# Rethinking RAFT for Efficient Optical Flow

Navid Eslami, Farnoosh Arefi, Amir M. Mansourian, Shohreh Kasaei  Department
of Computer Engineering
Sharif University of Technology
Tehran, Iran
Email: {navid.eslami, far.arefi, amir.mansurian<EMAIL_ADDRESS>
###### Abstract

Despite significant progress in deep learning-based optical flow methods,
accurately estimating large displacements and repetitive patterns remains a
challenge. The limitations of local features and similarity search patterns
used in these algorithms contribute to this issue. Additionally, some existing
methods suffer from slow runtime and excessive graphic memory consumption. To
address these problems, this paper proposes a novel approach based on the RAFT
framework. The proposed Attention-based Feature Localization (AFL) approach
incorporates the attention mechanism to handle global feature extraction and
address repetitive patterns. It introduces an operator for matching pixels
with corresponding counterparts in the second frame and assigning accurate
flow values. Furthermore, an Amorphous Lookup Operator (ALO) is proposed to
enhance convergence speed and improve RAFT’s ability to handle large
displacements by reducing data redundancy in its search operator and expanding
the search space for similarity extraction. The proposed method, Efficient
RAFT (Ef-RAFT), achieves significant improvements of 10% on the Sintel dataset
and 5% on the KITTI dataset over RAFT. Remarkably, these enhancements are
attained with a modest 33% reduction in speed and a mere 13% increase in
memory usage. The code is available at: https://github.com/n3slami/Ef-RAFT

###### Index Terms:

Optical Flow, Large Displacement, Repetitive Patterns, Attention Mechanism,
Deep Neural Networks

##  I Introduction

Optical Flow is a fundamental challenge in computer vision, focusing on
determining the displacement vector for each pixel between two consecutive
frames. This technique holds immense significance across downstream tasks,
like visual tracking[1], video segmentation[2], and robot navigation[3].
Traditionally, the problem has been tackled using different classical computer
vision methods such as correlation-based[4], block matching[5], and energy
minimization-based[6] techniques. However, these approaches have proven to be
computationally expensive, making them impractical for real-time applications.

In recent years, deep learning has emerged as a promising alternative to
conventional approaches. Deep learning techniques have the advantage of
bypassing the need to formulate optimization problems and instead train
networks to directly predict the optical flow. These methods[7, 8, 9] have
demonstrated comparable performance to the top traditional methods while
significantly reducing the inference time, making them faster and more
efficient. In general, neural network models take a pair of consecutive images
captured by a frame-based camera as input and generate predictions for the
optical flow that effectively warps pixels from one image to the other.

RAFT[10], which stands for Recurrent All-Pairs Field Transforms, is considered
one of the most successful learning-based methods for optical flow estimation.
It has gained significant popularity as a simple yet robust baseline approach
in the field. While this method exhibits efficient performance when evaluated
on benchmark datasets, it can still encounter challenges under specific
conditions. For instance, when dealing with significant displacements or
untextured/repetitive patterns, there is a possibility of encountering large
errors in the estimated optical flow. To enhance the performance of optical
flow estimation, advanced techniques have been developed specifically for
enhancing the RAFT method. These techniques encompass attention-based
operations[11, 12], graph models[13], and latent cost-volume augmentation[14].
However, it is important to note that these methods typically require
additional computational resources and introduce significant inference time,
which limits their practical application in real-world scenarios.

To tackle the challenges posed by large displacements and repetitive patterns
in the RAFT method, this paper introduces two novel mechanisms: 1) the
Amorphous Lookup Operator (ALO), and 2) the Attention-based Feature Localizer
(AFL). The former replaces the original lookup operator of RAFT, allowing the
network to vary the distribution of the correlation queries based on the input
frames, resulting in the ability to recognize larger displacements. The latter
transforms the resultant features of the feature encoders in such a way that
the network can differentiate and match the pixels in a poorly textured
region, reducing the ambiguities that stem from these regions and thus,
mitigating the resultant errors in estimation.

In summary, the main contributions of this work are as follows:

  * •

Proposing the Amorphous Lookup Operator (ALO) as a novel method for tackling
large displacement problem of the optical flow estimation methods.

  * •

Proposing the Attention-based Feature Localizer (AFL) as a novel method for
tackling repetitive patterns problem of the optical flow estimation methods.

  * •

Validating the effectiveness of the proposed methods by conducting extensive
experiments on the Sintel and KITTI datasets.

##  II RELATED WORK

Traditionally, optical flow was commonly approached using energy-based
methods[6, 15, 16] that employed a variational approach by defining a data
term and a regularization term. To address the challenge of large
displacements, improved estimation methods such as pyramid approaches[17] were
introduced. Additionally, feature-based methods emerged as a solution to
mitigate the large displacement problem[18, 19, 20, 21, 22]. These methods
defined a matching term and utilized dynamic programming and interpolation
techniques to enhance accuracy and robustness.

Significant progress has been made in the field of optical flow estimation,
thanks to recent advancements in deep learning. FlowNet[7] introduced the
first end-to-end network capable of tackling this task. Subsequently, a series
of methods based on neural networks[23, 24, 9, 8, 25, 26, 27, 28, 29] have
been developed, aiming to enhance performance through coarse-to-fine or
iterative approaches. These methods have contributed to a notable improvement
in optical flow estimation. Typically, these methods employ a correlation
matrix to capture the similarities between pixels in consecutive frames. This
correlation matrix takes the form of a 4D volume, which can be quite large
depending on the size of the frames being processed. Due to the limited
receptive field of Convolutional Neural Networks (CNNs), transformer-based
approaches have recently emerged[14, 30, 31] as a solution. These transformer
architectures have shown superior performance compared to CNNs. However, it is
worth noting that transformer-based approaches often require a substantial
amount of data and annotations to achieve their optimal performance.

RAFT[10], drawing inspiration from variational methods, incorporates recurrent
gate units and correlation matrices at multiple scales. It stimulates an
iterative optimization problem that improves the flow estimations through
iterative refinement. Following the utilization of large correlation matrices
in RAFT, numerous efforts have been made to enhance the performance of RAFT in
various aspects[11, 12, 13, 32, 33]. Many of the mentioned works focus on
leveraging the attention mechanism to improve RAFT. In particular,[32]
introduces an attention-based approach to estimate the similarity between
pairs of pixels. By incorporating vertical and horizontal correlations, it
enhances the estimation process. Notably,[14] replaces correlation matrices
with correlation memory, which consists of tokens describing pixels.
Correlation values are computed using attention within this memory structure.
Additionally,[33] employs the attention mechanism to address the impact of
noise on correlation matrices, resulting in the extraction of more globally
coherent and semantically stable features. Occlusion poses a significant
challenge in optical flow estimation, and[11] addresses this issue by
employing attention to generalize information to occluded areas. By utilizing
attention, it aims to provide a more comprehensive understanding of the scene,
even in the presence of occlusions. Repetitive patterns, on the other hand,
introduce difficulties in determining corresponding pixels due to their
similar feature vectors.[12] utilizes attention to identify pixels that are
highly likely to be correct correspondences, which proves beneficial in
handling scenarios with large displacements. By leveraging attention, it
enhances the ability to cope with repetitive patterns and improve the accuracy
of optical flow estimation.

In this paper, building upon the RAFT framework, two novel methods are
introduced to specifically tackle the challenges posed by large displacements
and repetitive patterns. These methods, referred to as Attention-based Feature
Localizer and Amorphous Lookup Operator significantly enhance the performance
of RAFT. Notably, these proposed techniques achieve notable improvements while
imposing a minimal burden on memory consumption and runtime.

##  III PROPOSED METHOD

In this section, first the optical flow baseline, RAFT[10], is reviewed as
this work is based on the RAFT architecture. Subsequently, a detailed
explanation of the two proposed methods for addressing large displacements and
repetitive patterns are provided.

Figure 1: Diagram of RAFT[10] encompasses three main components. 1) Feature
encoders are employed to extract per-pixel features from the input frames. 2)
A correlation layer constructs a correlation volume with dimensions $W\times
H\times W\times H$ by computing the inner product of feature vectors for all
pairs. 3) An update operator recurrently enhances the optical flow estimation
by leveraging the current estimate to retrieve values from the set of
correlation volumes.

###  III-A Optical Flow Estimation Baseline

Given a pair of consecutive frames, $I_{1}$ and $I_{2}$, a dense displacement
field $(a_{1},a_{2})$ is estimated. This field maps each pixel $(u,v)$ in
$I_{1}$ to its corresponding coordinates
$(u^{\prime},v^{\prime})=(u+a_{1}(u),v+a_{2}(v))$ in $I_{2}$. As illustrated
in Fig1, the RAFT model consists of some main components, each of which is
briefly explained below. For more information about the RAFT baseline, refer
to[10].

####  III-A1 Encoders

Two encoders with residual blocks are used in RAFT to extract features and
reduce noise. As depicted in Fig 1, the feature encoder, $g_{\theta}$, is
employed to extract features from both frames. On the other hand, the context
encoder, $h_{\theta}$, is applied to $I_{1}$ to extract features for
optimization and estimate the flow.

