DNOD: Deformable Neural Operators for Object Detection in SAR Images
Authors: GVS Mothish
J Rishi
Shobhit Kumar Shukla
Deepak Subramani
Venue: Accepted by TMLR ()

Abstract: We introduce a deep neural operator framework aimed at object detection in remotely sensed Synthetic Aperture Radar (SAR) images. Recent research highlights the impressive performance of the End-to-End Object Detection Transformer (DETR). Nonetheless, in domains like SAR imaging, managing challenges such as speckle noise and the detection of small objects continues to be problematic. To address SAR object detection issues, we present the Deformable Neural Operator-Based Object Detection (DNOD) framework, tailored for SAR tasks. We develop two neural operators: Multi-Scale Fourier Mixing (MSFM) for the encoder and Multi-scale, multi-input Adaptive Deformable Fourier Neural Operator (MADFNO) for the decoder. Detailed evaluations and ablation studies show that DNOD exceeds existing methods, delivering significantly better results with an improvement of +2.23 mAP on the SARDet-100k dataset, the largest SAR object detection compilation. The code is available at https://github.com/quest-lab-iisc/DNOD.

Published in Transactions on Machine Learning Research (11/2025)
DNOD: Deformable Neural Operators for Object Detection in SAR Images
GVS Mothish
mothishg@iisc.oce
Deportment of Computational and Data Sciences Indian Institute of Science, Bengaburu, India.
Rishi
rishij@issc.ac..
Department Computational and Data Sciences Indian Institute of Science, Bengaburu, India.
Shobhit Kumar Shukla
shabhitshhukk665355mmmaaac
Department of Computational and Data Sciences Indian Institute of Science, Bengaburu, India.
Deepak N. Subramani
deepakns @visc. in
Department Computational and Data Sciences Indian Institute of Science, Bengaluru, India.
Reviewed on OpenReview: https: // openreview. net/ form?? id= tjBqPJaQ72
Abstract
We introduce deep neural operator framework aimed at object detection in remotely sensed Synthetic A Aperture Radar (SAR) images. Recent research highlights the impressive perfor- mance of the End-to-End Object Detection Transformer (DETR). Nonetheless, in domains like SAR imaging, managing challenges such speckle noise and the detection of small objects continues to be problematic. To address SAR object detection issues, present the Deformable Neural Operator-Base Based Object Detection (DNOD) framework, tailored for SAR lasks. We develop two neural operators: Multi-Scale Fourier Mixing (MSFM) for the encoder and Multi-scale, multi-input Adaptive Deformable Fourier Neural Operator (MADFNO) for the decoder. Detailed evaluations and ablation studies show that DNOD excceds existing methods, delivering significantly better results with an improvement of +2.23 mAP on the SARDet-100k dataset, the largest SAR object detection compilation. The code is available at https:://githne com./uuett-aa--iiiccc///o/oocea
1 Introduction
Neural operators, emerging from computational physics, have demonstrated significant success in solving Partial Differential Equations (Kovachki et al., 2023). Rooted in operator theory, these neural operators learn mappings between function spaces of infinite dimensions, achieving notable success in numerous appli- cations while inherently maintaining discretization invariance. Neural operators comprise three fundamental parts: (1) lifting module, (2) iterative kernel integral module, and (3) projection module. Kernel integrals operations within the spatial domain that ascertain global interdependenciess crucial for learning infiiteeddimensional function maps. Based on different forms of kernel integral computation, different neural operators such as Fourier Neural Operator (Li et. al. 2020c), Graph Neural Operator (Li et al.) (020d). and Adaptive Fourier Neural Operator (Guibas et al., 2021) have been proposed. In a specific context, the attention mochanism utilized within transformers can be seen as spocial case of kernel integral operations [Kovachki al.) 2023). Recently, neural operators have demonstrated superior performance in computer vision applications such super-resolution (eii & Zhang
2023 Liu & Tang 2025 and inpainting (Guibas

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et al. 2021). However, neural operators have not been employed for the task of object detection in Synthetic Aperture Radar (SAR) imagery, gap this paper addresses.
SAR is an advanced active microwave sensing technology capable of acquiring high-resolution images regard-
less of weather conditions, illumination, or time of day (Trandaaz et al., 2020 Brown 1967
Moreira et all
2013 SAR images can provide much more useful information and be effective military reconnaissance.
marine surveillance, port management, and disaster response applications (Guan al. 2023
Zhang all
2022a; Chen et al., 2020 Zhang et al., 2022b). As modern satellites provide increasingly accessible high- resolution, large-scale SAR images, the demand for sophisticated methods to effectively process large data volumes has increased. Consequently, the precise detection of targets from complex terrestrial environments using SAR images is of great practical importance (Sharifzadeh et al. 2019
(a) Ground Truth
(b) DenoDet
(c) DINO
(d) DNOD (ours)
Figure 1: Qualitative evaluation of DNOD for SAR image object detection against recent state- of-the-art models: (a) Top row is the input SAR image and bottom row is the ground truth of bounding boxes. (b)-(d) The top row shows the learned representation by encoder operator, and bottom row shows detected objects by DenoDet, DINO and our DNOD. Notably, DNOD encoder operator (MSFM) effectively differentiates small objects and other features better than DenoDet. (Traanseno) and DINO (Deformable). underscoring the neural operator's efficiency and superior ability to detect small objects compared to other leading models.
