11institutetext: The Hong Kong University of Science and Technology
(Guangzhou), China 11email<EMAIL_ADDRESS>
22institutetext: The Hong Kong University of Science and Technology, China
33institutetext: University of Cambridge, UK 44institutetext: Imperial College
London, UK 55institutetext: The Hong Kong Polytechnic University, China

# LGRNet: Local-Global Reciprocal Network
for Uterine Fibroid Segmentation
in Ultrasound Videos

Huihui Xu* 11 Yijun Yang* 11 Angelica I Aviles-Rivero 33 Guang Yang 44 Jing
Qin 55 Lei Zhu(✉) 1122

###### Abstract

Regular screening and early discovery of uterine fibroid are crucial for
preventing potential malignant transformations and ensuring timely, life-
saving interventions. To this end, we collect and annotate the first
ultrasound video dataset with 100 videos for uterine fibroid segmentation
(UFUV). We also present Local-Global Reciprocal Network (LGRNet) to
efficiently and effectively propagate the long-term temporal context which is
crucial to help distinguish between uninformative noisy surrounding tissues
and target lesion regions. Specifically, the Cyclic Neighborhood Propagation
(CNP) is introduced to propagate the inter-frame local temporal context in a
cyclic manner. Moreover, to aggregate global temporal context, we first
condense each frame into a set of frame bottleneck queries and devise Hilbert
Selective Scan (HilbertSS) to both efficiently path connect each frame and
preserve the locality bias. A distribute layer is then utilized to disseminate
back the global context for reciprocal refinement. Extensive experiments on
UFUV and three public Video Polyp Segmentation (VPS) datasets demonstrate
consistent improvements compared to state-of-the-art segmentation methods,
indicating the effectiveness and versatility of LGRNet. Code, checkpoints, and
dataset are available at https://github.com/bio-mlhui/LGRNet

###### Keywords:

Uterine Fibroid Segmentation Ultrasound Videos Selective State Space Model
Video Polyp Segmentation

††footnotetext: $*$ Equal contributions.

##  1 Introduction

Uterine fibroids are the most common benign tumors in the female genital
tract, with approximately 70% of women at risk of experiencing such diseases
throughout their lifetime [20]. Consequently, regular screening and early
detection of uterine fibroids are essential for initiating timely life-saving
treatments. Since CT and MRI examinations are expensive and harmful to human
bodies, ultrasound is becoming a more popular imaging modality for clinical
diagnosis. Recently, automatic ultrasound detection and segmentation in videos
have attracted much attention from the medical community [17, 16, 31, 33]. For
example, FLA-Net[17] presents a frequency and location feature aggregation
network for ultrasound video breast lesion segmentation. UltraDet[33] proposes
to aggregate the negative temporal context to facilitate filtering out false
positive predictions in ultrasound video breast lesion detection. However,
automatic uterine fibroid segmentation in ultrasound videos remains
unexplored. Moreover, ultrasound video segmentation is challenged by several
factors including noisy temporal motions, blurry boundaries, and changing
lesion size over time.

In this paper, 1) we collect the first ultrasound video dataset for uterine
fibroid segmentation (UFUV), which contains 100 videos with per-frame expert
annotations. To handle aforementioned challenges in ultrasound segmentation,
2) we present Local-Global Reciprocal Net (LGRNet) to efficiently and
effectively aggregate the global temporal context using a set of frame
bottleneck queries. In LGRNet, 3) we incorporate Cyclic Neighborhood
Propagation (CNP) and Hilbert Selective Scan (HilbertSS) which reciprocally
propagate the crucial local-global temporal context through the bottleneck
queries. 4) We conduct extensive experiments to demonstrate that LGRNet can
both quantitatively and qualitatively outperform state-of-the-art segmentation
methods on UFUV and three publicly available Video Polyp Segmentation
datasets.

##  2 Method

Figure 1: LGRNet architecture. We introduce the notion of frame bottleneck
queries which efficiently condense the global temporal context and distribute
global context back for reciprocal local refinement.

As shown in Fig.1, given a video clip $\\{V^{t}\\}_{t=1}^{T_{\text{clip}}}$,
we first devise a backbone to extract its per-frame multi-scale features
$\\{\\{\mathbf{V}^{t,s}\in\mathcal{R}^{H_{s}W_{s}\times
c}\\}_{s=1}^{S}\\}_{t=1}^{T}$, where $c$ and $H_{s}W_{s}$ are dimension and
resolution of the $s$-th scale. Each scale is transformed into the common
dimension $c$ by a non-biased Conv2D with GroupNorm[29] layer. Then, clip
features of the last three scales are input to Cyclic Neighborhood Propagation
to propagate local inter-frame motion context in a cyclic manner. $CNP$ is
executed for each scale, with all scales sharing the same $CNP$ parameters.
Next, for each frame, the multi-scale features are input to a $Condense$ layer
and compressed into a short sequence of frame bottleneck queries
$\mathbf{L}^{t}$. Bottleneck queries of all frames, i.e.
$\\{\mathbf{L}^{t}\\}_{t=1}^{T_{\text{clip}}}$, are input to Hilbert Selective
Scan to efficiently path connect all frames. Then, for each frame, global-view
queries are input to a $Distribute$ layer to disseminate the global temporal
context back to multi-scale features. Finally, the reciprocally encoded multi-
scale features are input to a Mask2former [5] decoder for
foreground/background classification and mask prediction. Compared with
existing methods which only output single predicted mask, LGRNet can output a
set of different possible mask predictions associated with a foreground lesion
confidence score, which also facilities more comprehensive diagnosis.

Figure 2:  Cyclic Neighborhood Propagation. (Left) Instead of globally token-
mixing features of all frames, we enforce local cyclic inter-frame
dependencies (red). (Middle) In $CNP$, each query token takes the
corresponding nearest neighbors of the cyclic frame as attention keys. (Right)
Detailed implementation of $CNP$.

###  2.1 Local Cyclic Neighborhood Propagation (CNP)

Local CNP. Motion priories, such as optical flow [33], can be utilized as
pixel-wise guidance to propagate inter-frame temporal information. However,
they incorporate additional parameters of pretrained optical flow predictor
[7] and may not generalize to ultrasound videos due to noisy and monochromatic
color change. Recently, motivated by introducing locality inductive biases to
vanilla attention mechanism, Neighborhood Attention (NA) [12] demonstrates
that only involving nearest neighbors as attention keys can achieve comparable
performance in image tasks. In this paper, we interpret inter-frame locality
inductive biases as motion guidance and adapt NA to videos. We propose the
Cyclic Neighborhood Propagation. $CNP$ is executed for each scale and all
scales share the same set of $CNP$ parameters. We omit scale superscript $s$
in this subsection for simplicity. As shown in Fig.2, for a query token
$\mathbf{V}^{t}_{i}$ at $i$-th position of $t$-th frame, $CNP$ aggregates
local motion information from frame $\hat{t}$ to frame $t$ as:

$CNP(\mathbf{V}^{t}_{i},\mathbf{V}^{\hat{t}})=\sum_{m=1}^{M}\mathbf{W}_{m}\sum_{j\in\rho_{\hat{t}\rightarrow
t}^{k,d}(i)}A_{ij}^{\hat{t}\rightarrow
t}\mathbf{W}_{V}\mathbf{V}^{\hat{t}}_{j},$ (1)
$\hat{t}=\begin{cases}t-1,&t>1\\\ T,&t=1\end{cases}$ (2)

where $M$ denotes the number of attention heads, $\rho_{\hat{t}\rightarrow
t}^{k,d}(i)$ is the set of nearest neighbors w.r.t $\mathbf{V}^{t}_{i}$ at
$\hat{t}$-th frame, and the nearest neighbors are defined by a kernel with
size $k$ and dilation $d$. $\\{A_{ij}^{\hat{t}\rightarrow
t}\\}_{j=1}^{|\rho_{\hat{t}\rightarrow t}^{k,d}(i)|}$ denotes the attention
weights, which are the normalized dot product of query with each neighbor key,
i.e.
$\text{softmax}_{j}(\frac{{\mathbf{V}^{t}_{i}}^{T}\mathbf{W}_{Q}^{T}\mathbf{W}_{K}\mathbf{V}_{j}^{\hat{t}}}{\sqrt{c}})$.
Since $CNP$ is applied in each encoder layer, it enables the local inter-frame
temporal information to circulate within the clip. As shown in Fig.2, we do
not build fully connected inter-frame dependencies, since when motion changes
severely and is noisy, dense connections would connect a query point to
uninformative background tokens at distant frames with weak semantics, which
also leads to increased computation.