####  III-A2 Correlation Volume

This volume aims to represent the similarities between the points in the pair
of frames. Let’s suppose that $F_{1}=g_{\theta}(I_{1})$ and
$F_{2}=g_{\theta}(I_{2})$ are the extracted features from the feature encoder
for $I_{1}$ and $I_{2}$, respectively. The correlation volume, $C$, can be
calculated as follows[10]:

$\centering C^{l}_{i,j,x,y}=\begin{cases}\langle
F_{1}(i,j),F_{2}(x,y)\rangle&l=0\\\
AveragePooling_{x,y}(C^{l-1}_{i,j,x,y})&l\geq 1.\end{cases}\@add@centering$
(1)

Where $C^{l}$ denotes the correlation at the $l$-th level. In fact, the first
level ($l=0$) is created by calculating the dot product of each pair of points
in $F_{1}$ and $F_{2}$. The next three levels are formed by applying average
pooling on the last two dimensions of the correlation volume in the former
layer.

####  III-A3 Lookup Operator

Given a current estimate of optical flow ($a_{1}$, $a_{2}$), each pixel
$p=(u,v)$ in $I_{1}$ is mapped to its estimated correspondence in $I_{2}$:
$q=(u+a_{1}(u),v+a_{2}(v))$. Following that, a local grid is defined around
$q$:

$\centering\mathcal{N}(q)_{r}=\\{q+h|h\in\mathbb{Z}^{2},||h||_{1}\leq
r\\}\@add@centering$ (2)

where $r$ is the radius of the grid. This operator extracts similarity values
from the correlation volume as a vector for a pair of points, p and q.

####  III-A4 Recurrent Gate Units

This component simulates the optimization algorithm, which estimates a
sequence of flow estimates $\\{a_{1},a_{2},...,a_{N}\\}$ starting from an
initial flow $a_{0}=0$. It takes the current flow, correlations, and a hidden
state as input, and iteratively updates the flow and hidden state in output.

Finally, the loss function of the network is defined as follows:

$\centering\ell=\sum_{i=1}^{N}\gamma^{N-i}||a_{gt}-a_{i}||_{1}\@add@centering$
(3)

Where $a_{gt}$ represents the ground truth flows, $a_{i}$ denotes the
calculated flows in the $i$-th step of the algorithm and $\gamma$ is a
hyperparameter.

###  III-B Amorphous Lookup Operator

RAFT’s vanilla lookup operator has three major short-comings: 1) It’s
structure imposes a hard limit of 256 pixels on the distance that the network
can gather information from in each iteration. 2) In practice it often fails
to find meaningful correspondences up to this range. 3) Since the lookup
operator is used on all levels of the correlation volume, the data extracted
is pooled from the same regions, causing unneeded redundancy. The idea of the
Amorphous Lookup Operator (ALO) is to allow the network to change the lookup
operator so that, based on the input frames, it can choose to query farther
away similarities and reduce the redundancy of the extracted similarities.

$+d_{x}$$-d_{x}$$+d_{y}$$-d_{y}$

Figure 2: Structure of the grid used in the original lookup operator (left),
compared to the transformed lookup operator used in the ALO (right).

The ALO changes the original grids of the lookup operator by scaling and
translating parts of it. That is, it first calculates four scalar parameters
$s_{x},s_{y}\in[1,3]$, $d_{x},d_{y}\in[0,2]$ based on the input frames. It
then transforms each point $p_{old}=(i,j)$ in the old lookup operator to

$p_{new}=(s_{x}i+\textbf{sign}(i)d_{x},s_{y}j+\textbf{sign}(j)d_{y}),$ (4)

where $\textbf{sign}(\cdot)$ is a function returning $+1$ for positive inputs,
$-1$ for negative inputs, and $0$ for an input of zero. This transformation
breaks the grids into four sub-grids, which can be positioned and spaced
arbitrarily, while maintaining symmetry, as shown in Fig. 2. The parameters
$s_{x}$, $s_{y}$, $d_{x}$ and $d_{y}$ are calculated separately for each level
of the correlation volume, which enable the network to adaptively pool
similarity data from farther regions, while avoiding the extraction of too
much redundant information.

Frame 1Context Encoder

$H\times W$

$\frac{H}{2}\times\frac{W}{2}$

$\frac{H}{4}\times\frac{W}{4}$

4D Correlation VolumesHidden State$L^{\prime}$ ALO$+$$f$$\Delta f$ ConcatConv.
$1\times 1$ Global Max Pooling Global Min Pooling ConcatFeed Forward Network
Concat

Figure 3: Scalar parameter calculation network for the ALO.

The aforementioned scalars are calculated via the application of a light-
weight network module similar in structure to the BAM[34] and CBAM[35]
architectures. The structure of this module is depicted in Fig. 3. The network
first concatenates the current hidden state with the context features, and
enriches them with a $1\times 1$ convolution. Then, it generates a global and
compact descriptor of the algorithm’s state by doing a Max and a Min pooling
operation, and concatenating the resultant vectors. It then feeds this
descriptor into two single layer, fully connected networks, each with a
Sigmoid activation function. These networks then apply an appropriate affine
transform on the result of the Sigmoid function, outputting the scalar for all
levels of the correlation volume. The final scalars are then fed into the ALO,
and also concatenated with the context features of the network. Concatenation
with the context feature map is crucial, as the GRU will not know what type of
ALO is being used otherwise, and cannot reap the benefits of having this new
lookup operator.

Therefore, the light weight of this structure allows the network to calculate
the scalar parameters in a memory and compute efficient manner, adapting the
lookup operator based on the input image and the current stage of the
regression algorithm.

###  III-C Attention-based Feature Localizer

The vanilla RAFT architecture uses the outputs of the feature encoders to
calculate the correlation volumes. Due to the convolutional structure of these
encoders, the output features are inherently local, which may cause the
network as whole not being able to differentiate between points in poorly
textured regions of the frames. Consequently, the regression algorithm will
have a hard time estimating a correct flow value for these regions of the
frames, as it has to utilize the flow values outside of the regions to slowly
correct the prediction of the region itself.

$p$$x^{-}_{p}$$x^{+}_{p}$$y^{-}_{p}$$y^{+}_{p}$

Figure 4: Definition of the $x_{p}^{\pm}$ and $y_{p}^{\pm}$ values for a pixel
$p$ in a poorly textured region. The orange points depict the pixels that may
cause us to err in our estimation of $\Delta x_{p}$ and $\Delta y_{p}$.

To remedy this, the aforementioned features are augmented with a set of
“global features” that encapsulate some data from across the entire frames,
enabling the network to differentiate the poorly textures pixels by looking at
these global features. To achieve this, for each pixel $p$ of each frame, the
following four parameters are defined: $x_{p}^{+}$, $x_{p}^{-}$, $y_{p}^{+}$,
and $y_{p}^{-}$, which represent how many pixels must we move from $p$ to the
right (resp. left, up, and down) to exit the poorly textured region. An
example of these values is depicted in Fig. 4.

The idea then is to not estimate the values $x_{p}^{\pm}$ and $y_{p}^{\pm}$
themselves, but rather estimate $\Delta x_{p}=x_{p}^{+}-x_{p}^{-}$ and $\Delta
y_{p}=y_{p}^{+}-y_{p}^{-}$, which essentially represent the whereabouts of the
point $p$ when compared to the “middle-points” of the poorly textured region.
Therefore, $\Delta x_{p}$ and $\Delta y_{p}$ create something similar to a
coordinate system for each poorly textured shape. One can expect the values of
$\Delta x_{p}$ and $\Delta y_{p}$ to not change by much for corresponding
points, when poorly textured regions are translated only.

Suppose the values $\Delta x_{p}$ are calculated for the pixels in row $r$ of
the first frame. To this end, the multi-head attention mechanism is applied on
the features resulting from the feature encoders of this frame, which we shall
denote $f_{r,i}$ for the $i$-th pixel of this row. Then, the keys and values
for a query $q_{i}=f_{r,i}$ are defined as

$k_{j}=f_{r,j},v_{j}=pe(i-j),$ (5)

where $pe(\cdot)$ is the positional encoding function defined in[36] as
$pe(\cdot)_{2k}=sin(\frac{\cdot}{1000^{k/d}})$,
$pe(\cdot)_{2k+1}=cos(\frac{\cdot}{1000^{k/d}})$.

Since the keys $k_{i}$ and queries $q_{i}$ are defined in the same manner, in
the context of a poorly textured region, this mechanism will take a weighted
average of the relative positional encodings of the row, with the pixels in
the region having a much larger weight that the pixels outside. This will
result in an estimate of $\Delta x_{p}$ in the form of a positional encoding,
which gives the network some idea as to what the actual value of $\Delta
x_{p}$ is. The implementation uses four heads during this calculation, so that
it can take into account more complex similarities between the feature vectors
in the pooling procedure.

Notice that $v_{j}$ is a “relative positional encoding” dependent on both $i$
and $j$. To calculate this relative positional encoding, first the multi-head
attention mechanism is applied using $v^{\prime}_{j}=pe(j)$, which is the
standard positional encoding, and then exploit the linearity of the attention
mechanism along with well-known trigonometric identities to convert the input
$j$ into $i-j$. It is worth noting that the resulting estimation is not
without error. For instance, in Fig. 4, the points colored in orange will bias
the estimate of $\Delta x_{p}$ towards the left.