Numerous SAR object detection methods have been proposed, from traditional methods Nitzberg 2007
Migliaccio et al. 2012) to CNN-based methods (Gao et al. 2021). In recent developments, transformers have been introduced for object detection, explicitly known as DETR (Detection Transformers; Carion et al. 2020 ), and have shown superior performance compared to traditional hand-crafted feature engincering methods. Various iterations and modifications
DETR, such as Deformable DETR and DAB DETR. have exhibited outstanding results in the field of object detection, further enhancing their effectiveness and application (Zhu et al.) 2021 Liu et al.) 2022a Zhao et al., 2024 Zhang al.) 2023a Lin et al. 2023b Meng et al. 2021h. An extensive discussion of the different methodologies for object detection is available in the Appendix A.1 Even with the advent and introduction of multiple variants of the DETR model, its elfectiveness in SAR images has been less than satisfactory (Dai et al. 2024).. There have been different challenges associated with SAR images, specifically (i) speckle noise interference (Yue et al. 2020): and (ii) small target challenges (Wan et al., (201)). Given the DETR framework's notable success object detection, attributed its foundation on the transformer architecture, it feasible integrate neural operators into the DETR framework specifically for executing SAR object detection tasks.
2

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Published in Transactions on Machine Learning Research (11/2025)
This paper presents the Deformable Neural Operator for object Detection (DNOD) in SAR images. DNOD is trained and evaluated on the COCO-level large-scale multi-class SAR object detection dataset, SeRDet- 100k (Li et al. 2024b). For an illustrative example of the diversity of the dataset, refer to Appendix A.3 Our methodology employs neural operator architecture within the framework of End-to-End Object Detection using transformers (DETR). We rntuue two architectural components drawn from neural operator con- cepts: (i) The Multi-Scale Fourier Mixing (MSFM) Encoder and (ii) The Multi-Scale Adaptive Deformable Fourier Neural Operator (MADFNO) Decoder. There are two main advantages of using neural operators for SAR object detection: (i) Fourier component in the neural operator reduces the effect of speckle noise in SAR images; and (ii) the discretization invariance property of the neural operator reduces the challenges related to small target detection. Section 4.4 describes comprehensive visual insights.
In summary, our main contributions are as follows.
1. To the best of our knowledge, this is the first work to introduce neural operators for object dctection applications.
2. We develop two novel architectural components, MSFM and MADFNO, specifically designed to enhance object detection performance in SAR imagery. Visualizations of the learned representations highlight the capability of our new operators (Figure
3. We integrate our proposed neural operators within the DETR framework to achieve effective SAR object detection.
4. Through comprehensive empirical ovaluation, demonstrate that our method achieves SoTA per- formance for SAR object detection compared to existing object detection techniques.
2 Preliminaries: Neural Operator
Neural operators (Koaachii et al. 2023 are built to learn cinetion-to-function mappings. Initially introduced to solve PDEs, they have recently been applied to computer vision tasks. Consider an operator G: A + that acts between the function spaces and U. Neural operators are the parametric map G3: + U that approximates and is learned from empirical data or physical principles. Formally, the parametrized neural operator can be expressed as
g = Qo Qoa(WW + Kr bT) o o + K1 + op
(1)
where, P and serve as the lifting and projection operators. The lifting operator raises the codomain to higheer-imennsoonal representation space, while the projection operator reduces the codomain to the output dimension. These operators are typically parameterized multilayer perceptrons and act point-wisc on functions. The function a represents pointwise nonlinearity. Each layer L = 1,..., includes a local operator W, (usually parameterized by a point-wise neural network), a kernel integral operator KG, and a bias function b. Given an intermediate functional representation v, with domain in the t-th hidden layer, kernel integral operator K& is defined
(K)u))z)= = =/= (o()
(2)
where the kernel Ka is learnable neural network with paramcter d. Different neural operators (Equation are defined on the basis of their kernel integrals (Equation 2). Each of these operalor layers is expressed as {o:: D. + Rdv, + {1411: D.+1 + Rdeet1) } using
(x+1(x)= a4t1 (Woerto)+ + / (2 dve()) + bi(z)) Vx e D141.
(3)
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3 Methodology
Building on top of the DETR Iramework (see Appendix A.2 Carion et al. (2020) ) we develop our DNOD
model (Figure 2 by introducing two new neural operator architectural components: (i) The Multi-Scale Fourier Mixing (MSFM) Encoder (Sect. 3.1) and (ti) The Multi-Scale Adaptive Deformable Fourier Neural
Operator (MADFNO) Decoder (Sect. 3.2 Our proposed neural operator modules specifically designed
to learn multi-scale feature maps that have been shown to benefit modern object detection frameworks Liu
et al.
2020
Zhu et al. 2021
Multi Scale Features
Multi-Scale Positional Embedding
Loko
Lobm
Lete
Backbone
MSFM Encoder
MAD FNO Decoder
Initial proposals
Learnable Queries
Fourier enhanced Embeddings
Input Image
Figure 2: Overview of the DNOD framework: DNOD architecture processes input SAR images using a backbone and MSFM encoder to extract multi-scale, Fourier-enhanaee embeddings. These features are passed the MADENO decoder along with initial proposals and learnable queries. The decoder outputs are supervisod by three loss functions (i) classification (ii) one to one matching loss and (fii) one many matching, to guide robust SAR image detection.