###  2.2 Global Hilbert Selective Scan (HilbertSS)

Frame Queries as Information Bottleneck. Radiologists often need longer
temporal context [23] to not only decide whether a possible region is lesion
or not, but also refine their local per-frame predictions. Inspired by this
behavior, we devise a set of learnable frame bottleneck queries
$\mathbf{L}\in\mathcal{R}^{{\bar{N}\times c}}$ to first summarize each frame
into a query sequence:

$\mathbf{L}^{t}=Condense(\text{query}=\mathbf{L},\text{key}=\\{\mathbf{V}^{t,s}\\}_{s=1}^{S}),t=1,...,T_{\text{clip}},$
(3)

where the Condense layer is implemented as a Cross Attention Layer [25] with
Residual connection [13]. In the cross attention layer, the 2D multi-scale
features are flattened and concatenated. Specifically, the query length is
$\bar{N}$ and the key length equals to the sum of multi-scale shapes, i.e.
$\sum_{s=2}^{S}H_{s}\times W_{s}$. Frame queries can be seen as bottlenecks
extracting semantically rich lesion information and filtering out irrelevant
and redundant features of each frame, which facilitates later efficient and
effective global temporal information exchange.

Preliminary of Selective State Space Model.

Figure 3: Illustrations of $HilbertSS$. (Left) Comparison between (a) Zigzag
Flatten (b) Hilbert Flatten. (Right) Detailed implementation of global part of
LGRNet. Figure 4: Implementation of S6 block. (Left) Precomputation of input-
dependent weights. (Right) Scanning for iterative hidden state transformation.

Recently, Selective State Space Model (Mamba) [11] was proposed as a new
sequence model with linear-time complexity, which even outperforms
Transformers[25] on some language tasks.

Mamba is built based on the Selective Scan Block (S6). As shown in Fig.4, each
S6 block transforms the input sequence
$\\{\mathbf{x}^{s}\in\mathcal{R}^{c}\\}_{s=1}^{S}$ to
$\\{\mathbf{y}^{s}\in\mathcal{R}^{c}\\}_{s=1}^{S}$ through the precomputation
stage and the scanning stage. During precomputation, a set of linear layers
are used to precompute the input-dependent dynamic weights $\mathbf{\Delta
B},\mathbf{\Delta A}\in\mathcal{R}^{S\times c_{\text{state}}\times c}$,
$\mathbf{C}\in\mathcal{R}^{S\times c_{\text{state}}}$, where
$c_{\text{state}}$ denotes the dimension of the hidden state. The
precomputation stage can be formalized as:

$\displaystyle\mathbf{x}$
$\displaystyle=\text{SiLU}(\text{Conv1D}(\mathbf{x}))$ (4)
$\displaystyle\mathbf{C}$
$\displaystyle=\text{Linear}_{c_{\text{state}}}(\mathbf{x})\in\mathcal{R}^{S\times
c_{\text{state}}}$ $\displaystyle\mathbf{\Delta}$
$\displaystyle=\text{Softplus}(\text{Linear}_{c}(\text{Linear}_{c_{\text{rank}}}(\mathbf{x}))\in\mathcal{R}^{S\times
c}$ $\displaystyle\mathbf{\Delta A}$
$\displaystyle=\text{exp}\\{\otimes_{\text{o}}(\mathbf{\Delta},\mathbf{A})\\}\in\mathcal{R}^{S\times
c_{\text{state}}\times c}$ $\displaystyle\mathbf{\Delta B}$
$\displaystyle=\otimes_{\text{o}}((\mathbf{x}\odot\mathbf{\Delta}),\text{Linear}_{c_{\text{state}}}(\mathbf{x}))\in\mathcal{R}^{S\times
c_{\text{state}}\times c},$

where $\mathbf{A}$ denotes a learnable parameter of shape
$c_{\text{state}}\times c$, $c_{\text{rank}}$ denotes the dimension of low-
rank projection [11], $\text{Linear}_{*}$ denotes a linear layer transforming
input to dimension $*$, SiLU denotes the Sigmoid Linear Unit activation,
Conv1D denotes the depth-wise 1D convolution, $\otimes_{\text{o}}$ denotes the
outer product, and Softplus denotes the softplus activation. To some extent,
the pre-computation stage in S6 corresponds to weights computation (normalized
inner product of input queries and input keys) in attention [25]. Both two
procedures in S6 and attention generate input-dependent, i.e. dynamic, weights
for subsequent information aggregation. This characteristic enables S6 to
attend to the in-context information.

During scanning, the hidden state is zero-initialized, i.e.
$\mathbf{h}^{0}=\mathbf{0}\in\mathcal{R}^{c_{\text{state}}\times c}$.
Formally, each step of hidden state transformation can be formalized as:

$\displaystyle\mathbf{h}^{s}$
$\displaystyle=\mathbf{h}^{s-1}\odot\mathbf{\Delta A}^{s}\oplus\mathbf{\Delta
B}^{s}\in\mathcal{R}^{c_{\text{state}}\times c}$ (5)
$\displaystyle\mathbf{y}^{s}$
$\displaystyle=\otimes(\mathbf{C}^{s},\mathbf{h}^{s})\in\mathcal{R}^{c},$

where $\otimes$ denotes the inner product, $\oplus$ denotes the element-wise
sum, $\odot$ denotes the element-wise product. Although the scanning stage is
recurrent, the transition from $\mathbf{h}^{s-1}$ to $\mathbf{h}^{s}$ is
associative. As shown in [3][22], the scanning process can thus be
parallelized to logarithmic complexity w.r.t the sequence length.

Global HilbertSS. In LGRNet, we aim to token-mix the aggregated frame
bottleneck queries
$\\{\mathbf{L}^{t}\\}_{t=1}^{T_{\text{clip}}}\in\mathcal{R}^{\bar{N}\times
T_{\text{clip}}\times c}$ to efficiently propagate the global clip-level
temporal information. To apply S6 to 2D input, a direct approach is flattening
input into a sequence using Zigzag curve as in attention-based models.
However, Eq.5 implies the scanning stage is position-aware, in the sense that
different scanning orders would generate different sets of hidden states.
Moreover, the contextual information is propagated along the sequence only
through the hidden state $\mathbf{h}^{s}$. Although the selection mechanism
may enable boundary resetting [11] and context filtering [11] on some language
tasks, we believe locality is more important to vision tasks, in the sense
that for a local structured visual region, the hidden state should aggregate
them in a group, instead of corrupting their neighboring structure in the
flattened 1D sequence.

As an intuitive example shown in Fig.3, Zigzag scanning may corrupt the
original neighborhood structure, in the sense that two tokens which are
neighboring to each other on original 2D layout would be distant to each other
on the flattened 1D sequence. Formally, any flattening curve can be
generalized to a continuous Space Filling Curve (SFC) $\sigma$, which maps
each point $x\in$ [0, 1] to $\sigma(x)\in$ [0,1]$\times$ [0,1]. We also denote
$n$ as the curve order of the Hilbert curve, which, in our discrete case,
approximates to height and width of the filled grid. For any two points $x,y$
in [0,1], their Space to Linear Ratio (SLR) is defined as:

$\displaystyle\frac{|\sigma(x)-\sigma(y)|^{2}}{|x-y|}.$ (6)

The Dilation Factor (DF) of a SFC is defined as the upper bound of the SLR.
For same two points in [0, 1], if a SFC has lower DF, their mappings will also
be closer in the 2D grid, which accords with locality preserving intuition in
the scanning stage. As proved in [1][4], the DF of Hilbert curve is 6, while
the Zigzag curve is $4^{n}-2^{n+1}+2$, which diverges to $\infty$ as the curve
order $n$ increases. This shows that Hilbert curve could preserve the 2D
locality structure, which accords with our intuition that lesions spatial-
temporally close to each other should be scanned in groups and tracked
together.