This same procedure can be applied on all rows and columns of both frames to
derive estimates for all values $\Delta x_{p}$ and $\Delta y_{p}$, all using
the same module. The resulting estimates are then concatenated with the
features of their corresponding frames, effectively augmenting the old
features with a set of global features that span the entire row and column of
each pixel. This resulting estimate lends itself well to how the correlation
volume is calculated, as closer values of the input to $pe(\cdot)$ correspond
to a larger dot-product, and thus more similarity.

It is worth noting that this mechanism is applied only on the rows and
columns, enabling it to enjoy excellent speed and memory efficiency.

Figure 5: Qualitative comparison between the proposed method and RAFT. Frames
with orange and blue labels are from Sintel and KITTI datasets, respectively.

##  IV EXPERIMENTS

###  IV-A Datasets and Training Schedule

Based on prior works, the model is initially pretrained on the FlyingChairs
dataset[7] for 100k iterations using a batch size of 10. Following that, it
undergoes an additional pretraining phase on the FlyingThings dataset[37] for
200k iterations with a batch size of 6. In this work, no fine-tuning is
performed. The evaluation is conducted on both the train split of the
KITTI-15[38] dataset and the train split of the MPI Sintel[39] dataset,
considering both the Clean and Final passes of the data.

For the ablation results, fast configuration was employed for training and
evaluation purposes. In this configuration, the model underwent training for
25k steps using the train set of the MPI Sintel dataset. Following the
training phase, the model was evaluated on the train set of the KITTI-15
dataset.

The implementations of all the networks were done using the PyTorch framework.
Training and inference processes were conducted on a single NVIDIA GeForce RTX
3090 GPU. The training process involves the utilization of the AdamW optimizer
with a one-cycle learning rate scheduler. Additionally, the gradients are
clipped to the range of [-1, 1]. The hyperparameter $\gamma$ in the equation 3
is set to 0.8, similar to the approach used in[10].

###  IV-B Evaluation Metrics

The main evaluation metric for the MPI Sintel dataset is the end-point error
(EPE), which represents the average error of the flow estimation at each
pixel, measured in terms of the number of pixels. For the KITTI-15 dataset,
the evaluation metrics used are F1-EPE and F1-All. F1-EPE is the same as the
EPE metric described for the Sintel dataset. On the other hand, F1-All
measures the percentage of outliers, specifically pixels whose end-point error
exceeds either 3 pixels or 5% of the ground truth flow magnitude. This metric
is averaged over all pixels in the dataset. To validate the efficiency of the
proposed method, additional metrics such as the number of parameters in the
network, runtime, and memory usage are reported.

###  IV-C Experimental Results

Extensive experiments were conducted to validate the effectiveness of the
proposed method by comparing it with several existing methods, including RAFT
used as the baseline. The results of the proposed method in comparison to
these existing methods are presented in Table I. The table demonstrates that
the proposed method achieves great results in terms of EPE on the challenging
Sintel dataset, particularly on the Final pass. Moreover, on the KITTI dataset
and the F1-All evaluation metric, the proposed method achieves the third-best
performance. It is worth noting that GMFlowNet, being a transformer-based
method, requires a larger amount of training data and has a higher number of
parameters compared to our method. In light of this, the proposed method still
achieves remarkable results overall. Specifically, compared to RAFT, it
demonstrates more than a 10% improvement on the Sintel dataset and nearly a 6%
improvement on the KITTI dataset, which means the proposed method plays a
significant role in improving large errors.

Additionally, it is important to mention that CRAFT and GMA are two methods
that exhibit results comparable to the proposed method. These methods have
pursued independent approaches in their research and can even be combined with
the proposed method to further enhance performance.

Table II presents a runtime comparison between the proposed method and RAFT,
considering two different numbers of steps. The table illustrates that the
proposed method improves upon RAFT while introducing only a 33% runtime
overhead. This indicates a balanced trade-off between accuracy and efficiency,
as the proposed method achieves better results while maintaining reasonable
runtime performance.

Table III provides a comparison of RAFT, the proposed method, and GMFlowNet in
terms of the number of parameters and memory usage. As mentioned earlier, the
proposed method utilizes significantly less GPU memory compared to GMFlowNet,
making it feasible to run on the 24GB RAM of the Nvidia GeForce 3090 GPU.
Additionally, the table reveals that the proposed method introduces only a
slight increase in the number of parameters and memory usage (13%) compared to
RAFT, despite its significant improvement in flow estimation accuracy. This
highlights the efficiency of the proposed method in achieving impressive
results with relatively fewer parameters and memory requirements.

TABLE I: Comparison of the proposed method with existing techniques on the Sintel and KITTI datasets. Green, blue, and red colors denote the first, second, and third-best results. Method | MPI Sintel (Train) | KITTI-15 (Train)
---|---|---
EPE (Clean) | EPE (Final) | F1-EPE | F1-All
PWC-Net[9] | 2.55 | 3.93 | 10.35 | 33.7
LiteFlowNet[24] | 2.48 | 4.04 | 10.39 | 28.5
LiteFlowNet2[40] | 2.24 | 3.78 | 8.97 | 25.9
VCN[27] | 2.21 | 3.68 | 8.36 | 25.1
MaskFlowNet[25] | 2.25 | 3.61 | - | 23.1
FlowNet2[23] | 2.02 | 3.54 | 10.08 | 30.0
DICL[41] | 1.94 | 3.77 | 8.70 | 23.60
Flow1D[32] | 1.98 | 3.27 | 5.59 | 22.95
RAFT[10] | 1.43 | 2.71 | 5.02 | 17.46
FM-RAFT[42] | 1.29 | 2.95 | 6.98 | 19.3
CRAFT[33] | 1.27 | 2.79 | 4.88 | 17.50
GMA[11] | 1.30 | 2.74 | 4.69 | 17.10
AGFlow[13] | 1.31 | 2.69 | 4.82 | 17.0
KPA-Flow[43] | 1.28 | 2.68 | 4.46 | 15.90
S-Flow[29] | 1.30 | 2.59 | 4.60 | 15.90
GMFlowNet[12] | 1.14 | 2.71 | 4.24 | 15.40
EMD-S[28] | 1.31 | 2.67 | 5.0 | 17.0
Ef-RAFT (ours) | 1.27 | 2.60 | 4.83 | 16.45
TABLE II: Runtime comparison between the proposed method and RAFT. Method | #Steps | Runtime (s) | EPE (Clean) | EPE (Final)
---|---|---|---|---
RAFT | 32 | 87.81 | 1.46 | 2.69
Proposed | 32 | 117.64 | 1.29 | 2.62
RAFT | 22 | 60.39 | 1.48 | 2.69
Proposed | 22 | 85.91 | 1.28 | 2.60
TABLE III: Number of parameters and memory usage comparison. Method | #Steps | #Parameters | Memmory Used
---|---|---|---
RAFT | 12 | 5.3 M | 11988 (MiB)
GMFlowNet | 12 | 9.3 M |  $\geq$ 24000 (MiB)
RAFT | 12 | 5.7 M | 13520 (MiB)

###  IV-D Ablation

The effectiveness of the proposed components was further validated through an
ablation study. Quick training was conducted, where the lower steps method was
trained on the Sintel dataset and evaluated on the KITTI dataset. The results
in Table IV show that both ALO and AFL components contribute to the
improvement over RAFT. While the AFL alone achieves better results on the
KITTI dataset, the combined inclusion of both components is believed to offer
greater learning ability. Overall, the study demonstrates that the synergistic
effect of incorporating both components leads to enhanced performance compared
to individual components.

TABLE IV: Ablation for Ef-RAFT to validate the effectiveness of proposed ALO and AFL components. Method | MPI Sintel (Train) | KITTI-15 (Train)
---|---|---
EPE (Clean) | EPE (Final) | F1-EPE | F1-All
RAFT[10] | 1.73 | 2.40 | 8.14 | 22.60
w/o AFL | 1.57 | 2.19 | 6.40 | 18.81
w/o ALO | 1.81 | 2.41 | 3.15 | 11.36
Proposed | 1.57 | 2.13 | 6.02 | 18.75

###  IV-E Qualitative Assessment

Qualitative assessment was conducted to further validate the proposed method.
Figure 5 demonstrates that the proposed method is capable of estimating flows
that are significantly better than RAFT. The visual comparison clearly
showcases the improved accuracy and quality of the flow estimations achieved
by the proposed method.

##  V CONCLUSION AND FUTURE WORK

In this paper, Ef-RAFT was presented, a re-imagining of the renowned RAFT
network. The novel ideas of Amorphous Lookup Operator and Attention-based
Feature Localizer were explored, which enabled Ef-RAFT to improve upon its
predecessor while also keeping the computational complexity and memory
footprint low. Then experiments were conducted to showcase the improved
accuracy and efficiency of Ef-RAFT, as well as demonstrating the necessity of
the structures used ith an ablation study. It is believed that Ef-RAFT can be
extended in several directions: 1) The Attention-Based Feature Localizer is
not robust to rotations in the poorly textured regions. Applying the same
mechanism on rotated lines in the images, instead of the rows and columns
alone, may be a promising first step forward. 2) Ef-RAFT’s ideas are
orthogonal to what is used in many other papers, e.g. CRAFT and GMA. Combining
these methods in an efficient and practical manner may also open doors to
better accuracies with a light-weight network.

