# Separable Self and Mixed Attention Transformers for Efficient Object
Tracking

Goutam Yelluru Gopal Maria A. Amer
<EMAIL_ADDRESS><EMAIL_ADDRESS>
Department of Electrical and Computer Engineering, Concordia University,
Montréal, Québec, Canada

###### Abstract

The deployment of transformers for visual object tracking has shown state-of-
the-art results on several benchmarks. However, the transformer-based models
are under-utilized for Siamese lightweight tracking due to the computational
complexity of their attention blocks. This paper proposes an efficient self
and mixed attention transformer-based architecture for lightweight tracking.
The proposed backbone utilizes the separable mixed attention transformers to
fuse the template and search regions during feature extraction to generate
superior feature encoding. Our prediction head performs global contextual
modeling of the encoded features by leveraging efficient self-attention blocks
for robust target state estimation. With these contributions, the proposed
lightweight tracker deploys a transformer-based backbone and head module
concurrently for the first time. Our ablation study testifies to the
effectiveness of the proposed combination of backbone and head modules.
Simulations show that our Separable Self and Mixed Attention-based Tracker,
SMAT, surpasses the performance of related lightweight trackers on GOT10k,
TrackingNet, LaSOT, NfS30, UAV123, and AVisT datasets, while running at 37 fps
on CPU, 158 fps on GPU, and having 3.8M parameters. For example, it
significantly surpasses the closely related trackers E.T.Track and
MixFormerV2-S on GOT10k-test by a margin of 7.9% and 5.8%, respectively, in
the $AO$ metric. The tracker code and model is available at
https://github.com/goutamyg/SMAT

##  1 Introduction

The Siamese Network-based (SN) architecture is prevalent in visual object
tracking due to its simplicity and high speed [39, 34]. The SN architecture
consists of a backbone to generate robust feature representation of the target
template and search regions, a localization head module for target state
estimation, and an optional feature fusor module for relation modeling [37].
In recent years, the transformer-based [30, 11, 33] tracking methods have
unified feature extraction and relation modeling by deploying self and mixed
attention blocks in their backbone [34, 8] to simplify the SN architecture
further. Enabled by the computational power of GPUs, these transformer-based
SN trackers achieve high frames-per-second (fps) during inference. However,
the high computational complexity of these transformer-based tracking
algorithms severely impacts the fps on constrained hardware, e.g., CPUs,
limiting the utility of these algorithms towards applications with hardware
constraints.

The lightweight SN-based trackers [2, 35, 1], which are specifically designed
for resource-constrained environments, adopt efficient building blocks to
maintain real-time speed, i.e., $\geq 30$ fps. Therefore, these trackers
cannot fully leverage the modeling power of transformers in their architecture
because of the computational complexity of the standard transformers,
especially the expensive matrix multiplication while computing attention.
Mehta et al. [23] addressed this issue by replacing the costly matrix-to-
matrix multiplication with separable elementwise operations to present an
efficient Mobile Vision Transformer (ViT) block for vision-related tasks.
Leveraging these advances, we propose a separable self and mixed attention
transformer-based lightweight architecture for real-time tracking.

Figure 1: Proposed SMAT architecture. The separable mixed attention Vision
Transformer-based backbone jointly performs feature extraction and fusion of
template and search regions. The separable transformer-based head models long-
range dependencies within the fused features to predict accurate bounding
boxes.

The architecture of the proposed tracker is shown in Figure 1. We employ a
cascaded arrangement of convolutional neural network (CNN) and ViT blocks in
the proposed tracker backbone. Such a hybrid design [22] combines the merits
of convolutions (i.e., learning the spatially-local representations) and
transformers (i.e., modeling the long-range dependencies) with fewer
parameters compared to the fully transformer-based backbone architecture, such
as [8, 37]. Apart from generating a robust feature representation, the
proposed backbone facilitates the exchange of information between the target
template and search region by computing mixed attention [8] in the ViT block
without bloating the backbone latency. Our prediction head efficiently
performs global contextual modeling of encoded features using separable self-
attention units. Such transformer-based global modeling of encoded features
improves the localization accuracy compared to the fully convolution-based
methods, as shown in [1]. With these contributions, we propose a lightweight
self and mixed attention transformers-based tracker, SMAT, running beyond
real-time speed on a CPU.

##  2 Related Work

The introduction of transformer-based modeling has significantly improved the
performance of SN-based trackers in recent years [17]. These SN trackers [31,
34, 38, 6] exploited the global contextual modeling capabilities of the
transformer layers for relation modeling [37], i.e., to fuse the features
extracted from the target template and the search region. The deployment of
computationally expensive transformer-based backbones for SN tracking [37, 8,
19, 32, 5, 13] has improved the tracker performance further and achieved
state-of-the-art results on various challenging benchmarks [16, 25, 12].

In recent years, there have been several SN-based lightweight algorithms
proposed for efficient object tracking. LightTrack [35] used neural
architectural search [7] to design an efficient backbone and head modules
suitable for resource-constrained environments. FEAR [2] presented a compact
and energy-efficient SN-based tracking method running at real-time speed on a
smartphone. It uses the dual-template representation with a dynamic update
scheme to model the target appearance variations. Stark-Lightning [34]
proposed an efficient tracking method with a lightweight transformer-based
feature fusor module. HiFT [3] used hierarchical feature transformers to
achieve real-time speed on an embedded processor for aerial tracking. SiamHFFT
[10] introduced a hierarchical transformer-based feature fusion module for
efficient tracking on CPUs. HCAT [4] deployed a feature sparsification module
and a hierarchical cross-attention transformer-based architecture to achieve
real-time on edge devices. It should be noted that the transformer-based
tracker [34, 3, 10, 4] employ CNN-based backbones for feature extraction and
utilize the transformer layers only for relation modeling, i.e., to fuse the
feature representations generated by their backbones.

Fewer lightweight SN trackers use transformer modules in their backbone or
head architecture. MixFormerV2 [9] presented a fully transformer-based [11]
backbone for efficient tracking, based on knowledge-distillation [15] and
progressive model-depth pruning. E.T.Track [1] proposed an efficient Exemplar
Transformer-based prediction head for visual tracking. It utilized a single
instance-level attention layer in the transformer block to achieve real-time
on a CPU. Compared to the other related trackers, [9] and [1] are the closest
to our work.

Unlike the two-stream encoding approach by [2, 3, 10, 4, 1, 34, 35] (i.e., the
template and search region features are extracted independently), the proposed
backbone facilitates the exchange of information between template and search
regions during feature extraction. Different from [2, 3, 10, 4, 9, 34], we use
a transformer-based head for target localization. In contrast to the fully
transformer-based backbone by [9], we use a cascade of CNN [29] and ViT [23]
blocks in our backbone. Also, the iterative nature of knowledge-distillation
and model pruning by [9] requires multiple rounds of model training, whereas
our tracker model needs to be trained only once. Compared to the exemplar
transformer-based head module by E.T.Track [1], our separable self-attention
prediction head is $3\times$ compact in terms of parameters (5.7 Million for
E.T.Track versus 1.8 Million for our SMAT). As a post-processing step, the
related [35] and [1] refine the predicted bounding boxes by penalizing large
changes in target size and aspect ratio of the predicted boxes between
consecutive frames. We do not employ any bounding-box refinement techniques
during post-processing.

To summarize our contributions, we are the first to propose:

  * •

A separable mixed attention ViT-based backbone for joint feature extraction
and information fusion between the template and search regions. Our approach
combines the merits of CNNs and efficient ViTs to learn superior feature
encoding for accurate tracking without bloating the backbone latency.

  * •

A separable self-attention transformer-based prediction head to model the
global dependencies within the fused feature encoding for accurate bounding-
box prediction.

Compared to related work, for the first time, our method concurrently deploys
an efficient transformer-based backbone and prediction head for lightweight
tracking.

