, Leonardo Mendonça, Christian Júnior de Oliveira, Vitor Balestro, Lucas dos Santos Fernandez, Daniel Yukimura, Pavel Petrov, João Pereira, Tiago Novello, and Nissenbaum Lucas. Neuro-spectral architectures for causal networks. arXiv preprint arXiv:2509.04966, 2025.
[6] Arthur Bizzi, Lucas Nissenbaum, and João M. Pereira. Neural conjugate flows: A physics- informed architecture with flow structure. Proceedings of the AAAI Conference on Artificial Intelligence, 15576--5555---58558 Apr. 2025. doi: 10. 10..166//aai..//99/11 v39115.33710. URL https //ojs. //ojs.aaai.or///n/aaooe iidex.iddxxxxe php/ Chap:/AAT/aaaa htwaa//ariiceeeve////3113
[7] Fred I. Bookstein. Principal warps:
Thin-plate splines and the decomposition of deformations. IEEE Transactions on pattern analysis and machine intelligence, 11(6):667-585, 1989.
[8] Daniel J Butler, Jonas Wulff, Garrett B Stanley, and Michael J Black. A naturalistic open source movie for optical flow evaluation. In Computer -ision-ECCV 2012: 12th European Conference on Computer Vision, Florence, Italy, October 7-13, 2012, Proceedings, Part 12, pages 611-6116666 1-155.5. Springer, 2012.
[9] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Neurips, 31, 2018.
[10] Lisa DeBruine and Benedict Jones. Face research lab london set, May 2017.
[11] Laurent Dinh, Jascha Sohl-Dickstein and Samy Bengio. Density estimation using real nvp. In International Conference on Learning Representations, 2017.
[12] Jonas Gomes, Lucia Darsa, Bruno Costa, and Luiz Velho. Warping & morphing of graphical objects. Morgan Kaufmann, 1999.
[13] Marcel Grimmer and Christoph Busch. Ladimo: Face morph generation through biometric template inversion with latent diffusion, 2024. URL httttttss//ariii org//ogggl abs/aab//000//////////////////// 2410. 07988
[14] Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regularization for learning shapes. In 37th International Conference on Machine Learning, ICML 2020, pages 3747-3757, 2020.
[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 770-778, 2016. doi: 10. 10.11099CVPR.22116.90
[16] Martin Heusel. Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. GANs trained by a two time-scale update rule converge to a local Guyon, U. Von Luxburg, S. Bengio, gus, S. Vishwanathan, and R. Garnett, editors, Neurips, volume 30. Curran Associates, Inc., 2017. URL http::: htttps::/proceeednngg hewpeeeeedddiinnnennneeeeeeeeeee neurips.comm wttpcc/paaaa http:c/ppeeelllee/epppeeeeeeeeee wwt.fflle///pprr/0/////eeee 8a.dd9977774400ffff6681116666660
nash equilibrium.
Wallach, R. Fer-
Paper.

---

[17] Zhewei Huang, Tianyuan Zhang, Wen Heng, Boxin Shi, and Shuchang Zhou. Real-time inter- mediate flow estimation for video frame interpolation. In European Conference on Computer Vision, pages 624 642. Springer, 2022.
[18] Tero Karras, Samuli Laine, Mika Aittala, Janne Hellsten, Jaakko Lehtinen, and Timo Aila. Analyzing and improving the image quality of stylegan. In 2020 IEEE/CVT Conference on Computer Vision and Pattern Recognition, pages 8
Computer Vision Foundation /
IEEE, 2020. doi: 10.1 N0.11099CVPP--2600220000000
[19] Vahid Kazemi and Josephine Sullivan. One millisecond face alignment with an ensemble of regression trees. In CVPR, June 2014.
[20] Bernhard Kerbl, Georgios Kopanas, Thomas Leimkühler, and George Drettakis. 3d gaussian splatting for real-time radiance field rendering. ACM Transactions on Graphics. 42(4), July 2023. URL atttps://repe sam inria. fr wrriiiafffffuaaia ht/fungraph////// 3d
www.:appiidddaasiaa-aaaaaaaaaaea gaussian spllatttnn
[21] Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization, 2017. URL
httpp:///arxiv.or//abbbss 1412 6980
[22] Tobias Kirschstein, Shenhan Qian, Simon Giebenhain, Tim Walter, and Matthias Nießner. Nersemble: Multi-view radiance field reconstruction of human heads. ACM Transactions on Graphics (TOG), 42(4):1-14, 2023.
[23] Lingtong Kong, Boyuan Jiang, Donghao Luo, Wenqing Chu, Xiaoming Huang, Ying Tai, Chengjie Wang, and Jie Yang. Ifmet: Intermediate feature refine network for efficient frame interpolation. In CVPR, pages 1969-1978, 2022.
[24] Tianye Li, Timo Bolkart, Michael Black, Hao Li, and Javier Romero. Learning a model of facial shape and expression from 4d scans. ACM Trans. Graph., 16 2017.
[25] Zhengqi Li and Noah Snavely. Megadepth: Learning single-view depth prediction from internet photos. In CVPR, pages
2018.
[26] Jing Liao, Rodolfo Lima, Diego Nehab, Hugues Hoppe, Pedro V Sander, and Jinhui Yu. Automating image morphing using structural similarity on a halfway domain. ACM Transactions on Graphics (TOG), 33(5):1-12, 2014.
[27] Ishit Mehta, Manmohan Chandraker, and Ravi Ramamoorthi. A level set theory for neural implicit evolution under explicit flows. In Computer Vision ECCV 2022, pages 711-729, Cham, 2022. Springer Nature Switzerland. ISBN 978-3-0311-2086-11
[28] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng. Nerf: Representing scenes as neural radiance fields for view synthesis. In European Conference Computer Vision, pages 405-421. Springer, 2020.
[29] Richard D Neidinger. Introduction to automatic differentiation and matlab object-oriented programming. SIAM review, 5263):555-563, 2010.
[30] Tiago Novello, Guilherme Schardong, Luiz Schirmer, Vinícius Da Silva, Hélio Lopes, and Luiz Velho. Exploring differential geometry in neural implicits. Computers & Graphics, 108:49-60, 2022.
[31] Tiago Novello, Vinicius Da Silva, Guilherme Schardong, Luiz Schirmer, Helio Lopes, and Luiz Velho. Neural implicit surface evolution. In Proceedings of the IEEE/CVF international conference on computer vision, pages 14289, 2023.
[32] Tiago Novello, Diana Aldana, Andre Araujo, and Luiz Velho. Tuning the frequencies: Robust training for sinusoidal neural networks. In Proceedings of the Computer Vision and Pattern Recognition Conference, pages 3071-3080, 2025.
[33] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang. Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala, Pytorch: An imperative style, high-performance deep learning library, 2019. URL httts::///rxii www./rgg///ooeoooooooooeoeeeeeeo 1ttp//b//111120033
12