3.1 Multi-Scale Fourier Mixing (MSFM Encoder)
Next Encoder Layer
3D Fourier Transform using FFT
Adaplive Frequency Multiplication
3D Inverse Fourier Transform using IFFT
Channel Mixing
Multi-Scale
F(H, W, S)
Spatial Mixing
(H, W,S)
Positional
Embedding
Num Encoder layers
Figure MSFM Encoder: Multi-scale features combined with positional embedding are fed into the encoder. Initially, Fourier transform exccuted across scale, height, and width. This is succooded by spatial mixing and then an inverse transform. Subsequently, channel mixing is applied, and the resulting output is passed to the succeeding encoder layer. This entire sequence is repeated for the specified number of encoder layers, ultimately yielding Fourier-enhanced embeddings.

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Within the DETR framework, multiple encoders have been utilized, such Vision Transformer CCarion et al. 2020j and Deformable Transformer (Zhu et al. 2021 However, these encoders work in the image domain, thereby failing to segregate coherent speckle noise in SAR images that are intermixed with features Dai et al. 2024). Removal of noise before detection can result in missing crucial details for subsequent tasks, rendering an ill-posed problem (Sun el. al.] 2022,, thus necessitating an architecture adept at handling noisy features. We introduce a neural operator framework, named Multi-Scale Fourier Mixing (MSFM) adept at effectively handling multi-scale features and speckle denoising in the frequency domain (Figure 3 Our MSFM is motivated by the success of the spectral convolutions used in the Fourier Neural Operator (FNO) (Li et al.) 2020c) and the efficient token mixer introduced in the Adaptive Fourier Neural Operator (AFNO) Guubas et al. 2021p. These operators employ the convolution theorem to transform convolutions in the spatial realm to elernent-wise multiplications with block diagonal structure in the Fourier domain. The main distinction between AFNO and our MSFM is that MSFM is specifically designed to manage multi-scale features in the Fourier domain, essential for tasks like denoising and object detection.
The MSFM kernel integral for continuous multi-scale variable X D with a kernel function k at a specific token can be expressed as
(x) (F(R)) (8) &A E,
(4)
where. denotes matrix multiplication, and F,F-1 denotes the contimuous Fourier transform and its inverse.
In practice, each MSFM encoder block begins with spatial mixing across multiple scales via the Fourier transform (Zm,n where (m,n) is the index per token), which is followed by channel mixing (=m,)) with a block diagonal structure and the inverse Fourier transform (8m,a). The final Fourier-enhanced embeddings (F (Figure are obtained after multiple encoder blocks. Mathematically, each encoder block can be expressed as
m,, 2m. [X) m,
=(t)
(0) m, 1- Um,n (
(5)
The above formulation improves efficiency, generalization, and speckle noise removal through block-diaggnal channel mixing, shared MLP weights, and soft-thresholding. The pseudo code for the MSFM encoder is provided in Appendix A.5
3.2 Multi-Scale Adaptive Deformable Fourier Neural Operator (MADFNO Decoder)
DETRs face challenges in settings with limited feature resolution, resulting in below-average performance in detecting objects across varying scales. This constraint impairs the model's ability to detect smaller objects prevalent in SAR imagery. Furthermore, employing a transformer decoder hinders the convergence speed. To mitigate these issues, deformable attention modules (Zhu et al. 2021) have been implemented in generic object detection. However, as mentioned earlier (Section 3.1), successful SAR object detection demands addressing both speckle noise and multi-scale features, calling for neural operator. Moreover, in DETRs, the decoder's task is to query objects based on the features produced by the encoder. This requires neural operator that can handle multiple inputs. Recent studies (Jiang et al. 2024 Lehmann et al.) 2025) have explored multi-input neural operators; however, none incorporate deformable methods that are essential for multi-scale feature extraction.
Thus, we introduce the MADFNO (Figure 4 a novel neural operator designed for handling multiple inputs, multi-scale scenarios, and incorporating deformable technique with Fourier mixing.
MADENO takes object queries (Q) input along with features of the encoder (Eo). First, object queries undergo self-Fourier mixing (M) with kernel function a at a specific token s, defined as
= = (FW))))) (8) Vs
(6)
Subsequently, these object queries are combined with encoder embeddings (E.) through the deformable operator (D) obtain the refined object queries, D(M(Q). E.), evaluated as
D(Z, (s) = [B(T,, Fak), ... B(T,, F.r }}; where z= =QQ)..
(7)
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MAD FNO
Channel Moxing
Multi scale Adaptive Deformable
(ngg Rajion)
Fourier Neural Operator
Decoder Block
Moan Pooling Layer
Adaptive Spatial Mixing
Multi Scale Deformable Selection
Slice
Slice
Slice 3
(Rgys Tays Pddio))
concat
FFT
AFM IIFF
Fourier enhanced Embeddings
(Thg: Re
Adaptive Fourier Mixing
(ng Myoincs Tlievls)
FFT and IF FT along
Sampling Location Layer
(72ge hagim)
1D Adaptive Fourier Neural Operator
Object Queries
encoder embeddings serve as input to the decoder. The deformable operator extracts features from the encoder embeddings through sampling locations determined by the object qucries. These extracted fcatures are subscquently subjected to Fourier mixing, resulting in the final object queries for the subsequent decoder layer.
