EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
Authors: Jiawei Cao
Chaochen Gu
Hao Cheng
Xiaofeng Zhang
Kaijie Wu
Changsheng Lu
Venue: ICML 2025 poster (2025)
Keywords: Instance Segmentation
Learnable Elliptical Fourier Descriptor
Contour Prediction
Primary area: applications->computer_vision
TL;DR: EFDTR: A polygon-based instance segmentation framework using Elliptic Fourier Descriptors (EFDs) for precise contour modeling, outperforming polygon methods and rivaling mask-based approaches.

Abstract: Polygon-based object representations efficiently model object boundaries but are limited by high optimization complexity, which hinders their adoption compared to more flexible pixel-based methods. 
In this paper, we introduce a novel vertex regression loss grounded in Fourier elliptic descriptors, which removes the need for rasterization or heuristic approximations and resolves ambiguities in boundary point assignment through frequency-domain matching.
To advance polygon-based instance segmentation, we further propose EFDTR (\textbf{E}lliptical \textbf{F}ourier \textbf{D}escriptor \textbf{Tr}ansformer), an end-to-end learnable framework that leverages the expressiveness of Fourier-based representations. 
The model achieves precise contour predictions through a two-stage approach: the first stage predicts elliptical Fourier descriptors for global contour modeling, while the second stage refines contours for fine-grained accuracy. Experimental results on the COCO dataset show that EFDTR outperforms existing polygon-based methods, offering a promising alternative to pixel-based approaches. Code is available at \url{https://github.com/chrisclear3/EFDTR}.

EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
Jiawei Cao 12 Chaochen Gu
Hao Cheng 12 Xiaofeng Zhang' 12
Kaijie Wu
Changsheng Lu
Abstract
Polygon-based object representations efficiently model object boundaries but are limited by high optimization complexity, which hinders their adoption compared to more flexible pixel-based methods. In this paper, we introduce a novel vertex regression loss grounded in Fourier el- liptic descriptors, which removes the need for rasterization heuristic approximations and re- solves ambiguities in boundary point assignment through frequency-domain matching. To advance polygon-basse based instance segmentation, we further propose EFDTR (Elliptical Fourier Descriptor Transformer), an end-to-end learnable framework that leverages the expressiveness of fourier-based representations. The model achieves precise con- tour predictions through two-stage approach: the first stage predicts elliptical Fourier descrip- tors for global contour modeling, while the second stage refines contours for fine-grained accuracy. Experimental results on the COCO dataset show that EFDTR outperforms existing polygon-base based methods, offering promising alternative to pixel- based approaches. Code is available at ht attps: T/..TTR.
y
(8)
278.5'
32nd order
1st order
337.2
(a
(b)
278 57 337.2
Figure 1. (a) Pixel-wise instance mask from the COCO dataset. (b) First-order reconstruction (black line) compared with 32nd- order EFD reconstruction (gray dashed line). (c) Dense contour sampling with 128 points, initially equidistant, followed by target snapping, where sampling points are adjusted to the nearest target points.
/ //github
1. Introduction
Instance segmentation is a fundamental task in computer vision, focusing on the detection and delineation of indi- vidual objects within an image by assigning unique masks each detected instance. This capability is essential for applications in autonomous driving, medical image process-
ing (Chen al., 2018: 2023), and robotics (Ma et al., 2024; Liu al., 2025; Zhang al., 2024a), where precise object boundaries are critical to decision making and interaction. Most existing instance segmentation methods rely on pixel- wise masks to delineate objects (He et al., 2017; Kirillov et al., 2020; Wang et al., 2020; Cheng et al., 2022b;a; Li et al., 2023a). However, contour- or polygon-based approaches have gained attention due their ability repre- sent object shapes with fewer parameters while retaining key topological properties. Notably, these methods output contour coordinates or keypoints (Lu et al., 2024; Lu & Koniusz, 2024; 2022; Lu, 2024; Lu et al., 2023) without post- post-processing, making them particularly well suited for tasks such as remote sensing (Zhang et al., 2024b) and BEV vector map generation (Liao et al., 2024) in autonomous
'Department of Automation, Shanghai Jiao Tong University, Shanghai, China Key Laboratory of System Control and Infor- mation Processing. Ministry of Education of China, Shanghai, China "School of Computing. The Australian National University, Canberra, Australia 'Australian Institute for Machine Learning, University of Adelaide. Correspondence to: Kaijic Wu & Chang- sheng Lu <kaijiewu @ sjjutedu.ccm changshenglun gmail.com>>
directly
Proceedings of the ndnd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
driving. Since contours typically represented collec- tions of polygons, these methods collectively referred to as polygon-based approaches in this paper to highlight their focus on polygon regression.
Polar Typ
Sampling
Sampling
In recent years, polygon-basee methods have significantly advanced polygon regression accuracy and performance. Representative approaches include Curve GCN (Ling al., 2019), Poly Transfooom (Liang al., 2020), Deep Snake (Peng et al., 2020), PolarMask (Xie et al., 2020), DANCE (Liu et al., 2021), E2EC (Zhang et al., 2022b), and Bound- aryFormer (Lazarow et al., 2022). These methods can be broadly categorized into single-stage and two-stage ap- proaches.
Missing
(a)
Ambiguous
Regress on
Single-stage models directly predict contour coordinates in an end-to-end manner. While efficient, their precision is often limited, making two-stage methods preferable. Two- stage methods typically generate a coarse initial contour, followed by refined polygon regression through displace- ment or offset learning. However, both approaches depend on the regression of 2D vertex coordinates, which requires precise target assignment. Apart from differentiable ras- terization methods like PolyTransform (Liang et al., 2020) and Boundary (Lazarow et al., 2022), which perform supervised learning via rendered pixel maps, existing target assignment strategies can be broadly categorized into three types:
Predit dion
(C)
(b)
Figure 2. Limitations of regression matching between Polar Cartesian coordinate systems. (a) Original polygon segmentation. (b) Inadequate sampling of target points system. (c) Ambiguous regression in the Cartesian coordinate system due to minimum Euclidean distance assignment.
polar coordinate
address these issues, we introduce Elliptic Fourier Descrip- tors (EFDs) as compact, reliable representation for polyg- onal contours. Any closed 2D contour can be expressed as Fourier series with ellipses as basis functions (Kuhl & Giardina, 1982). As illustrated in Figure 1b, the inverse transformation of first-order EFD (solid) captures a rough global shape represented by an ellipse (Lu al., 2019; 2017), while thirty-sccondorrder EFD (dashed) restores fine-grained details. Furthermore. the phase information encoded in EFDs establishes a bijective mapping between contour points and their positions along the shape. This facilitates precise regression target assignment and resolves the inherent ambiguity found in Euclidean distance- based methods. For example, as shown in Figure 1c, points that are spatially close but topologically distinct are clearly dis- ambiguated through their EFD phase values. To fully lever- age the advantages of EFDs, we propose EFDTR (Elliptic Fourier Descriptor-basee Transformer), a two-stage regres- sion model that addresses the limitations existing tar- get assignment strategies through phase-domain sampling. In the first stage, EFD parameter regression captures the global coarse topology, while the second stage refines the regression through phase- based target assignment to achieve precise boundary localization. The framework effectively harnesses the Transformer's capability for sequence model- ing, with deformable attention facilitating flexible feature aggregation.