To execute Hilbert scan on a 2D $\bar{N}\times T_{\text{clip}}$ grid, we use
the implementation of [34] to generate the pseudo Hilbert curve. In all, the
HilbertSS module can be formalized as:

$\\{\mathbf{L}^{t}\\}_{t=1}^{T_{\text{clip}}}=HilbertUnFlatten(S6(HilbertFlatten(\\{\mathbf{L}^{t}\\}_{t=1}^{T_{\text{clip}}}))),$
(7)

where $HilbertFlatten$ means we use the torch.index_select to flatten 2D input
into a 1D sequence using the generated Hilbert indices. $HilbertUnFlatten$
means we use torch.Tensor.scatter_add_ to add the transformed features back to
the original input according to the same Hilbert indices.

It should also be noted that existing S6-based vision models, including
Vim[37], Vmamba[18], Segmamba[30], and Vivim[32], scan the multi-scale feature
maps, which include uninformative background tokens with weak semantics and
thus may affect the recurrent hidden state transformation in Eq.5. LGRNet
devises a set of queries to filter out noisy information and extract the
semantically rich lesion information, which would help the scanning process to
effectively attend to more informative lesion regions.

Reciprocal Local-Global Refinement. Radiologists may use global view to refine
their per-frame diagnosis. We use a $Distribute$ layer to disseminate the
global temporal context back to the multi-scale features for local refinement:

$\\{\mathbf{V}^{t,s}\\}_{s=1}^{S}=Distribute(\text{query}=\\{\mathbf{V}^{t,s}\\}_{s=1}^{S},\text{key}=\mathbf{L}^{t}),t=1,...,T_{\text{clip}}.$
(8)

Same to the $Condense$ layer, the Distribute layer is also implemented as a
Cross Attention Layer [25] with Residual connection [13], where the query
length is $\sum_{s=2}^{S}H_{s}\times W_{s}$ and the key length is $\bar{N}$.

###  2.3 Decoder

Our decoder uses the same architecture with Mask2Former[5]. A set of learnable
temporal queries $\hat{\mathbf{L}}\in\mathcal{R}^{\hat{N}\times c}$ are used
to cross-attend to different scale features at each cross attention layer,
where the query length is $\hat{N}$ and the key length is
$T_{\text{clip}}\times H_{s}\times W_{s}$. The final mask is the dynamic
convolution between the stride 4 scales ($s$=1) and the temporal queries. The
bipartite matching loss is composed of classification (foreground/background)
cross entropy loss, mask dice loss and binary cross entropy loss:
$\lambda_{class}L_{class}+\lambda_{dice}L_{dice}+\lambda_{ce}L_{ce}$. During
inference, the foreground classification probability can be interpreted as the
lesion confidence of the corresponding predicted mask region. LGRNet can
generate multiple, i.e. $\hat{N}$, mask predictions, each with a lesion
confidence score. We choose the mask with the highest foreground probability
as final output to compare with other methods in our experiments.

##  3 Experiments

Dataset. We collect and annotate the first ultrasound video uterine fibroid
segmentation dataset (UFUV). Our UFUV dataset contains 100 videos and each
video has 50 frames. The ultrasound videos were collected using Mindray Resona
8 and Supersonic Alxplorer. The dataset encompasses a cohort of female
subjects aged between 20 to 45 years. We chose video that showcases at least
one clearly delineated hypoechoic region (indicative of a fibroid) within the
uterine wall, with a diameter exceeding 1 cm. The annotation process was
rigorously conducted by two experienced gynecological ultrasound
diagnosticians with over five years of professional experience. To ensure the
accuracy and reliability of the annotations, the collected data underwent a
cross-annotation procedure between the two diagnosticians. We randomly select
83 videos for training, and the remaining 17 videos are utilized for testing.

Compared Methods.We compare LGRNet against 9 recent state-of-the-art
segmentation methods, including five image-based methods and four video-based
methods. These image-based methods are UNet++[36], PraNet[9], LDNet[35],
WeakPolyp[26], and BUSSeg[28], while video-based methods are PNS-Net[14],
DPSTT[16], FLA-Net[17], and MS-TFAL[6]. For each compared method, we utilize
the hyperparameters settings from the original paper or their official codes
for fair comparisons.

Evaluation Metrics. For quantitative comparison, we utilize five common
metrics, including Dice Coefficient (Dice), Intersect of Union (IoU),
Sensitivity, Mean Absolute Error (MAE), and S-Measure (structural similarity
[8]). We also compute the inference Multiply-Accumulate Counts (MACs, GFLOPS)
and the number of parameters (Params) for efficiency comparisons.

Implementation Details. We set $T_{clip}=6$ in our experiments. Each frame is
resize to $352\times 352$. The training augmentation contains the horizontal
flip, the vertical flip, and the perspective transform with magnitude 0.12. We
use AdamW [19] and set initial learning rate as 1e-3 with a backbone
multiplier of 0.1. The multistep scheduler the learning rate by 0.5 every 3
epochs. Gradient clipping with square norm value 1e-2 is used. We use a point
sampling [5] with an oversampling ratio of 3.0 and importance of 0.76 for the
bipartite matching mask loss computation. Both the number of encoder and
decoder layers are set to 3. We use Res2Net-50[10] as backbone, and
empirically set $\lambda_{class}=2$, $\lambda_{dice}=5$, $\lambda_{ce}=2$,
$c=64$, $k=5$, $d=2$, $\bar{N}=20$, $\mathcal{\mu}=3$, and $\hat{N}=10$. More
ablation studies on hyperparameters are demonstrated in Sec. 3.2.

Table 1: Quantitative comparisons on our UFUV dataset.

Method | Publication | Type |  Dice$\uparrow$ |  IoU$\uparrow$ |  Sensitivity $\uparrow$ |  S-Measure$\uparrow$ |  MAE $\downarrow$ |  GFLOPs$\downarrow$ |  Params$\downarrow$
---|---|---|---|---|---|---|---|---|---
UNet++[36] | TMI’19 | image | 0.681 | 0.540 | 0.611 | 0.728 | 0.081 | 22.6 G | 29.8 M
PraNet[9] | MICCAI’20 | image | 0.724 | 0.583 | 0.709 | 0.751 | 0.076 | 20.3 G | 16.1 M
LDNet[35] | MICCAI’22 | image | 0.738 | 0.588 | 0.707 | 0.753 | 0.068 | 12.6 G | 15.8 M
WeakPolyp[26] | MICCAI’23 | image | 0.725 | 0.579 | 0.682 | 0.747 | 0.075 | 5.2 G | 25.8 M
BUSSeg[28] | TMI’23 | image | 0.740 | 0.612 | 0.711 | 0.770 | 0.066 | 23.8 G | 28.6 M
PNS-Net[14] | MICCAI’21 | video | 0.735 | 0.601 | 0.685 | 0.750 | 0.065 | 19.5 G | 15.7 M
DPSTT[16] | MICCAI’22 | video | 0.738 | 0.609 | 0.707 | 0.769 | 0.065 | 24.8 G | 30.2 M
FLA-Net[17] | MICCAI’23 | video | 0.741 | 0.615 | 0.710 | 0.773 | 0.066 | 18.4 G | 87.6 M
MS-TFAL[6] | MICCAI’23 | video | 0.748 | 0.625 | 0.714 | 0.781 | 0.063 | 12.2 G | 24.6 M
Ours | – – | video | 0.775 | 0.658 | 0.776 | 0.793 | 0.060 | 13.2 G | 26.6 M