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sense of [Whi00, Section 3]. Since $\partial M$ constitutes itself a (steady)
MCF, the well-known [Ilm92, Inclusion Property 5.3] ensures that the possibly
singular evolving sets $\partial\Omega_{t}\setminus\partial M$ remain
homologous to the horizon. By [Whi00, Theorem 11.1], $\partial\Omega_{t}$ must
converge smoothly to a minimal hypersurface $\Sigma$, necessarily homologous
to $\partial M$. We show that $\Sigma$ can be the horizon only. Indeed, if
this were not the case, $\Sigma$ would be detached from $\partial M$ by the
Maximum Principle, and $(iii)$ in Proposition 2.7 would apply, foliating an
outer neighbourhood of $\Sigma$ with hypersurfaces of nonpositive mean
curvature. But this is a contradiction, through the Maximum Principle for the
mean curvature operator, applied on tangency points, with the smooth mean-
convex Mean Curvature Flow smoothly approaching $\Sigma$. Then, the Mean
Curvature Flow of $\Omega$ converges smoothly to $\partial M$. Observe that
this also implies that no minimal hypersurface homologous to $\partial M$
contained in $\Omega$ can exist. Indeed, if there were one, it would obviously
remain fixed under MCF, and thus $\Omega_{t}$ converging to $\partial M$ would
eventually go beyond it, contradicting [Ilm92, Inclusion Property 5.3]. The
nonminimal outward minimizing sets forming an exhaustion, we in particular
proved that $\partial M$ is outermost.

Finally, recall that the outward minimizing property of the initial set is
preserved along the flow, as it can be easily checked applying [Whi00, One-
Sided Minimization Theorem 3.5] (see [HI01, Lemma 5.6] for a proof in the
smooth flow setting). Then, $\partial M$ being one-sided limit of outward
minimizing hypersurfaces homologous to the boundary, is outward minimizing as
well. ∎

As already pointed out in the Introduction, our proof of Theorem A ultimately
builds on the application of the Willmore-type inequality (4.14) on
hypersurfaces homologous to $\partial M$ bounding a set that is isoperimetric
with respect to the volume weighted by $f$. In order to bypass the lack of
existence, we will consider _constrained_ isoperimetric sets. We find
convenient to extend $(M,g)$ over the horizon, letting $(N,g_{N})$ be the
extended Riemannian manifold. This can be obtained through gluing another copy
of $M$ along its boundary, and endowing it with a smooth metric that coincides
with $g$ on the original manifold. The existence of such metric is ensured by
[PV20, Theorem A]. Let $S$ be homologous to $\partial M$ and disjoint from it,
and let $\Omega\subset M$ have boundary $S\sqcup\partial M$. Extend $\Omega$
too, so to find $\Omega_{N}\subset N$ satisfying $\Omega_{N}\cap M=\Omega$,
and let

$B^{N\setminus M}_{\varepsilon}(\partial M)=\\{p\in N\setminus
M\,|\,\mathrm{d}_{N}(p,\partial M)\leq\varepsilon\\},$ (5.3)

for $\varepsilon>0$ such that $B^{N\setminus M}_{\varepsilon}(\partial
M)\subset\Omega_{N}$ , where $\mathrm{d}_{N}$ is the distance induced by the
metric $g_{N}$. We are going to consider sets of finite perimeter $E_{V}$ in
$(N,g_{N})$ satisfying

$\lvert E_{V}\cap M\rvert_{f}=V\qquad\qquad
P(E_{V})=\inf\left\\{P(F)\,|\,B^{N\setminus M}_{\varepsilon}(\partial
M)\subset F\subset\Omega_{N},\lvert F\cap M\rvert_{f}=V\right\\}$ (5.4)

for $V<\lvert\Omega\rvert_{f}$, where we recall that given $E\subset M$ we
defined

$\lvert E\rvert_{f}=\int_{E}fd\mu.$ (5.5)

The following result gathers the main properties these constrained
isoperimetric sets satisfy.

######  Theorem 5.2 (Existence and structure of constrained $f$-isoperimetric
sets).

Let $(M,g,f)$ be a substatic triple with horizon boundary, of dimension $n\leq
7$. Let $S$ be a strictly mean-convex outward minimizing hypersurface
homologous to $\partial M$, and let $(N,g_{N}),\Omega,\Omega_{N}$ and
$B^{N}_{\varepsilon}(\partial M)$ as above, for $\varepsilon>0$. Then, for any
$V<\lvert\Omega\rvert_{f}$, there exists $E_{V}\subset\Omega_{N}$ satisfying
5.4. Moreover,

  * $(i)$

$\partial E_{V}\cap\partial M=\mathchar 31\relax$. Moreover,
$\partial(E_{V}\cap M)=\Sigma\sqcup\partial M$, where $\Sigma$ is a
$\mathscr{C}^{1,1}$-hypersurface.

  * $(ii)$

The set $\Sigma\setminus S$ is a smooth hypersurface. Moreover, there exists a
_positive_ constant $\lambda$ such that ${\rm H}(x)=\lambda f(x)$ for any
$x\in\Sigma\setminus S$.

  * $(iii)$

We have

$\lambda\geq\frac{{\rm H}}{f}(x)>0$ (5.6)

for $(n-1)$-almost any $x\in\Sigma$.

###### Proof.

The existence of $E_{V}$ directly follows from the Direct Method. Indeed, let
$(F_{j})_{j\in\mathbb{N}}$ be a minimizing sequence for 5.4. Then, by
compactness, up to subsequences it converges to a set $E_{V}\subset\Omega_{N}$
in $L^{1}$. In particular, we have

$\lim_{j\to+\infty}\int_{M}f\lvert\chi_{E_{V}}-\chi_{F_{j}}\rvert
d\mu\leq\lim_{j\to+\infty}\sup_{\Omega}f\lvert(E_{V}\bigtriangleup F_{j})\cap
M\rvert=0.$ (5.7)

So, $\lvert E_{V}\cap M\rvert=V$. By the convergence almost everywhere one
also deduces that $B_{\varepsilon}^{N\setminus M}(\partial M)\subset
E_{V}\subset\Omega_{N}$ is satisfied too. The lower semicontinuity of the
perimeter also ensures that the infimum in 5.4 is attained by $E_{V}$.

As far as the regularity of $\partial(E_{V}\cap M)$ is concerned, let us first
crucially observe that $E_{V}\cap M$ is (constrained) isoperimetric in $M$
endowed with the conformal metric $\overline{g}=f^{2}g$ with respect to a
perimeter and volume with the same weight, namely with respect to

${P}(E)=\int_{\partial^{*}E}f^{1-n}d\sigma_{\overline{g}},\quad\lvert
E\rvert_{f}=\int_{E}f^{1-n}d\mu_{\overline{g}},$ (5.8)

where $d\sigma_{\overline{g}}$ and $d\mu_{\overline{g}}$ denote the area and
volume measure induced by $\overline{g}$ respectively. In particular, away
from $S$ and $\partial M$, where $\overline{g}$ becomes singular, we have that
classical regularity for the weighted isoperimetric problem applies [Mor03,
Section 3.10], and implies the claimed smoothness. In order to prove the
global $\mathscr{C}^{1,1}$-regularity, we mainly follow the nice exposition in
[MS17, Section 6], taking advantage also of [Mag12, Section 17]. We first show
that $E_{V}\cap M$ is an almost minimizer for the perimeter. This amounts to
say that there exists $r_{0}$ such that for every $x\in\Omega$ and every
$r<r_{0}$

$P(E_{V}\cap M)\leq P(F)+\mathrm{C}r^{n}$ (5.9)

holds for any $F$ such that $(E_{V}\cap M)\bigtriangleup F\Subset B(x,r)$, for
some constant $\mathrm{C}$ independent of $x$ and $r$. Observe that $B(x,r)$
can intersect $\Omega_{N}\setminus\Omega$, and this is the main reason we did
extend our substatic manifold. Let for simplicity $E=E_{V}\cap M$, consider
two small enough balls $B_{1}$ and $B_{2}$ centered on $\partial
E\setminus\partial M$ with $B_{1},B_{2}\Subset M\setminus\partial M$, and let
$X_{1}$ and $X_{2}$ be variation vector fields compactly supported in $B_{1}$
and $B_{2}$ respectively. Let $E_{t}^{i}=\psi_{t}^{i}(E)$, where
$\psi_{t}^{i}$ is the flow of $X_{i}$ at time $t$, for $i=1,2$. By [Mag12,
Proposition 17.8], we have

$\lvert E_{t}^{i}\rvert_{f}=\lvert E\rvert+t\int_{\partial E}f\langle
X_{i}|\nu_{E}\rangle d\sigma+O(t^{2})$ (5.10)

as $t\to 0$, where $\nu_{E}$ is a unit normal for $E$. If $f>c>0$ uniformly on
$B_{1}\cup B_{2}$, we deduce

$\left|\lvert E_{t}^{i}\rvert-\lvert E\rvert\right|\geq\mathrm{C}\lvert
t\rvert$ (5.11)

for $t$ in some small neighbourhood of $0$ and for some uniform constant
$\mathrm{C}$. Moreover, the perimeter satisfies the usual expansion [Mag12,
Theorem 17.5]

$P(E^{i}_{t})=P(E)+t\int_{\partial E}\mathrm{div}_{\partial
E}X_{i}d\sigma+O(t^{2})$ (5.12)

as $t\to 0$, where $\mathrm{div}_{\partial E}$ denotes the tangential
divergence $\mathrm{div}_{\partial E}X_{i}=\mathrm{div}(X_{i})-\langle
X_{i}|\nu_{E}\rangle$. Thus,