---

[34] Hallison Paz, Daniel Perazzo. Tiago Novello, Guilherme Schardong, Luiz Schirmer, Vinicius da Silva, Daniel Yukimura. Fabio Chagas, Helio Lopes, and Luiz Velho. Mr-net: Multiresolution sinusoidal neural networks. Computers & Gruphics, 2023.
[35]
Hallison Paz, Tiago Novello, and Luiz Velho. Spectral periodic networks for neural rendering. In ACM SIGGRAPHI 2024 Posters, pages 1-2. 2024.
[36] Vitor Pereira Matias, Daniel Perazzo, Vinicius Silva, Alberto Raposo, Luiz Velho, Afonso Paiva, and Tiago Novello. From volume rendering to 3d gaussian splatting: Theory and applications. In SIBGRAPI Conference on Graphics, Patterns and Images, 2025.
[37] Patrick Pérez, Michel Gangnet, and Andrew Blake. Poisson image editing. In Seminal Graphics Pupers: Pushing the Boundaries, Volume 2, pages 577-582. 2023.
[38] Guilherme Potje, Felipe Cadar, André Araujo, Renato Martins, and Erickson Nascimento. Xfeat: Accelerated features for lightweight image matching. In CVPR, pages 2682-2691, 2024.
[39] Shenhan Qian, Tobias Kirschstein, Liam Schoneveld, Davide Davoli, Simon Giebenhain, and Matthias Nießner. GautsianAvatars: Photorealistic head avatars with rigged 3d gaussians, 2024.
URL
22220000999
[40] Christos Sagonas, Epameinondas Antonakos, Georgios Tzimiropoulos, Stefanos Zafeiriou, and Maja Pantic. 300 faces wn-the-wild challenge: database and results. Image and Vision Computing, 47:3-18, 2016. ISSN 0262-8856. doi: https://doi.orr//00. 10. 1016/j 2016.. 1.0.00............ URL https::/ //ww:.//wwwwww./// sciencedireett.............iii.. www. wwt.c////ccccceecceeeeeee aataac//articee//ee wtteeee/iiiiie000 httpa//aaa.aaaappo////0/006o0110 ///0//2222885666000104 300-W, W, the First Automatic Facial Landmark Detection in-the- in-the-Wild Challenge.
[41] Lu Sang, Zehranaz Canfes, Dongliang Cao, Florian Bernard, and Daniel Cremers. Implicit neural surface deformation with explicit velocity fields, 2025. URL https://axxiv org/
abs/2501. 140//8
[42] Lu Sang, Zehranaz. Canfes, Dongliang Cao, Riccardo Marin, Florian Bernard, and Daniel Cremers. 4deform: Neural surface deformation for robust shape interpolation. In Proceedings of the Computer Vision and Pattern Recognition Conference, 2025.
[43] Eklavya Sarkar, Pavel Korshunov, Laurent Colbois, and Sébastien Marcel. Vulnerability analysis of face morphing attacks from landmarks and generative adversarial networks, October 2020.
URL luttp:///axxix owgprxii.ooooroeeoooeoo 2012.. 05344
[44] Guilherme Schardong, Tiago Novello, Hallison Paz, Turii Medvedev, Vinícius Da Silva, Luiz Velho, and Nuno Gonçalves. Neural implicit morphing of face images. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition, pages 7321-7330, 2024.
[45] Luiz Schirmer, Tiago Novello, Vinícius da Silva, Guilherme Schardong, Daniel Perazzo, Hélio Lopes, Nuno Gonçalves, and Luiz Velho. Geometric implicit neural representations for signed distance functions. Computers & Graphics, 125:104085, 2024.
[46] Jag Mohan Singh and Raghavendra Ramachandra. 3-d face morphing attacks: Generation, vulnerability and detection. IEEE Transactions on Biometrics, Behavior, and Identity Science, (1):103-117, 2024. doi: 10. 101109//BIOOM.222..32666
[47] Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. Neurips, 33, 2020.
[48] Douglas Smythe. A two-pass mesh warping algorithm for object transformation and image interpolation. Technical Report 1030, ILM Technical Memo, Computer Graphics Department, Lucasfilm Ltd., 1990.
[49] Olga Sorkine and Mare Alexa. As-rigid-as-posse surface modeling. In Proceedings of the Fifth Eurographics Symposium on Geometry Processing, SGP '07, page 109-116, Goslar, DEU, 2007. Eurographies Association. ISBN 9783905673463.
[50] Shanlin Sun, Kun Han, Deying Kong, Hao Tang, Xiangyi Yan, and Xiaohui Xie. Topology- preserving shape reconstruction and registration via neural diffeomorphic flow. In CVPR, 2022.
13

---

[51] Shanlin Sun, Kun Han, Chenyu You, Hao Tang, Deying Kong, Jumayed Naushad, Xiangyi Yan, Haoyu Ma, Pooya Khosravi, James S Duncan, al. Medical image registration via neural fields. Medical Image Analysis, 97:103249, 2024.
[52] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Kriiivich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Neurips, 33:7537-75477, 2020.
[53] Zachary Teed and Jia Deng. Raft: Recurrent all-pairs field transforms for optical flow. In Computer -CcionECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part II 16, pages 402-419. Springer, 2020.
[54] Marcelo Viana and José M Espinar. Differential equations: a dynamical systems approach to theory and practice, volume 212. American Mathematical Society, 2021.
[55] George Wolberg. Digital image warping, volume 10662. IEEE computer society press Los Alamitos, CA, 1990.
[56] George Wolherg. Image morphing: survey. The visual computer, 14(8-9):360-37,, 1998.
[57] Yifan Wu, Tom Z Jiahao, Jiancong Wang. Paul Yushkevich, M Ani Hsieh, and James C Gee. Nodeo: A neural ordinary differential equation based optimization framework for deformable image registration. In CVPR, 2022.
[58] Guandao Yang. Serge Belongie, Bharath Hariharan, and Vladlen Koltun. Geometry processing with neural fields. Neurips, 34:22483 22497, 2021.
[59] Kaiwen Zhang, Yifan Zhou, Xudong Xu, Bo Dai, and Xingang Pan. Diffmorpher: Unleashing the capability of diffusion models for image morphing. In CVPR, pages 7912-7921, 2024.
[60] Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreason- able effectiveness of deep features as perceptual metric. In CVPR, 2018.
A Forward differentiation
We implement forward differentiation (FD) to speed up the calculation of Jacobian or Hessian terms, for thin-plate regularization. Forward mode differentiation is alternative way implement automatic differentiation, opposite to backward (or reverse mode) differentiation. FD is typically implemented using tangent (or dual) numbers [29]. Let illustrate FD with an example. Suppose want to evaluate dzn. where
24= = f( i i= 1,...,n.
(9)
Using the chain rule, we obtain that
Ozn
Oz, Oc.-1
II (((-1)))
Oro
i=1
To implement FD, we introduce the tangent (or dual) variables in. ht O,..., n, defined by the recursion in 8us 8mp
= 1, and
4 = f1(x4-1) 1= 1=
(10)
Using induction, follows that
Jzk Oxo
in = (((
II ((x-1))
I....,n.
Finally, the last element yields the desired derivative i,, replace every function fü by an FD version, that calculates both æ; and é; the same time.
Ura Dar
To implement this approach, we
14