Polar Target Assignment: This strategy aligns targets and predictions using the same polar angle. For multi- polygon instances, selects the largest polygon and defines its centroid as the origin for uniform ray sam- pling (see Figure 2b). While conceptually simple, this approach is limited to star-convex shapes and struggles with non-convex structures.
This strategy mini- mizes the Euclidean distance between predicted and ground-truth vertices to determine assignments. How- ever, it lacks global context, leading to ambiguities in matching (Figure 2c). Additionally, it disregards the se- quential topology contours, often causing structural inconsistencies in polygon regression.
Cartesian Target Assignment:
This strategy segment the con- tour into regions based on polar angles and apply Fu- clidean distance-based assignment within each sub- region. While this mitigates certain topological errors, it does not fundamentally resolve the limitations of distance-basee regression.
Hybrid Strategies:
Despite their merits, existing methods impose restrictive geometric assumptions, struggle with topological consis- tency, lack robust global contour representation. To
By integrating EFDTR into an end-to-end differentiable
2

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
segmentation pipeline, we achieve superior polygon regres- sion accuracy, surpassing prior polygon-based approaches. EFDTR emerges as promising alternative to pixel-based approaches, offering a structured and efficient solution for shape representation in learning-based segmentation.
and deformable attention to predict point offsets, incorpo- rating a differentiable rasterization loss into the objective function for effective supervision.
Recent studies in geometric representation provide addi- tional insights. PolygonGINN (Yu et al., 2024) introduces a heterogeneous graph structure for polygon modeling, and its spanning tree sampling strategy inspires our MST-based contour merging. PolyhedronNet (Yu al., 2025), though focused on 3D polyhedra, offers useful ideas for future ex- tensions of surface-based contour representation.
2. Related Work
2.1. Instance Segmentation with Polygon
Various methods have been proposed for contour prediction and instance segmentation using deep learning techniques due to their advantage of capturing object shapes (Shi et al., 2024). Curve-GCN (Ling et al., 2019) employs graph con- volutional network for prediction, fine-tuning the process with cyclic matching loss and a differentiable rasterization loss. PolyTransform (Liang et al., 2020) extracts initial con- tours via a segmentation network and regresses points using a deforming network. DeepSnake (Peng et al., 2020) adapts the ExtremeNet framework (Zhou et al., 2019), initially detecting an octagonal contour, which is then iteratively refined via ring convolution. Point-Set Anchors (Wei et al., 2020) regresses sampled points on anchor points placed on bounding boxes, facilitating instance segmentation. Dance (Liu et al., 2021) extracts boundary features using a con- tour extraction network and applies attentive deformation to regress the boundaries. These methods typically rely predicting target points from object boundaries, but the re- sulting point allocation tends to be uneven, limiting their ability to model complex contours effectively.
2.2. DEtection with TRansformer (DETR)
The decoding mechanism of our model is inspired by key developments in the DETR series, which has significantly influenced the field of object detection (Wang et al., 2022a), Carion et al. (Carion et al., 2020) introduced DETR, an end- to-end object detection framework powered by Transform- ers, which surged waves due to its novel approach. Com- pared to traditional approaches. the standout characteristic of DETR is eliminating the need for hand-crafted anchor boxes and the step of Non-Maximum Suppression (NMS) commonly used in earlier detection paradigms. Since its inception, several variants of DETR have been proposed to tackle its limitations and further improve performance. For example, Deformable-DE.TR (Zhu et al., 2020) enhances training efficiency by incorporating multi-scale features and optimizing the attention mechanism. Conditional DETR (Meng et al., 2021) and Anchor DETR (Wang et al., 2022b) address the challenges of query design optimization. DAB- DETR (Liu et al., 2022) introduces the concept of 4D refer- ence points and refines the model by optimizing prediction boxes in successive layers. DINO (Zhang et al., 2022a), building upon these methods, achieves state-of-the-an per- formance, establishing a solid foundation for further ad- vancements in DETR-based detection.
PolarMask (Xie et al., 2020) represents contours in polar coordinates through their center, radius, and angle, utiliz- ing FOCS-based (Tian et al., 2022) architecture and an IoU loss function for contour regression. FourierNet (Riaz et al., 2021) predicts contours by transforming contour pa- rameters into Fourier space and directly regressing Fourier coefficients. These representation methods struggle to accu- rately model general object contours due to their structural constraints.
Incorporating insights from DINO (Zhang et al., 2022a), RT- DETR (Zhao et al., 2024) combines YOLO-inspired tech- niques to enable real-time performance for DETR-based detection systems. BoundaryFormer takes a step further by incorporating Deformable Attention and applying pro- gressive upsampling in the decoder improve the sequen- tial regression of polygon points. These innovations lay the groundwork for the two-stage decoder architecture proposed in this paper, contributing to the evolution of DETR-style object detectors.
E2EC (Zhang et al., 2022b) aligns points within intervals us- ing multi-directional alignment and supervises point regres- sion through dynamic matching loss, effectively combin- ing the allocation strategies of DANCE and PolarMask. C- AOI (Zhu et al., 2023), an improvement of E2EC, introduces a Transformer into the model architecture and enhances the matching loss with additional constraints, proposing an adaptive matching loss to reduce allocation ambiguity.
PolyFormer (Liu et al., 2023) is referring task model that can also be applied to instance segmentation, treat- ing polygons as natural language statements, where each token corresponds to predicting a point on the contour. Fi- nally, BoundaryFormer (Lazarow et al., 2022), a hybrid Transfrranfrrmer model, utilizes iterative upsampling
3. Method
3.1. Multiple Polygon Connection
Elliptical Fourier descriptors designed for single closed polygons. For instances containing multiple polygonal con- tours, these must be converted into single polygon by
3

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
introducing auxiliary edges. Intuitively, the objective is to minimize the total length of the connecting edges, shorter edges help reduce the error in Intersection over Union (loU) when predicted auxiliary edges do not perfectly align with the ground truth.
with the harmonic number are computed as follows:
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Azp and Аy, indicate the displacement along the x- and y-axes, respectively, between points p and p + 1. Atp is the step length between points and + 1.
Figure 3. The illustration of multiple polygon connection.
As Fourier coefficients are sensitive to the choice of the initial point, reparameterization strategies are commonly adopted maintain consistency across different applica- tions. To achieve a standardized representation, we enforce phase constraint at t = O, which corresponds to the starting point of the contour. Specifically, we require that the first- order elliptical Fourier coefficient aligns with the positive x-axis relative to the centroid.