##  3 The Proposed Method

This section discusses the architecture and training details of the proposed
SMAT tracker. Section 3.1 presents our tracker backbone and Section 3.2
describes the architecture of the proposed head module. Section 3.3 has
details of the loss function used during training.

Figure 2: Proposed Separable Mixed Attention ViT block. The qkv-proj denotes
the set of three $1\times 1$ convolutional filters to generate the Query, Key,
and Value for attention computation. The mixed attention output is passed
through a $1\times 1$ convolutional ffn-out block to generate the output of
the transformer layer.

###  3.1 Proposed Mixed Attention ViT-based Backbone

The backbone of the proposed tracker receives two images as its input; one is
the target template $Z_{in}$, and the other is the search region for target
localization, $X_{in}$. First, we apply the CNN-based Inverted Residual (IR)
[29] blocks on $Z_{in}$ and $X_{in}$. These IR blocks generate spatially local
feature representations of $Z_{in}$ and $X_{in}$ while being efficient
compared to the regular convolutional blocks [29]. In addition, these IR
blocks reduce the spatial dimensionality of input images by the pooling
operation to generate low-dimensional feature representations for our
separable mixed attention ViT block.

The architecture of the proposed mixed attention ViT block is shown in Figure
2. Let $Z\in R^{W_{z}\times H_{z}\times C}$ and $X\in R^{W_{x}\times
H_{x}\times C}$ denote the template and search region feature representations,
respectively. Inside the proposed ViT block, we initially pass $Z$ and $X$
through a series of $3\times 3$ and $1\times 1$ CNN layers with shared weights
to project the number of channels in $Z$ and $X$ from $C$ to $d$. We tokenize
[22] the output of CNN blocks and concatenate them to generate a total of $k$
tokens to learn mixed attention [8] between the template and search regions.
Inside the transformer layer, we first apply a set of three $1\times 1$
convolutional filters (denoted as qkv-proj in Figure 2) to generate the query
$\mathcal{Q}\in R^{k\times 1}$, the key $\mathcal{K}\in R^{k\times d}$, and
the value $\mathcal{V}\in R^{k\times d}$. Then, we apply the softmax operation
on the query vector $\mathcal{Q}$ and broadcast along its column (i.e., the
element in $i^{th}$ row is repeated $d$ times along the column dimension) to
generate $\mathcal{\tilde{Q}}\in R^{k\times d}$. Using $\mathcal{\tilde{Q}}$
and $\mathcal{K}$, the context vector $\mathcal{A}\in R^{1\times d}$ is
computed as

$\mathcal{A}=\sum_{k}\mathcal{\tilde{Q}}\odot\mathcal{K},$ (1)

where $\odot$ denotes the element-wise multiplication and $\sum_{k}$ indicates
summation across the rows. The context vector $\mathcal{A}$ is broadcasted
along its rows to create $\mathcal{\tilde{A}}\in R^{k\times d}$, which is used
to compute the mixed attention $\mathcal{M}\in R^{k\times d}$ as

$\mathcal{M}=\mathcal{\tilde{A}}\odot\text{ReLU}(\mathcal{V}).$ (2)

The elementwise multiplication operations in Eq. 1 and Eq. 2 reduce the
latency of the separable transformer layer, shown in Figure 2, when compared
to the dense matrix-to-matrix multiplication-based attention computation in
standard transformers [30]. Also, computing the mixed attention on the
concatenated features concurrently models the global interactions within
(i.e., self) and between (i.e., cross) the target template and the search
area. Therefore, mixed attention requires fewer transformer block evaluations
than separately computing the self and cross-attention. Similar to [30], we
employ a residual connection [14] around the attention computation block. We
pass the output of the residual connection through a $1\times 1$ convolutional
feedforward network (denoted as ffn-out in Figure 2) to generate the output of
the separable transformer layer. We split the resulting feature map to
separate the template and search region features with $d$ channels. Finally,
we re-project the number of channels from $d$ to $C$ by applying a shared
$1\times 1$ convolutional filter on the separated feature maps to generate
$\tilde{X}$ and $\tilde{Z}$ as the output.

The computation of mixed attention in the ViT block facilitates implicit
relation modeling during feature extraction, thereby generating superior
features compared to the two-stream encoding approach. Such feature fusion
also avoids needing a parameter-heavy module for the subsequent relation
modeling between the template and search region features. For a parameter-
efficient relation modeling (or feature fusion) between the features
$\tilde{X}$ and $\tilde{Z}$ generated by the proposed backbone, we use the
parameter-free pixel-wise cross-correlation [36] operation (cf. Figure 1). The
fused feature encoding $F$ is computed as,

$F=PWCorr(\tilde{X},\tilde{Z}).$ (3)

where $PWCorr$ denotes the pixel-wise cross-correlation operation. We apply a
$1\times 1$ convolution filter on the fused encoding to transform the number
of channels to $C_{h}$. To perform target localization, we pass the resulting
encoding $F$ to the proposed prediction head in Section 3.2.

###  3.2 Proposed Self-Attention Prediction Head

The pipeline of the proposed predictor head is shown in Figure 3. Our
prediction head has two branches: target classification and bounding-box
regression. Unlike a fully CNN-based approach, we implement these branches via
a convolutional and transformer layers cascade. The CNN layers are suitable
for modeling the local relationships within the fused feature representation
but are limited in effectively capturing the non-local associations. On the
other hand, the transformer layers explicitly model the long-range global
interactions within the features by processing the tokenized feature encoding.
Such a global modeling scheme is beneficial for localization under scenarios
of drastic target shape variations and heavy occlusion [34]. The cascade of
convolutional and transformer-based layers combines the best of their modeling
strengths to produce high-quality tracking results on challenging videos.

Figure 3: Proposed Separable Self-Attention Transformer-based Predictor Head.
It utilizes two branches for target classification and bounding-box
regression.

First, we apply a $3\times 3$ convolutional filter for the proposed predictor
head on the fused encoding $F$ from Eq. 3 to extract spatially local feature
representations. We then tokenize the filter output and pass it through a
stack of $L_{cls}$ and $L_{reg}$ separable self-attention transformer layers
in classification and regression branches, respectively. For these transformer
blocks, the pipeline of computing the attention is similar to the transformer
layer in the ViT block from Section 3.1; except, here, we calculate self-
attention to model long-range dependencies within the fused feature encoding
$F$ for robust target state estimation. The output of the self-attention
transformer layer is passed through a $3\times 3$ CNN layer to generate a
score map $\mathcal{R}$ for the classification branch, and the local offset
and the normalized bounding-box size for the regression branch, as in [37]. A
combination of local and long-range contextual modeling by our predictor head
improves tracker performance without significantly increasing the overall
model latency.

###  3.3 Loss function for Training

We use loss functions on the classification and regression output generated by
the proposed head module while training our model. We use the weighted focal
loss for the output of the classification branch; for the output of the
regression branch, we use the $\ell_{1}$ loss and generalized Intersection-
over-Union ($IoU$) loss, as in [37]. The overall loss function
$\mathcal{L}_{total}$ is defined as

$\mathcal{L}_{total}=\mathcal{L}_{focal}+\lambda_{\ell_{1}}\cdot\mathcal{L}_{\ell_{1}}+\lambda_{IoU}\cdot\mathcal{L}_{IoU},$
(4)

where $\mathcal{L}_{focal}$, $\mathcal{L}_{\ell_{1}}$, and $\mathcal{L}_{IoU}$
represent the focal loss, $\ell_{1}$ loss and the generalized $IoU$ loss
functions, respectively. $\lambda_{\ell_{1}}$ and $\lambda_{IoU}$ are the
hyperparameters determining the relative impact of the respective loss
functions.