Figure 4: MADFNO Decoder: Object queries along with
Deformable operator (D): Rather than selecting features directly from the encoder output, we partition
the encoder embeddings (Eo) into & different slices such that E, = = and ni Esi = 6, which implies that E, Eo1, E.2. Eos]. where [.] denotes the concatenation operator. This sliced sampling facilitates feature selection by concentrating each slice on distinct relations within E,, similar to the multi- head attention mochanism in traditional transformer models. Note that cach slice contains multi-scale features encoder embeddings.
The sampling locations (T') necessary for the deformable operator obtained from the initial reference points (Rp), which are in turn obtained from encoder embeddings (E,) and sampling residuals (r), such that T Rd + where is learnable and Rd is estimated from E,, The sampling residuals are learned via the sampling location layer (SL), that is,
T1. - (T(s) -...)))-T(())))) SL())(s) where Z Z=M(Q).
(8)
where ry represents the sampling residuals for the 8th slice. To sample features from each slice, sampling location layer takes object queries M(Q) as input and outputs sampling residuals per slice, which are further added to reference points per slice Ryj to obtain final sampling locations per slice Tj, i.e., T3 Rys + rj.

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The final sampling locations derived are continuous; consequently, bilinear interpolation is used to extract features from the encoder embeddings, denoted as B(Tj, Faj) for the 5th slice. All these slices are then concatenated into a single slice of sampled encoder embeddings with dimensions (Mp, hamm, where n = Rpoints * naccaee with Apoints as a hyperparameter denoting the required number of sampling points per leature scale. These embeddings are then concatenated with object queries to form combined embeddings per object query with dimensions (Mgs Np: hdim). Next, a Fourier transform is performed across these sampled features (nz), followed by spatial mixing and an inverse Fourier transform leading Fourier mixing of queries with sampled encoder embeddings. This process is followed by mean pooling operation along the selected features and subsequent channel mixing to produce the final refined object queries. Overall, the kernel integral of the multi-input neural operator MADFNO can be expressed as
K(Q, Eo)ss Mo F (F(K)
(s) Vs z =
(9)
4 Experiments
We first present the datasets used for our experimental analysis and the implementation procedure. We then proceed to evaluate the performance of DNOD in comparison to baseline models. Finally, performed an ablation study to illustrate the significance of each component and its impact on overall effectiveness.
4.1
Experimental Setup
4.1.1 Datasets
The SARDet-100k dataset used our experiments focused on object detection. This dataset comprises 116,598 images and 245,653 instances classified into six categories: Aircraft, Ship, Car, Bridge, Tank and Harbor. As the first extensive SAR object detection dataset, SARDet-100K similar in scale to the widely recognized COCO dataset (118K images) (Lin et al. 2014), benchmark for general object detection. SARDet-100k is constructed by integrating nine different datasets focused on SAR object detection. These datasets exhibit varied polarities and encompass a wide range of resolutions, ranging from 0.1 to 25 meters. The data is collected by utilizing six different satellites, each operating within four diverse frequency bands. The extensive scope and diversity of the SARDet- 100K dataset, as outlined in Table [ accurately represent the real-world obstacles encountered in deploying SAR object detection models across different data sources. We employ intensity images in the present work.
Table 1: Source datasets in SARDet-100K Li al. 2024b Target categories S: ship, A: aircraft, C: car, B: bridge, H: harbour, T: tank.
Datasets
Target
Res, (m)
Band
Polacization
Satellites
License
AIR SARShip Xian et al.
2019
S
1.3mm
VV
GF-3
HHSID (ee et al.
20/20
S
0.5--Itn
C/X
HH. HV. VH, vV
S-1B, TeraSAH-X, TanDEM-X
GNU General Public
MSAR
Xia et sl.
2022
A. T. B.
S 1m
HH. HV, VH, VV
HISEA-1
CC BY-NC 1.0
SADD
Zhang
2022c
8
0,5++3m
HH
TerraSAR-X
SAR- AAR-AIReraft
(Zhicui ml.
2023
A
Im
0
Unl-polar
GF-3
CC BY-XC 4.0
Shipoot (Wang et al 2015
8
3--25m
HH, VV, VH, HV
S-1, GF-3
SSDD Zhang et al.
2021b
S
1--15m
C/X
HH. VV, VH, HV
S-1, RadarSat-2, TemaSAR-X
Apache 2.0
OGSOD
Wong
2023
T
3m
VV/VH
GF-3
SIVED
Ln et al.