Figure 3 illustrates an example of connecting four polygons. The distance lag between two polygons P; and Pj. with ver- tex sets V; {vi, ug...., " and V3 = {vf. vg...., 1/2) respectively, is given by:
4i min SEV.
11 r/1l-
(1)
Furthermore, we adhere to the right- hand rule, where coun- terclockwise motion is considered positive in the standard Cartesian coordinate system, while clockwise motion is pos- itive in the image coordinate system. Consequently, the first-order Fourier coefficients must satisfy:
Each polygon is treated as node in a fully connected graph, with edge weights Ly- Finding the optimal auxiliary edges reduces to solving the minimum spanning tree (MST) of this graph. We adopt Prim's algorithm to compute the MST, efficiently merging multiple polygons into a single closed contour.
A > 0 C =
(8)
D1 > 0
3.2. Elliptic Fourier Phase Assignment
This guarantees that the zero-phase frequency component in the Fourier domain is properly aligned with the positive x-axis of the centroid, as depicted in Figure 1d.
The contour coordinates of a single closed polygon with m vertices {v of),, (v2, uy),............. (1i7) } can be de- scribed by an elliptical Fourier series, as defined by the following equation:
The elliptical Fourier series computed for each polygon should then be multiplied by the following rotation matrix to satisfy the normalization requirements of the first-order Fourier series. Here, represents the phase shift required to align the standardized starting point with that of the input polygon:
(t)t] v()
xt yo
An
Bn
cos
2 T
> (2)
Cn
Dn
sin
2
1=1
where (Ie, Ue) represents the centroid of polygon, and (a(t), v()) denotes the coordinates of point along the polygon, parameterized by its cumulative arc-length f. The arc-length parameterization is defined as follows:
0 o- arctan2(-Ci, A > BiC, A $ B,C,
(9)
(C,
Then apply the rotation matrix to the original elliptical Fourier coefficients, yielding:
an
bn d,
An Bn
[[
cosn8 sin nd
sin n0 cos na
(10)
tp
M++1 - vell2. to f0=0. O, Im = T,
(3)
On
Cn
D.
The phase O, of point p is computed as:
where T denotes the perimeter of the polygon and Vm++1
2nt p
V1. The Fourier coefficients An, Bay Gny Dn associated
0
(11)
T

---

EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
Here, 0p represents a normalized arc-length parameter, map- ping each point along the contour to a phase in the Fourier representation. Consequently, there exists a bijective rela- tionship between the phase and the contour points of the closed polygon, ensuring a unique correspondence.
The input features are first encoded by the Self-Attention module, where positional embeddings are incorporated, and the resulting representations serve as queries to retrieve rele- vant flattened features. These queries are then processed by the Deformable Attention module, which adaptively attends to spatially significant regions to refine the feature represen- tations. The output features are subsequently passed through two prediction heads: the Class Head, which predicts the in- stance class, and the EFD Head, which estimates the Fourier descriptor parameters. The predicted EFD parameters serve as reference information for the next layer, enabling iterative refinement. The total number of parameters at each stage follows the formulation 4n + 1, as illustrated in Figure 4.
As illustrated in Figure Ic, the cyan and orange points are spatially close in Cartesian coordinates, making Euclidean distance an ambiguous matching criterion. However, their distinct phase values in the frequency domain enable precise differentiation. Similarly, the green and magenta points, which originate from overlapping vertices introduced by auxiliary edges, further highlight the importance of phase information in resolving maaee-n and ambiguity in polygonal contour regression.
The Fourier inverse transform is applied to reconstruct the 2D coordinates corresponding to the elliptic Fourier de- scriptors predicted in the first stage. To facilitate precise feature extraction, we employ the grid_sample function for feature sampling. Given the need for accurate contour regression in the subsequent stage, we integrate multi-scale fusion information from feature maps P3 to Ps. where four- scale normalized coordinate sampling produces four sets of feature vectors.
3.3. Model Structure with Leanable Elliptic Fourier
The model architecture proposed in this paper is shown in Figure 4. follows the design principles of two-stage ob- ject detection models and incorporates key elements from the DETR series in the decoder. Specifically, in the fea- ture extraction network, we primarily draw inspiration from RT-DETR (Zhao et al., 2024), which uses an improved Pyramid Attention Network (PAN) (Li et al., 2018) called Hybrid Feature Fusion module to obtain multi-scale feature maps P2, P3, P4; Ps. In our case, since we aim to predict instance contours, we retain feature maps with up to 4x downsampling P2- This module performs feature fusion using self-attention on the 32x downsampled feature map Ps. while applying convolutional layers for fusion in the other feature map paths. The final output consists of feature vectors with a size of 85W H. where W and H represent the 1024
To reduce computational overhead, we adopt group-based feature sampling strategy, where instead of sampling all 128 predicted points individually, only 128 points are sampled, where y denotes the group size, set to 4 in our design. Addi- tionally, explicit EFD parameters and class information are leveraged to predict the fusion weights for the four scales, enabling adaptive weighting across hierarchical features. The fused single-scale feature representation is then used as the query input for the Polygon Decoder, refining the polygonal structure prediction in an iterative manner.
width and height of the input image, respectively.
In the EFD decoder, the input comprises multi-scale fea- tures from Ps. Pa. and Ps, striking balance between com- putational efficiency and prediction accuracy. Since the EFD decoder primarily predicts Fourier parameters to cap- ture a coarse global topology, utilizingP's P3. Pa. and Ps is sufficient. Incorporating P2would introduce a significant computational overhead without a proportional gain in per- formance.
3.4. Loss
We utilize the VFL loss (Zhang et al., 2021) in the classifi- cation loss function to correlate the target value of positive samples with the loU prediction. For the elliptical Fourier descriptor (EFD) regression, we apply the LI loss, while the Smooth-L loss is used for polygons under phase alignment.
Based on the information in Section 3.2, the phase set for the target polygon points is defined as:
For the first EFD decoder layer, Score Encoder ranks the flattened feature representations and selects the top-k features based on their assigned scores. These selected fea- tures are then concatenated with the noised target, forming the primary input to the decoder. The added noise facili- tates denoising, aiding in convergence acceleration and im- proving accuracy. Furthermore, the EFD Encoder extracts the Fourier parameters corresponding to the top-k features, which serve as reference information to refine the decoding process. Consequently, each decoder layer operates with two key inputs: the concatenated top-k features with the noised target, and the EFD reference parameters.
2tt T
P. =
t f1m.:
(12)
where T represents the total perimeter of the contour, and is the number of vertices of the target polygon.