---

Effectively, the FD version of f: maps ordered pairs (21-1, to ordered pairs (z₁, 24), so that both and (TPp are satisfied:
(4) f.D (#i-1, = (fi(z-1)) (((=-))))) i
Since we need calculate Hessians, develop multivariate FD implementation involving ordered triplets (z, 2, 2). Here, & contains intermediary gradients, and ž contains intermediary Hessians, with respect variable More specifically, if Rm and I", implementation ensures that 2 = V E Rmx" and _ = Hess, (2) E mxxxx.
To calculate the Hessian of the model fa(z) with respect 2, we first write it composition of simpler functions (linear operators, non-linear activations, flow operator of affine flows), for which have implemented forward differentiation: fo fi f2 o...o fm- Then, noting that V I,
(the n x n identity matrix) and Hess, (z) = OxxxXn we calculate
fo (x. In, (xxnxn) = fi
f (x, In Onxnxn) = (u, ú, ū)
and set Hess (fo(x)) ü.
Followingly, we provide the details for the FD implementation of linear transformations, non-linear activations, and the flow operator of an affine flow. By composing these, we can construct all architectures we consider in the paper.
Linear operators: Consider the linear operator Az + Az + b. Since derivatives are also linear operators, the FD implementation is straightforward.
ADD (z, 2, 2) (Az + b, Aě, A x1 2);
where Ax denotes multiplication by in the first dimension.
Non-linear activations: chain rule, one obtains:
Suppose that E R. g. R::RR + RR R and E E R Then using the
HD (z, 2) (o(=), Vg(z), Hess (g(2))),
[g(()) o (2)2, v(d(2))))
(g(2), gf (z)2, (2)= (=))
Here, the terms g(2), and g=(=) are the derivatives of the activation function. For instance, for SIRENs, we have g(z)
sin (w)2), g'(=) = cos(w))) and g=(z) ww sin(wo=) wZg()).
wag
Flow operator of an Affine Flow:
The formula for an affine flow provided in (6)
@ = - [
bbb
where the affine integral term is calculated exactly using an augmentation trick [6]. This flow has some nice properties: the map linear on and the derivatives of the exponential in terms of t are easier to calculate. Suppose we want to evaluate PFD (x, x, x) (t,t, f)), where x E IRd x E RRdxn kdxxxn t R. H. and paxx Letting 4xx, t), Af b and E At obtain, using the chain rule:
HD ((x, x) (1,1,)) = (#(x, (), V TT V x V T 0 tae ii,
= (f, Ex PiT Ex (AEX
2 i (Af) 8i)
B Ablation studies
NODE integration steps. Neural ODEs require solving differential equations through numerical integration, both during training and inference, to associate each point with its corresponding source and target coordinates by integrating forward and backward in time. Thus, it might appear that the trajectories need to approximated using a large number of steps, making the process computationally expensive. However, we find that this is not the case: high-quality solutions can
15

---

be obtained with only a few steps. We sampled several initial conditions from the [-1, 1]2 grid and displaced the points using the trained NODE model with the same number integration steps used during training. We then approximated the reference (ground-truth) trajectories by computing high-resolution baseline with 1000 integration steps. As a quality metric, we measured the maximum squared error between the predicted trajectory points and the linearly interpolated baseline points; a lower error indicates that the numerical integration used during training accurately captures the underlying dynamics. Table 4 shows the results of this experiment. We observe that only small number of integration steps is sufficient to approximate the flow with high fidelity. Moreover, applying Jacobian regularization during training further stabilizes the dynamics, enabling reliable results even with very few integration steps.
Table 4: Comparison of trajectory quality (maximum squared enror) of NODEs with different integration steps against a baseline of 1000 steps. NODE-based morphing needs very few steps in order to obtain high quality approximations of the associated flow.
Int. steps
MegaDepth Sample
FRLL Sample
No regularization
With regularization
No regularization
With regularization
3
5,44E-05
6,43E-07
1.36E-07
7. 7.16E-10
5
3.39E-08
3.27E-09
2.08E-10
6.191111
7
1.88E-09
5.24E-10
5.46E-11
6.00E-11
15
8.88E-12
5.18E-10
5.46E-11
6,10E-11 11
Forward differentiation (FD). We conducted an experiment to evaluate the computational speed-up achieved by our FD scheme for Hessian computation. An NCF model was initialized using the same config- uration as in our main experiments. For this model, computed the Hessian 1,000 domain points using both PyTone PyTorch's 's autograd and our FD imple- mentation, then evaluated the thin-plate loss and per- formed backpropagation. Each method was run 100 times, and we report the average computation time per iteration. On an NVIDIA GeForce RTX 4090 GPU, our FD implementation achieved an average iteration time of 6.93 x 10-3 seconds, compared to 1.62 10-1 seconds using autograd, corresponding to x speed-up. This demonstrates the substan- tial efficiency gains of our FD approach over standard automatic differentiation.
ifmorph
NCF
NODE
w/O TP
w/ TP
Figure 8: Effect of TP regularization on if- morph (left) and our methods, NCF (center) and NODE (right). The top row shows results without TP regularization, while the bottom row shows the same models with TP applied. TP regularization promotes smoother defor- mations and enhances structural coherence across the morphing process.
Thin-plate regularization. We perform an ablation study to evaluate the effect of thin- plate (TP) regu- larization on deformation smoothness and structural consistency. As illustrated in Figure 8 TP regular- ization significantly improves the coherence of the deformation field, reducing distortions and enforcing smoother transitions between source and target landmarks. The effect is most pronounced for the baseline ifmorph, which exhibits severe artifacts in the absence of regularization. In contrast, our models (NCF and NODE) already demonstrate stable deformation behavior due to their flow-based formulation, yet still benefit from TP regularization, yielding even more consistent and visually coherent morphings.
C Additional 3DGS morphing experiments
Flow evaluation Figure Iillustrates the 3D flow streamlines corresponding to landmark trajectories from the source image for three methods used in 3DGS morphing: ifmorph, and FLOWING with NCF and NODE backbones. As shown, ifmorph produces flow fields that are less stable and exhibit pronounced curvature, while NCF and NODE yield smoother, more coherent, and regularized flows.
16

---

This difference is particularly evident in regions not directly constrained by landmarks- as the hair-where ifmorph introduces noticeable distortions. In contrast, the FLOWING variants preserve both structural continuity and spatial consistency across the field.
IFMORPH
NCF
NODE
Figure Visualization of flow fields generated by ifmorph, NCF, and NODE the 3DGS morphing task. The red arch highlights instability in the hair region for ifmorph, whereas NCF and NODE preserve structure more reliably.
Training details Our 3DGS morphing configuration extends the 2D morphing pipeline while maintaining the same loss weights. To accommodate the higher dimensionality, we use slightly larger learning rates (LRs) and an LR scheduler for both NODE and NCF to ensure convergence within 20,000 steps. The initial LRs are set to 0.001 for NODE, 0.002 for NCF, and 0.0001 for ifmorph. Early stopping employed with patience 500 epochs for NCF and NODE, and 1,000 ifmorph. If the loss plateaus for 100 epochs, the LR is reduced, and training terminates once the patience threshold reached. The best-performing model selected based on lowest loss.
Additional results Figure T0 presents additional qualitative results for Gaussian morphing framework. FLOWING consistently achieves smooth and coherent transitions across diverse subjects, effectively handling challenging regions such facial and head hair (third row). Furthermore, demonstrates strong generalization when morphing between subjects of different genders, maintaining both geometric structure and photometric consistency.
D Additional comparisons using generative blending
Figure I1 shows additional qualitative results combining FLOWING with DiffMorpher as generative blending strategy. These examples further confirm that introducing a warping stage prior to generative blending significantly improves the quality of the final morphs. In particular, FLOWING enhances spatial alignment and structural coherence, producing smoother intermediate representations, even in simple scenarios such morphing between spherical objects.
17