2023a
C
0.1, 0.3t
Ka, Ku, X
VV/HH
Aitborne SAR syuthetic slice
4.1.2 Implementation Details
In DNOD and other baselines, the same ResNet-50 backbone (pre-trained on the ImageNet-IK dataset) is utilized for fair comparison of object detection models. DNOD contains MSFM encoder and MADFNO decoder cach having 3 layers, utilizing a hidden fcature dimension of 256. With 1200 decoder object queries, training is accomplished through both one-to-one (Zhang et al. 2023a) and one-to-many matching (Zhao
et al. 2024 losses. Based on Zhang et al. (2023a). we use L. and GloU losses for the regression of the

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bounding box and adopt focal loss with 0.25 and y 2 for classification. Additionally, techniques such Look Forward Twice and Mixed Query Selection are integrated (Zhang et al. 2023a Following the DETR framework, auxiliary losses are introduced after each decoder layer. The model underwent 56 epochs of training on Nvidia RTX A6000 GPUs, with a cumulative batch size of 16. Initially, the learning rate was configured at 10-1, which was reduced by a factor of after 52 epochs. We utilized the AdamW optimizer, with a weight decay rate of x 10-4, Further implementation details are the Appendix A.6
4.2 Results
Table 2: Comparison of SAR Object detection methods on the SARDee--100 dataset. Bold indicates it is better than all models, and an underline is the second best. All the models used a ResNet-50 backbone pretrained (Pre.) on ImageNet. Baseline results are from (Dai et al. 2024
Method
Pre. FLOPs #Params mAP
AP@50 APê75
APs
APM
APL
One-stage
FCOS
IN
51.57G
32.13M
52.52
85.82
54.93
47.01
66.13
57.82
GFL
IN
52.36G
32.27M
55.01
85.16
58.87
49.44
67.29
60.45
RepPoints
IN
48.49G
36.82M
51.66
86.43
53.99
46.66
63.26
53.78
ATSS
IN
51.57G
32.13M
54.95
87.60
58.25
49.89
67.94
58.97
CenterNet
IN
51.55G
32.12M
53.91
86.17
57.31
48.88
66.22
57.74
PAA
IN
51.57G
32.13M
52.20
85.71
54.80
46.00
63.90
57.61
PVT-T
IN
42.19G
21.43M
46.10
77.55
49.00
38.01
59.53
53.35
RetinaNet.
IN
52.77G
36.43M
46.48
77.74
48.94
40.25
59.35
50.26
TOOD
IN
50.52G
30.03M
54.65
86.88
58.41
50.20
66.72
58.60
DDOD
IN
45.58G
32.21M
54.02
86.64
57.23
49.33
64.70
58.02
VFNet
IN
48.38G
32.72M
53.01
84.32
56.32
47.37
65.39
57.99
AutoAssign
IN
51.83G
36.26M
53.95
89.58
55.96
50.14
63.40
54.73
YOLOF
IN
26.32G
42.46M
42.83
74.95
43.18
33.73
56.19
53.57
YOLOX
IN
8.53G
8.94M
34.08
66.77
31.31
28.49
43.06
28.95
Two-stage
Faster R-CNN
IN 63.2G
41.37M
39.22
70.04
39.87
32.55
47.23
42.02
Cascade R-CNN
IN
90.99G
69.17M
53.55
87.33
56.81
49.09
62.89
48.68
Dynamic R-CNN
IN
63.2G
41.37M
49.75
80.96
53.91
43.12
59.72
54.77
Grid R-CNN
IN
0.18T
64.47M
50.05
80.58
53.49
42.43
62.01
52.70
Libra R-CNN
IN
64.02G
41.64M
52.09
83.54
55.81
45.85
63.52
55.40
Ne
IN
63.84G
45.07M
53.15
85.52
57.28
45.67
64.55
58.61
VonVNeXtV2
IN
0.12T
0.11G
53.91
86.01
58.90
47.63
64.67
59.57
LSKNet
IN
53.73G
30.99M
52.39
85.07
56.96
45.15
63.59
59.16
End2End
DETR.
IN
24.94G
41.56M
45.73
78.57
46.87
37.01
58.16
55.58
Deformable DETR
IN
51.78G 40.10M
52.00
88.77
54.03
46.99
63.58
58.55
DAB-DETR
IN
28.94G 43.70M
43.31
78.14
43.10
34.82
56.34
52.62
Conditional DETR
IN
28.09G
43.45M
44.04
77.88
44.40
35.25
56.47
52.86
DINO
IN
81.41G
46.67M
53.40
87.82
56.15
47.05
66.19
61.98
DenoDet
IN
52.69G
65.78M
55.88
85.81
60.16
50.63
68.47
60.96
DNOD (ours)
IN
40.36G
35.67M
56.96
90.36
59.69
52.94
71.22
65.43
Promotion
+1.08
+0.78
+2.31
+2.75
+4.47
DNOD large (ours)
IN
78.01G
43.10M
58.11
91.44
61.31
55.31
70.30
64.36
Promotion
+2.23
+1.86
+1.15
+4.68
+1.83
+3.40

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To assess the effectiveness of our model, conducted comparative analysis with 28 baseline methods (details in Appendix A.4), encompassing variety of categories, including one-stage, two-stage, and end-to- end approaches. This selection, which includes convolution- based models, wransformerbbasee models, and single-shot detectors such as YOLO, ensures a robust evaluation our proposed model. The baseline results were obtained from DenoDet (Dai al. 2024). We report average precision (AP) calculated using slandard COCO (Lin al., 2014) evaluation metrics. We report AP at different IOU thresholds and on different object scales, small (APs), medium (APM) and large (APr).. We report our primary metric, COCO mAP, which calculates the mean of AP scores on 10 IoU thresholds from 0.50 to 0.95 with step size of 0.05.