In contrast, the initial phase for the predicted polygon sam- pling points corresponds to uniform sampling in the interval [0, 2T):
Pr = 2nT
t EE 2,, .... 11}. (13)
5

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
EFD Decoder Layer XN
Score Eneeder
Layero
Hend
Class
Top-k Festura
Self- Attention
Deformable Attention
Backbone
Neck
Feature
Noival Thrget
EFD
Pred lnpua for
Next Layer
Encodler
On
Last Lager Pred LFD
Polygon Layer x
Layorg
J.P.Moraan
J.P.Moraa
PILLO
lspui for Neat Layer TTeature
Deformable Attention
Self- Attention
Points &xr Feture Sampling
Prod Polygon
Polygon Head
Weigh
Ref
128 Equidistnt Sampling
Points
Figure 4. Overview EFDTR: It consists of feature extraction, learnable elliptical fourier descriptors (EFDs) decoder and polygon deocder three parts.
where n is the number of vertices of the predicted polygon and n 2 m. To better align with the target points, we apply a snapping operation S, which defined as follows:
4. Experiments
4.1. Dataset and Pre-processing
arg min 10p- P.
11,
(14)
The COCO dataset (Lin et al., 2014) widely used bench- mark in computer vision, supporting tasks like object detec- tion, segmentation, and captioning. It contains over 330,000 images across 80 categories with detailed annotations re- flecting complex real-world object interactions.
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otherwise.
COCO offers pixel-Ievel annotations Run-Length coding (RLE) and Polygon formats. This work focuses on Polygon annotations, for which adjust the original la- bels during preprocessing to adhere to the right-hand rule. This ensures consistency, as our elliptical Fourier series is sensitive to orientation conventions.
After applying the snapping operation, the phase set of the predicted points is updated as:
{ 2nt s
Pp
1 20 1}}. (16)
Let J represent the inverse elliptical Fourier transform. The polygon vertex regression loss under phase alignment is defined as follows:
4.2. Implementation Detail
Li (pred, gt)
smooth (1(,,(()) P, (17)
The query number in the EFD decoder is set to 300, with adjacent 4 points grouped together. The EFDTR model is trained using the Adam optimizer, with different learning rates for each model component and a multi-step learning rate scheduler. Additionally. Exponential Moving Average (EMA) is employed during training stabilize the process. Data augmentation includes RandomFlip, RandomloUCropp and multi-scale training. During inference, the input image scale is fixed at 800 x 800.
Cz(pred, gf)
smooth ((FFF(()) 0 E Pt (18)
C (pred, gt) (2(rred, gt)
Epolygon
(19)
The overall loss is as follows:
Loverall Lets + aLesd + 3 ppolyonn:
(20)
where a=6 and 3=10.
6

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
Table Quantitative Results on MS COCO. We compare our EFDTR with state-of-the-art models va12017.
Method
Epoch
Output
Supervision
AP
AP
AP
APs
APM
APL
ResNet-50 backbone
Mask R-CNN(ICCV17)
400
masks
masks
42.5
23.8
45.0
60.0
DynaMask(CVPRR23
24
masks
edges+masks
38.2
58.1
41.5
20.5
40.8
52.7
Mask DINO(CVPR23)
50
masks
masks
46.0
69.0
50.7
26.1
49.3
66.1
MixingMask<(PR22)
36
masks
polygons+masks
42.2
65.2
45.9
19.8
46.5
65.7
DeepSnake(CVVPR22)
160
polygons
polygons
30.5
PolarMaskkCCVPR20)
24
polygons
polygons
32.1
53.7
33.1
14.7
33.8
45.3
DANCE(WACV21)
12
polygons
edgestpolygons
34.5
55.3
36.5
17.2
37.5
48.0
ShampContour(CVPPR2)
12
polygons
maints+masks
37.8
24.3
49.4
59.1
BoundaryFormer(CVPR2)
12
polygons
masks
36.1
56.7
EFDTR(Ours)
36
polygons
polygons
43.6
64.8
47.2
23,4
46.5
64.1
ResNet- 101 backbone
Mask2Former(CVPR222
50
masks masks
masks
44.2
23.8
47.7
66.7
DynaMask (CVPR23)
24
edges+masks
39.0
59.1
42.2
20.9
42.1
53.3
BEIS(ECAI24)
36
masks
moints+masks
42.1
25.0
45.4
55.4
SharpContou((CPPR22)
36
polygons
maints+
40.8
EFDTR(Ours)
36
polygons
polygons
45.1
66.6
49.3
24.1
48.2
66.4
4.3. Comparison with State-of-the-Arts
Order of EFD Prediction. To assess the impact of different orders of elliptical Fourier descriptors (EFDs) segmenta- tion, we compare the performance of 1tt, 2nd, and 4th order harmonics. The first-order EFDs, which focus on predicting the ellipse shape, yield the best performance. higher-order terms introduce noise due to their complexity, impairing coarse localization and hindering polygon con- vergence. This suggests that simpler, first-order EFDs more effective for our task.
Table I compares EFDTR model with mask-based meth- ods (e.g., Mask R-CNN (He et al., 2017), Mask DINO (Li et al., 2023a)) and polygon-based approaches (e.g., Deep- Snake (Peng et al., 2020), PolarMask (Xie et al., 2020)) Our model significantly outperforms prior polygon-based methods, with an AP of 43.6, while remaining competi- tive with mask-based models like Mask R-CNN (AP 42.5) and Mask DINO (AP 46.0). EFDTR excels in both small and large object categories, and, when using ResNet-101 backbone, further improves performance to 45.1 AP, sur- passing Fark2Former and DynaMask (Cheng et al., 2022a; Li al., 2023b). This highlights EFDTR as a promising alternative to pixel-based methods, advancing the state of polygon-based segmentation.
contrast,
Number of Vertices in Group. We conducted experiments with different vertex groupings: 2, 4, 8, and 16 vertices per group. The results, shown in Table 4, indicate that using vertices per group strikes the best balance between performance and computational cost. Although using 2
4.4. Ablation Study
Table 2. Effect of the number of EFD decoders (Ng) and polygon decoders (Np) instance segmentaion.
In this section, we conduct ablation studies evaluate the key components of our proposed EFDTR method and their impact performance, validated on the COCO va12017 dataset. For fairness and efficiency, all experiments are trained for 12 epochs.
(Ng, Np)
(4,3)
(4, 4)
(5.3)
(5, 4)
(6,3)
AP
39.4
39.4
40.2
40.0
40.6
40.4
APso
58.7
58.6
60.2
59.8
60.8
60.5
AP7s
42.7
42.7
43.5
43.3
43.7
43.7
Params
49.6M
50.8M
50.8M
51.9M
51.9M
53.0M
Number of Decoder Layers. We investigate the effect of varying number of EFD decoders (Ng) and polygon decoders (Np) on instance segmentation performance. The experimental results, shown in Table 2, indicate that the best performance is achieved with Ng 6 and Np 3, yielding an AP of 40.5.
Table 3, Effect of the order of EFD on instance segmentaion.
order
AP
AP
AP75
APs
AРM
AP L
1
40.6
60.8
43.7
20.5
43.5
61.4
2
29.4
46.2
31.0
11.1
33.4
48.7
31.2
50.3
32.4
10.7
32.7
53.8
7

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
Figure 5. EFDTR visualization results COCO va12017. The grey points contour predicted vertices of instances.