##  4 Experimental and Ablation Results

###  4.1 Implementation Details

We set the size of the template and search region images, i.e., $Z_{in}$ and
$X_{in}$ from Section 3, to $128\times 128$ and $256\times 256$, respectively,
during training and inference. We deploy two CNN-based IR [29] blocks and two
ViT-blocks in our backbone. The sequential ordering of the IR and ViT blocks
is the same as MobileViTv2 [23] pipeline. During feature extraction, our
backbone performs four downsampling operations, each by a factor of 2;
therefore, the spatial dimension of the features generated by our backbone is
$8\times 8$ and $16\times 16$ for the template and search regions,
respectively. For the proposed head module, we set the number of channels in
the fused encoding, i.e., $C_{h}$ in Figure 3, to 128. We set the number of
transformer layers $L_{cls}$ and $L_{reg}$ in the classification and
regression branches to 2 and 4, respectively. The reason for defining
$L_{reg}$ two times the value of $L_{cls}$ is because the regression head
predicts twice the variables, i.e., local offset and bounding-box size,
compared to the classification branch predicting the target center.

###  4.2 Training and Inference Details

We use the GOT10k [16], LaSOT [12], TrackingNet [25], and COCO [20] datasets
to train our tracker. GOT10k, LaSOT, and TrackingNet have a non-overlapping
train-test split ratio of 9335-to-180, 1120-to-280, and 30132-to-511 videos,
respectively. Also, GOT10k provides 180 additional videos as the validation
split. We apply data augmentation (horizontal flip and scale jittering) to
generate training image pairs for the still images in the COCO train dataset.
We use the combined training splits of these four datasets to train our model.

We train our model for 300 epochs, and each epoch uses $6\times 10^{4}$ image
pairs uniformly sampled from the training dataset. The initial learning rate
(lr) is set to 0.0004 and is reduced by a factor of 10 after 240 epochs. The
lr for the backbone parameters is set 0.1 times the lr for the remaining
trainable parameters of our model. We use AdamW [21] as the network optimizer
and set the weight decay to $10^{-4}$. The values of hyperparameters
$\lambda_{\ell_{1}}$ and $\lambda_{IoU}$ from Eq. 4 are set to 5 and 2,
respectively. These hyperparameter values used for training are derived from
[37] with no additional finetuning. We initialize our backbone weights using a
pretrained MobileViTv2 model provided by the authors[23]. We use PyTorch [27]
for developing the tracker code. Our model is trained on a single NVidia Telsa
V100 GPU (32GB memory) with a batch size of 128. We monitor the possibility of
overfitting by periodically evaluating the values of loss functions
$\mathcal{L}_{focal},\mathcal{L}_{\ell_{1}}$, and $\mathcal{L}_{IoU}$ from Eq.
4 using the GOT10k validation videos.

During inference, we use the annotation from the first frame in the video as
the target template and do not perform model update. To define the search
region at frame $t$, we crop a region around tracker output at frame $t-1$,
four times the target size. This image is resized to $256\times 256$ and
utilized as the search image at frame $t$. As a post-processing step, we apply
a Hanning window on the classification score map $\mathcal{R}$ to penalize
large target displacement predictions.

###  4.3 Comparison to the Related Work

To assess the performance of the proposed SMAT, we evaluate our tracker on the
test-split of GOT10k [16], LaSOT [12], TrackingNet [25], NfS30 [18], UAV123
[24], and AVisT [26] datasets. GOT10k-test has 180 test videos having non-
overlapping object classes compared to their training videos, mainly to
promote the generalization of tracking algorithms towards unseen object
categories. LaSOT-test has 280 videos with 14 different attributes and
balanced class categories. With an average length of 2500 frames per video,
LaSOT-test is effective in accessing long-term tracking capabilities.
TrackingNet-test contains 511 challenging videos curated from the large-scale
YouTube-BB [28] dataset with 15 attribute annotations. NfS30 has 100 test
videos, predominantly containing fast-moving objects with significant motion
blur. UAV123 is a low-altitude UAV tracking benchmark and has 123 videos with
12 attribute annotations. AVisT dataset has 120 challenging videos with a wide
range of atmospheric adverse scenarios such as rain, fog, fire, low-light,
snow, tornado, and smoke impacting the target appearance in the test videos.
The datasets NfS30, UAV123, and AVisT have no training split videos.

We use the metrics recommended by the corresponding dataset authors to
quantify the tracker performance during our evaluation. GOT10k uses the
Average of Overlap ($AO$) based on the Intersection-of-Union (IoU) value
between the groundtruth and predicted bounding boxes, averaged over all the
test videos. It also uses Success Rate ($SR$), computing the fraction of
frames having an $IoU$ value greater than a threshold $\tau$, with values of
$\tau$ as 0.5 and 0.75. TrackingNet uses Area-Under-the-Curve ($AUC$),
Precision ($P$), and Normalized-Precision ($P_{norm}$) as the tracker
evaluation metrics. $AUC$ is equivalent to $AO$ [40], and $P$ is computed
based on the distance between the groundtruth and predicted bounding-box
centers, measured in pixels. The metric $P_{norm}$ is similar to $P$; however,
$P_{norm}$ uses normalized bounding boxes while measuring the distance between
their centers. LaSOT uses the same evaluation metric as TrackingNet, whereas
NfS30 and UAV123 use $AUC$ and $P$ for tracker evaluation. Along with the
$AUC$, AVisT uses OP$50$ and OP$75$ as its evaluation metrics, which are
equivalent to $SR$ at thresholds 0.5 and 0.75, respectively, from GOT10k. To
ensure fair tracker evaluation and avoid finetuning of parameters on the test
data, GOT10k and TrackingNet sequester the ground-truth annotations for their
test videos. Therefore, we generate the metrics for these datasets by
submitting the raw tracker results to the remote evaluation server. The
groundtruth annotations are available for the LaSOT-test, NfS30, UAV123, and
AVisT datasets.

We compare the results of the proposed SMAT against the related lightweight
trackers: LightTrack [35], Stark-Lightning [34], FEAR-XS [2], HCAT [4],
E.T.Track [1], and MixFormerV2-S [9], evaluated using the pretrained models
provided by their authors. From Table 1, we can see that the proposed SMAT
outperforms the related trackers on all six test datasets: GOT10k-test,
TrackingNet-test, LaSOT-test, NfS30, UAV123, and AVisT.

Tracker | GOT10k-test [16] | TrackingNet-test [25] | LaSOT-test [12] | NfS30 [18] | UAV123 [24] | AVisT [26] | fps
---|---|---|---|---|---|---|---
| $AO$ | $SR_{0.50}$ | $SR_{0.75}$ | $AUC$ | $P_{norm}$ | $P$ | $AUC$ | $P_{norm}$ | $P$ | $AUC$ | $P$ | $AUC$ | $P$ | $AUC$ | OP$50$ | OP$75$ | (CPU)
LightTrack [35] (CVPR’21) | 0.582 | 0.668 | 0.442 | 0.729 | 0.793 | 0.699 | 0.522 | 0.583 | 0.517 | 0.565 | 0.692 | 0.617 | 0.799 | 0.404 | 0.437 | 0.242 | 42
Stark-Lightning [34] (ICCV’21) | 0.596 | 0.696 | 0.479 | 0.727 | 0.779 | 0.674 | 0.578 | 0.660 | 0.574 | 0.596 | 0.710 | 0.620 | 0.820 | 0.394 | 0.431 | 0.223 | 50
FEAR-XS [2] (ECCV’22) | 0.573 | 0.681 | 0.455 | 0.715 | 0.805 | 0.699 | 0.501 | 0.594 | 0.523 | 0.486 | 0.563 | 0.610 | 0.816 | 0.370 | 0.421 | 0.220 | 42
HCAT [4] (ECCV’22) | 0.634 | 0.743 | 0.558 | 0.763 | 0.824 | 0.726 | 0.590 | 0.683 | 0.605 | 0.619 | 0.741 | 0.620 | 0.805 | 0.418 | 0.481 | 0.263 | 45
E.T.Track [1] (WACV’23) | 0.566 | 0.646 | 0.425 | 0.740 | 0.798 | 0.698 | 0.589 | 0.670 | 0.603 | 0.570 | 0.694 | 0.626 | 0.808 | 0.390 | 0.412 | 0.227 | 44
MixFormerV2-S [9] (arXiv’23) | 0.587 | 0.672 | 0.482 | 0.767 | 0.812 | 0.714 | 0.610 | 0.694 | 0.614 | 0.610 | 0.722 | 0.634 | 0.837 | 0.396 | 0.425 | 0.227 | 30
SMAT (ours) | 0.645 | 0.747 | 0.578 | 0.786 | 0.842 | 0.756 | 0.617 | 0.711 | 0.646 | 0.620 | 0.746 | 0.643 | 0.839 | 0.447 | 0.507 | 0.313 | 37