---

Figure 10: Additional results of Gaussian morphing across multiple timesteps (first and last columns show the targets). FLOWING achieves smooth and coherent transitions across diverse subjects while preserving structure and appearance.
source
DiffMorpher NODE + DiffMorphar
target
source
DiffMorpher
NODE DiffMorphor
target
Figure 11: Additional examples comparing generative blending with and without FLOWING as a warping prior. Incorporating FLOWING before applying DiffMorpher (NODE+DiffMorphe) ours) yields superior structural preservation and smoother transitions.
E
Applications in video interpolation
Video interpolation aims to synthesize intermediate frames between two given frames, key compo- nent in applications such as video compression and frame rate upscaling. Most deep learning-base -baseg approaches rely on estimating optical flow between consecutive frames, which describes pixel motion as vector field 17 233 Assuming temporal proximity between frames, optical flow can be leveraged -ndeeeeneendnnt ODE defined by the flow field.
approximate intermediate frames by integrating
Inspired by this principle, we explore the use of FLOWING to learn a compact representation of optical flow from sparse samples and to reconstruct intermediate frames. For this experiment, we employ a cropped sequence from the Sintel dataset ( and use RAFT [53] to generate dense optical flow between frames. We then sample sparse set of image points and construct landmark pairs (p" p²) by displacing them according to the optical flow. FLOWING is trained to model continuous flow field aligning these points over time. To evaluate interpolation quality, we warp and linearly blend the input frames at = 0.5 using the learned flow and compare the resulting image against the ground-trut middle frame. We further compare against three baselines: (1) linear frame blending, (2) dense optical-flow warping followed by linear blending, and (3) ifmorph [44]. As shown Table FLOWING achieves lower MSE than all baseline interpolation methods, indicating improved accuracy in predicting intermediate frames. These preliminary results suggest that learning a continuous optical flow representation from sparse correspondences is promising direction for video interpolation.
18

---

Table 5: Comparison of interpolation accuracy against the ground-trutt intermediate frame on Lower MSE indicates better reconstruction.
Sintel
Method
MSE (4)
Linear Blend
2.6E-4
Optical Flow Warp
4.9E-5
ifmorph
5.8E-5
NCF (Ours)
4.0E-5
NODE (Ours)
3.9E-5
F Comparison with TPS warping
Another standard morphing algorithm thin-plate-spline warping. While TPS warping does not involve an iterative process, the method does require solving linear system of equations in order to find the appropriate linear combination of the basis functions. The matrix such a system can, in some cases, be badly conditioned. When using the scikit-image implementation of the algorithm, we find that this problem leads to crashes or low-quality warpings approximately 10% FRLL samples. This can negatively impact metrics such the LPIPS. FLOWING, the other hand, does not present these problems as it does not rely on solving a similar sudly-conditioned system. For comparison, using TPS warping combined with linear blending for the FRLL dataset, obtain LPIPS T) = 0.371 and LPIPS(TL, T) = 0.374.
19

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: We have described the method and performed experiments to demonstrate the paper's contributions.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: We have testes our method presenting ablation studies on its parameters to show some limitations.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (c.g.. independence assumptions, noiseless settings, model sell-speciiiatiinn asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect the scope of the claims made, e.g. if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-tex, system might not be used reliably to provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and faimesss.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community, Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
20

---

Answer: [Yes]
Justification:
We present a clear statement of our definitions with experiments to motivate
them.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proof' in the paper should be numbered and cross referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental Result Reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: We presented all the details and, in case of the paper being accepted, we'll 'lll make available all experiments and code.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reprodueibility may be tricky in some cases, in which authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
21

---

reproducible, they Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: We presented all the details and, in case of the paper being accepted, we'll make available all experiments and code.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, so "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
/
Cooee//mdeeeSbmmmcia ssi onPolicy) for more details.
The authors should provide instructions data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended to the paper) recommended, but including URLs to data and code is permitted.
6. Experimental Setting/Details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: For all experiments have presented the detailed information about relevant hyperparameters and evaluation.
Guidelines:
The answer NA means that the paper does include experiments.
The experimental setting should be presented in the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment Statistical Significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer:
[Yes]
Justification:
We presented statistical information in our main results.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
22

---

The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful figures symmetric error bars that would yield results that are out of range (e.g. negative rates).
show in tables or
If error bars are reported in tables or plots, The authors should explain the text how they were calculated and reference the corresponding figures tables in the text.
8. Experiments Compute Resources
For each experiment, does the paper provide sufficient information on the puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
The information related is given at beginning of Section
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary failed experiments that didn't It make it into the paper).
9. Code Of Ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurlPS Code of Ethics
Answer: [Yes]
Justification:
We read the Code Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is special consid- cration due to laws or regulations in their jurisdiction).
10. Broader Impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [NA]
Justification:
We mostly deal with theoretical and technical issues of implicit neural net-
works.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
23

---

Examples of negative societal impacts include potential malicious unintended uses (e.g., disinformation, generating fake profiles, surveillance), faireess considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper deseribe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: Our work poses no such risks.
Guidelines:
The answer NA means that the paper poses no such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification:
We reference the authors of the code used in Section 4.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website). the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in package should be provided. For popular datasets, paperswithcode com/ datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
24

---

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if has changed) should be provided.
If this information is not available online, the authors are encouraged to reach out to the asset's s creators.
13. New Assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: Our work doesn't 't release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details the dataset/code/moodl as part their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and Research with Human Subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: Our work doesn't involve crowdsourcing nor researach with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional Review Board (IRB) Approvals or Equivalent for Research with Human Subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: Our work doesn't involve crowdsourcing nor researach with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
25