Table Effect of number vertex group size on segmentation
Table Impact the IoU type varifocal loss function.
performance.
ToU type
AP50
AP7s
APs
AP
AP,
number
AP
AP 50
AP78
APs
AP M
APL
axis-aligned box
41.1
62.5
44.1
20.5
44.1
61.4
40.8
60.7
44.2
19.8
44.0
61,8
rotated box
60.8
43.7
20.5
40.6
60.8
43.7
20.5
43.5
61.4
8
40.1
60.6
43.3
20.0
43.1
60.8
16
39.1
60.3
42.0
18.8
42.3
59.3
vertices yields the highest accuracy, the improvement in performance is not justified by the increased computational cost.
Multi-scale Feature Fusion Module Table shows that the weighted fusion method outperforms both mean and "only P2" fusion, achieving the highest AP and improved performance across all metrics, highlighting its effectiveness in enhancing multi-scale feature integration.
Figure 6. Segmentation of multiple polygon instance based on our EFDTR.
that using axis-aligned IoU (AP 41.1) outperforms rotated IoU (AP 40.6). The simpler axis-aligned method provides higher loU values, offering better supervisory signals and improving the overall performance.
IOU Type in varifocal loss. varifocal loss O. the target value for positive samples depends on IoU, Table shows
4.5. Qualitative Analysis
Table Effect of Multi-scale Fusion Methods on Segmentation Performance.
As shown in Figure 5, our model effectively segments fine- grained boundaries, such as umbrella handles and suitcase straps, and also performs well segmenting internal bound- aries of objects like bicycles and motorcycles. For relatively dense objects, EFDTR still demonstrates strong segmenta-
method
AP
APso
AP7s
APs
APM
AP,
mean
60.1
43.6
18.9
43.5
61.2
only Ps
40.1
60.0
43.2
19.1
43.1
61.0
weighted fusion
40.6
60.8
43.7
20.5
43.5
61.4
8

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EFDTR: Learnable Elliptical Fourier Descriptor Transformer for Instance Segmentation
tion performance.
References
Additionally, Figure 6 illustrates the segmentation results for instances containing multiple polygons. The model can predict and connect contours, merging multiple polygons into a single output. However, there are still issues with the auxiliary boundaries not being brought close enough to each other, which remains a limitation of our current model.
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5. Conclusion
In this paper, we propose EFDTR, a novel polygon- based framework for instance segmentation that leverages learn- able Elliptic Fourier Descriptors (EFDs) to model object contours. Our method introduces several key innovations, including a multi-polygon connection strategy based on minimum spanning trees, phase-aligned Fourier repre- sentation for more accurate contour parameterization, and two-stage decoding architecture for polygon refinement. These contributions enable both precise boundary localiza- tion and scalable feature encoding. Experimental results on the COCO dataset demonstrate that EFDTR outperforms existing polygon- based approaches while remaining com- petitive with mask-based methods. We hope that the phase assignment of the Elliptic Fourier Descriptors will inspire further ad advnnements in polygon contour learning and ap- plications.
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Impact Statement
This work introduces EFDTR, an instance segmentation framework using learnable Elliptical Fourier Descriptors for accurate and compact contour representation. By resolving regression ambiguity in the frequency domain, it improves both precision and efficiency. EFDTR shows potential in applications like autonomous driving and medical imaging. As it relies on public datasets and avoids sensitive data, societal risks are minimal, though caution is advised in safety-critical deployments.
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Reviews:
[{"review_id": "vMCzjslkMS", "reviewer": "Reviewer_tj3E", "summary": "The paper proposes an instance segmentation method for images, based on the prediction of a polygon as a series of connected points instead of a more commonly used pixel mask. The contour of a polygon capturing an instance is decomposed with a Fourier decomposition. The authors propose a transformer-like architecture that extracts multi-resolution features from input images, predicts coefficients of the Fourier decomposition, samples the obtained contour, and refines the sampled point (polygon vertices) with several additional transformer-like layers to predict the final positions of the vertices. The approach is evaluated on COCO dataset and compared to the other mask-based and polygon-based instance segmentation baselines. A series of ablation experiments empirically justifies the choice of some important hyperparameters.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer tj3E\nWe sincerely appreciate your time and effort in reviewing our paper and are glad to provide detailed responses to your insightful questions and suggestions.\n\n### Clarification on Method\n\nWhile higher-order EFDs can offer finer contour approximations, the concern regarding \"same phase but different amplitudes\" may stem from a misunderstanding of the concept of phase in the EFD frequency domain versus angular coordinates in polar representations.\n\nIn our approach, Elliptic Fourier Descriptors establish a **bijective mapping** between each contour point \\( p(\\theta) \\) and a unique phase value \\( \\theta \\in [0, 2\\pi) \\). This bijection guarantees a one-to-one correspondence between contour points and their associated phase values in the frequency domain.\n\nOur point regression strategy exploits this property to avoid ambiguities present in Cartesian and polar systems. For example, in polar coordinates, a ray from the centroid may intersect the contour at multiple points sharing the same angle but differing in radius. In contrast, under EFD, such points are associated with distinct phase values, as illustrated in the figure below.\n\n[img](https://upload.cc/i1/2025/04/01/rE6OP8.png)\n\n\n### Response to Vanilla Ellipse\n\nTo further validate the effectiveness of our two-stage architecture and the role of the first-stage EFD prediction, we conducted additional experiments on the SBD dataset in response to the reviewer’s concerns:\n\n- **Type 1**: Identical to the original model, except the first stage predicts a naive inscribed ellipse.\n- **Type 2**: Based on Type 1, the number of layers in the first stage is reduced to 2, while the second-stage polygon decoder is deepened to 6 layers.\n\n| Model     | 1st Stage Layers | 2nd Stage Layers | AP\\(_{vol}\\) |\n|-----------|------------------|------------------|--------------|\n| Original  | 6                | 3                | **70.2**     |\n| Type 1    | 6                | 3                | 59.5         |\n| Type 2    | 2                | 6                | 62.7         |\n\nThe results demonstrate that the EFD-based first-stage prediction significantly outperforms the naive ellipse initialization. Moreover, increasing the depth of the second-stage decoder alone does not compensate for the lack of EFD-based guidance.\n\nWe attribute the superior performance of our design to the following:\n\n1. Although both are ellipses, the 1st-order EFD behaves more like a PCA-based initialization, better capturing the object's principal orientation and shape distribution.\n2. EFD provides a more principled framework for assigning regression targets than heuristic elliptical approximations.\n\n### Response to Larger Backbone\n\nYou raised an important point regarding the potential benefit of a stronger backbone in enhancing the learning of high-order EFDs. We conducted experiments with a Swin-L backbone using a 1× training schedule on the COCO *val2017* set:\n\n| EFD Order | Schedule | Backbone  | mAP(val2017)  |\n|-----------|----------|-----------|---------------|\n| 1         | 1×       | Swin-L    | 44.1          |\n| 4         | 1×       | Swin-L    |  33.2   |\n| 1         | 1×       | ResNet-50 | 40.6          |\n\nInterestingly, even with a more powerful backbone, the 1st-order EFD still achieves the highest mAP. We believe that, under the current architecture, high-order EFD parameters are difficult to optimize effectively, introducing instability in the second-stage point sampling. Conversely, the 1st-order EFD serves as a robust and easily learnable target, similar to rotated object detection, which helps stabilize the overall training.