4.2.1 Quantitative Results
We have benchmarked DNOD with 28 diverse baselines (Table 2 Our DNOD achieved SoTA performance across all metrics at different. ToU thresholds and on all object scales: small, medium, and large. DNOD uses scales with 32 x 32 resolution scale derived from the backbone as the primary feature map. Compared to the previous SoTA model (DenoDet 4 Scale (Dai et al. (0244) on SARDet-100k; our model demonstrated +1.08 mAP improvement, with a 45.7% reduction in parameters and 23.4% fewer GFLOPs. This computational efficiency is due to the neural operator-based encoder and decoder introduced. We also developed larger version of our model called 'DNOD Large' to further enhance the performance SAR object detection, In this design, we utilized four scales with a 64 x 64 resolution scale derived from the backbone as primary feature maps. This enhancement led to an increase in mAP to 58.11, marking an improvement +2.23 over the previous SoTA.
4.2.2 Qualitative Results
(a) Ground Truth
(b) DenoDet
(c) DINO
(d) DNOD (ours)
Figure 5: Qualitative Assessment: (Row 1): DNOD effectively identified partially visible object, which
was not achieved by the other models. (Row 2): DNOD accurately detected an aircraft, which the other leading models failed to recognize. (Row 3): DNOD succeeded in detecting all ships, unlike other models. This analysis shows the superior ability of DNOD to detect objects in diverse settings.

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We present the qualitative analysis of DNOD compared with two leading object detection models, (1) DenoDet (Dai et al. 2024), which is leading model in SAR object detection, and (2) DINO (Zhang et al. 2023a), a leading model for generic object detection. This ensures a diverse evaluation of our model against both SAR object detection and generic object detection models. Figure presents 4 different scenarios, (1) Partially occluded: DNOD successfully identified a partially occluded image, a feat not accomplished by the other models. (2) Similar objects close proximity: DNOD precisely detected two different aircrafts that the other leading models had missed. (3) Smaller objects & (4) Crowded scenario of smaller objects: In both the third and fourth rows, DNOD able to identify almost all of the ships which are much smaller objects compared to the image size, in contrast to the other leading counterparts. All predictions were assessed with a classification confidence greater than 0.5. This qualitative assessment underscores the superiority of DNOD in SAR object detection compared to other leading models, altributed to the discrete invariance property of neural operators (Kovachki et al. 2023
4.3 Ablation Study
Table 3: Ablation comparison of (MSFM) encoder and (MADFNO) decoder
Encoder
Decoder
mAP
AP
AP75
APs
APM
AP,
Deformable
Deformable
52.00
88.77
54.03
46.99
63.58
58.55
MSFM
Deformable
54.47
88.58
57.47
50.73
68.94
62.04
Promotion
+2.47
+33.44
+33.74
+5.36
+3.49
MHSA
MADFNO
55.80
89.67
58.51
51.78
69.86
63.42
Promotion
+3.80
+0.90
+4.48
+44.79
+6.28
4.887
MSFM
MADFNO
56.96
90.36
59.69
52.94
71.22
65.43
Promotion
+4.96
+1.59
4-5.6
+-5.5
+7.64
+.888
To evaluate the effectiveness of the proposed architecture, we perform ablation study comparing various encoder-decoder configurations. As shown in Table 3 replacing the conventional deformable encoder with our MSFM encoder consistently improves performance across all AP metrics. Especially mean average precision (mAP) improved by +2.47 with our encoder on SAR imagery. Furthermore, incorporating the MADFNO decoder in place of traditional deformable decoding significantly improves detection accuracy and has shown +3.80 improvement in mAP. The combination of the MSFM encoder and the MADFNO decoder achieves the best performance, achieving an AP of 56.96, a promotion of +4.96, with notable improvements in A55(92.94) (+5.95) and APM (71.22) (+7.64), demonstrating the effectiveness of the two neural operators.
Effect of MSFM and MADFNO:
Table 4: Ablation comparison of number of feature scales
Feature Scales
mAP
AP50
AP7s
APş
APa
AP,,
DenoDet 4 Scale (Previous SoTA)
55.88
85.81
60.16
50.63
68.47
60.96
DNOD Scale
55.56
89.75
57.66
51.36
70.63
63.49
Promotion wrt SoTA
+3.94
+0.73
+2.16
+2.53
DNOD Scale
56.96
90.36
59.69
52.94
71.22
65.43
Promotion wrt SoTA
+1.08
+4.55
+2.31
+2.75
+4.47
DNOD Scale
58.11
91.44
61.31
55.31
70.30
64.36
Promotion wrt SoTA
+22.3
+5.63
+1.15
+4.68
+1.83
+3.40
We examine the effect of varying the number of feature scales in our architec- ture, adjusting it from 2 to 4. Table 4 reveals that increasing the feature scales consistently improves the overall detection performance. Transitioning to three scales significantly boosts the average precision (AP) 56.97 from 55.56 at two scales, and also improves AP7s to 61.31 and APs of 55.31, highlighting the effectiveness of rich comprehensive multi-scale representations for detecting objects of differing sizes. These findings confirm the crucial role of integrating more scales in our framework.