###  3.1 Comparisons with SOTA methods

Figure 5: Visual comparisons on UFUV of our network and compared SOTA methods.
Table 2: Quantitative comparisons CVC-612[2] and CVC-300[24] for video polyp
segmentation.

|  |  UNet++[36] |  PraNet[9] |  PNS-Net [14] |  LDNet[35] |  FLA-Net[17] |  MS-TFAL[6] | Ours
---|---|---|---|---|---|---|---|---
| Metrics | TMI’19 | MICCAI’20 | MICCAI’21 | MICCAI’22 | MICCAI’23 | MICCAI’23 | –
CVC-612-V |  maxDice$\uparrow$ | 0.684 | 0.869 | 0.873 | 0.870 | 0.885 | 0.911 | 0.933
maxIoU$\uparrow$ | 0.570 | 0.799 | 0.800 | 0.799 | 0.814 | 0.846 | 0.877
$S_{\alpha}\uparrow$ | 0.805 | 0.915 | 0.923 | 0.918 | 0.920 | 0.961 | 0.947
maxSpe$\uparrow$ | 0.952 | 0.983 | 0.991 | 0.987 | 0.992 | 0.994 | 0.995
$E_{\phi}\uparrow$ | 0.830 | 0.936 | 0.944 | 0.941 | 0.963 | 0.971 | 0.977
$MAE\downarrow$ | 0.025 | 0.013 | 0.012 | 0.013 | 0.012 | 0.010 | 0.007
CVC-300-TV |  maxDice$\uparrow$ | 0.649 | 0.739 | 0.840 | 0.835 | 0.874 | 0.891 | 0.916
maxIoU$\uparrow$ | 0.539 | 0.645 | 0.745 | 0.741 | 0.789 | 0.810 | 0.852
$S_{\alpha}\uparrow$ | 0.796 | 0.833 | 0.909 | 0.898 | 0.907 | 0.912 | 0.937
maxSpe$\uparrow$ | 0.944 | 0.993 | 0.996 | 0.994 | 0.996 | 0.997 | 0.997
$E_{\phi}\uparrow$ | 0.831 | 0.852 | 0.921 | 0.910 | 0.969 | 0.974 | 0.986
$MAE\downarrow$ | 0.024 | 0.016 | 0.013 | 0.015 | 0.010 | 0.007 | 0.005
CVC-612-T |  maxDice$\uparrow$ | 0.740 | 0.852 | 0.860 | 0.857 | 0.861 | 0.864 | 0.875
maxIoU$\uparrow$ | 0.635 | 0.786 | 0.795 | 0.791 | 0.795 | 0.796 | 0.814
$S_{\alpha}\uparrow$ | 0.800 | 0.886 | 0.903 | 0.892 | 0.904 | 0.906 | 0.907
maxSpe$\uparrow$ | 0.975 | 0.986 | 0.992 | 0.988 | 0.993 | 0.995 | 0.998
$E_{\phi}\uparrow$ | 0.817 | 0.904 | 0.903 | 0.903 | 0.904 | 0.910 | 0.915
$MAE\downarrow$ | 0.059 | 0.038 | 0.038 | 0.037 | 0.036 | 0.038 | 0.035

Our UFUV dataset. As shown in Table1, although our network does not have the
smallest inference time and the smallest number of parameters, our method
achieves superior performance over all compared SOTA image/video segmentation
methods on all segmentation metrics. Moreover, our method also achieves
significant efficiency improvement compared with most methods. Besides, Fig.5
compares the visual results produced by different methods. Our network can
more accurately segment uterine fibroid than SOTA methods, and our
segmentation results are most consistent with the ground truth. More visual
comparison results are presented in the supplementary material.

Video Polyp Segmentation (VPS) benchmark datasets. To further demonstrate the
effectiveness of our method, we compare our network against SOTA methods on
three public Video Polyp Segmentation (VPS) benchmark datasets, which are
CVC-612[2], CVC-300[24], and SUN-SEG[15]. For CVC-612[2] and CVC-300[24], we
follow PNS-Net[14] to use the same training setting and test datasets. For
SUN-SEG[15], we follow WeakPolyp [26] to combine the "Hard (Easy) Seen" and
"Hard (Easy) Unseen" split into "Hard (Easy) Testing". As shown in Table2 & 3,
our network also achieves better metric performance than all compared methods
on all three benchmark datasets, which indicates that our network has the best
video polyp segmentation performance.

Table 3: Quantitative comparisons on SUN-SEG[15] for video polyp segmentation.

Model | Publication | Backbone | Easy Testing | Hard Testing
---|---|---|---|---
Dice | IoU | Dice | IoU
PraNet [9] | MICCAI’20 | Res2Net-50 | 0.689 | 0.608 | 0.660 | 0.569
2/3D [21] | MICCAI’20 | ResNet-101 | 0.755 | 0.668 | 0.737 | 0.643
SANet [27] | MICCAI’21 | Res2Net-50 | 0.693 | 0.595 | 0.640 | 0.543
PNS+ [15] | MIR’22 | Res2Net-50 | 0.787 | 0.704 | 0.770 | 0.679
DPSTT[16] | MICCAI’22 | Res2Net-50 | 0.804 | 0.725 | 0.794 | 0.709
WeakPolyp[26] | MICCAI’23 | Res2Net-50 | 0.792 | 0.715 | 0.807 | 0.727
PVTv2-B2 | 0.853 | 0.781 | 0.854 | 0.777
FLA-Net[17] | MICCAI’23 | Res2Net-50 | 0.805 | 0.723 | 0.811 | 0.730
PVTv2-B2 | 0.856 | 0.784 | 0.858 | 0.781
MS-TFAL[6] | MICCAI’23 | Res2Net-50 | 0.822 | 0.742 | 0.826 | 0.751
PVTv2-B2 | 0.859 | 0.792 | 0.862 | 0.788
Ours | – – | Res2Net-50 | 0.843 | 0.765 | 0.843 | 0.774
Ours | – – | PVTv2-B2 | 0.875 | 0.810 | 0.876 | 0.805

Table 4: Component Analysis.

$CNP$ | $HilbertSS$ | Dice$\uparrow$ | IoU$\uparrow$ | S-Measure$\uparrow$ | MAE $\downarrow$
---|---|---|---|---|---
|  | 0.722 | 0.581 | 0.750 | 0.074
|  | 0.753 | 0.633 | 0.784 | 0.062
|  | 0.747 | 0.626 | 0.776 | 0.067
|  | 0.775 | 0.658 | 0.793 | 0.060

Table 5: Hyperparameter Ablations

Component | Version | Dice$\uparrow$ | IoU$\uparrow$ | S-Measure$\uparrow$ | MAE $\downarrow$
---|---|---|---|---|---
$CNP$ | k=3, d=1 | 0.768 | 0.652 | 0.786 | 0.062
k=3, d=2 | 0.771 | 0.656 | 0.789 | 0.061
k=5, d=2 | 0.775 | 0.658 | 0.793 | 0.060
k=7, d=2 | 0.766 | 0.647 | 0.784 | 0.064
$HilbertSS$ | Zigzag Scan | 0.761 | 0.639 | 0.788 | 0.060
Hilbert Scan | 0.775 | 0.658 | 0.793 | 0.060
$\bar{N}=10$ | 0.764 | 0.643 | 0.786 | 0.062
$\bar{N}=20$ | 0.775 | 0.658 | 0.793 | 0.060
$\bar{N}=30$ | 0.771 | 0.657 | 0.792 | 0.060

###  3.2 Ablation Study

We conduct ablation analysis on $CNP$ and $HilbertSS$ by removing them or
ablating their hyperparameters. As shown in Tab.5, removing either component
leads to a performance drop. Removing both components causes the model to
ignore the vital temporal context information. Moreover, the global HilbertSS
(0.722$\rightarrow$0.753) achieves more improvement than local $CNP$
(0.722$\rightarrow$0.747), which validates the design of frame bottleneck
queries and reciprocal local-global learning. For the hyperparameter ablation,
we set different kernel size $k$ and dilation $d$ for $CNP$, different
selective scan strategy, and number of frame bottleneck queries $\bar{N}$ for
$HilbertSS$. As shown in Fig.5, for $CNP$, both bigger kernel size and larger
dilation may lead to improvement, but when much bigger kernel size is used
(k=7 compared with k=5), the performance saturates. For $HilbertSS$, using
Zigzag scan leads to lower performance. Moreover, using more bottleneck
queries increases performance, but the performance also saturates after some
threshold.