$\left|\lvert P(E^{i}_{t})\rvert-\lvert P(E)\rvert\right|\leq\mathrm{C}\lvert
t\rvert,$ (5.13)

again for $t$ in some small neighbourhood of $0$ and for some uniform constant
$C$. We can now conclude the proof of 5.9 for the suitable competitors $F$ as
done for the proof of [MS17, Lemma 6.3]. Namely, letting $F$ as above, we
recover the possibly lost or gained $f$-volume $\delta$ by the competitor
$F\cap\Omega$ by slightly deforming $E$ inside $B_{i}$ with $i$ chosen so that
$(F\bigtriangleup E)\cap B_{i}=\mathchar 31\relax$. Observe that
$\lvert\delta\rvert\leq\mathrm{C}r^{n}$ for some suitable constant. Exploiting
5.11 and 5.13 we get a set $\tilde{F}$ with $\lvert
E\rvert_{f}=\lvert\tilde{F}\rvert_{f}$ such that

$P(\tilde{F})\leq P(F\cap\Omega)+\mathrm{C}\lvert\delta\rvert\leq
P(F\cap\Omega)+\mathrm{C}r^{n}$ (5.14)

for some suitable $\mathrm{C}>0$, uniform for any $r<r_{0}$, with $r_{0}$
small enough. Since $E$ is constrained $f$-isoperimetric, we have then

$P(E)\leq P(\tilde{F})\leq P(F)+P(\Omega)-P(F\cup\Omega)+\mathrm{C}r^{n}.$
(5.15)

Observe that $F\cup\Omega$ may intersect $\Omega_{N}\setminus\Omega$. On the
other hand, it is easy to notice that sets with smooth boundary are in fact
automatically almost minimizers for the perimeter (see e.g. the derivation of
[MS17, (6-9)]), and so $P(\Omega)\leq P(F\cup\Omega)+\mathrm{C}r^{n}$.
Plugging it into 5.15 concludes the proof of $E=E_{V}\cap M$ being almost
minimizing. From this crucial property, one deduces that $\partial E$ is
$C^{1,1/2}$ in a neighbourhood of $\partial E\cap\partial\Omega$ exactly as
exposed in [MS17, Proof of Proposition 6.1].

To establish the optimal $\mathscr{C}^{1,1}$ regularity, we first take
advantage of $(ii)$, which we proceed to prove. In order to make the
comparison with the references easier, along this proof we are going to assume
that $\nu_{E}$ is the interior unit normal to $E$, in the extended manifold.
Let $Y$ be a vector field supported in some small ball centered at some point
of $\partial E$, with a flow $\psi$ such that $\psi_{t}(E)\subset\Omega$, for
$t$ small enough. Let now $X$ satisfy the same assumptions, with the
additional requirement to be supported around the smooth part of $\partial
E\setminus\partial\Omega$ and such that
$\mathrm{supp}X\cap\mathrm{supp}Y=\mathchar 31\relax$. Assume also that the
composition of the two flows gives a $f$-volume preserving diffeomorphism.
Then, the first variation formula gives

$0\leq\int_{\partial E}\mathrm{div}_{\partial E}(Y+X)d\sigma=\int_{\partial
E}\mathrm{div}_{\partial E}Yd\sigma-\int_{\partial E}{\rm H}\langle
X|\nu_{E}\rangle d\sigma.$ (5.16)

Assume for the time being that $Y$ is supported around a point where $\partial
E\setminus\partial\Omega$ is smooth. Then we can integrate by parts also in
the integrand involving $Y$, and obtain

$0\leq-\int_{\partial E}{\rm H}\langle Y+X|\nu_{E}\rangle d\sigma.$ (5.17)

Repeating the argument with $-X$ and $-Y$ in place of $X$ and $Y$, that is
possible in the present case since these vector fields are supported away from
$\partial\Omega$, we actually get

$0=\int_{\partial E}{\rm H}\langle Y+X|\nu_{E}\rangle d\sigma.$ (5.18)

Moreover, the $f$-volume being preserved entails, by 5.10,

$0=\int_{\partial E}f\langle Y+X|\nu_{E}\rangle d\sigma.$ (5.19)

Choose now $X$ and $Y$ so that $X=\alpha\nu_{E}$ and $Y=-\beta\nu_{E}$ on
$\partial E$, with $\alpha$ and $\beta$ being smooth functions compactly
supported on $\partial E$. Combining (5.19) with 5.18 with this choice of $X$
and $Y$, we obtain

$\frac{\int_{\partial E}\left(\frac{{\rm H}}{f}\right)f\alpha
d\sigma}{\int_{\partial E}f\alpha d\sigma}=\frac{\int_{\partial
E}\left(\frac{{\rm H}}{f}\right)f\beta d\sigma}{\int_{\partial E}f\alpha
d\sigma}=\frac{\int_{\partial E}\left(\frac{{\rm H}}{f}\right)f\beta
d\sigma}{\int_{\partial E}f\beta d\sigma}.$ (5.20)

Then, since the support of $\alpha$ and $\beta$ on $\partial E$ can be chosen
arbitrarily close to any two points in the smooth part of $\partial
E\setminus\partial\Omega$, we conclude that there exists
$\lambda\in\mathbb{R}$ such that ${\rm H}=\lambda f$ in the smooth part of
$\partial E\setminus\partial\Omega$. We plug this information into 5.16, for a
vector field $Y$ that is now supported in a ball centered on a point of
$\partial E\cap\partial\Omega$. Coupling with 5.19, this yields

$\int_{\partial E}\mathrm{div}_{\partial E}Yd\sigma+\lambda\int_{\partial
E}f\langle Y|\nu_{E}\rangle d\sigma\geq 0.$ (5.21)

Writing 5.21 in local coordinates in a neighbourhood of a point in $\partial
E\cap\partial\Omega$, where $\partial E$ is given by the graph of a function
$u:B\to\mathbb{R}$ and $\partial\Omega$ as the graph of a function
$\psi:B\to\mathbb{R}$ with $B\subset\mathbb{R}^{n-1}$, and $Y$ as a normal
vector field cut off with a function $\varphi$, it is a routine computation to
check that, for some quasilinear elliptic operator $L$, it holds

$\int_{B}\varphi[-Lu+\lambda f(\cdot,u(\cdot))]d\mu\geq 0,$ (5.22)

where $\varphi\in\mathscr{C}^{\infty}_{c}(B)$ and $\varphi\geq 0$ in
$\\{u=\psi\\}$. We address the interested reader to [MS17, Section 6C] and
[FGS17, Section 4.1] for details of these computations. In particular, the
function $u$ succumbs to the regularity theory for obstacle problems, that is
$u\in\mathscr{C}^{1,1}$, see [FGS17, Theorem 3.8]. As a consequence, $\partial
E$ has a notion of mean curvature defined almost everywhere. Observe that the
quasilinear elliptic operator $Lu$ provides the mean curvature of $\partial E$
at the point $(x,u(x))$, with $x\in B$. We are going to take advantage also of
the basic step leading to the regularity result recalled above. Namely, as
nicely presented in [FGS17, Proposition 3.2], the variational property 5.22
implies that $u$ is also solution to the Euler–Lagrange equation

$\int_{B}\varphi[-Lu+\lambda f(\cdot,u(\cdot))]d\mu=\int_{B}\xi\,d\mu,$ (5.23)

given any $\varphi\in\mathscr{C}^{\infty}_{c}(B)$, where

$0\leq\xi\leq[-L\psi+\lambda f(\cdot,\psi(\cdot))]^{+}\chi_{\\{u=\psi\\}}.$
(5.24)

We now proceed to show that $\lambda>0$, that $\partial E_{V}$ is disjoint
from $\partial M$ and that 5.6 holds, completing thus the proof. Let $Y$ in
5.21 be supported on a neighbourhood of a point $\partial E\cap S$.
Integrating by parts the first summand in 5.21, and letting $Y=\alpha\nu_{E}$
for some compactly supported nonnegative test function $\alpha$, we get

$\int_{\partial E}(\lambda f-{\rm H})\alpha\,d\sigma\geq 0.$ (5.25)

The arbitrariness of $\alpha$ implies that

$\lambda\geq\frac{{\rm H}}{f}(x)\qquad\text{for $(n-1)$-almost any
$x\in\partial E\setminus\partial M$}.$ (5.26)

Since $S$ is mean-convex, if the $(n-1)$-induced measure of $\partial E\cap S$
is strictly positive, then the (weak) mean curvature of such region is
strictly positive [MS17, Lemma 6.10], and consequently 5.26 directly implies
$\lambda>0$ and 5.6. If instead the intersection is $(n-1)$-negligible, then
5.23 implies that $Lu=\lambda f$ holds in the weak sense outside of $\partial
M$. In particular, classical regularity theory implies that $\partial
E\setminus\partial M$ is smooth and that its mean curvature is given by
$\lambda f$. Assume by contradiction that $\lambda<0$. Hence, by the Maximum
Principle, $\partial E\cap S=\mathchar 31\relax$. Then, $S$ being outward
minimizing acts as a barrier to minimize the perimeter among sets homologous
to $\partial M$ containing $E$ [FM22, Theorem 2.10]. We call $E^{*}$ such a
minimizer. The boundary of such set is $\mathscr{C}^{1,1}$ [SZW91], and
obviously it is outward minimizing, so that its (weak) mean curvature is
nonnegative. Having assumed that $\lambda<0$, and since $\partial M$ is
minimal, we deduce that $\partial E^{*}$ is minimal itself, and by the Maximum
Principle disjoint from $\partial M$. However, this is a contradiction with
$\partial M$ being outermost, proved in Proposition 5.1. We established that
$\lambda\geq 0$, and that $\lambda>0$ if $\partial E\cap S$ has positive
$(n-1)$-measure. We focus our attention to 5.23 again, considering a
neighbourhood of a point where $\partial E$ meets $\partial M$. Crucially
observe that by the minimality of $\partial M=\\{f=0\\}$ the right hand side
of 5.23 in this case vanishes, and that thus by classical regularity $\partial
E$ is smooth in a neighbourhood of $\partial M$. Since its mean curvature
${\rm H}=\lambda f\geq 0$, the Maximum Principle implies that $\partial E$ is
disjoint from $\partial M$. The possibility that $\lambda=0$ in the case of
negligible intersection $\partial E\cap S$ is finally ruled out by Proposition
5.1 again, and 5.6 becomes simply 5.26. ∎

The Willmore-type inequality 1.13 and the description of $f$-isoperimetric
sets provided by Theorem 5.2 allow to carry out the proof of the Isoperimetric
Inequality of Theorem A.