Table 1: Comparison of proposed SMAT with the related lightweight SN trackers
on GOT10k-test (server), TrackingNet-test (server), LaSOT-test, NfS30, UAV123,
and AVisT datasets. The best and second-best results are highlighted in red
and blue, respectively.

No related tracker performs consistently second best across the six datasets.
HCAT exhibits the second-best results in 10 out of 16 metrics across all
datasets, while MixFormerV2-S scores the second-best in 6 out of 16 cases.
Regarding fps under CPU, our tracker is relatively faster than MixFormerV2-S
by 19% and slower than HCAT by 17%. Considering the GOT10k-test dataset
(server-based evaluation), our tracker is better than MixFormerV2-S and HCAT
on average by 3.5% in $AO$, 4% in $SR_{0.50}$, and 5.8% in $SR_{0.75}$.

Since GOT10k-test dataset has videos with target object categories unseen
during training, it is well-suited for evaluating tracker generalization. Our
SMAT results on the GOT10k-test demonstrate its superior generalization
capability compared to the related lightweight trackers. Since E.T.Track has a
transformer-based predictor head and MixFormerV2-S has a fully transformer-
based backbone, these trackers are closely related to our work. By
concurrently deploying a transformer-based backbone and head module, our SMAT
outperforms both E.T.Track and MixFormerV2 on average by a significant margin
of 6.8% in $AO$, 8.8% in $SR_{0.50}$, and 12.4% in $SR_{0.75}$. On
TrackingNet-test benchmark, the proposed SMAT has a better performance than
all the related trackers by at least 1.9% in $AUC$, 1.8% in $P_{norm}$, and 3%
in $P$. No single tracker consistently exhibits the second-best performance on
the TrackingNet dataset. The results of our SMAT on the LaSOT-test videos show
that our tracker has better long-term tracking performance than other
lightweight trackers. The related E.T.Track employs a post-processing step to
refine the predicted bounding boxes, which reduces the chances of target loss
or drift during long-term tracking. The other related tracker, MixFormerV2,
utilizes an online template update scheme to improve long-term tracking
performance. Despite the absence of such bounding-box refinement step and
template update scheme, our SMAT performs better than E.T.Track by 2.8% and
MixFormerV2 by 0.7% in $AUC$. The results on NfS30 dataset show that our SMAT
is resilient to motion blur and is better-suited for tracking fast-moving
objects than the related trackers. Similarly, proposed SMAT has a better $AUC$
by at least 2.8% and 0.9% on AVisT and UAV123 datasets, respectively, compared
to the other lightweight trackers. It shows that our tracker performs better
than the related trackers on videos affected by adverse visibility conditions
and is robust to the challenges of airborne scenarios.

The last column of Table 1 lists the fps of the proposed SMAT and related
trackers, evaluated on a 12th Gen Intel(R) Core-i9 CPU. The proposed SMAT
tracker achieves a real-time speed of 37 fps by leveraging the computational
efficiency of the separable self and mixed attention blocks used in its model
architecture. Compared to standard transformer-based MixFormerV2-S, our
tracker is faster by 19% since the proposed SMAT uses separable attention-
based transformers in its backbone for efficient computation of attention.
However, in comparison to the related lightweight trackers with a two-stream
pipeline [35, 34, 2, 4, 1], our SMAT has a 17% lower fps on average. It is
mainly due to the coupling of features in our tracker backbone, requiring
evaluation of template and search region features at every frame during
inference. On the other hand, trackers with a two-stream pipeline compute the
template features only once since they do not perform feature fusion in their
backbone.

Our tracker achieves a speed of 158 fps upon its evaluation on an Nvidia RTX
3090 GPU. With nearly 3.8 Million parameters, our model has a size of 15.3MB
on disk.

###  4.4 Ablation Study

In this section, we present ablation study results quantifying the role of
different components of our tracker. For this study, we use GOT10k-test [16]
and LaSOT-test [12] due to their suitability towards gauging tracker
generalization and long-term tracking performance, respectively.

Comparing feature fusion techniques: In Section 3.1, we mentioned that the
proposed mixed attention efficiently approximates the explicit modeling of
interaction within and between the target template and search regions. In this
section, we experimentally verify the efficacy and efficiency of the proposed
approach by comparing its results with other feature fusion methods that
deploy explicit computation of self or cross-attention (or both).

Figure 4: Comparing different feature fusion techniques (A, B and C) with the
proposed mixed attention-based method shown in D.

For the variant-A shown in Figure 4, we use a shared separable self-attention
transformer block for the template and search regions. This approach
facilitates independent computation of the template and search features for
high tracking speed; however, it restricts the information exchange between
the two regions (i.e., no feature fusion). For the variant-B, we deploy shared
separable cross-attention transformers for inter-region feature fusion but no
intra-region feature modeling (i.e., no self-attention). A similar cross-
feature blending approach for relation modeling has shown excellent tracking
results [6]. Lastly, we implement cascaded self and cross-attention
transformers for the third variant-C to explicitly model inter and intra-
region feature fusion. The proposed mixed attention block, shown as variant-D
in Figure 4, approximates explicit self and cross-attention computation by
applying a separable transformer block on the concatenated features from the
template and search regions.

Attention | GOT10k-test [16] | LaSOT-test [12] | fps
---|---|---|---
Mechanism | $AO$ | $SR_{0.50}$ | $SR_{0.75}$ | $AUC$ | $P_{norm}$ | $P$ | (CPU)
A | 0.631 | 0.726 | 0.578 | 0.604 | 0.689 | 0.629 | 39
B | 0.645 | 0.743 | 0.590 | 0.609 | 0.696 | 0.631 | 30
C | 0.654 | 0.761 | 0.598 | 0.621 | 0.717 | 0.656 | 24
D (ours) | 0.645 | 0.747 | 0.578 | 0.617 | 0.711 | 0.646 | 37

Table 2: Summarizing the feature fusion-based ablation study results for the
proposed SMAT tracker. The best and second-best results are highlighted in red
and blue, respectively.

Table 2 summarizes the performance of these variants on GOT10k and LaSOT test
datasets. From Table 2, we can see that variant-A has a $1.05\times$ higher
fps than the proposed method; however, the lack of information flow between
the template and search region impacts its performance. Hence, compared to the
proposed tracker, it has a lower $AUC$ score by 1.4% and 1.3% on GOT10k and
LaSOT, respectively. Performance of the pure cross-attention-based variant-B
is lower than our SMAT by 0.8% in $AUC$ on the LaSOT dataset, and both
approaches have comparable performance on the GOT10k dataset. However,
variant-B has a relatively lower fps by 18.9% than our SMAT, which indicates
that separately computing cross-attention for the template and search regions
is slower than our mixed-attention computation on concatenated features.
Variant-C achieves the best results on both datasets; however, explicit
computation of self and cross-attention significantly impacts its tracking
speed. Therefore, it has the lowest fps value compared to the other variants
and is relatively slower than our approach by 35% on a CPU. Compared to the
variant-C, our method approximates the inter and intra-region feature fusion
by a single attention operation. Therefore, as seen from Table 2, our mixed
attention-based approach provides the best trade-off between performance and
speed compared to other feature fusion variants shown in Figure 4.