##  4 Conclusion

This paper collects and annotates the first ultrasound video uterine fibroid
segmentation (UFUV) dataset, which contains 100 videos with 5,000 annotated
video frames. We further propose the Local-Global Reciprocal Net (LGRNet) to
efficiently aggregate global temporal context information for ultrasound video
segmentation. The $Condense$ and $Distribute$ layers with our proposed frame
bottleneck queries bridge the local $CNP$ and global $HilbertSS$, and
facilitate reciprocally propagating the crucial local-global temporal context
information. Experimental results on UFUV dataset and other three public Video
Polyp Segmentation (VPS) datasets show that LGRNet quantitatively and
qualitatively outperforms existing state-of-the-art image and video
segmentation methods.

####  4.0.1 Acknowledgments

This work is supported by the Guangzhou-HKUST(GZ) Joint Funding Program (No.
2023A03J0671), the Guangzhou Municipal Science and Technology Project (Grant
No. 2023A03J0671), and the InnoHK funding launched by Innovation and
Technology Commission, Hong Kong SAR.

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††footnotetext: Email addresses<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Almost everywhere convergence of Fourier series on compact connected Lie
groups

David Grow and Donnie Myers  Missouri University of Science and Technology
Department of Mathematics and Statistics
Rolla, Missouri 65409-0020, USA.

###### Abstract.

We consider the open problem: Does every square-integrable function $f$ on a
compact, connected Lie group $G$ have an almost everywhere convergent Fourier
series? We prove a general theorem from which it follows that if the integral
modulus of continuity of $f$ is $O(t^{\alpha})$ for some $\alpha>0$ then the
Fourier series of $f$ converges almost everywhere on $G$. In particular, the
Fourier series of any $\alpha$-Hölder continuous function on $G$ converges
almost everywhere. On the other hand, we show that to each countable subset
$E$ of $G=SU(2)$ and each $0<\alpha<1$ there corresponds an $\alpha$-Hölder
continuous function on $SU(2)$ whose Fourier series diverges on $E$.

###### Key words and phrases:

Fourier series, compact Lie groups

###### 2010 Mathematics Subject Classification:

22E30, 43A50

##  1\. Introduction

The Peter-Weyl theorem suggests the study of the formal Fourier series
${\sum d_{\lambda}(\chi_{\lambda}\ast f)}$ of a square-integrable function $f$
on a compact, connected Lie group $G$. Here the sum is over the equivalence
classes of continuous irreducible unitary representations of $G$,
$d_{\lambda}$ is the degree of the representation, and $\chi_{\lambda}$ is its
character. The vast literature of Fourier analysis on $G$ is primarily
concerned with mean convergence or divergence of the Fourier series of $f\in
L^{p}(G)$ (e.g.[12, 15, 19, 27, 28]), uniform or absolute convergence of the
partial sums if $f$ is smooth (e.g.[7, 15, 17, 21, 22, 24, 25, 29, 30]),
almost everywhere convergence or divergence of the partial sums if $f$ is a
central function in $L^{p}(G)$ (e.g.[9, 13, 20]), and uniform, mean, or almost
everywhere summability of the partial sums if $f$ belongs to various subspaces
of $L^{1}(G)$ (e.g.[5, 6, 8, 15, 32]). The aim of this work is to advance the
study of almost everywhere convergence or divergence of Fourier partial sums
of nonsmooth, possibly noncentral functions in $L^{2}(G)$.

Relying on Jackson’s theorem for compact, connected Lie groups [4] and a
general version of the Rademacher-Menshov theorem [18], we show that if $f$ in
$L^{2}(G)$ has an integral modulus of continuity $\Omega(f,\cdot)$ satisfying

$\int_{0}^{1}\frac{\Omega^{2}(f,t)}{t}dt<\infty,$

then the sequences of polyhedral Fourier partial sums
$\\{S_{N}f(x)\\}_{N=1}^{\infty}$ and spherical Fourier partial sums
$\\{\widetilde{S}_{N}f(x)\\}_{N=1}^{\infty}$ converge to $f(x)$ almost
everywhere on $G$. In particular, if $f$ is an $\alpha$-Hölder continuous
function on $G$ for some $\alpha>0$, or more generally if
$\Omega(f,t)=O(t^{\alpha})$ for some $\alpha>0$, then these Fourier partial
sums of $f$ converge to $f(x)$ almost everywhere on $G$.

On the other hand, consider the two-dimensional special unitary group
$G=SU(2)$. We show that to each $\alpha$ in $(0,1)$ and each countable subset
$E$ of $SU(2)$ there corresponds an $\alpha$-Hölder continuous function on
$SU(2)$ whose Fourier partial sums diverge at each $x$ in $E$. Since it is
possible to arrange that such a set $E$ is dense in $SU(2)$, the Fourier
partial sums of the corresponding function are divergent at infinitely many
points in every nonempty open subset of $SU(2)$, despite the fact that the
Fourier partial sums of such a function converge almost everywhere on $SU(2)$.
It is worth noting in the case $\alpha=1$, i.e. when $f$ is Lipschitz
continuous on $SU(2)$, the Fourier partial sums of $f$ converge uniformly to
$f$ on $SU(2)$ [21, 22].

It is an open problem whether

(1.1) $\underset{N\rightarrow\infty}{\lim}S_{N}f(x)=f(x)$

holds almost everywhere for every $f$ in $L^{2}(G)$. A general theorem of
Stanton and Tomas [28] for compact, connected, semisimple Lie groups $G$ shows
that if $p<2$ then there correspond an $f$ in $L^{p}(G)$ and a subset $E$ of
$G$ of full measure such that (1.1) fails for all $x$ in $E$. However,
Carleson’s celebrated proof of Lusin’s conjecture [3] guarantees (1.1) holds
almost everywhere when $f$ is square-integrable on the circle group
$\mathbb{T}$, and this was extended to $\mathbb{T}^{n}$ for $n\geq 2$ [11, 26,
31]. Furthermore, it follows from a result of Pollard [23] on Jacobi series
that if $f$ is a central function in $L^{p}(SU(2))$ for some $p>4/3$, then
(1.1) holds almost everywhere. Finally, the Peter-Weyl theorem implies that to
every compact, connected Lie group $G$ and each $f$ in $L^{2}(G)$ there
corresponds an increasing sequence $\\{N_{j}\\}$ of positive integers such
that

$\underset{j\rightarrow\infty}{\lim}S_{N_{j}}f(x)=f(x)$

for almost every $x$ in $G$. This lends some hope for a positive answer to the
almost everywhere convergence problem (1.1), as do the results in this paper.