###### Proof of Theorem A.

If there is more than one end, then by Lemma 4.7 all ends have vanishing
asymptotic volume ratio. In this case, 1.3 reduces to the just proved 5.2.
Obviously, the same is true for the one-ended case if $\mathrm{AVR}(M,g,f)=0$.
The core of the Theorem then lies in the one-ended case with
$\mathrm{AVR}(M,g,f)>0$.

We first carry out the proof in the more involved and relevant case of
nonempty boundary. Let $S$ be one of the outward minimizing hypersurfaces in
the outward minimizing exhaustion, and let $\Omega$ be such that
$\partial\Omega=\partial M\sqcup S$. For $V<\lvert\Omega\rvert_{f}$, consider
an $f$-isoperimetric set $E_{V}$ constrained in $\Omega$ with $f$-volume equal
to $V$, that is, satisfying 5.4. $E_{V}$ exists and is subject to the
properties described in Theorem 5.2. In particular, $\partial E_{V}=\partial
M\sqcup\Sigma_{V}$, with $\Sigma_{V}$ a $\mathscr{C}^{1,1}$ hypersurface.
Varying $V$, we also define $I_{f}:(0,\lvert\Omega\rvert_{f})\to(0,+\infty)$
by $I_{f}(V)=\lvert\Sigma_{V}\rvert$, the $f$-isoperimetric profile of
$\Omega$. It is argued as in the classical case that $I_{f}$ is continuous.
Indeed, $E_{V+\varepsilon}$ converges in $L^{1}$ to some $\tilde{E}$, that in
particular satisfies $\lvert\tilde{E}\rvert_{f}=V$. We have that, for a fixed
$\delta>0$, by lower semicontinuity, for any $\varepsilon$ close enough to
zero such that

$I_{f}(V)\leq P(\tilde{E})\leq
P(E_{V+\varepsilon})+\delta=I_{f}(V+\varepsilon)+\delta\leq
P(E_{V})+P(B_{\varepsilon})+\delta=I_{f}(V)+P(B_{\varepsilon})+\delta.$ (5.27)

In the above inequality, $B_{\varepsilon}$ is chosen so that $\lvert F\cup
B_{\varepsilon}\rvert_{f}=V+\varepsilon$ or $\lvert F\setminus
B_{\varepsilon}\rvert_{f}=V+\varepsilon$, according to the sign of
$\varepsilon$. Letting first $\varepsilon\to 0$, and then $\delta\to 0^{+}$
establishes the continuity of $I_{f}$. Let now $\varepsilon>0$. Let
$\Sigma^{\varepsilon}$ be an inward variation of $\Sigma_{V}$ supported in
$\Sigma_{V}\setminus S$ such that $\lvert
E_{V}^{\varepsilon}\rvert_{f}=V-\varepsilon$, where $E_{V}^{\varepsilon}$ is
such that $\partial
E_{V}^{\varepsilon}=\partial\Sigma^{\varepsilon}\sqcup\partial M$. We have

$\liminf_{\varepsilon\to
0^{+}}\frac{I_{f}^{\frac{n}{n-1}}(V)-I_{f}^{\frac{n}{n-1}}(V-\varepsilon)}{\varepsilon}\geq\liminf_{\varepsilon\to
0^{+}}\frac{\lvert\Sigma\rvert^{\frac{n}{n-1}}-\lvert\Sigma^{\varepsilon}\rvert^{\frac{n}{n-1}}}{\varepsilon}$
(5.28)

Assume now that $\Sigma^{\varepsilon}$ is obtained through a normal variation
field coinciding with $\varphi\nu$ on $\Sigma$, with
$\varphi\in\mathscr{C}^{\infty}_{c}(\Sigma)$. Since the first variation of
$f$-volume is given by the $f$-weighted area, the right hand side is computed
as

$\liminf_{\varepsilon\to
0^{+}}\frac{\lvert\Sigma\rvert^{\frac{n}{n-1}}-\lvert\Sigma^{\varepsilon}\rvert^{\frac{n}{n-1}}}{\varepsilon}=\frac{n}{n-1}\lvert\Sigma_{V}\rvert^{\frac{1}{n-1}}\frac{\int_{\Sigma}{\rm
H}\varphi d\sigma}{\int_{\Sigma}f\varphi
d\sigma}=\frac{n}{n-1}\lvert\Sigma_{V}\rvert^{\frac{1}{n-1}}\lambda,$ (5.29)

where ${\rm H}=\lambda f$ on the support of $\varphi$ is due to Theorem 5.2.
Letting now $W$ be the infimum of $\int_{\Sigma}{\rm H}/fd\sigma$ taken among
strictly mean-convex smooth hypersurfaces $\Sigma$ homologous to $\partial M$,
we actually have, by the Substatic Willmore-type inequality 1.13

${\rm AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\leq
W\leq\frac{1}{(n-1)^{n-1}}\int_{\Sigma_{V}}\left(\frac{{\rm
H}}{f}\right)^{n-1}d\sigma\leq\frac{1}{(n-1)^{n-1}}\lvert\Sigma_{V}\rvert\lambda^{n-1},$
(5.30)

since $0<{\rm H}/f\leq\lambda$ by Theorem 5.2 for $(n-1)$-almost any point on
$\Sigma_{V}$. The above inequality holds for the $\mathscr{C}^{1,1}$
$\Sigma_{V}$ since we can approximate it with smooth, strictly mean-convex
hypersurfaces through Mean Curvature Flow, see [HI01, Lemma 5.6]. Observe that
this is possible since, by Theorem 5.2, $\Sigma_{V}$ is disjoint from
$\partial M$. Combining 5.28, 5.29 and 5.30 yields

$\liminf_{\varepsilon\to
0^{+}}\frac{I_{f}^{\frac{n}{n-1}}(V)-I_{f}^{\frac{n}{n-1}}(V-\varepsilon)}{\varepsilon}\geq
n\left[{\rm AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}.$
(5.31)

Comparing with the reference warped product $f$-isoperimetric profile given by

$J_{f}(V)=n^{\frac{n-1}{n}}\left[{\rm
AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n}}V^{\frac{n-1}{n}},$
(5.32)

whose derivative equals the right-hand side of 5.31, we deduce at once that
the continuous function $I_{f}^{{n}/{(n-1)}}-J_{f}^{{n}/{(n-1)}}$ has
nonnegative Dini derivative, and is thus monotone nondecreasing. Hence, for
$V_{0}<V$, we get

$I_{f}^{\frac{n}{n-1}}(V)\geq n\left[{\rm
AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}V+I_{f}^{\frac{n}{n-1}}(V_{0})-n\left[{\rm
AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}V_{0}.$ (5.33)

Recall now that $I_{f}(V_{0})=\lvert\Sigma_{V_{0}}\rvert$ for some
$\Sigma_{V_{0}}$ homologous to $\partial M$. The boundary being minimizing, by
Proposition 5.1, implies then that $I_{f}(V_{0})\geq\lvert\partial M\rvert$.
Plugging it into 5.33, and then letting $V_{0}$ go to $0$, leaves us with

$I_{f}^{\frac{n}{n-1}}(V)-\lvert\partial M\rvert^{\frac{n}{n-1}}\geq
n\left[{\rm AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}V.$
(5.34)

By definition of Isoperimetric profile, the above inequality implies 1.3 for
any hypersurface homologous to $\partial M$ inside $\Omega$, enclosing a set
of volume $V$. But the volumes being arbitrary and the outward minimizing
envelopes forming an exhaustion, the proof of 1.3 is actually complete.