Reviews:
[{"review_id": "6sc9o4LvnS", "reviewer": "Reviewer_Qbyp", "summary": "The paper proposes a structure preserving morphing mechanism which is based on Neural ODEs. The reason for using Neural ODE is because it naturally provides features like invertibility and temporarily smooth. This line of thinking has been used in several works such as in shape deformation so having a similar line o attack for the problem of morphing makes sense. The theoretical development presented in the paper is sound and rather straightforward. In addition to the flow based morphing approach, the paper leverages specialised implicit neural representations (INRs) to represent complex data such as images, etc. The paper includes several morphing experiments including extensions such as morphing of 3dGS and latent space morphing which open up exciting research directions.", "questions": "1. A section on limitations would be helpful. \n\n2. I am not clear why a vanilla NODE applied on the features (where t=0 is source and t=1 is target) won't work for morphing? Can the authors elucidate this point in more detail?", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. The simplicity of the framework, I argue is a major strength. NODEs are a very powerful class of models which come with several advantages such as invertibility by construction. Using them for the problem of morphing is an interesting idea. \n\n2. Instead of naively applying NODE to morphing, the paper uses vision encoders to first map images in feature space and then learn the flows between source and target images. I think this underlying idea can very well be generalisable to other problems in computer vision and seeing its promise for the problem of morphing is interesting. \n\n3. The paper shows both qualitatively and quantitatively superiority over prior art and includes interesting applications such as with 3DGS morphing and use of generative blending. \n\n\nWeakness. \n\n1. As with most NODE based approaches, I suspect there would be complications due to numerical errors in solving the initial value problem as well as slower speed. The paper hasn't sufficiently discussed these aspects.", "was_revised": true, "final_justification": "I thank the authors for their response. I believe that my concerns are validated. I disagree with my fellow reviewers and find the paper contributions significant enough. Additionally I see that concerns regarding performance gains over prior methods are also substantial as noted in the paper as well as in the rebuttal. Thus, I vote to accept this paper.", "rebuttal": "We thank the reviewer for the detailed comments. We are encouraged by the positive remarks regarding the simplicity of the framework, the use of flows for the problem of morphing as an interesting idea, the fact that the underlying idea can be generalizable to other problems, the qualitative and quantitative superiority over prior art, and interesting applications in 3DGS morphing / generative blending. We tried our best to address the reviewer’s concerns by performing additional experiments regarding numerical precision and computational requirements.\n\n## Weakness\n\n> **1. As with most NODE based approaches, I suspect there would be complications due to numerical errors in solving the initial value problem as well as slower speed. The paper hasn't sufficiently discussed these aspects.**\n\nWe use a fourth-order Runge-Kutta integrator with 10 steps, which should lead to errors on the order of $O(10^{-4})$.\nIn practice, we observed no significant numerical error accumulation. In any case, this was one of the reasons why we included Neural Conjugate Flows as an integration-free alternative to Neural ODE. Additionally, we include an ablation study in which we compare inference trajectories with different number of steps against a 1000 steps baseline.  We perform this experiment using a FRLL sample (faces) and a Megadepth sample (monument views)\n\nFRLL sample\n| Steps | Max Squared Error |\n| ----- | ----------------- |\n| 2     | 5.12e-07          |\n| 5     | 6.19e-11          |\n| 10    | 6.37e-11          |\n| 20    | 6.19e-11          |\n\nMegadepth sample\n| Steps | Max Squared Error |\n| ----- | ----------------- |\n| 2     | 2.17e-05          |\n| 5     | 3.27e-09          |\n| 10    | 5.51e-10          |\n| 20    | 5.51e-10          |\n\nThe results indicate that trajectories obtained using 10 and 1000 integration steps are nearly identical.\n\nWe would also like to mention the difference between the number of integration steps during training and the number of integration steps during inference. The training integration steps are used to train a NODE with the desired properties. During inference, we need to integrate accurately enough to reflect the dynamics of the trained NODE model. Thus, the training integration steps and the inference integration steps do not need to coincide.\nWe are happy to provide additional clarifications as needed, during the discussion period.\n\n## Questions\n> **1. A section on limitations would be helpful.**\n\nWe included a discussion of limitations in the supplementary material due to space constraints. For the final version, we will reorganize the manuscript to incorporate this section into the main paper, making it more visible to readers. Currently, one of the weaknesses of the proposed approach is its reliance on feature correspondences, as pointed out by other reviewers.\n\n> **2. I am not clear why a vanilla NODE applied on the features (where t=0 is source and t=1 is target) won't work for morphing? Can the authors elucidate this point in more detail?**\n\nWe believe that “vanilla” in this context refers to NODEs without the sinusoidal activations and the Jacobian-based regularization. We do evaluate such NODEs, with results reported in Table 1 (NODE Sigmoid). Likewise, they are further explored in the ablation studies in the supplementary material (Section 2, Figure 1). These configurations yield inferior alignment and morphing quality, confirming the importance of our architectural and regularization choices. For the final paper, we will ensure that the distinction is clearer."}, {"review_id": "3Iy8KUZy3F", "reviewer": "Reviewer_Bay7", "summary": "This paper proposes FLOWING, a framework for morphing between 2D images and 3D Gaussian splats using flow-based implicit neural representations. The method leverages Neural ODEs and Neural Conjugate Flows to ensure continuity, invertibility, and temporal coherence by construction, avoiding expensive regularizations used in prior work like ifmorph. The approach demonstrates applications in face morphing, image interpolation, and 3D Gaussian Splatting morphing.", "questions": "- How does FLOWING perform on datasets lacking clearly defined landmark structures, such as natural scenes with heavy occlusion or clutter, or on domains outside conventional faces (e.g., animals, medical images)? Is it possible to relax landmark reliance?\n- Could you quantify and compare compute/memory requirements and wall-clock times for the main approaches (FLOWING vs. ifmorph vs. classical), especially as input/image sizes or landmark counts scale?\n- How does FLOWING compare to classical, non-learning-based flow warping (e.g., dense optical flow, thin-plate spline warping), both quantitatively and qualitatively on the main datasets?", "limitations": "Yes; limitations are discussed—primarily in acknowledgment of ablation experiments and practical constraints—but further explicit analysis of error/failure cases and dataset diversity would strengthen the discussion.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "**Strengths**:\n- Quantitative results in Table 1 and Table 2 show striking improvements: for example, mean squared alignment errors drop by orders of magnitude on the FRLL and MegaDepth datasets when using the SIREN-activated NCF/NODE models compared to ifmorph. LPIPS and FID scores for image blending in Table 2 further highlight the superiority of FLOWING for perceptual quality.\n- FLOWING converges much faster and with more stability than ifmorph, which is susceptible to singularities and unstable warps.\n- Versatility for 2D and 3D\n- The method can be augmented by generative blending (e.g., DiffMorpher) after flow-based alignment, yielding visually superior morphs in diverse scenarios.\n\n**Weaknesses**\n- Flow-based morphing is widely applied in other domains, such as medical image registration (Voxelmporph-diff). Neural ODE-based solutions are also well-explored, such as NODEO (CVPR22), Neural Diffeomorphic Flow (CVPR22) and NMF (NeurIPS 2020). It is unclear what novel theoretical contributions are beyond prior works.\n- Though the landmark-based formulation is effective for faces or some designed datasets, the approach's reliance on accurate correspondences may limit applicability to scenarios where dense or reliable landmark annotations are unavailable or ambiguous, such as unconstrained scenes or many non-human objects.