##  2\. Notation and Preliminaries

We use primarily the notation of [2]. Let $G$ be a compact, connected Lie
group with Haar measure $\mu$, let $T$ be a fixed maximal torus in $G$, and
let $\mathfrak{g}$ and $\mathfrak{t}$ be the Lie algebras of $G$ and $T$,
respectively. Choose an inner product $\langle\cdot,\cdot\rangle$ on
$\mathfrak{g}$ which is invariant under the adjoint action of $G$ on
$\mathfrak{g}$. (When $G$ is semisimple we may choose
$\langle\cdot,\cdot\rangle$ to be $-B(\cdot,\cdot)$ where $B$ is the Killing
form.) This provides inner products on $\mathfrak{t}$ and $\mathfrak{t}^{*}$
which are invariant under the Weyl group $W=N(T)/T$. Let
$I=\\{H\in\mathfrak{t}:\exp(H)=1\\}$ be the integral lattice, let
$I^{*}=\\{\lambda\in\mathfrak{t}^{*}:\lambda(H)\in\mathbb{Z}\text{ for all
}H\in I\\}$ be the lattice of integral forms, and let $P\subset I^{*}$ be the
set of real roots of $G$ with basis $\\{\alpha_{1},...,\alpha_{l}\\}$, $P_{+}$
the set of positive roots, and $C\subset\mathfrak{t}^{*}$ the corresponding
Weyl chamber. The dual object $\widehat{G}$ of $G$ is in one-to-one
correspondence with $\overline{C}\cap I^{*}$. (When $G$ is semisimple and
simply connected, there exist integral forms
$\rho_{1},...,\rho_{l}\in\mathfrak{t}^{*}$ satisfying
$2\langle\rho_{j},\alpha_{k}\rangle/\langle\alpha_{k},\alpha_{k}\rangle=\delta_{jk}$
for $j,k\in\\{1,...,l\\}$ and $\overline{C}\cap
I^{*}=\\{n_{1}\rho_{1}+...+n_{l}\rho_{l}:\text{ each }n_{j}\in\mathbb{Z}\text{
and }n_{j}\geq 0\\}$.)

Fix $\omega\in C\cap I^{*}$ and for $N=1,2,3,...$ set
$\Lambda_{N}=\\{\lambda\in\overline{C}\cap I^{*}:\lambda\leq N\omega\\}$. The
associated polyhedral Dirichlet kernel on $G$ is
$\boldsymbol{D}_{N}=\sum_{\lambda\in\Lambda_{N}}d_{\lambda}\chi_{\lambda}$
where $N=1,2,3,...$, and the convolution products
$S_{N}f=\boldsymbol{D}_{N}\ast f$ define the sequence of associated polyhedral
partial sums for the Fourier series of $f\in L^{2}(G)$.

Using the Weyl integration formula on $G$ [2], the $N$th Fourier polyhedral
partial sum of $f$ at $x\in G$ is given by

$\displaystyle S_{N}$ $\displaystyle f(x)=(\boldsymbol{D}_{N}\ast f)(x)$
$\displaystyle=\intop_{G}\boldsymbol{D}_{N}(y)f(y^{-1}x)d\mu(y)$
$\displaystyle=\frac{1}{|W|}\intop_{T}\det(E_{G/T}-\textrm{Ad}_{G/T}(\theta^{-1}))\intop_{G}\boldsymbol{D}_{N}(y{\theta}y^{-1})f(y\theta^{-1}y^{-1}x)d\mu(y)d\theta$
$\displaystyle=\frac{1}{|W|}\intop_{T}\eta(\theta)\boldsymbol{D}_{N}(\theta)[Q_{x}f](\theta)d\theta$

where $\eta(\theta)=\det(E_{G/T}-\textrm{Ad}_{G/T}(\theta^{-1}))$ and
$[Q_{x}f](\theta)=\intop_{G}f(y\theta^{-1}y^{-1}x)d\mu(y)$.

Let $\pi(\lambda)$ denote the continuous, irreducible, unitary representation
of $G$ corresponding to $\lambda\in\overline{C}\cap I^{*}$, let
$H_{\pi(\lambda)}$ be the finite dimensional subspace of $L^{2}(G)$ generated
by the coordinate functions of $\pi(\lambda)$ [16, p.24], and let
$P_{\lambda}f=d_{\lambda}(\chi_{\lambda}\ast f)$ be the orthogonal projection
of $L^{2}(G)$ onto $H_{\pi(\lambda)}$. Then

$S_{N}f(x)=\sum_{\lambda\in\Lambda_{N}}(P_{\lambda}f)(x)=\sum_{j=0}^{N}(\Gamma_{j}f)(x)$

with $\Gamma_{n}f=\sum P_{\lambda}f$ where the sum is over all
$\lambda\in\overline{C}\cap I^{*}$ such that $(n-1)\omega<\lambda\leq
n\omega$. Let $\delta_{h}f(x)=f(x)-f(h^{-1}x)$ be the difference operator on
$L^{2}(G)$ and let the integral modulus of continuity of $f\in L^{2}(G)$ be
given by

$\Omega(f,t)=\sup\\{\|\delta_{h}f\|_{L^{2}(G)}:h=\exp(X),X\in\mathfrak{g},\textrm{
and }0<\|X\|\leq t\\};$

here $\|\cdot\|$ denotes the norm on $\mathfrak{g}$ generated by the inner
product $\langle\cdot,\cdot\rangle$.

An alternate theory for almost everywhere convergence of Fourier series of
$f\in L^{2}(G)$ can be based on spherical partial sums (cf. [4] and [14]). Let

$\rho=\frac{1}{2}\sum_{\alpha\in P_{+}}\alpha$

and for $N=1,2,3,...$ define
$\widetilde{\Lambda}_{N}=\\{\lambda\in\overline{C}\cap
I^{*}:\|\lambda-\rho\|\leq N\\}$, the spherical Dirichlet kernel
$\widetilde{\boldsymbol{D}}_{N}=\sum_{\lambda\in\widetilde{\Lambda}_{N}}d_{\lambda}\chi_{\lambda}$,
and the spherical partial sums

$\widetilde{S}_{N}f=\widetilde{\boldsymbol{D}}_{N}\ast
f=\sum_{\lambda\in\widetilde{\Lambda}_{N}}(P_{\lambda}f)=\sum_{j=0}^{N}(\widetilde{\Gamma}_{j}f)$

with $\widetilde{\Gamma}_{n}f=\sum P_{\lambda}f$ where the sum is over all
$\lambda\in\overline{C}\cap I^{*}$ such that $n-1<\|\lambda-\rho\|\leq n$. We
see no essential difference between the behavior of polyhedral and spherical
partial sums in the results of this work.

##  3\. Almost everywhere convergence results

######  Theorem 3.1.

Let $G$ be a compact, connected Lie group and $f\in L^{2}(G)$. If

$\intop_{0}^{1}\frac{\Omega^{2}(f,t)}{t}dt<\infty,$

then $S_{N}f(x)\rightarrow f(x)$ and $\widetilde{S}_{N}f(x)\rightarrow f(x)$
for almost every $x\in G$.

###### Proof.

We give the proof for polyhedral Fourier partial sums of $f$; the case of
spherical Fourier partial sums is completely analogous. Note that by Jackson’s
theorem for compact, connected Lie groups [4], there is a constant $B>0$ such
that

$\displaystyle\intop_{0}^{1}\frac{\Omega^{2}(f,t)}{t}dt$
$\displaystyle=\sum_{k=0}^{\infty}\intop_{2^{-(k+1)}}^{2^{-k}}\frac{\Omega^{2}(f,t)}{t}dt$
$\displaystyle\geq\log(2)\sum_{k=1}^{\infty}\Omega^{2}(f,2^{-k})$
$\displaystyle\geq B\log(2)\sum_{k=1}^{\infty}(E_{2^{k}}^{2}f)$

where

$E_{M}(f)=\inf\\{\|f-P\|_{L^{2}(G)}:P\in\bigoplus_{\lambda\in\Lambda_{M}}H_{\pi(\lambda)}\textrm{
}\\}$

is the best approximation of $f$ in
$\bigoplus_{\lambda\in\Lambda_{M}}H_{\pi(\lambda)}$. Since

$\displaystyle E_{2^{k}}^{2}(f)$
$\displaystyle=\|f-S_{2^{k}}f\|_{L^{2}(G)}^{2}$
$\displaystyle=\sum_{j=2^{k}+1}^{\infty}\|\varGamma_{j}f\|_{L^{2}(G)}^{2},$

it follows that

$\displaystyle\sum_{k=1}^{\infty}E_{2^{k}}^{2}(f)$
$\displaystyle=\sum_{j=2}^{\infty}\left\lfloor\log_{2}(j)\right\rfloor\|\varGamma_{j}f\|_{L^{2}(G)}^{2}$
$\displaystyle\geq\frac{\log_{2}(e)}{2}\sum_{j=2}^{\infty}\log(j)\|\varGamma_{j}f\|_{L^{2}(G)}^{2},$

and therefore

$\infty>\intop_{0}^{1}\frac{\Omega^{2}(f,t)}{t}dt\geq\frac{B}{2}\sum_{j=2}^{\infty}\log(j)\|\varGamma_{j}f\|_{L^{2}(G)}^{2}.$

Consequently, a general version of the Rademacher-Menshov theorem [18] implies

$\lim_{N\rightarrow\infty}S_{N}f(x)=\lim_{N\rightarrow\infty}\sum_{j=0}^{N}(\varGamma_{j}f)(x)=f(x)$

for almost every $x\in G$. ∎

It should be noted that in [10], Dai established almost everywhere convergence
results for Fourier-Laplace series of functions on spheres which are analogous
to the preceding theorem.