\n\n### Response to Model Design Question\n\n> *Remove the EFD decoder and just use a polygon decoder with more layers.*\n\nThis experiment is in the **Response to Vanilla Ellipse** section.\n\n### Additional Remarks\n\nOur method is polygon-based, where both the output and the supervision signals are sequences of points. Within the domain of polygonal contour prediction, our method surpasses all prior work. Although in terms of AP, pixel-based methods outperform ours, we emphasize that our vector-based formulation offers a unified and end-to-end framework that holds practical value in applications such as vectorized HD map construction and remote sensing vectorization. We believe our contribution provides an effective approach for tasks requiring structured, vectorized outputs.\n\nFor further comparisons on additional datasets and inference efficiency, please refer to our responses to other reviewers. Once again, we sincerely thank you for your valuable feedback.", "claims_and_evidence": "1. The paper claims to obtain the best instance segmentation results among polygon-based methods. To support this claim they compare their method to a series of polygon-based baselines (the latest are from CVPR’22).\n2. The paper claims that their approach is a competitive alternative to pixel mask-based methods. To verify, it is compared to some mask-based methods.", "methods_and_evaluation_criteria": "The proposed method that decomposes the contour of the polygon using Fourier decomposition and uses a transformer-based architecture to infer the coefficients of the decomposition looks valid from a theoretical point of view. However, I suspect the method does not work well in practice. In theory, high-order decompositions are desirable because the instance contours have non-trivial topology, and low-order approximations are not able to distinguish the contour point samples with the same phase but different amplitudes. However, the ablation in Table 3 shows that the method works best with the first-order approximation (which approximates the contour with an ellipsoid). That fact raises the question of whether the decomposition is required at all. Given that, the method could have just sampled the initial points on a circle and just used the polygon decoder with more layers to infer the final positions of the vertices.", "theoretical_claims": "There are no theoretical claims in the paper.", "experimental_designs_or_analyses": "Overall, the experiments are designed well; the ablations make sense. However, the authors only used relatively old ResNet-50/101 backbones for their method, which makes the approach significantly weaker compared to the overall state-of-the-art methods.", "relation_to_broader_scientific_literature": "I am not sure about the subfield of polygon-based instance segmentation, but overall the authors do not include several important prior works. For example, even on the simplest ResNet-50 backbone, Co-DINO-Deformable-DETR++ [1] achieves AP of 52.1 (vs 43.6 reported for the presented method) when trained for the same number of epochs. The corresponding paper also contains multiple references to other methods that produce better results.", "essential_references_not_discussed": "[1] Z. Zong, G. Song, Y. Liu. DETRs with Collaborative Hybrid Assignments Training. In ICCV’23.", "other_strengths_and_weaknesses": "The authors try to make a case for their method by carving a niche for polygon-based instance segmentation methods and trying to push their results, but it is hard to justify the niche if the difference between the state-of-the-art and the proposed methods is that large (AP = 45.1 vs 65.9 best for [1] with ViT-L backbone). The fact that the high-order decompositions do not work well does not help the case.", "other_comments_or_suggestions": "I suggest the authors consider using other backbones and try to make the high-order approximations work properly.\n\n## Update after rebuttal\n\nFirst of all, I would like to apologize for my initial misunderstanding of the proposed method. Indeed, the segmentation contour is projected on the approximated decomposition in a bijective manner, so I withdraw this concern and, as a result, am willing to improve my rating.\n\nAfter the rebuttal, my concerns mostly stayed the same:\n\n1. The proposed method does provide a way to project segmentation contours on ellipsoids (1st order approximations), which is of some value and likely helps the method, but the fact that the use of any high-order approximations lowers the performance is concerning and might indicate that the method is either severely limited in its potential of is technically not correct.\n\n2. The authors operate in a subfield of the polygon-based segmentation and only choose very weak mask-based baselines for comparison. As I mentioned in the review, there are mask-based instance segmentation methods that outperform the proposed method using the same back-bones by a large margin (Co-DINO-Deformable-DETR++ achieves AP of 52.1 vs 43.6 reported for the presented method on ResNet50). The gap further widens with the use of the more up-to-date backbones. I agree that a direct comparison of these classes of methods is not entirely fair, but it is hard to ignore the existence of these related methods and not diminish the importance of the presented method. To the very least, the paper should contain a baseline that takes modern superior mask-based segmentations and converts them to the polygon-based format. This seems like a much fairer competition compared to outdated mask-based methods with worse performance. If the masks can be converted to polygons without a significant loss in quality, the presented method loses its significance.\n\nOverall, it is still hard for me to recommend acceptance due to these concerns. There are some merits in the presented work but to me, the approach seems to be too far from the current state of the art (or to the very least it is hard to properly assess that).", "questions_for_authors": "Have the authors tried to remove the EFD decoder and just use a polygon decoder with more layers?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7bYt5OjM3e", "reviewer": "Reviewer_3kZq", "summary": "This paper devises a method for regressing vertex positions using Elliptic Fourier Descriptors (EFDs). Furthermore, it proposes a learnable transformer architecture to incorporate these EFDs. The transformer pipeline consists of two stages: 1) a transformer predicting EFDs to get coarse instance regions and 2) a transformer to decode the EFDs into more precise polygon instance segmentations. In order to represent multiple polygons as a single EFD, this approach connects polygons together. To determine the connectivity, this method computes distances between polygons and uses a minimum spanning tree of the resulting graph. To supervise the EFD regression, an L1 loss is used whereas a smooth-L1 loss is used to supervise the polygons. The overall approach is compared with both polygon and pixel-based methods showing improved performance compared to polygon methods and comparable performance with pixel-based methods. Key components of the method (number of decoder layers, EFD prediction order, and others) are thoroughly ablated justifying their importance.\n\n## update after rebuttal\nI have read the rebuttal and it addresses my main concerns. I am maintaining my score of accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer 3kZq\n\nThanks a lot for the time and effort you invested in providing the detailed reviews. Regarding the current weaknesses you pointed out, we are glad to give our responses.\n\n### Summary and Strengths\n\nWe are pleased that you found our method well motivated, the design sound, and the paper clearly written. We're also glad that you appreciated our comprehensive ablation studies and performance improvements over existing polygon-based methods.