Convolutional vs Transformer-based head: To quantify the significance of our
separable self-attention transformer-based predictor head from Section 3.2, we
replace the proposed module with a fully-convolutional predictor head for
classification and bounding-box regression. As seen from Table 3, this
replacement decreases the performance of the proposed SMAT tracker by 3.5% in
$AO$ for the GOT10k-test dataset and 2.3% in $AUC$ for the LaSOT-test dataset,
highlighting the impact of global contextual modeling of encoded features by
the proposed predictor head.

Predictor Head | GOT10k-test [16] | LaSOT-test [12]
---|---|---
| $AO$ | $SR_{0.50}$ | $SR_{0.75}$ | $AUC$ | $P_{norm}$ | $P$
Fully-Convolutional | 0.610 | 0.709 | 0.540 | 0.594 | 0.682 | 0.612
Transformer-based (ours) | 0.645 | 0.747 | 0.578 | 0.617 | 0.711 | 0.646

Table 3: Ablation study results for the proposed transformer-based prediction
head. Best results are highlighted in red.

Standard vs Separable Attention mechanism: To evaluate the efficiency of the
separable attention mechanism deployed in the proposed SMAT, we retrain our
model by replacing the separable transformer blocks in the tracker backbone
with the standard transformer-based MobileViT block [22].

Mixed Attention | GOT10k-test [16] | LaSOT-test [12] | fps
---|---|---|---
Mechanism | $AO$ | $SR_{0.50}$ | $SR_{0.75}$ | $AUC$ | $P_{norm}$ | $P$ | (CPU)
Standard | 0.659 | 0.760 | 0.605 | 0.600 | 0.687 | 0.630 | 32
Separable (ours) | 0.645 | 0.747 | 0.578 | 0.617 | 0.711 | 0.646 | 37

Table 4: Comparison of tracker performance using the standard vs separable
mixed attention mechanism in the backbone. The best results are highlighted in
red.

From Table 4, we observe that the proposed method has a higher $AUC$ of 1.7%
on the LaSOT dataset compared to the standard attention-based tracker. On the
other hand, our approach has a lower $AUC$ of 1.4% on the GOT10k dataset.
However, the efficient attention evaluation enhances the speed of the proposed
tracking approach by 15.6% compared to the standard transformer-based
tracking, as seen from Table 4.

###  4.5 Attribute-based analysis

Tracker | ARC | BC | CM | DEF | FM | FOC | IV | LR | MB | OV | POC | ROT | SV | VC | Overall
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
LightTrack [35] | 0.503 | 0.434 | 0.539 | 0.577 | 0.334 | 0.386 | 0.550 | 0.407 | 0.457 | 0.441 | 0.497 | 0.519 | 0.523 | 0.502 | 0.522
Stark-Lightning [34] | 0.572 | 0.491 | 0.613 | 0.594 | 0.471 | 0.505 | 0.610 | 0.516 | 0.568 | 0.557 | 0.554 | 0.577 | 0.582 | 0.581 | 0.578
FEAR-XS [2] | 0.488 | 0.437 | 0.528 | 0.505 | 0.389 | 0.403 | 0.506 | 0.421 | 0.473 | 0.425 | 0.478 | 0.489 | 0.506 | 0.487 | 0.501
HCAT [4] | 0.587 | 0.524 | 0.639 | 0.619 | 0.460 | 0.507 | 0.606 | 0.520 | 0.579 | 0.538 | 0.568 | 0.592 | 0.600 | 0.567 | 0.590
E.T.Track [1] | 0.573 | 0.526 | 0.590 | 0.619 | 0.404 | 0.480 | 0.612 | 0.484 | 0.545 | 0.519 | 0.562 | 0.588 | 0.594 | 0.576 | 0.589
MixFormerV2-S [9] | 0.603 | 0.519 | 0.642 | 0.626 | 0.507 | 0.539 | 0.619 | 0.556 | 0.604 | 0.574 | 0.586 | 0.603 | 0.617 | 0.630 | 0.610
SMAT (ours) | 0.610 | 0.553 | 0.656 | 0.663 | 0.466 | 0.517 | 0.662 | 0.523 | 0.592 | 0.570 | 0.600 | 0.621 | 0.624 | 0.597 | 0.617

Table 5: Comparing the $AUC$ values of the proposed SMAT with the related
lightweight trackers for 14 attributes of the LaSOT dataset. The best and
second-best results are highlighted in red and blue, respectively. The last
column indicates the mean $AUC$ across all videos.

In this section, we compare the per-attribute performance of the proposed SMAT
tracker with related lightweight trackers on the LaSOT dataset. Table 5
summarizes the tracker evaluation results on various challenging factors (or
attributes) of the LaSOT dataset, namely Aspect Ratio Change (ARC), Background
Clutter (BC), Camera Motion (CM), Deformation (DEF), Fast Motion (FM), Full
Occlusion (FOC), Illumination Variation (IV), Low Resolution (LR), Motion Blur
(MB), Out-of-View (OV), Partial Occlusion (POC), Rotation (ROT), Scale
Variation (SV), and Viewpoint Change (VC).

From Table 5, we can see that our tracker has the best performance on 8 out of
14 attributes, and it has the second-best performance in 5 cases. For the
attributes IV and DEF, the proposed SMAT performs significantly better than
the second-best tracker MixFormerV2-S[9], with a higher $AUC$ of 4.3% and
3.7%, respectively. In comparison to the related trackers that do not perform
feature fusion in their backbone, i.e., [35, 34, 2, 4, 1], our tracker has a
higher $AUC$ of 4.4% for DEF and 2.3% for ARC than the second best trackers,
[4, 1] and [4], respectively. It indicates the effectiveness of transformer-
based feature fusion in our tracker backbone, producing accurate bounding
boxes under drastic target appearance variations. Also, in comparison to these
trackers, our SMAT is resilient to tracking failures under POC and BC, with a
higher $AUC$ of 3.2% and 2.7%, respectively, than the second-best results. On
the other hand, our SMAT has an inferior performance than the related trackers
[9] and [34] for the attribute FM; we are working to improve our SMAT
performance under FM.

###  4.6 Visualizing the attention maps

To showcase the interpretability of the proposed SMAT tracker, we visualize
the tracker output and the corresponding attention maps in Figure 5 for four
sequences chosen from the LaSOT test dataset. The images on the left contain
the target template (top-left corner), the search region at frame $\\#t$, and
the tracker output. The images in the center and right indicate the attention
maps corresponding to the transformer blocks of our tracker backbone at the
spatial resolution of $32\times 32$ and $16\times 16$, respectively. For the
examples shown in Figure 5, the target object is impacted by a challenging
factor described in Section 4.5, i.e., ARC for bicycle-18, IV for drone-2, DEF
for drone-2, and POC for microphone-16. Despite the influence of these
attributes, our SMAT successfully locates the target object. For the example
shown in the last row of Figure 5, the target is partially occluded by an
external object. In this case, the tracker focuses on the visual cues around
the target object, as seen from the attention maps in the center. This
information is processed by the subsequent transformer blocks of our tracker
backbone to produce stronger attention values in the target center and
generate an accurate bounding box.

Figure 5: Visualizing the bounding box output (left) and the corresponding
attention maps (center and right) for the proposed SMAT tracker. The larger
values in the attention map are denoted by red color, while the smaller values
are represented by blue color.