\n- The claim that minimizing Jacobian/Hessian norms produces plausible, semantically reasonable morphs is plausible but not universally justified, especially in settings where \"wiggly\" but valid transitions may be preferred in some tasks.\n- No Comparison with Learning-Free Flow Methods: Method comparisons seem limited to ifmorph and OpenCV warping. The inclusion of established, learning-free or classical flow-based approaches (e.g., dense optical flow with smoothing) would help contextualize FLOWING’s place among \"classic methods\" and modern INR-based approaches.", "was_revised": true, "final_justification": "In authors' final remark, they said:\n\"Novelty of Sinusoidal INRs + Neural ODEs – Acknowledged the employment of these techniques in Sun et al. (2024), in the context of medical registration; highlighted that our work is the first to adapt NODEs and NCFs with sinusoidal INRs to morphing of images and 3DGS, addressing morphing-specific constraints and demonstrating strong results.\"\n\nI didn't see substantial technical contributions from adapting the same paradigm into images and 3DGS given Sun et al. (2024) has already dealt with 3D image registration problem. \n\nI acknowledge their efforts in comprehensive experiments. My final rating is made based on the limited technical novelty.", "rebuttal": "We would like to thank the reviewer for the detailed and constructive comments on our work. We are encouraged by the remarks regarding the quantitative metrics, as well as by the mentions of perceptual quality, fast convergence and versatility. We hope our response can address the reviewer’s concerns. We conducted additional experiments regarding comparisons with classical/non-learning-based flow warping, and also regarding compute/memory requirements. If there are any remaining questions, we are more than happy to address them.\n\n## Weaknesses\n\n> **It is unclear what novel theoretical contributions are beyond prior works (Voxelmporph-diff, NODEO, Neural Diffeomorphic Flow, NMF).**\n\nWe appreciate these references and intend to add them to the final paper. Although these architectures can be said to share our objectives of using flow-like transformations that preserve geometrical properties, they do so in different contexts with different tools.\n\nThe main theoretical tools we develop are tailored specifically to the interplay of implicit neural representations (INRs) and morphing. In particular, we demonstrate how the group properties of flows translate to desirable properties of morphing transformations. To the best of our knowledge, this connection has not been made explicit before.\n\nIn contrast, Voxelmorph and NODEO generally deal with the carrying of signals in high-dimensional spaces with CNNs, in a setup more akin to optimal transport. Among other things, this makes them inadequate for temporal feature matching, as well as blending between signals with distinct color content. Likewise, the approaches of NDF and NMF are built on different latent representations and therefore cannot be easily used in our applications.\n\nOur contributions are also methodological: To the best of our knowledge, combining sinusoidal INRs and Neural ODEs has not been explored in prior work.\n\n> **Though the landmark-based formulation is effective for faces or some designed datasets, the approach's reliance on accurate correspondences may limit applicability to scenarios where dense or reliable landmark annotations are unavailable or ambiguous, such as unconstrained scenes or many non-human objects.**\n\nWe acknowledge this limitation and discuss it in the supp mat. However, even generative methods often implicitly rely on alignment of features—for example, DiffAE requires pre-aligned inputs, and DiffMorpher struggles with misaligned cases (see Fig. 4 in the supplementary material). Easing these restrictions is a major avenue for future work.\n\n> **The claim that minimizing Jacobian/Hessian norms produces plausible, semantically reasonable morphs is plausible but not universally justified, especially in settings where \"wiggly\" but valid transitions may be preferred in some tasks.**\n\nOur TPS regularization reduces distortion and enforces smoothness, but it is optional. As shown in Fig. 1 of the supplementary material, disabling the TPS term in FLOWING still produces plausible morphs with wiggly deformations while preserving keypoint alignment. This flexibility is not present in prior INR methods like ifmorph, where TPS is essential due to the absence of true flow structure. Since we are not able to use figures in the rebuttal, we will include in the final version an experiment illustrating how varying the TPS weight affects the deformation behavior.\n\nThe reviewer’s comment suggests a promising avenue for future work: designing regularizations that impose physical or task-specific constraints on the vector field, such as deformations with inertia (“jelly-like”).\n\n> **No Comparison with Learning-Free Flow Methods: Method comparisons seem limited to ifmorph and OpenCV warping. The inclusion of established, learning-free or classical flow-based approaches (e.g., dense optical flow with smoothing) would help contextualize FLOWING’s place among \"classic methods\" and modern INR-based approaches.**\n\nThank you for the suggestion. Classical optical flow methods typically assume small motion between consecutive frames of the same scene, which differs from our morphing setting involving large geometric and semantic changes. Nonetheless, we agree that classical baselines help contextualize FLOWING, and we include a comparison with thin-plate spline (TPS) warping on a subset of the FRLL dataset. TPS is applied using landmark interpolation followed by linear blending, and we report LPIPS metrics below for comparison:\n\n| Morphing Type | LPIPS(I₀, I)↓ | LPIPS(I, I₁)↓ |\n| -------------------- | ---------------- | ---------------- |\n| TPS warping    | 0.311           | 0.314            |\n| NODE (Ours)   | **0.210**      | **0.213**      |\n\nThese results show that FLOWING performs favorably compared to TPS warping.\n\nWhile dense optical flow is not directly applicable to the FRLL dataset, it raises an interesting direction for applying FLOWING to frame interpolation. We conducted a preliminary experiment on a single video with slow motion. Using RAFT [1] to compute dense optical flow, we trained an NCF model with sampled points displaced according to the flow. We then evaluated the quality of the reconstructed intermediate frame using MSE against the ground-truth frame:\n\n| Method | MSE↓ |\n| -------------------- | ---------------- |\n| Linear Blend    | 0.000264     |\n| Optical flow warp | 0.000049  |\n| NCF (Ours)      | **0.000040** |\n\nWhile preliminary, these results suggest that FLOWING is also competitive in optical-flow-based settings and may generalize to temporal interpolation tasks. Further experiments are underway to validate these findings.\n\n[1] Teed, Zachary, and Jia Deng. \"Raft: Recurrent all-pairs field transforms for optical flow.\" European Conference on Computer Vision. Cham: Springer International Publishing, 2020.\n\n## Questions\n\n> **How does FLOWING perform on datasets lacking clearly defined landmark structures, such as natural scenes with heavy occlusion or clutter, or on domains outside conventional faces (e.g., animals, medical images)? Is it possible to relax landmark reliance?**\n\nOur experiments in the paper do cover domains beyond conventional faces, including animals, clothing and general pictures; these can be found in the supplementary material. Nevertheless, relaxing this reliance is an important direction for future work. While FLOWING currently relies on landmarks for supervision, even generative methods such as DiffMorpher may struggle in cases where landmark structure is not exploited, as highlighted in Fig. 4 of the supplementary material.\n\n> **Could you quantify and compare compute/memory requirements and wall-clock times for the main approaches (FLOWING vs. ifmorph vs. classical), especially as input/image sizes or landmark counts scale?**\n\nMemory requirements for FLOWING and ifmorph training are not significantly affected by the number of landmarks, with usage around 2.5 GB for FLOWING and 1.5 GB for ifmorph. For compute requirements, we conducted experiments on an RTX 4090 GPU for both warp training and morphing (i.e., warping inference + blending). We report results for NODE trained with both 2k and 20k steps, as it converges quickly. Training times for NODE and ifmorph remain relatively stable across different landmark counts, unlike NCF, whose training time increases with more landmarks. Morphing time depends on image resolution: NODE offers faster training but slower morphing, while NCF has slower training and intermediate morphing speed.