\n\n### Weakness: Comparison to Pixel-Based Methods\n\n> *\"The main weakness would be that this method still performs worse than some pixel-based approaches. While these polygon methods might be more efficient, it seems likely that for many applications, performance would be more important and pixel-based methods would be used instead.\"*\n\nThank you for this valuable observation. We agree that in high-accuracy applications, pixel-based methods often remain the preferred choice. However, our goal is to provide a segmentation model that outputs structured data, which is particularly useful in tasks such as bird's-eye view vector map construction and remote sensing vector map generation. We believe our method can contribute to these areas by offering a novel solution.\n\nAdditionally, we provide inference performance comparisons in our submission, which highlight the efficiency of our method. For further context and elaboration, we also encourage referring to our responses to the other reviewers.\n\nOnce again, thank you for your valuable comments and support.", "claims_and_evidence": "The claims made are supported by sufficient evidence.", "methods_and_evaluation_criteria": "Yes. The authors compare to both polygon-based methods which are most similar and pixel-based methods which exhibit state of the art performance.", "theoretical_claims": "N/A", "experimental_designs_or_analyses": "The experimental design appears sound.", "relation_to_broader_scientific_literature": "The findings in this paper advance the state of polygon-based instance segmentation which provides a more lightweight representation than pixel-based methods. The authors demonstrate this advancement by showing that their method archives superior performance as compared to other polygon-based methods such as SharpContour and BounrdaryFormer.", "essential_references_not_discussed": "None that I am aware of.", "other_strengths_and_weaknesses": "Strengths:\n- This paper presents a new method for polygon-based segmentation using EFDs.\n- The proposed method is evaluated against multiple baselines. Results show that this method leads to improved performance over existing polygon-based methods and comparable performance against pixel-based methods.\n- Paper is well written and the method is easy to follow.\n\nWeaknesses:\n- The main weakness would be that this method still performs worse than some pixel-based approaches. While these polygon methods might be more efficient, it seems likely that for many applications, performance would be more important and pixel-based methods would be used instead.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "reICnFbUsr", "reviewer": "Reviewer_GvJY", "summary": "This paper proposes EFDTR, an instance segmentation framework that leverages Elliptical Fourier Descriptors (EFDs) to represent object contours. The approach employs a two-stage Transformer decoder: the first stage predicts low-order (particularly first-order) elliptical Fourier coefficients to capture global shape information, and the second stage refines the polygon with many vertices aligned by phase matching. The authors argue that EFDTR retains the flexibility of polygon-based methods while using frequency-domain matching to resolve ambiguities in vertex assignment, achieving strong segmentation performance in a variety of challenging cases.\n\n## update after rebuttal\nMy concerns have been fully addressed.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer GvJY\n\nThank you very much for your valuable and constructive feedback. We truly appreciate your time and effort. Below, we provide point-by-point responses to the concerns and suggestions you raised.\n\n\n### Response to More Dataset\n\nWe have conducted additional experiments on the **SBD** and **Cityscapes** datasets. The results are as follows:\n\n**SBD Results:**\n\n| Method       | Venue     | AP$_{vol}$ | AP$_{50}$ |\n|--------------|-----------|--------------|-------------|\n| E2EC[1]      | CVPR 2022 | 59.2         | 65.8        |\n| PolySnake[2] | TCSVT 2024| 60.0         | 66.8        |\n| Ours (R50)   | -         | **70.2**     | **78.2**    |\n\n**Cityscapes Results (Val):**\n\n| Method       | Venue     | AP (Val)     |\n|--------------|-----------|--------------|\n| E2EC[1]      | CVPR 2022 | 39.0         |\n| PolySnake[2] | TCSVT 2024| 40.2         |\n| Ours (R50)   | -         | **43.4**     |\n\nOur method outperforms all other polygon-based instance segmentation methods on both datasets, achieving the highest AP. Notably, PolySnake, which is recognized as the best contour prediction model according to Paperwithcode, is surpassed by our approach by 10 AP$_{vol}$\n\n\n### Response to EFD with Higher Order\n\nWe evaluated the impact of different EFD orders on SBD and Cityscapes:\n\n| Dataset     | EFD Order | AP$_{vol}$ / AP (Val) |\n|-------------|-----------|--------------------------|\n| SBD         | 1         | 70.2                     |\n| SBD         | 2         | 66.2                     |\n| SBD         | 4         | 58.6                     |\n| Cityscapes  | 1         | 43.4                     |\n| Cityscapes  | 4         | 32.6                     |\n\nWe observe that the first-order EFD consistently delivers the best performance. We hypothesize that, within the current model architecture, higher-order EFDs introduce optimization challenges, resulting in unstable point sampling during the second stage. In contrast, the first-order EFD acts as a stable and learnable geometric representation, analogous to rotated bounding boxes. Although higher-order EFDs are capable of capturing more detailed shape distributions, they tend to be unstable, which adversely affects the performance of the downstream polygon decoder. Below, we present samples of the fitting results using first- and fourth-order EFDs.\n\n[img](https://upload.cc/i1/2025/04/01/XyUFoC.png)\n\n\n### Response to Self-intersection and Degraded IoU\n\nOn the COCO dataset, we apply a minimum spanning tree (MST) algorithm to connect multiple polygons into a single contour. Among the COCO instances annotated with multiple polygons (9.71% of the total), we sample up to four polygons per instance, which accounts for 99.67% of all multi-polygon annotations. With a maximum of four polygons, the likelihood of self-intersection is extremely low. Given the small proportion of instances with more than four polygons, we believe the impact on training is minimal.\n\n\n| polygon number | instance number | proportion |\n|----------------|-----------------|------------|\n|1               | 767315          | 90.28%     |\n|[2, 5)          | 79842           | 9.39%      |\n|[5, 10)         | 2589            | 0.30%      |\n|[10, ∞)         | 203             | 0.02%      |\n\n\n### Response to Holes and EFD Order\n\nIn COCO, instance masks are annotated using simple polygons without regard to orientation, and thus do not include holes. However, the SBD dataset does contain instances with holes. As discussed in the **Response to EFD with Higher Order**, we observed that higher-order EFDs do not lead to better performance. Visualization shows that the 1st-order model captures the outer contour more robustly.\n\nAlthough some hole-containing instances exist in SBD, they represent a small portion of the dataset. Our analysis suggests that the model’s strength lies in accurate outer contour detection rather than specific handling of holes. This could be a limitation of the current model architecture, which we aim to address in future work.\n\n[1] Zhang T, Wei S, Ji S. E2ec: An end-to-end contour-based method for high-quality high-speed instance segmentation[C]//Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. 2022: 4443-4452.\n\n[2] Feng H, Zhou K, Zhou W, et al. Recurrent generic contour-based instance segmentation with progressive learning[J]. IEEE Transactions on Circuits and Systems for Video Technology, 2024.", "claims_and_evidence": "The paper presents quantitative comparisons with prior polygon-based methods (e.g., DeepSnake, PolarMask, DANCE, BoundaryFormer) and some mask-based methods (e.g., Mask R-CNN, Mask DINO). The reported AP values show an improvement in boundary precision.