##  5 Conclusion

This paper proposed a separable self and mixed attention transformer-based
architecture for lightweight tracking. The proposed backbone utilized the
separable mixed attention transformer layer to facilitate the exchange of
information between the target template and the search region and generate
improved encoding compared to the two-stream tracking pipeline. The proposed
separable self-attention transformer-based predictor head efficiently modeled
long-range dependencies within the fused encoding to generate superior target
classification and bounding-box prediction results. Our ablation study
analyzed the accuracy-speed tradeoffs using different feature fusion methods,
showcased the effectiveness of the proposed head module for accurate tracking,
and demonstrated the efficiency of the separable mixed attention compared to
the standard attention-based tracking. Our SMAT performed better than related
lightweight trackers on six challenging benchmarks. The computational
efficiency of the proposed architecture assisted our tracker, with 3.8M
parameters, to exceed real-time speed on a CPU, while running at 158 fps on
GPU.

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# Mechanism of stochastic resonance in viscoelastic channel flow

Yuke Li  Victor Steinberg  Department of Physics of Complex Systems, Weizmann
Institute of Science, Rehovot 7610001, Israel

###### Abstract

Our experiment discloses mechanism of occurrence of stochastic resonance (SR)
discovered in viscoelastic channel flow above non-modal elastic instability
onset, $Wi_{c}$, and characterized at various channel locations, $l/h$, where
$h$ is its height. Thus, $l/h$ is the control parameter besides Weissenberg
number, $Wi$. We find that SR appearance depends on $l/h$ at fixed $Wi$, and
it vanishes close to inlet, where streamwise velocity fluctuations, $u_{rms}$,
are large and steeply reduce to smaller value persisting till outlet. At
larger $l/h$ and smaller $u_{rms}$, flat spanwise velocity, $w$, power
spectrum, $E_{w}$, is noticed at $Wi>Wi_{c}$. Then interaction of chaotic $u$
with noisy $w$ power spectra and elastic waves causes SR, whereas chaotic
$E_{w}$ prevents it. Similar mechanism is suggested for autonomous dynamical
model, where chaotic power spectrum and limit cycle interacting with external
noise generate SR. Further, SR frequency, $f_{p}$, and elastic wave frequency,
$f_{el}$, show the same scaling with $Wi-Wi_{c}$, indicating their close
relation, though $f_{p}$ is shifted down 4-5 times relatively to $f_{el}$.

viscoelastic channel flow, elastically driven instability, supercritical non-
modal instability, stochastic instability mechanism

††preprint: APS/123-QED

Recently, we disclosed the route from laminar to sustained chaotic flow in
inertia-less viscoelastic channel flow externally perturbed by finite-
amplitude white noise [1]. In the limited range above the onset of
supercritical, non-normal-mode elastic instability, $Wi_{c}$, in a lower
subrange of a transition flow regime, the laminar flow becomes chaotic only in
the streamwise direction, $x$, while in the spanwise direction, $z$, the flow
continues to exhibit flat white noise spectrum of the spanwise velocity, $w$,
and very low intensity peaks of elastic waves. Moreover, in the lower
subrange, the time series of the streamwise velocity, $u(t)$, shows nearly
periodic spikes on the top of the chaotic flow. These spikes are characterized
by a narrow, low frequency, high energy peak on the top of the chaotic
streamwise velocity power spectrum, $E_{u}$. Here, $Wi=U\lambda/d$ is the main
control parameter in inertia-less viscoelastic fluid flows that denotes the
ratio of elastic stress to relaxation dissipation of elastic stress, where $U$
is the average flow velocity defined by fluid discharge rate (see the
Supplemental Material [2] and also references [3, 4] therein), $d$ is the
characteristic vessel size, and $\lambda$ is the longest polymer relaxation
time. For inertia-less, viscoelastic channel flow above $Wi_{c}$, Reynolds
number $Re=Ud\rho/\eta\ll 1$ and $Wi\gg 1$, where $\rho$ and $\eta$ are the
solution density and dynamic viscosity, respectively.

As discussed in Ref. [1], the almost periodic sharp spikes resemble stochastic
resonance (SR), observed typically as the response of a bistable system to a
periodic external force in the presence of noise [5]. However, as shown in
chaotic nonlinear dynamical systems, SR may be generated via interaction of
chaotic attractor with external white noise in the presence of a limit cycle,
at fixed values of the controlled parameter [6, 7, 8]. Thus, SR, the periodic
spikes, are predicted to appear in autonomous, nonlinear dynamical model as
the result of solely internal dynamics, namely the interaction of chaotic
attractor and a limit cycle of coherent motion with external white noise
perturbations. For the best of our knowledge, such SR has been never observed
experimentally (see for example [5]).

Recently we reported in Ref. [1] the discovery of such type of SR in polymer
solution of inertia-less, straight channel flow. Indeed, in the lower subrange
at $Wi_{c}=150<Wi\leq 300$ of the transition flow regime, interaction of the
one-dimensional (1D) chaotic attractor of $u$ with the 1D white noise $w$
power spectrum, $E_{w}$, displaying also transverse, streamwise propagating
elastic waves, results in the periodic spikes associated with SR. Further on,
in the upper subrange at $400\leq Wi\leq 1000$ of the transition regime, SR
disappears, and we observe the two-dimensional (2D) chaotic attractor of both
$u$ and $w$ and elastic waves of up to two orders higher intensity [1]. Thus,
the mechanism of a stochastic path from laminar to persistent chaotic flow is
based on the low frequency and high energy SR discovered in the lower subrange
above $Wi_{c}$.

Thus, as mentioned above, in the lower subrange, $E_{w}$ remains flat that
characterizes white noise. On the top of $E_{w}$, broad and noisy peaks of low
intensity elastic waves, $I_{el}$, are also observed. Since $I_{el}$ is much
lower than the SR peaks, $I_{p}$, in $E_{u}$, the $I_{p}$ takes over the role
of elastic waves to drive the viscoelastic channel flow to the sustained
chaotic flow that emerges in the upper subrange of the transition regime.
Firstly, in the lower subrange, SR strongly modifies the Gaussian probability
density function (PDF) of the normalized streamwise velocity fluctuations,
$u^{\prime}/u_{rms}$, where $u^{\prime}=u-u_{mean}$, $u_{mean}$ is the mean
streamwise velocity averaged over narrow spanwise extent around the channel
central line, and $u_{rms}$ is the root mean square of velocity fluctuations.
Further on, by pumping energy extracted from the main flow into random streaks
and wall-normal vorticity fluctuations, SR initiates and amplifies them. On
the other hand, in the upper subrange of the transition regime, $E_{w}$
becomes chaotic, and now presence of both $E_{u}$ and $E_{w}$ breaks the
mechanism of SR generation. Moreover, high-energy elastic waves are now able
to amplify the wall-normal vorticity fluctuations and to self-organize streaks
into cycle synchronized at the elastic wave frequency, $f_{el}$ [9].
Therefore, SR is the key factor of the stochastic pathway to the sustained
chaotic flow at the early stage of the flat, white noise $E_{w}$ just above
$Wi_{c}$. However, this finding is based only on the study of viscoelastic
channel flow at the fixed spatial position, $l/h=380$ [1].

Here, we present experimental results obtained in the same inertia-less
viscoelastic channel flow and with the same approach as in Ref. [1], where SR
is discovered and characterized in the lower subrange of the transition
regime, but at the multiple spatial locations $l/h$ along the channel length,
where $l/h$ plays a role of the second controlled parameter besides $Wi$.

The main goal of the experiment is to find out either the upper bound of the
lower subrange for $Wi$ remains constant along the channel or depends on
$l/h$. If such a dependence exists, we are going to establish a role of the
second control parameter, $l/h$, to define the range of SR existence along the
channel at given value of $Wi$.