We are left to characterize the situation when some smooth $\Sigma$ homologous
to $\partial M$ fulfils the equality in 1.3. Let $V$ be the $f$-volume
subtended by $\Sigma$. Let $\Omega=\partial M\sqcup S$, with $S$ strictly
mean-convex outward minimizing and $\Sigma\subset\Omega$. As above, let
$I_{f}$ be the $f$-isoperimetric profile of $\Omega$ and $J_{f}$ the reference
warped product $f$-isoperimetric profile defined in 5.32. By approximation, we
observe that any other $f$-isoperimetric set constrained in $\Omega$ of
$f$-volume $V_{0}$ satisfies the Isoperimetric inequality 1.3. Hence, by 5.33,
we have

$\lvert\partial
M\rvert^{\frac{n}{n-1}}=I_{f}^{\frac{n}{n-1}}(V)-J_{f}^{\frac{n}{n-1}}(V)\geq
I_{f}^{\frac{n}{n-1}}(V_{0})-J_{f}^{\frac{n}{n-1}}(V_{0})\geq\lvert\partial
M\rvert^{\frac{n}{n-1}}.$ (5.35)

As a consequence, any $f$-isoperimetric set of volume $V_{0}\leq V$ satisfies
the equality in the $f$-Isoperimetric inequality 1.3. Observe now, that by
approximation with smooth sets in the possibly extended Riemannian manifold
$(N,g_{N})$, this implies that such constrained $f$-Isoperimetric sets of
volume $V_{0}$ are in fact _globally_ $f$-Isoperimetric, and consequently the
regularity observed in Theorem 5.2 implies that any $\Sigma_{V_{0}}$ is
smooth. Retracing the steps that lead to 5.33, we have that the smooth
hypersurface $\Sigma_{V_{0}}$ satisfies the equality in the Willmore-type
inequality in Theorem D. This triggers the rigidity stated there, and yields,
for $\Omega_{V_{0}}$ the domain enclosed between $\Sigma_{V_{0}}$ and
$\partial M$, the isometry between $(M\setminus\Omega_{V_{0}},g)$ and
$[{s}_{0},+\infty)\times\Sigma$ endowed with

$g=f^{2}d\rho\otimes d\rho+\eta^{2}g_{\Sigma_{V_{0}}}.$ (5.36)

In particular, since $M$ has one end, the hypersurface $\Sigma_{V_{0}}$ is
necessarily connected. We now observe that, again due to the global
$f$-isoperimetry of $\Sigma_{V_{0}}$, the value of ${\rm H}/f$ is constant on
such hypersurface. But then, retracing the computations that lead to the
isometry with 5.36, more precisely coupling 2.41 with 4.15, we deduce that $f$
and $\eta$ in 5.36 depend only on $\rho$. Introduce now a new coordinate $s$
defined by $f^{2}(\rho)d\rho=ds$. Recall that $\eta$ satisfies
$\partial_{\rho}\eta=f^{2}$, and thus $\partial_{s}\eta=1$. Possibly
translating the variable $s$, we thus have

$g=\frac{ds\otimes ds}{f^{2}(s)}+s^{2}g_{\Sigma_{V_{0}}},$ (5.37)

for any $V_{0}\leq V$, and $s\geq s_{0}$ for some $s_{0}>0$.

Since we have proved that $f$ is a function of the distance $\rho$ from
$\Sigma_{V_{0}}$ only, in particular we have shown that $f$ must be constant
on $\Sigma_{V_{0}}$. This must hold for all $V_{0}\leq V$. Moreover, the
(Hausdorff) distance between $\Sigma_{V_{0}}$ and $\partial M$ goes to $0$ as
${V_{0}}\to 0$, because otherwise the volume enclosed along the (sub)sequence
would be necessarily bounded away from $0$. But then, the level sets of $f$
forming a regular foliation of a neighbourhood of $\partial M$, we deduce that
$\Sigma_{V_{0}}$ must actually be a level set of $f$, in particular
diffeomorphic to $\partial M$, for $V_{0}$ small enough. Letting $V_{0}$ to
zero we thus extend the expression 5.37 to the whole manifold, that is 1.4.
The connectednedness of $\partial M$ is again a consequence of $(M,g)$ being
one ended.

As far as the characterization of $\Sigma$ is concerned, we already showed
that $f$ is constant on it. If, by contradiction, $s$ were not constant on
$\Sigma$, then, letting $s_{\mathrm{min}}=\min\\{s(p):p\in\Sigma\\}$ and
$s_{\mathrm{max}}=\max\\{s(p):p\in\Sigma\\}$, $\Sigma$ would lie in the region
$[s_{\mathrm{min}},s_{\mathrm{max}}]\times\partial M$, where, $f$ being
constant, by 1.4 has the metric of a truncated cone. By 5.37 for $V_{0}=V$,
$\Sigma$ is a totally umbilical constantly mean-curved hypersurface in such
cone. Moreover, the constancy of $f$ on such region makes the substatic
condition simplify to nonnegative Ricci curvature. By [MR02, Lemma 3.8],
$\Sigma$ could then only be a level set of $s$ or bound a flat round ball. The
first possibility gives a contradiction with the initial assumption that $s$
were not constant on $\Sigma$, the second one with $\Sigma$ being homologous
to $\partial M$. This concludes the proof of $\Sigma$ being a level set of
$s$, and of Theorem A in the nonempty-boundary case.

We finally discuss the empty-boundary-case. It is immediately checked that the
$f$-Isoperimetric inequality 1.3 follows with a pure simplification of the
proof given above. When a hypersurface $\Sigma$ satisfies with equality the
$f$-Isoperimetric inequality, arguing as done above for 5.37 we reach for an
isometry between $(M\setminus\Omega_{\Sigma},g)$ and
$I=[\overline{s},+\infty)\times\Sigma$ endowed with

$g=\frac{ds\otimes ds}{f^{2}(s)}+s^{2}g_{\Sigma},$ (5.38)

for $\Omega_{\Sigma}$ enclosed by $\Sigma$. Again, $\Sigma$ must be connected,
since $M$ is one ended by Lemma 4.7. Now, we claim that $\Sigma$ satisfies

$\frac{n-1}{n}\int_{\Sigma}\frac{f}{{\rm
H}}d\sigma=\int_{\Omega_{\Sigma}}fd\mu,$ (5.39)

in fact saturating the substatic Heintze-Karcher inequality [LX19, Theorem
1.3] (see also [FP22, Theorem 3.6]) in boundaryless substatic manifolds. The
analysis of the equality case worked out in [BFP23, Theorem 3.1-$(ii)$] then
provides us with an isometry between $(\Omega_{\Sigma},g)$ and
$I\times\mathbb{S}^{n-1}$ endowed with

$g=\frac{ds\otimes
ds}{f^{2}(s)}+\left(\frac{s}{f(x)}\right)^{2}g_{\mathbb{S}^{n-1}},$ (5.40)

for $x\in\Omega_{\Sigma}$ with $\Sigma$ becoming a level set of $s$. Coupled
with 5.38 on the complement of $\Omega$, this yields the desired rigidity
statement.

In order to check 5.39, just observe that, since as above one has that
$\Sigma$ is $f$-isoperimetric, ${\rm H}/f$ _is constant on such hypersurface_.
Moreover, since it satisfies equality in the Willmore-type inequality 1.13,
one has

$\frac{{\rm
H}}{f}\,\lvert\Sigma\rvert^{\frac{1}{n-1}}=(n-1)\left[\mathrm{AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}.$
(5.41)

Coupling with

$\lvert\Sigma\rvert^{\frac{n}{n-1}}=n\left[\mathrm{AVR}(M,g,f)\lvert\mathbb{S}^{n-1}\rvert\right]^{\frac{1}{n-1}}\lvert\Omega_{\Sigma}\rvert_{f},$
(5.42)

it is straightforwardly seen that 5.39 holds, completing the proof. ∎

##  Appendix A Comments on the substatic condition

###  A.1. Physical motivation

Here we give a physical interpretation of substatic triples, following [WWZ17,
Lemma 3.8]. Let

$L=\mathbb{R}\times M\,,\qquad\mathfrak{g}\,=\,-f^{2}dt\otimes dt+g$

be a static spacetime satisfying the Einstein Field Equation

${\rm
Ric}_{\mathfrak{g}}\,+\,\left(\Lambda-\frac{1}{2}{\mathrm{R}}_{\mathfrak{g}}\right)\mathfrak{g}\,=\,T\,,$

where $T$ is the stress-energy tensor and $\Lambda\in\mathbb{R}$ is the
cosmological constant. Using standard formulas to express the Ricci tensor of
a warped product, we find out that

${\rm Ric}_{\mathfrak{g}}(\partial_{t},\partial_{t})\,=\,f\Delta
f\,,\qquad{\rm Ric}_{\mathfrak{g}}(\partial_{i},\partial_{j})\,=\,{\rm
Ric}(\partial_{i},\partial_{j})-\frac{1}{f}\nabla^{2}f(\partial_{i},\partial_{j})\,.$

In particular a simple computation gives

${\mathrm{R}}_{\mathfrak{g}}\,=\,{\mathrm{R}}-\frac{2}{f}\Delta f\,.$

Putting these pieces of information inside the Einstein Field Equation, we get

$\displaystyle T_{tt}\,$
$\displaystyle=\,\left(-\Lambda+\frac{{\mathrm{R}}}{2}\right)f^{2}\,,$ (A.1)
$\displaystyle T_{it}\,$ $\displaystyle=\,0\,,$ (A.2) $\displaystyle T_{ij}\,$
$\displaystyle=\,{\mathrm{R}}_{ij}-\frac{1}{f}\nabla^{2}_{ij}f+\left(\Lambda-\frac{{\mathrm{R}}}{2}+\frac{\Delta
f}{f}\right)g_{ij}\,.$ (A.3)