######  Corollary 3.2.

Let $f\in L^{2}(G)$. If $\Omega(f,t)=O(t^{\alpha})$ for some $\alpha>0$, then
$S_{N}f(x)\rightarrow f(x)$ and $\widetilde{S}_{N}f(x)\rightarrow f(x)$ for
almost every $x\in G.$

######  Corollary 3.3.

If $f\in\textrm{Lip}_{\alpha}(G)$ for some $\alpha>0$, then
$S_{N}f(x)\rightarrow f(x)$ and $\widetilde{S}_{N}f(x)\rightarrow f(x)$ for
almost every $x\in G$.

##  4\. Divergence of Fourier partial sums on a countable subset

To begin this section, we review notation and results of [2, p.84f] and [16,
p.125f]. Equip the two-dimensional special unitary group $SU(2)$ with the left
and right translation invariant metric $d$ given by

$d(x,y)=\sqrt{\frac{1}{2}\textrm{tr}\big{(}(x-y)(x-y)^{*}\big{)}}.$

Let $0<\alpha<1$ and let $f$ be a real function on $SU(2)$. If there exists a
number $M>0$ such that

$|f(x)-f(y)|\leq Md^{\alpha}(x,y)$

for all $x,y\in SU(2)$, then $f$ is an $\alpha$-Hölder continuous function on
$SU(2)$ and we write $f\in\textrm{Lip}_{\alpha}(SU(2))$. A real function $f$
on $SU(2)$ is central if, for $\mu$-almost every $x\in SU(2)$,
$f(yxy^{-1})=f(x)$ for all $y\in SU(2)$. In particular, since every $x\in
SU(2)$ is diagonalizable via a similarity transformation:

$yxy^{-1}=\begin{bmatrix}&e^{i\theta}&0\\\
&0&e^{-i\theta}\end{bmatrix}\equiv\omega(\theta),$

where $y\in SU(2)$ and $e^{\pm i\theta}$ are the eigenvalues of $x$, it
follows that if $f$ is central then for $\mu$-almost every $x\in SU(2)$,

$f(x)=f(\omega(\theta))$

with $\theta\in[0,\pi]$. With this notation, the Weyl integral formula is
explicitly

(4.1)
$\int_{SU(2)}g(x)d\mu(x)=\frac{2}{\pi}\int_{0}^{\pi}g(\omega(\theta))\sin^{2}(\theta)d\theta$

when $g$ is a central function in $L^{2}(SU(2))$.

Let $\\{\pi_{n}\\}_{n=0}^{\infty}$ denote the family of all (inequivalent)
continuous, irreducible, unitary representations of $SU(2)$; here $\pi_{n}$
has dimension $n+1$ and its character $\chi_{n}=\textrm{trace}(\pi_{n})$ is
the continuous central function on $SU(2)$ given by

$\chi_{n}(x)=\chi_{n}(\omega(\theta))=\frac{\sin((n+1)\theta)}{\sin(\theta)}$

where $e^{\pm i\theta}$ are the eigenvalues of $x$. It follows that the
Dirichlet kernel $\\{\mathbf{D}_{N}\\}_{N=0}^{\infty}$ on $SU(2)$ is the
sequence of continuous central functions given by

(4.2)
$\mathbf{D}_{N}(x)=\mathbf{D}_{N}(\omega(\theta))=\sum_{n=0}^{N}(n+1)\chi_{n}(\omega(\theta))=\frac{-1}{2\sin(\theta)}D_{N+1}^{\prime}(\theta)$

where

$D_{n}(t)=1+2\sum_{j=1}^{n}\cos(jt)=\frac{\sin((2n+1)t/2)}{\sin(t/2)}$

is the Dirichlet kernel on $[-\pi,\pi]$.

######  Theorem 4.1.

Let $\alpha\in(0,1)$ and let $\\{x_{i}\\}_{i=1}^{\infty}$ be any countable
subset of $SU(2)$. Then there exists a function
$f\in\textrm{Lip}_{\alpha}(SU(2))$ such that

$\underset{N\geq 1}{\sup}|S_{N}f(x_{i})|=\infty$

for all $i=1,2,3,\ldots$.

###### Proof.

Observe that for $\alpha\in(0,1)$, $\textrm{Lip}_{\alpha}(SU(2))$ is a Banach
space with norm

$\|f\|_{\textrm{Lip}_{\alpha}(SU(2))}=\underset{x\in
SU(2)}{\sup}|f(x)|+\underset{\underset{x\neq
y}{x,y\in}SU(2)}{\sup}\frac{|f(x)-f(y)|}{d^{\alpha}(x,y)}.$

Fix $x\in SU(2)$ and $n\in\mathbb{N}$, and set $\Phi_{n}^{x}(f)=S_{n}f(x)$.
Each $\Phi_{n}^{x}$ is a bounded linear functional on
$\textrm{Lip}_{\alpha}(SU(2))$ of norm

$\displaystyle\|\Phi_{n}^{x}\|$
$\displaystyle=\sup\left\\{|S_{n}f(x)|:f\in\textrm{Lip}_{\alpha}(SU(2)),\;\|f\|_{\textrm{Lip}_{\alpha}(SU(2))}\leq
1\right\\}$ $\displaystyle\leq\|\boldsymbol{D}_{n}\|_{L^{1}(SU(2))}.$

Specializing to the case when $x=e$, the identity matrix in $SU(2)$, and $f\in
L^{2}(SU(2))$ is central, we have

$\displaystyle\Phi_{n}^{e}(f)=(f\ast\boldsymbol{D}_{n})(e)$
$\displaystyle=\int_{SU(2)}f(y)\boldsymbol{D}_{n}(y)d\mu(y)$
$\displaystyle=\frac{2}{\pi}\int_{0}^{\pi}f(\omega(\theta))\boldsymbol{D}_{n}(\omega(\theta))\sin^{2}(\theta)d\theta$

by (4.1). It then follows from (4.2) that

$\displaystyle\Phi_{n}^{e}(f)=$
$\displaystyle\frac{1}{\pi}\int_{0}^{\pi}f(\omega(\theta))\cos^{2}\left(\frac{\theta}{2}\right)D_{n+1}(\theta)d\theta$
$\displaystyle\quad-\frac{(2n+3)}{\pi}\int_{0}^{\pi}f(\omega(\theta))\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\cos\left(\frac{\theta}{2}\right)d\theta.$

For $n\geq 2$, the absolute maxima and minima of the function
$h_{n}(\theta)=\cos\left(\left(n+\frac{3}{2}\right)\theta\right)$ on $[0,\pi]$
occur at the endpoints of the intervals

$I_{k}=\left[\frac{2k\pi}{2n+3},\frac{2(k+1)\pi}{2n+3}\right]$

where $k\in\\{0,1,2,...,n\\}$. Let $g_{n}$ be the sawtooth function on
$[0,\pi]$ determined by $g_{n}\left(\frac{2k\pi}{2n+3}\right)=(-1)^{k}$ for
$0\leq k\leq n+1$, $g_{n}(\pi)=0$, and $g_{n}$ is piecewise linear between
these points. Define a central function $f_{n}$ on $SU(2)$ by
$f_{n}(\omega(\theta))=g_{n}(\theta)$ for $\theta\in[0,\pi]$. It is easy to
see that each $f_{n}$ belongs to $\textrm{Lip}_{\alpha}(SU(2))$; in fact,

$\frac{|f_{n}(x)-f_{n}(y)|}{d^{\alpha}(x,y)}\leq\left(\frac{\pi}{2}\right)^{\alpha}\left(\frac{2\pi}{2n+3}\right)^{1-\alpha}\leq\pi$

for all distinct matrices $x$ and $y$ in $SU(2)$.