We conducted the experiment in a long channel of dimensions 500 ($L$, $x$:
streamwise) $\times$ 0.5 ($h$, $y$: wall-normal) $\times$ 3.5 ($W$, $z$:
spanwise) mm3 (see Fig.S1 in [2]), the same as used in Ref. [1]. The channel
has two sources to initiate finite-size perturbations, namely a non-smoothed
inlet and two holes of 0.5 mm diameter in the top plate close to the inlet and
outlet, used for pressure measurements. A dilute aqueous solution of 64%
sucrose concentration and 80 ppm of high molecular weight polyacrylamide
(PAAm) is the same as used in Ref. [1]. The longest polymer relaxation time
($\lambda$) is 13 s. The average velocity $U$, used in both $Wi$ and $Re$, is
obtained via the flow discharge rate, $Q$, as $U=Q/\rho Wh$. In SM Methods, we
present the details of the experimental setup, preparation and
characterization of polymer solution, and Particle Image Velocimetry (PIV)
technique to measure 2D velocity field in $x-z$ horizontal plane.

A possible reason for the dependence of the upper $Wi$ bound of the lower
subrange on $l/h$, where SR is detected, is the fact that $u_{rms}/U$ and the
normalized elastic wave energy, $I_{el}/w^{2}_{rms}$, where $w_{rms}$ is the
root mean square $w$ fluctuations, decrease with $l/h$ at all $Wi$ in the
transition regime. This decline has been first observed in a channel with six
small holes in a top plate, shown in Fig. 6 of Ref. [10], and for the current
channel in Fig. S2 in [2]. Firstly, since both $u_{rms}/U$ and
$I_{el}/w^{2}_{rms}$ persist up to the channel outlet, the non-normal elastic
instability, caused by continuous perturbations at the inlet, is absolute (or
global), as first pointed out in Ref. [10]. It means that a convective (or
local) instability is absent, contrary to Newtonian open flows, where first
convective instability followed by absolute one are explained theoretically
Refs. [11, 12, 13] and observed experimentally, e.g. in Refs. [14, 15, 16, 17,
18]. Moreover, in a contrast to Newtonian open flows, the viscoelastic flow
properties depend on $l/h$ above the absolute elastic instability onset,
$Wi_{c}$, (see Figs. S2(a)(b) in [2]). Furthermore, the $u_{rms}/U$ dependence
on $l/h$ is the same for the lower and upper subranges (see Figs. S2(a) in
[2]), whereas for $I_{el}/w^{2}_{rms}$, the $l/h$ dependence differs in both
subranges (see Fig. S2(b) in [2]). In addition, both $u_{rms}/U$ and
$I_{el}/w^{2}_{rms}$ exhibit two distinct dependencies on $l/h$ in the same
way. For $u_{rms}/U$ in both subranges, close to the inlet a steep decline due
to external perturbations from about $14\%$ down to $\sim 4\%$ at $l/h\approx
90$ and a gentle decrease further away in intrinsic elastic instability region
are observed. Whereas, for $I_{el}/w^{2}_{rms}$, transition locations from
steep to gentle slope takes place for the lower subrange at $\sim 140$ and for
the upper subrange at $l/h\approx 90$, and as the result, the decline
steepness for $I_{el}/w^{2}_{rms}$ differs for the lower and upper subranges
(see Fig. S2 in [2]). As already suggested in Ref. [10], these discrepancies
between the properties of Newtonian and viscoelastic open flows are caused by
the different nature of the nonlinear terms: Only one advection term in
velocity field in the former versus three nonlinear terms, including advection
one, in the elastic stress field in the latter.

In Fig. 1 on the left column, we present three plots of temporal evolution of
$u^{p}_{rms}(t)$, the partially averaged over consecutive 1s (100 samples)
streamwise velocity fluctuations, at three positions, $l/h$: (a) 90 at
$Wi=235$; (c) 180 at $Wi=228$; (e) 960 at $Wi=172$. On the right column, one
finds $u(t)$ power spectra, $E_{u}$, in lin-log presentation at the
corresponding $l/h$ and various $Wi$ values in the lower subrange of the
transition regime: (b) 90 at $Wi=160,235$; (d) 180 at $Wi=164,228$ (f) 960 at
$Wi=155,172,250$. In Fig. 1(a),(c),(e), the $u^{p}_{rms}(t)$ temporal
evolution discloses almost periodic spikes of the partially smoothed data,
whose $E_{u}$ in Fig. 1(b),(d),(f) reveal sharp, high energy peaks associated
with SR.

Figure 1:  Partially averaged over every consecutive 1s (100 samples) stream-
wise velocity fluctuations , $u^{p}_{rms}(t)$, and $u$ energy spectra,
$E_{u}$, at various $l/h$ and $Wi$ values. At $l/h$=90: (a) $u^{p}_{rms}(t)$
at $Wi$=235 and (b) $E_{u}$ at $Wi$= 160 and 235; at $l/h$=180: (c)
$u^{p}_{rms}(t)$ at $Wi$=228 and (d) $E_{u}$ at $Wi$= 164, and 228; and at
$l/h$=960: (e) $u^{p}_{rms}(t)$ at $Wi$=172 and (f) $E_{u}$ at $Wi$= 155, 172,
and 250.  Figure 2: Probability density functions $P(u^{\prime}/u_{rms})$, at
various $l/h$ and $Wi$ values: (a) at $l/h$=150 and $Wi$=62, 150, 167, and
200; and (b) at $l/h$=720 and $Wi$=91, 153, 173, 198, and 260. The dashed
black lines present Gaussian PDF. Figure 3:  Spanwise velocity energy spectra
in log-log scales measured at channel center far from the inlet: $E_{w}$ at
$l/h$=635 and $Wi$=168, 233, 413, 612, and 748.

In Fig. 2, we show PDFs of $P(u^{\prime}/u_{rms})$ at $l/h=150,720$ and
several $Wi$ values in the lower subrange of the transition regime. As in
Ref.[1], strong deviations from Gaussian PDF appear for negative values of
$u\prime/u_{rms}$ for all $Wi$ values in the lower subrange. Remarkably,
further downstream, deviations of the left tail from the Gaussian PDF become
closer to exponential than another shifted Gaussian PDF, as observed at
$l/h=380$ in Fig. 2 in Ref. [1]. The characterization of the deviation of PDF
shape from Gaussian is quantified by its third and fourth moments, called
skewness, $S$, and flatness, $F$, respectively, presented in Fig. 4(c) as
function of $Wi$ values in the lower subrange in a wide range of $l/h$. At
$l/h=380,720,960$, sufficiently far away from the inlet, both $S$ and $F$
exhibit a non-monotonic dependence on $Wi$ with a smoothed maximum at the
middle of the lower subrange, therewith the absolute values of $S$ and $F$
decrease with increasing $l/h$. Meanwhile, at $l/h=150$, $F$ abruptly grows
above $Wi_{c}$ and then drops significantly toward $Wi\approx 200$.

As discussed in the introduction and also presented in Ref. [1], SR is
observed only in the lower subrange at $l/h=380$, where remaining flat
spectrum $E_{w}$ that denotes white noise versus $\lambda f$ in log-log
coordinates exhibits a weak growth with $Wi$. According to the suggested
explanation of the SR existence, based on the internal dynamics of the
autonomous, nonlinear dynamical model [6, 7, 8], the interaction of chaotic
attractor with external white noise in the presence of limit cycle is the
necessary condition for the SR appearance. To verify the dependence of $E_{w}$
on $\lambda f$ at multiple positions, $l/h$, and various $Wi$ in the lower
subrange of the transition regime are presented in log-log coordinates for
$l/h=635$ and $Wi=168,233,413,612,748$ in Fig. 3 and additionally for
$l/h=22,62,90,180,840$ and various $Wi$ values in Fig. S3 [2]. The results,
presented in Fig. 3 and Fig. S3(c),(d),(e) [2] at $l/h$ from 90 up to 840,
confirm the observation reported in Ref. [1] that SR is observed only for
chaotic $E_{u}$ interacting with flat, white noise power spectrum $E_{w}$
incorporating elastic waves. However, close to the inlet at $l/h=22$, $E_{w}$
remarkably disclose chaotic spectra for all $Wi$ values in the whole
transition regime in Fig. S3(a) [2], whereas at $l/h=62$, $E_{w}$ is close to
flat only at $Wi=162$ in Fig. S3(b) [2]. Thus, the SR exists only in the lower
subrange, where the flat, low intensity $E_{w}$ presents, as shown by the
phase diagram in $Wi$ and $l/h$ coordinates (Fig. S4 [2]), the key finding
first reported at $l/h=380$ in Ref. [1]. Moreover, $E_{w}$ remains flat and
grows up to $\sim 10^{-7}(mm^{2}/s^{2})$, close to the resolution limit.