\n\n| Method | Landmark count | Training steps | Warp training time↓ | \n| ---------- | --------------------- | ------------------- | -------------------------- |\n| ifmorph | 68                      | 20k                  | 05m26s                    |\n| NCF (Ours) | 68                      | 20k                  | 08m48s                    |\n| NODE (Ours) | 68                      | 2k                    | 00m16s                    |\n| NODE (Ours) | 68                      | 20k                  | 02m39s                   |\n\n| Method | Landmark count | Training steps | Warp training time↓ | \n| ---------- | --------------------- | ------------------- | -------------------------- |\n| ifmorph | 130                      | 20k                  | 05m22s                    |\n| NCF (Ours) | 130                      | 20k                  | 15m05s                    |\n| NODE (Ours) | 130                      | 2k                    | 00m15s                    |\n| NODE (Ours) | 130                      | 20k                  | 02m38s                   |\n\n| Method | Image resolution | Morphing time (s)↓ | \n| ---------- | --------------------- | -------------------------- |\n|OpenCV | 256 x 256          | 0.01                   |\n| ifmorph | 256 x 256           | 0.05                    |\n| NCF (Ours) | 256 x 256    | 0.02                    |\n| NODE (Ours) | 256 x 256 | 0.03                   |\n\n| Method | Image resolution | Morphing time (s)↓ | \n| ---------- | --------------------- | -------------------------- |\n|OpenCV | 1350 x 1350          | 0.06                   |\n| ifmorph | 1350 x 1350           | 0.10                    |\n| NCF (Ours) | 1350 x 1350    | 0.31                    |\n| NODE (Ours) | 1350 x 1350 | 1.48                  |"}, {"review_id": "pNgQpGCY9b", "reviewer": "Reviewer_12rU", "summary": "Given an annotated set of corresponding points in a fixed and moving image, this paper trains a neural network to interpolate between these corresponding points.", "questions": "The main comparison that I'm interested in is the comparison to the OpenCV baseline in table 2. To convince me to increase my score, the main avenue would be to convince me that OpenCV's warping algorithm is seriously well tuned for VPIP, and so this method is impressive for edging it out- for example, if you show that you did some hyperparameter search for the OpenCV baseline on this task, or that this is a benchmark task that the OpenCV authors were aware of and were optimizing for.", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 1, "originality": 1, "strengths_and_weaknesses": "It's interesting to visually see the flow field from when the pre-existing NODE and NCF algorithms are applied to 2-D points, although NCF in particular is being used as something of a chainsaw cutting tissue paper- the work in NCF was needed for interpolating in extremely high dimensional spaces such as pixel intensities of RGB images.\n\nThe practicality of this approach seems dubious to me. It seems like an enormous amount of machinery and computation to replace a thin plate spline or delauney triangulation: the improvement over the opencv baseline in table 2 seems marginal. \n\nIt would be more convincing if the authors had picked one of NCF or NODE- the two proposed approaches seem to have little in common to make this a single paper.\n\nThe improvement over the INR baseline is dramatic and real- if you already have a correspondence oracle but are absolutely determined that the correspondence between the remaining points in the image will be achieved by training an MLP, then this approach is correct.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the detailed comments. We are encouraged by positive remarks regarding dramatic and real improvement over the INR baseline. We performed additional experiments regarding comparisons with the OpenCV baseline in challenging scenarios.\n\n## Strengths and Weaknesses\n\n> **[...]NCF in particular is being used as something of a chainsaw cutting tissue paper- the work in NCF was needed for interpolating in extremely high dimensional spaces such as pixel intensities of RGB images.**\n\nWe believe the reviewer refers here to Continuous Normalizing Flows (CNF) [2,3], which do take as inputs high-dimensional pixel data. However, the method we use in the paper are Neural Conjugate Flows (NCF) [1], which had first been used in the context of low-dimensional dynamical systems, for example 2D/3D systems.\n\nThough the names (and especially acronyms) of each architecture are similar, their scope is quite different. NCFs, at least as originally proposed, usually take as input coordinates in state or image space, like we use in the paper.\n\n\n[1] Bizzi, Arthur, Lucas Nissenbaum, and João M. Pereira. \"Neural Conjugate Flows: A Physics-Informed Architecture with Flow Structure.\" Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 39. No. 15. 2025. \n\n[2] Chen, Ricky TQ, et al. \"Neural ordinary differential equations.\" Advances in neural information processing systems 31 (2018). \n\n[3] Grathwohl, Will, et al. \"FFJORD: Free-Form Continuous Dynamics for Scalable Reversible Generative Models.\" International Conference on Learning Representations. 2019.\n\n> **[...] It would be more convincing if the authors had picked one of NCF or NODE- the two proposed approaches seem to have little in common to make this a single paper.**\n\nAs mentioned above, both NCFs and Neural ODEs operate quite similarly - NCFs are originally introduced as “integration-free” alternatives to Neural ODEs. Our intention by including them was to demonstrate that the framework is, in fact, flexible and does not necessarily depend on the specific architecture, as long as it displays the structure of flows.\n\n> **The practicality of this approach seems dubious to me. It seems like an enormous amount of machinery and computation to replace a thin plate spline or delauney triangulation: the improvement over the opencv baseline in table 2 seems marginal.**\n\nUnlike OpenCV, our method avoids strictly straight paths, resulting in better feature alignment. Importantly, it can enforce the uniqueness of trajectories, avoiding ghosting; an artifact common when straight lines intersect during interpolation. While we cannot include a figure illustrating this, we refer to Figure 9 of the ifmorph paper, which depicts a similar issue.\n\nLikewise, classical methods like thin-plate splines and Delaunay triangulation work in 2D but do not generalize well to higher dimensions, nor do they integrate with neural pipelines. In contrast, our flow-based model is fully differentiable, dimension-agnostic (2D/3D), and integrates seamlessly with neural modules like INRs and generative tools.\n\n## Questions\n\n> **The main comparison that I'm interested in is the comparison to the OpenCV baseline in table 2. To convince me to increase my score, the main avenue would be to convince me that OpenCV's warping algorithm is seriously well tuned for VPIP, and so this method is impressive for edging it out- for example, if you show that you did some hyperparameter search for the OpenCV baseline on this task, or that this is a benchmark task that the OpenCV authors were aware of and were optimizing for.**\n\nOpenCV’s warping method is based on Delaunay triangulation and, to the best of our knowledge, does not have tunable hyperparameters for morphing. However, due to its maturity and widespread use, it serves as a strong classical baseline. To evaluate performance in more challenging scenarios, we conducted an experiment using unaligned faces from Figs. 7 and 9 of the ifmorph paper. In this case, our proposed NCF and NODE models both outperform OpenCV and ifmorph, with NODE achieving the best results:\n\n| Morphing Type | LPIPS(I₀, I)↓ | LPIPS(I, I₁)↓ |\n| -------------------- | ---------------- | ---------------- |\n| OpenCV           | 0.408           | 0.421            |\n| ifmorph             | 0.421           | 0.408           |\n| NCF (Ours)      | 0.391           | 0.392           |\n| NODE (Ours)   | **0.375**      | **0.377**      |\n\n We are happy to provide additional clarifications as needed, during the discussion period."}, {"review_id": "0gD8kGJ8vR", "reviewer": "Reviewer_h25A", "summary": "This paper introduces a flow-based implicit neural representation to estimate the vector flow to morph a source into a target one. By casting the problem as learning a time-dependent diffeomorphism, this method avoids expensive regularization overheads and naturally guarantees continuity, invertibility and temporal coherence. This method can generalize to both 2D and 3D inputs, exhibiting its huge generalization ability of the framework. The training target consists of two simple terms, a data penalty to ensure temporal coherence and invertibility, and a regularization term on flow Jacobian to ensure continuity and smoothness. At inference, the learned flow warps images or 3D Gaussian‐splat representations, followed by linear or generative blending for smooth interpolation. The authors adapt Neural ODE and Neural Conjugate Flows to solve their defined problem, and both achieves reasonable results.", "questions": "- How much does this method rely on the accuracy of the correspondence pair?