Since $g_{n}(\theta)\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\geq
g_{n}^{2}(\theta)\geq 0$ on each interval $I_{k}$ and on
$\left[\frac{2(n+1)\pi}{2n+3},\pi\right]$, and since the function
$\theta\mapsto\cos(\theta/2)$ is positive and decreasing on $[0,\pi)$,

$\displaystyle\int_{I_{k}}f_{n}(\omega(\theta))\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\cos(\theta/2)d\theta$
$\displaystyle\geq\int_{I_{k}}g_{n}^{2}(\theta)\cos(\theta/2)d\theta$
$\displaystyle\geq\cos\left(\frac{(k+1)\pi}{2n+3}\right)\int_{I_{k}}g_{n}^{2}(\theta)d\theta$
$\displaystyle=\frac{2\pi}{3(2n+3)}\cos\left(\frac{(k+1)\pi}{2n+3}\right)$

for all $k\in\\{0,1,2,...,n\\}$ and

$\int_{\frac{2(n+1)\pi}{2n+3}}^{\pi}f_{n}(\omega(\theta))\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\cos(\theta/2)d\theta\geq
0.$

Adding these inequalities we obtain

$\displaystyle\intop_{0}^{\pi}f_{n}(\omega(\theta))\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\cos\left(\frac{\theta}{2}\right)d\theta$
$\displaystyle\geq\frac{2}{3}\left(\frac{\pi}{2n+3}\right)\sum_{k=1}^{n+1}\cos\left(\frac{k\pi}{2n+3}\right)$
$\displaystyle=\frac{2}{3}\left(\frac{\pi}{2n+3}\right)\left\\{D_{n+1}\left(\frac{\pi}{2n+3}\right)-1\right\\},$

and hence

$\biggl{|}\frac{(2n+3)}{\pi}\int_{0}^{\pi}f_{n}(\omega(\theta))\cos\left(\left(n+\frac{3}{2}\right)\theta\right)\cos\left(\frac{\theta}{2}\right)d\theta\biggr{|}\geq\frac{2}{3}\left\\{D_{n+1}\left(\frac{\pi}{2n+3}\right)-1\right\\}.$

Because the function $\theta\mapsto f_{n}(\omega(\theta))\cos^{2}(\theta/2)$
is uniformly bounded by $1$ on $[0,\pi]$,

$\displaystyle\biggl{|}\frac{1}{\pi}\int_{0}^{\pi}f_{n}(\omega(\theta))\cos^{2}(\theta/2)D_{n+1}(\theta)d\theta\biggr{|}$
$\displaystyle\leq\frac{1}{\pi}\intop_{0}^{\pi}|D_{n+1}(\theta)|d\theta$
$\displaystyle=\frac{4}{\pi^{2}}\log(n+1)+o(1)$

as $n\rightarrow\infty$. Consequently

$\frac{|\Phi_{n}^{e}(f_{n})|}{\|f_{n}\|_{\textrm{Lip}_{\alpha}(SU(2))}}\geq\frac{\frac{2}{3}\left\\{D_{n+1}\left(\frac{\pi}{2n+3}\right)-1\right\\}-\left(\frac{4}{\pi^{2}}\log(n+1)+o(1)\right)}{1+\pi}.$

But
$D_{n+1}\left(\frac{\pi}{2n+3}\right)=\left(\sin\left(\frac{\pi}{2(2n+3)}\right)\right)^{-1}\geq\frac{2(2n+3)}{\pi}$
and hence

$\|\Phi_{n}^{e}\|=\sup\left\\{\frac{|\Phi_{n}^{e}(f)|}{\|f\|_{\textrm{Lip}_{\alpha}(SU(2))}}:\quad
f\in\textrm{Lip}_{\alpha}(SU(2)),\quad f\neq 0\right\\}$

is asymptotically bounded below by

$\frac{\frac{2}{\pi}(2n+3)-\frac{4}{\pi^{2}}\log(n+1)}{1+\pi}$

as $n\rightarrow\infty$. Thus the sequence of bounded linear functionals

$\Phi_{n}^{e}(f)=S_{n}f(e)$

is not uniformly bounded on the Banach space $\textrm{Lip}_{\alpha}(SU(2))$ as
$n\rightarrow\infty$. By the uniform boundedness principle

$\underset{n\geq 1}{\sup}|S_{n}f(e)|=\infty$

for all $f$ belonging to some dense $G_{\delta}$ set in
$\textrm{Lip}_{\alpha}(SU(2))$.

If $z\in SU(2)$, define the left translation operator $L_{z}$ on
$\textrm{Lip}_{\alpha}(SU(2))$ by $L_{z}f(y)=f(zy)$ for all $y\in SU(2)$. For
each element of the countable subset $\\{x_{i}\\}_{i=1}^{\infty}$ of $SU(2)$
and each $n\geq 1$, observe that

$\frac{|\Phi_{n}^{x_{i}}\left(L_{{x_{i}}^{-1}}f_{n}\right)|}{\|L_{{x_{i}}^{-1}}f_{n}\|_{\textrm{Lip}_{\alpha}(SU(2))}}=\frac{|\Phi_{n}^{e}\left(f_{n}\right)|}{\|f_{n}\|_{\textrm{Lip}_{\alpha}(SU(2))}},$

so there corresponds a dense $G_{\delta}$ subset $E_{x_{i}}$ of
$\textrm{Lip}_{\alpha}(SU(2))$ such that

$\underset{n\geq 1}{\sup}|S_{n}f(x_{i})|=\infty$

for all $f\in E_{x_{i}}$. By the Baire category theorem
$E=\bigcap_{i=1}^{\infty}E_{x_{i}}$ is dense in
$\textrm{Lip}_{\alpha}(SU(2))$. In particular, $E$ is nonempty and any $f\in
E$ gives the desired conclusion. ∎

A general theorem of Belen’kii [1] guarantees that if
$F:[-1,1]\rightarrow\mathbb{R}$ satisfies a Dini-Lipschitz condition and the
Fourier-Jacobi partial sums

$\\{s_{N}^{(\alpha,\beta)}(F;\pm 1)\\}_{N=0}^{\infty}$

at $\pm 1$ converge for some $\alpha>-1$ and $\beta>-1$, then the
corresponding Fourier-Jacobi series of $F$ converges uniformly to $F$ on
$[-1,1]$. This leads directly to the following result on $SU(2)$.

######  Theorem 4.2.

Let $f$ be a central function in $\textrm{Lip}_{\alpha}(SU(2))$ for some
$\alpha\in(0,1)$. Then:

  1. (a)

$S_{N}f(x)\rightarrow f(x)$ uniformly outside any open set containing
$\\{e,-e\\}$;

  2. (b)

$S_{N}f(x)\rightarrow f(x)$ uniformly on $SU(2)$ if $\\{S_{N}f(\pm
e)\\}_{N=0}^{\infty}$ converge.

This suggests the question: Can the word “central” be deleted from the
hypothesis of Theorem 4.2 and still obtain conclusions (a) and (b)? Theorem
4.1 shows that there is no possibility of such an analogue of Theorem 4.2 for
general functions in $\textrm{Lip}_{\alpha}(SU(2))$ for some $\alpha\in(0,1)$.
The points of divergence for the Fourier partial sums of such a noncentral
function need no longer be at the “poles” $\pm e$ of the “sphere” $SU(2)$.
According to Theorem 4.1, points of divergence for
$\textrm{Lip}_{\alpha}(SU(2))$ functions can be dense in $SU(2)$.

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