Furthermore, the SR energy peak, $I_{p}$, and normalized SR frequency,
$\lambda f_{p}$, observed exclusively in the lower subrange of the transition
regime at various $Wi$ values and ten locations, $l/h=$ 62, 90, 150, 180, 270,
380, 635, 720, 840, 960, are shown in Fig. 4(a),(b) and also provided in Table
S1 [2]. The SR normalized frequency, $\lambda f_{p}$, grows with $Wi$ at all
locations, and the fit of all data reveals scaling relation $\lambda
f_{p}\sim(Wi-Wi_{c})^{\gamma}$ with $\gamma=0.45\pm 0.5$, about the same as
found at $l/h=380$ in Ref. [1]. The $I_{p}$ exhibits non-monotonic behavior
with $Wi$ for all $l/h$ values that decreases towards $Wi\approx 300$, close
to the upper bound of the lower subrange.

Further on, we are going to compare $\lambda f_{p}$ and $I_{p}$ with $\lambda
f_{el}$ and $I_{el}$, presented in Fig. S5(a)(b) [2] as functions of $Wi$ at
eleven locations $l/h=$22, 62, 90, 150, 180, 270, 380, 635, 720, 840, 960 in
both subranges of the transition flow regime. In the lower subrange, $E_{w}$
remains flat, as discussed above, and $I_{el}$ slowly increases up to the
upper bound at $8\times 10^{-7}(mm^{2}/s^{2})$, marked by red dotted line for
all positions, and $I_{el}$ displays two orders lower energy than $I_{p}$
(compare Fig. 4 and Fig. S5(b) [2]). Only for $l/h=22$ and $l/h=62$, $I_{el}$
values are found above the upper bound at $Wi=162,235$ and $Wi=162$,
respectively, and persist to be larger than those at the rest locations,
though they vary with about the same rate, as at other $l/h$ values. In Fig.
S5(a) [2], we display the growth of $\lambda f_{el}$ with $Wi$ at all
positions along the channel, though at $l/h=22,62,90$, $\lambda f_{el}$
remains lower than the rest in both subranges. The power-law fit of the data
solely in the lower subrange gives $\lambda\cdot{f_{el}}\sim(Wi-150)^{\beta}$
with $\beta=0.4\pm 0.1$ (Fig. S5(a) [2]) that is close to the exponent
$\gamma=0.45\pm 0.5$ obtained for $\lambda f_{p}$. In the upper subrange,
$E_{w}$ becomes chaotic with the power-law decay at high frequencies (see Fig.
3 and Fig. S3(a),(b),(c)[2]), and SR disappears. The latter is accompanied by
a steep growth of $I_{el}$, which reaches or even exceeds the $I_{p}$ values,
displayed in the lower subrange.

Figure 4: Properties of spikes near threshold instability at two sub ranges.
(a) Normalized SR frequency, $\lambda\cdot{f}_{p}$, versus $Wi$ at various
downstream positions, $l/h$. The black dash-dotted line is the power-law fit
$\lambda\cdot{f}_{p}\sim(Wi-Wi_{c})^{\gamma}$ with $\gamma=0.45\pm 0.05$. (b)
SR peak energy, $I_{p}$, obtained from streamwise velocity spectra versus $Wi$
at the same downstream positions, $l/h$. (c) Skewness, $S$, and flatness, $F$,
of $P(u^{\prime}/u_{rms})$ versus $Wi$ at four positions, $l/h$. The dashed
lines in colors are guides to show the trend. The vertical black dashed lines
in three plots mark $Wi_{c}=150$ and $Wi$=300, the approximate upper bound of
the lower subrange.

Summarizing, we observe a strong dependence of the $Wi$ upper bound of the
lower subrange of the SR appearance as a function of $l/h$ close to the inlet,
where $u_{rms}/U$ is up to 4 times larger at $l/h=22$ for $Wi=162$ and 235,
and at $l/h=62$ for $Wi=237$ about twice larger than far away from the inlet.
Such large $u_{rms}/U$ at these values of $l/h$ causes chaotic $E_{w}$ and
much larger $I_{el}$ values even for $Wi$ slightly larger $Wi_{c}$. For such
reasons, both the SR and lower subrange are absent at $l/h=22$, and for
$l/h=62$ the upper bound is located up to $W\leq 237$, lower than $Wi\geq
300$, the upper bound for larger $l/h$ values. It means that the existence
range of SR depends on $u_{rms}$ value, which determines whether $E_{w}$ is
flat, noting white noise spectrum, or chaotic, when the mechanism of SR
excitation is ceased to be effective. To verify the connection between the
elastic waves and SR, we found that the exponents $\beta$ and $\gamma$ in the
scaling relations of $f_{el}$ and $f_{p}$ on $Wi-Wi_{c}$ at all $l/h$ have
close values $0.45\pm 0.5$ and $0.4\pm 0.1$, respectively, inside the error
bars that evidences their direct relation. However, the range of $\lambda
f_{p}$ is significantly shifted down for four-five times compared to the
existence range of $f_{el}$ (Fig. 4(a) versus Fig. S5(a) [2]), that is
contrary to the dynamical model with $f_{p}$ shifted up relatively to
$f_{el}$, and its value is determined by the noise strength [6, 8].

To conclude, SR peaks appear exclusively in the lower subrange of the
transition flow regime just above $Wi_{c}$. The first key factor of SR
existence is the presence of flat $E_{w}$, noting white noise spectra. If
$E_{w}$ becomes chaotic, SR ceases to exist, since the mechanism of SR
generation stops to be effective. The next crucial observation is uncovering
the same scaling dependence of both frequencies, $f_{p}$ and $f_{el}$, on $Wi-
Wi_{c}$ that indicates their direct relation, where elastic waves is the
second key factor for the mechanism SR generation to be active. Therewith,
$f_{p}$ is shifted down 4-5 times relatively to $f_{el}$. Thus the two key
findings are the critical ingredients for comparison with the autonomous
nonlinear dynamical model, where chaotic attractor interacts with external
noise in the presence of limit cycle (oscillations) revealing SR. Of course,
our experimental observations are obtained in much more complicated system
than dynamical one. In the lower subrange above $Wi_{c}$, SR takes a role of
elastic waves, whose energy peak $I_{el}$ is by more than two orders lower
than $I_{p}$. Thus, SR initiates appearance of random streaks and wall-normal
vortex fluctuations by transfer energy from the main flow to velocity
fluctuations. When $E_{w}$ becomes chaotic at some $Wi$ value, $I_{el}$ also
grows significantly to take over the role of the driving factor for
development self-organized streaks cycles synchronized by elastic wave
frequency, $f_{el}$, and to amplify wall-normal vortex fluctuations via the
mechanism described and verified by us earlier [9].

We thank Guy Han and Rostyslav Baron for their help with the experimental
setup. This work was partially supported by the Israel Science Foundation
(ISF, grant #784/19).

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