\n- Due to the implicit formulation, I wonder how efficient this method is compared to other methods?\n- In the 3D case, how many Gaussian blobs are used? What if the source Gaussians have a significantly different number of blobs from the target one?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n- The method applies flow-based implicit neural representation to estimate the flow for warping. Since the formulation satisfies an ODE, the method naturally overcomes the difficulty of temporal coherence and continuity.\n- Great versatility. This method can be solved by different architectures, and be applied to both 2D and 3D data, as well as both feature space and data (images or 3D points) space.\n### Weakness\n- No discussion on the computational efficiency.\n- The definition of $\\mathcal{F}^0$ at L162 seems wrong. For $i = 0$, the parameter should be $t$.\n- Likewise to Equation. 8", "was_revised": true, "final_justification": "The authors have addressed my questions and concerns. While the inference time is higher than other methods, I believe it can be optimized. Methodologically, framing morphing into INRs provides a new insight in this field. Moreover, the versatility of this method may push the method even further. I raise my score to accept.", "rebuttal": "We thank the reviewer for the detailed comments. We are encouraged by the positive remarks regarding the fact that it naturally overcomes the difficulty of temporal coherence/continuity, and also regarding its great versatility with respect to architectures, 2D/3D data and feature/data space. We will try our very best to address the reviewer’s concerns by performing additional experiments regarding computational efficiency and Gaussian morphing quality with respect to the number of blobs.\nWe are happy to provide additional clarifications as needed, during the discussion period.\n\n## Weaknesses\n> **No discussion on the computational efficiency.**\n\nWe did some additional experiments regarding memory/compute requirements, which we describe next. \nMemory requirements for FLOWING and ifmorph training are not heavily affected by landmark count, being around 2.5GB for FLOWING and around 1.5GB for ifmorph. Regarding compute requirements, we present next some experiments on a RTX 4090 GPU for both warp training and blending. We show results for NODE trained using both 2k and 20k training steps, since it converges quickly. We can see that NODE and ifmorph training times do not change significantly with respect to landmark count, which is not true for NCF. The blending times will depend on image resolution. NODE presents faster training and slower blending, while NCF has slower training and intermediate blending time.\n\n| Method | Landmark count | Training steps | Warp training time↓ | \n| ---------- | --------------------- | ------------------- | -------------------------- |\n| ifmorph | 68                      | 20k                  | 05m26s                    |\n| NCF (Ours) | 68                      | 20k                  | 08m48s                    |\n| NODE (Ours) | 68                      | 2k                    | 00m16s                    |\n| NODE (Ours) | 68                      | 20k                  | 02m39s                   |\n\n| Method | Landmark count | Training steps | Warp training time↓ | \n| ---------- | --------------------- | ------------------- | -------------------------- |\n| ifmorph | 130                      | 20k                  | 05m22s                    |\n| NCF (Ours) | 130                      | 20k                  | 15m05s                    |\n| NODE (Ours) | 130                      | 2k                    | 00m15s                    |\n| NODE (Ours) | 130                      | 20k                  | 02m38s                   |\n\n| Method | Image resolution | Morphing time (s)↓ | \n| ---------- | --------------------- | -------------------------- |\n|OpenCV | 256 x 256          | 0.01                   |\n| ifmorph | 256 x 256           | 0.05                    |\n| NCF (Ours) | 256 x 256    | 0.02                    |\n| NODE (Ours) | 256 x 256 | 0.03                   |\n\n| Method | Image resolution | Morphing time (s)↓ | \n| ---------- | --------------------- | -------------------------- |\n|OpenCV | 1350 x 1350          | 0.06                   |\n| ifmorph | 1350 x 1350           | 0.10                    |\n| NCF (Ours) | 1350 x 1350    | 0.31                    |\n| NODE (Ours) | 1350 x 1350 | 1.48                  |\n\n> **The definition of F^0 at L162 seems wrong. For i=0, the parameter should be t. Likewise to Equation 8**\n\nThe relationship between the features, which flow forward in time for the source and backwards for the target, and the resulting pixel values of the warped image, which flow in the inverse direction, is somewhat unintuitive and will be better covered for the paper's final version.\n\nIn essence, to construct a warped image, we reverse the direction of the flow, so as to identify the source of the pixel at hand. Take, for example, $i = 0$ and $t=0.5$. The pixel value at a given coordinate $x$ will be the pixel value of the point $x’$ in the original image which landed on $x$ after flowing for $0.5$ units of time. This can be written as follows:\n$$ \\mathscr{I}^0(x,t) = I^0(x’), \\text{ where } x = \\Phi(x’,t) $$\n\nNow, because our flows are autonomous, the latter relationship may be inverted to express x’ in terms of x: \n$$x’ = \\Phi(x,-t)$$ \n\nCombining these expressions together, we land at the formula at L162:\n\n$$ \\mathscr{I}^0(x,t) = I^0(\\Phi(x, -t)) $$\n\nAn analogous argument may be derived for the target image ($i = 1$).\n\nWith the final version’s additional content page, we expect to be able to describe this relationship more clearly and intuitively. \n\n## Questions \n\n> **How much does this method rely on the accuracy of the correspondence pair?**\n\nThe method suffers from a significant, albeit not critical, reliance on the accuracy of corresponding pairs. In preliminary experiments, we have verified that small amounts of noise distributed along all features or large amounts of noise applied to few features can be mostly handled by the network, in part due to the regularizations implicit to the framework. In particular, uniqueness should “force” noisy features into their place when surrounded by accurate ones. The same should apply in the presence of small amounts of mislabeled pairs. This is, in some sense, the best possible outcome for a fundamentally feature-based method.\n\n> **Due to the implicit formulation, I wonder how efficient this method is compared to other methods?**\n\nThe answer to this question depends on what type of efficiency is being discussed. One of the main features of INRs is the compact representations they provide for geometrical objects, meaning it is exceptionally memory efficient. Likewise, NODE is much more computationally efficient than other implicit representations, such as ifmorph.\n\n> **In the 3D case, how many Gaussian blobs are used? What if the source Gaussians have a significantly different number of blobs from the target one?**\n\nModels from Gaussian avatars typically have between 60,000 and 120,000 Gaussians, which represents a significant change in magnitude. Since alignment relies on 3D landmarks, it is independent of the number of Gaussians in the source and target. For example, in the supplementary video, the source has 108,142 Gaussians and the target 89,021—a considerable ~20k difference. We also conducted an experiment with the target downsampled to 73,939 Gaussians, increasing the difference to around 35k. The results were similar, though we can’t show them here due to image submission limits. They will be included in the paper’s Supplementary Material."}]

Author rebuttal:

Meta-review: This submission initially received mixed ratings: 2xBR and 2xA. On the positive side, reviewers appreciated effectiveness and efficiency of the proposed method, which promises to be generalizable and powerful. The experimental results back up these claims. At the same time, one of the reviewers raised concerns that a similar approach has been previously applied by Sun et al. (2024) in the domain of medical imaging. The other reviewer with a BR rating raised concerns about baselines, which were addressed in the rebuttal (but the rebuttal was not acknowledged by the reviewer and their final rating was not updated).
AC acknowledges certain similarities of the proposed method with Sun et al. (2024), but agrees with the authors that extended and a more generalizable framework with applications in generic image and 3D GS morphing will likely be of interest for the community. The final recommendation is to accept, with a strong recommendation to the authors to discuss this prior work in the camera-ready version of the paper.
Decision: Accept (poster)