\n\nWhile the results on COCO appear convincing, the experiments are predominantly limited to this single dataset. Additional evaluation on other datasets would further strengthen the claims regarding robustness to multi-polygon or highly complex shapes.", "methods_and_evaluation_criteria": "The approach is methodologically sound, however, broader testing on additional datasets or tasks would better illustrate general applicability.", "theoretical_claims": "There are no formal proofs of new theorems.", "experimental_designs_or_analyses": "The main experiments are on COCO, comparing the proposed method to polygon-based approaches and selected mask-based baselines, focusing on standard AP metrics.", "relation_to_broader_scientific_literature": "The paper compares to standard polygon-based methods like DeepSnake, PolarMask, PolyTransform, BoundaryFormer, as well as to conventional mask-based methods like Mask R-CNN and Mask DINO.", "essential_references_not_discussed": "One key contribution is a Multiple Polygon Connection strategy (via minimum spanning tree) that merges all polygons of an instance into a single closed contour. \n\nResearch on multi-polygon (PolygonGNN: Representation Learning for Polygonal Geometries with Heterogeneous Visibility Graph [https://dl.acm.org/doi/abs/10.1145/3637528.3671738]) and work where multiple surfaces are combined into a closed 3D manifold (PolyhedronNet: Representation Learning for Polyhedra with Surface-attributed Graph) are relevant.", "other_strengths_and_weaknesses": "Strengths\n1. Introducing novel elliptical Fourier descriptors to segment polygons addresses the vertex-matching issue via the frequency-domain phase.\n2. On COCO, it substantially outperforms prior polygon-based methods and approaches certain mask-based baselines.\n\nWeaknesses\n\n1. Experiments are primarily on COCO; more challenging or domain-specific datasets are not tested.\n2. While the paper shows first-order EFD is most beneficial, there is limited discussion on whether certain shapes might need higher orders.\n3. The two-stage design, along with multi-scale feature fusion, may increase computational load; speed comparisons to baseline mask-based methods are not thoroughly explored.", "other_comments_or_suggestions": "1. Evaluating on additional datasets would better demonstrate the method’s robustness.\n2.\tExtending the discussion around the MST-based multi-polygon connection could strengthen the justification and highlight potential edge cases.\n3. Including more analyses of the performance/accuracy trade-off with higher-order Fourier terms might clarify when they can be helpful.", "questions_for_authors": "1. When connecting multiple polygons to form a single loop, is there any risk of self-intersection or artifacts that degrade IoU? How do you handle such scenarios?\n2. How well do higher-order EFDs cope with extreme non-convex boundaries or shapes containing holes? Could second or higher orders be beneficial in such situations?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ETGUXKfnb1", "reviewer": "Reviewer_jXM9", "summary": "The paper presents EFDTR as an innovative solution for instance segmentation, combining the strengths of polygon-based representations with advanced deep learning techniques. The proposed framework not only enhances segmentation accuracy but also paves the way for future advancements in contour learning and applications in computer vision.\n\n## update after rebuttal\nThe rebuttal addressed my concerns, and I updated my score accordingly.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "## Response to Reviewer jXM9\n\nThank you for your constructive comments and insightful suggestions, which have helped us improve the quality and clarity of our manuscript. We address your points in detail below.\n\n\n### Response to Hole Case\n\nThe COCO dataset primarily uses polygon annotations for instance masks, with only 1.2% of annotations using RLE. Polygon annotations in COCO do not explicitly account for holes during the annotation process, meaning that hole structures are generally not represented in the ground truth masks. Therefore, during training on the original dataset, the model has limited exposure to true hole instances.\n\nTo further investigate this, we conducted experiments on the SBD dataset, which includes explicit annotations for hole regions. Upon qualitative analysis, we found that the current 1st-order EFD formulation struggles to accurately capture hole structures. However, using higher-order EFDs—while helpful for capturing fine-grained shapes—led to a drop in overall performance. This trade-off suggests a limitation in the current design, and we consider hole representation as a promising direction for future improvement.\n\n\n### Response to Balance between Sharp and Other Regions\n\nThis is an excellent point. Our current approach samples points uniformly from the EFD-predicted contours in the first stage. While this ensures consistent coverage, it may under-sample sharp corners or highly curved regions. A more adaptive sampling strategy—allocating denser points around such regions—could yield more accurate reconstructions. However, this would require an additional module to learn the sampling distribution dynamically. We greatly appreciate this suggestion and plan to explore it in future work.\n\n\n### Response to Reconstruction Performance\n\nTo evaluate the balance between reconstruction accuracy and inference speed, we tested different vertex group settings on the SBD dataset. All experiments were conducted at an input resolution of 512×512 using an NVIDIA RTX 3090 GPU, with CUDA 11.8, PyTorch backend, and FP32 precision. The results are as follows:\n\n| Model       | Vertex Group | FPS (mean ± std) | AP$_{vol}$ |\n|-------------|--------------|------------------|--------------|\n| Ours (R50)  | 16             | 22.26 ± 0.30       |  67.3    |\n| Ours (R50)  | 8               | 21.29 ± 0.28       |  69.8    |\n| Ours (R50)  | 4               | 18.19 ± 0.16       | 70.2     |\n| Ours (R50)  | 2               | 13.76 ± 0.08       | 70.9     |\n| MaskDINO | -                | 6.45 ± 0.03         | -           |\n| E2EC$^*$  | -                | 36                       | 59.2     |\n\n$^*$ E2EC is tested at an NVIDIA A6000.\n\nThese results demonstrate that our method maintains a favorable trade-off between accuracy and inference speed, especially compared to pixel-based methods like MaskDINO.", "claims_and_evidence": "Most claims are supported.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "Yes", "relation_to_broader_scientific_literature": "No", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Pros:\n+ The idea is interesting.\n+ The paper is well-written and well-structured.\n\n\nCons:\n- Hole Case: The proposed method demonstrates suboptimal performance in scenarios where holes are present in the mask. This is illustrated by the zebra example in Figure 5, where the foot of the right zebra appears to be merged.\n- Details. The reviewer noticed a few missed details when there are sharp changes in boundary, where some SAM-based methods can perform well in such case (the connection point between the motorcycle wheel and the rod). But the reviewer also acknowledges that it is a common case even for mask-based methods. So, is there some balance or trade-off between sharp and other concerns?\n- The reconstruction performance. For the elliptical representation, the reviewer would be pleased to see the reconstruction error raised by the downsampling and speed.", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The submission got 1 negative and 3 positive recommendations eventually. The reviewers were mainly concerned about the performance on real 3D data, evaluations, and technical details. The authors did a good job in their rebuttal and addressed most of these concerns. The reviewers did not actively engage in a discussion although the AC tried to initiate one. The AC read through the submission, all reviews, rebuttals, and the confidential comments to the AC. The AC agreed with most of the reviewers and values the idea. Per this, the AC made a decision of acceptance. The decision was also approved by the senior AC.
Decision: Accept (poster)