Effect of Different Scales:
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We have also benchmarked DNOD with ResNet-18 backbone, analyzed its robustness by varying scales and the number of queries, and provided detailed computational analysis with varying batch sizes, including latency and energy consumption. Furthermore, FLOPs vs. mAP comparison is included to assess DNOD's effectiveness against baselines, and experiments with the DNOD encoder using Non-Neural Operator (NO) spectral mixers are presented justify the impact of MSFM. All these results are provided in Appendix
4.4 DNOD in Action: Visual Insights
SAR Image
DINO
DemoDes
(4) DNOD
Figure 6: Comparative Evaluation of Encoder Outputs: DNOD vs. Other leading Models. For images impacted by spockle noise (rows 1 and 2) and those fcaturing small objects (row 3), we obtained the encoder outputs at the same scales from each model. The feature maps generated by DNOD demonstrate robustness speckle noise (rows and 2) and capability in representing small objects (row 3). This evaluation underscores the DNOD's representational strength in challenging scenarios.
For visual comparison purposes and to provide insight into the representations learned by DNOD model, visualize the feature maps obtained from both the MSFM and DINO encoders, logether with the object queries derived from the MADENO and DINO decoders.
Encoder Feature Maps: Two major challenges in SAR object detection are speckle noise and small object size (compared to image resolution). We have visualized our MSFM feature maps in both cases (Figure G). It is evident that our MSFM operator robust tackling both challenges. Furthermore, ensures distinct separation between the object and the background, which facilitates the detection ability of the decoder.
Decoder Query Locations: We selected object queries with classification scores exceeding 50% and projected these queries onto the fcature maps (from encoder) to cxamine the deformable query locations, thus assessing detection lypes in each instance (Figure 7 In both instances (lop and boltom images), MADENO decoder, through its utilization of Fourier components and operator properties, able to target each query on individual objects, using dense representations the boundaries and on the object's surface (top image). In addition, it accurately outlines boundaries with query locations, aiding in bounding-box detection.
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In summary, while DINO's attention is scattered, DNOD effectively concentrates its focus, resulting in sharper boundary localization. From the foregoing discussion, it becomes clear that the integration of our novel neural operators (MSFM and MADENO) into the DETR framework has markedly enhanced the detection of SAR objects. For additional examples, refer to the Appendix A.10.3)
(a) Ground Truth
(b) DINO (DETR. based)
(d) DNOD (Ours)
Figure 7: Visualization of feature maps and object queries derived from MSFM-DING encoders and MADENO DINO decoders. MSFM exhibits effective feature extraction and clear object background sep- aration. MADFNO decoder utilizes Fourier components to densely position queries object surfaces and edges, facilitating more precise boundary detection than DINO, whose attention remains more dispersed.
5 Conclusion and Future Work
We developed DNOD, the first-ofiits-kind neural operator-based encoder called MSFM and a decoder called MADENO within the DETR framework for object detection, showcasing its implementation on SAR datasets. These are new multi-input, multi-scale deformable neural operators. Experimental results and ablation stud- ies show that DNOD offers notable advances over the current leading methods in achieving SoTA perfor- mance. Although SAR object detection was the focus here, the utility of our novel architecture with some modifications might have broader implication for generic object detection and other computer vision tasks using neural operators. Further, SLC/comples phase SAR images would need modifications of both MSFM and MADFNO, which could be attempted in the future. On the whole, the present paper opens exciting new directions for future research.
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A Appendix
A.1 Related Work
CNN-based methods: Convolutional Neural Networks (CNNs) have become particularly successful in computer vision tasks. R-CNN is the breakthrough method that effectively integrated CNNs with region proposals for object dctection (Girshick et al. 2014). Advancements include Fast R-CNN (Ginshick 2015
which employs single-stage training with a multi-task loss, and Faster R-CNN (Reu et al.) 2015), which
integrates the region proposal network for a streamlined end-to-end approach. RetinaNet (Lin al.) 2017)
introduced focal loss for effective dense object detection, while (Tian al.
2019) advanced these approaches
using anchor- and proposal-free strategies within per-pixel framework. Further studies have suggested
new training techniques and objectives to improve object detection (Li et al. 2020a Zhang et. al.] 2020b;
2021a; Zhu et al. 2020 Zhang et al. 2020b). An alternative line of investigation, as demonstrated by YOLO
Redmon et al.) 2016), approaches object detection through a one-step process for the prediction of bounding
boxes. Its popularity was driven by its efficiency in real-time applications. Various developments, such as
(Chen et al.
2021a) and (Ge et al. 2021), have been built on the YOLO framework.
DETRs: With the recent success of transformers (Vaswani et al., 2017) in language modeling, a new paradigm has emerged in object detection, namely DETRs (Carion et al., 2020), , which opened new possi- bilities for integrating encoder-decoder frameworks into object detection tasks. Although this work was not state-of-the-art at the time and suffered from slower convergence problems, it established a new pathway for the field. Subsequently, several works have improved the DETR framework for improved performance and efficiency. Conditional DETR (Meng et al.) 2021) introduced a conditional spatial query technique for the docoder, which addressed the convergence problem DETR. Inspired by dcformable convolutions (Dai et al. 2017) in computer vision, Deformable DETR introduced multi-scale deformable attention- encoders and decoders for improved convergence and spatial resolution. DAB-DETR (Liu et al. 2022a)
employed a different query formulation using dynamic anchor boxes. DINO (Zhang et al. 2023a combined approaches from Zhu et al. (2021) and Liu et al. (2022a), further incorporating denoising queries with a con- trastive denoising strategy, achieving superior performance compared to previous models. Various research
initiatives, such as Zhao et al. (2024); Lin et al. (202bb;; Zang et al. (2022 1 Li et al. (2023a); Chen et al.