We now assume that the Null Energy Condition is satisfied. Namely, for any
vector $X=\partial_{t}+Y^{i}\partial_{i}$ with $\mathfrak{g}(X,X)=0$ (that is,
$g(Y,Y)=f^{2}$), we require $T(X,X)=T_{tt}+T_{ij}Y^{i}Y^{j}\geq 0$. Using the
above identities, this hypothesis tells us

$0\,\leq\,T_{tt}+T_{ij}Y^{i}Y^{j}\,=\,\left(-\Lambda+\frac{{\mathrm{R}}}{2}\right)f^{2}+\left({\rm
Ric}-\frac{1}{f}\nabla^{2}f+\frac{\Delta
f}{f}g\right)(Y,Y)+\left(\Lambda-\frac{{\mathrm{R}}}{2}\right)g(Y,Y)\,.$

Recalling that $g(Y,Y)=f^{2}$, we have obtained

$\hbox{ NEC holds}\ \ \Leftrightarrow\ \ \left({\rm
Ric}-\frac{1}{f}\nabla^{2}f+\frac{\Delta f}{f}g\right)(Y,Y)\geq 0\ \hbox{ for
all $Y$ with $g(Y,Y)=f^{2}$.}$

By rescaling of $Y$, we then conclude that the Null Energy Condition on static
spacetimes is equivalent to

${\rm Ric}-\frac{1}{f}\nabla^{2}f+\frac{\Delta f}{f}g\,\geq\,0\,.$

In other words, a static spacetime satisfies the Null Energy Condition if and
only if its spacelike slices are substatic.

Finally, we briefly discuss the physical interpretation of the conformal
metric $\tilde{g}=g/f^{2}$. In the context of static spacetimes, this metric
is usually referred to as optical metric and has the property that
$\tilde{g}$-geodesics lift to null geodesics in the spacetime metric
$\mathfrak{g}$. This follows easily from the fact that the trajectories of
null geodesics do not change under a conformal change of metric, hence the
null geodesics of $\mathfrak{g}$ are the same as the null geodesics of
$f^{2}\mathfrak{g}=-dt\otimes dt+\tilde{g}$.

###  A.2. Relation between ${\rm CD}(0,1)$ and substatic condition

Let $(M,g,f)$ be a substatic triple and let $\tilde{g}=g/f^{2}$. We want to
show that $(M,\tilde{g},\psi)$ satisfies the ${\rm CD}(0,1)$ condition, where
$\psi=-(n-1)\log f$. To this end, we need to rewrite the substatic condition
in terms of the conformal metric. We start from the following formulas:

$\displaystyle\widetilde{\nabla}^{2}f\,$
$\displaystyle=\,\nabla^{2}f+\frac{1}{f}\left(2df\otimes df-|\nabla
f|^{2}g\right)\,,$ (A.4) $\displaystyle\Delta_{\tilde{g}}f\,$
$\displaystyle=\,f^{2}\Delta f-(n-2)f|\nabla f|^{2}\,,$ (A.5)

In particular

$\displaystyle\frac{1}{f}\nabla^{2}f-\frac{1}{f}\Delta fg\,$
$\displaystyle=\,\frac{1}{f}\widetilde{\nabla}^{2}f-\frac{1}{f^{2}}\left(2df\otimes
df-|\nabla
f|^{2}g\right)-\frac{1}{f}\Delta_{\tilde{g}}f\,\tilde{g}-(n-2)|\nabla
f|^{2}\tilde{g}$ (A.6)
$\displaystyle=\,\frac{1}{f}\widetilde{\nabla}^{2}f-\frac{2}{f^{2}}df\otimes
df-\frac{1}{f}\Delta_{\tilde{g}}f\tilde{g}-(n-3)\frac{1}{f^{2}}\big{|}\widetilde{\nabla}f\big{|}_{\tilde{g}}^{2}\,\tilde{g}$
(A.7)

On the other hand, it is well known that the Ricci tensor ${\rm Ric}$ of $g$
and the Ricci tensor ${\rm Ric}_{\tilde{g}}$ of the conformal metric
$\tilde{g}$ are related as follows

${\rm Ric}\,=\,{\rm
Ric}_{\tilde{g}}-\frac{n-2}{f}\widetilde{\nabla}^{2}f+\frac{2(n-2)}{f^{2}}df\otimes
df-\left(\frac{1}{f}\Delta_{\tilde{g}}f+\frac{n-3}{f^{2}}|\widetilde{\nabla}f|^{2}_{\tilde{g}}\right)\tilde{g}\,.$

Putting together the above formulas, we get

$\displaystyle{\rm Ric}-\frac{1}{f}\nabla^{2}f+\frac{1}{f}\Delta fg\,$
$\displaystyle=\,{\rm
Ric}_{\tilde{g}}-\frac{n-1}{f}\widetilde{\nabla}^{2}f+\frac{2(n-1)}{f^{2}}df\otimes
df$ (A.8) $\displaystyle=\,{\rm
Ric}_{\tilde{g}}-(n-1)\widetilde{\nabla}^{2}(\log f)+(n-1)d\log f\otimes d\log
f$ (A.9) $\displaystyle=\,{\rm
Ric}_{\tilde{g}}+\widetilde{\nabla}^{2}\psi+\frac{1}{n-1}d\psi\otimes
d\psi\,.$ (A.10)

It follows then that, if $(M,g,f)$ satisfies the substatic condition, then
$(M,\tilde{g},\psi)$ satisfies the ${\rm CD}(0,1)$ condition.

###  A.3. Li–Xia connections

In [LX17], Li and Xia consider the family of connections ${\rm
D}^{u\alpha\gamma}$, where $u\in\mathscr{C}^{\infty}(M)$,
$\alpha,\gamma\in\mathbb{R}$, defined by

${\rm
D}^{u\alpha\gamma}_{X}Y=\nabla_{X}Y+\alpha\left[X(u)Y+Y(u)X\right]+\gamma\,g(X,Y)\nabla
u\,.$

They then compute the Ricci tensor ${\rm Ric}^{u\alpha\gamma}$ induced by a
connection ${\rm D}^{u\alpha\gamma}$, showing that it is related to the usual
Ricci tensor by

${\rm Ric}^{u\alpha\gamma}={\rm
Ric}-\left[(n-1)\alpha+\gamma\right]\nabla^{2}u+\left[(n-1)\alpha^{2}-\gamma^{2}\right]du\otimes
du+\left[\gamma\Delta u+\left(\gamma^{2}+(n-1)\alpha\gamma\right)|\nabla
u|^{2}\right]g\,.$

When $\alpha=0$ and $\gamma=1$, in particular we have

${\rm Ric}^{u01}\,=\,{\rm Ric}-\nabla^{2}u-du\otimes du+\left[\Delta u+|\nabla
u|^{2}\right]g\,,$

which can be rewritten as follows by setting $u=\log f$:

${\rm Ric}^{u01}\,=\,{\rm Ric}-\frac{1}{f}\nabla^{2}f+\frac{\Delta f}{f}g\,.$

It is then clear that the condition ${\rm Ric}^{u01}\geq 0$ is equivalent to
the substatic condition.

Choosing instead $\alpha=1/(n-1)$, $\gamma=0$, setting $v=-u$ one gets

${\rm Ric}^{u\frac{1}{n-1}0}={\rm Ric}+\nabla^{2}v+\frac{1}{n-1}dv\otimes
dv\,,$

hence ${\rm Ric}^{u\frac{1}{n-1}0}\geq 0$ gives the ${\rm CD}(0,1)$ condition.
In fact, the connection ${\rm D}^{u\frac{1}{n-1}0}$ had already been
considered in the work [WY16] that was focused on the ${\rm CD}(0,1)$ case
only.

Here we show that the two connections ${\rm D}^{u01}$ and ${\rm
D}^{u\frac{1}{n-1}0}$ are in fact conformally related: let $(M,g)$ be a
Riemannian manifold and let $\nabla,\widetilde{\nabla}$ be the Levi-Civita
connections corresponding to the metrics $g$, $\tilde{g}=g/f^{2}$,
respectively. It is easy to show that $\nabla$ and $\widetilde{\nabla}$ are
related a follows

$\nabla_{X}Y\,=\,\widetilde{\nabla}_{X}Y+\frac{1}{f}\left[X(f)Y+Y(f)X-g(X,Y)\nabla
f\right]\,=\,\widetilde{\nabla}_{X}Y+X(u)Y+Y(u)X-g(X,Y)\nabla u\,,$

hence, setting $\psi=-(n-1)u$:

$\displaystyle{\rm D}_{X}^{u01}Y\,$ $\displaystyle=\,\nabla_{X}Y+g(X,Y)\nabla
u$ (A.11) $\displaystyle=\,\widetilde{\nabla}_{X}Y+X(u)Y+Y(u)X$ (A.12)
$\displaystyle=\,\widetilde{\nabla}_{X}Y+\frac{1}{n-1}\left[X(-\psi)Y+Y(-\psi)X\right]\,.$
(A.13)

This is then precisely ${\rm D}^{-\psi\frac{1}{n-1}0}$ using
$\widetilde{\nabla}$ as the Levi-Civita connection in place of $\nabla$.

Notice that in this subsection $\psi$ and $f$ have been introduced as the
functions satisfying $u=\log f$ and $\psi=-(n-1)u$, hence they are related by
$\psi=-(n-1)\log f$, in agreement with Subsection A.2.

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