(2023): : Dai et al. (2021). ,, among others, have proposed several modifications to the initial DETR model.
SAR Object detection: Tn the literature, SAR object detectors are predominantly developed by adapting current stato-of-tho-art object detection frameworks. Specifically, two-stage approaches, such as Kang et al.) (017)), implement modified R-CNN architectures for object detection in SAR imagery. A variely of faster R-CNN adaptations have been presented (Li et al. 2017]] 2020bb, alongside methodologies derived from RetinaNct (Miao et al. 2022). The Dense Attention Pyramid Networks utilized by DAPN (Cui ct al. 2019) facilitated the detection of objects at multiple scales. The LMSD-YOLO framework (Guo el al. 2022) was enhanced with depthwise separable convolutions, batch normalization, and ACON activation functions. YOLO-FA (Zhang et al., (023b) introduced frequency-adaptive learning components into the YOLO architecture. In line with the advances of DETR in general computer vision, mumerous variants of
DETR tailored for object detection have emerged in SAR images (Zhang et al. 2024 Feng et. al. 2023
Another direction of research has introduced novel methodologies specifically tailored for object detection in SAR images. Li et al. (2024a) proposed space-frequency selection convolution layers specifically designed for SAR object detection. (Zhang et al., 2025b) proposed a benchmark for rotated SAR object detection. Very recently [Zhang et al.) (0255a) have proposed Gamma Distribution PCA enhancod fcature Icarning for SAR larget recognition. Li al. (024bb developed a Multi-Stage with Filter Augmentation (MSFA) pretraining framework for SAR object detection that adapted existing state-of-the-art methods for SAR applications. DenoDet (Dai et al. 2024) employed dynamic frequency domain attention module that performs soft thresholding operations in a transformed domain to enhance object detection performance under high speckle noise conditions.
Neural Operators: Neural operators (Kovackii et al. 2023 differ from conventional neural networks by learning mappings from functions to functions. Initially proposed for PDE solutions, they have subsequently been applied in computer vision tasks. Super Resolution Neural Operator (SRNO) (Wei & Zhang 2023)
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introduces neural operator for computer vision tasks. Later, (Guibas al. 2021 mixing for transformers to improve vision transformers. Very recently DO (Liu & Tang 2025 integrated diffusion models with neural operators and achieved SoTA results in super resolution.
proposed efficient token
A.2 DETR framework
DETRs, as initially introduced (Carion et al. 2020). comprise a CNN backbone for Teature extraction, followed by transformer encoder and decoder Figure The backbone is typically ResNet-50 (He et al, 2016) pre-trained ImageNet (Deng et al. 2009). The backbone takes image 7 input and outputs feature representations F backbone(/). Positional embeddings are added to these backbone features, and a 1x1 convolution layer reduces the channel dimension d before feeding into the encoder. The spatial dimensions H and W are flattened to create d x HW feature map, where HW serves as the sequence length and d as the feature dimension for token mixing in the encoder layer. The encoder outputs refined features X encoder PF+positionn embedding); which serve as keys and values for the cross-allention mechanism in the decoder.
The decoder receives two imputs: (1) object queries Qinit that serve queries, and (2) content from the encoder X that provides keys and values. Each decoder layer queries objects within the encoder content to produce final object queries Qinal decoder(X, Qinit). These object queries are then passed to predic- tion heads, two fully connected networks (FFNs) that output class probabilities C' and bounding boxes B, respectively. The entire framework is trained end-to-end using a bipartite matching loss.
prediction heads
backbone
encoder
decoder
Positional Encoding
FFN
class, box
transformer encoder
transformer decoder
FFN
no object
CNN
FFN
class, box
FFN
no object
object queries
Figure 8: Overview of the DETR framework: DETR integrates CNN backbone with transformer encoder-decoder to perform object detection. The decoder's attention is directed by position-encoded features from encoder and object queries toward relevant image regions. The final class labels and bounding boxes are obtained through feed-forward networks.
A.3
Visual examples from SARDet100k dataset
A.4 Baselines
To evaluate the effectiveness of our model performance, we have conducted comparative analysis against three distinct categories of object detection models.
(i) One-Stage Methods These methods perform localization and classification in single pass, i. directly predict bounding boxes and class probabilitics from image pixels. Such FCOS (Tian et al.] 2019). GFL
[Li et al. 2020a). RepPoints (Yang et al. 2019), ATSS (Zhang et al. 2020b), CenterNet (Zhou el. al.
2019
PAA Kim & Lee 2020), PVT-T Zhou et al., 2022 RetinaNet (Lin et al. 2017), TOOD (Feng et al.
2021
DOOD [Chen et al. 2021b), VFNet (Zhang et al. 2021a), AutoAssign Zhu al. 2020 YOLOF (Chen
et al. 2021a), YOLOX (Ge al. (021)).
(M) Two-Stage methods: A sequential pipeline is used in these methods, initially candidate object regions
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bank
Bank
Tank
tank
aircraft
bricdge
Bank
Tank
lank
Lank
Bank
Barik
Barii
Lamil
Sank
Airrrntt
airernft
Harrbrr
Tierbar
harbar
Figure 9: SAR images from the SARDet100k dataset illustrating six object classes: tank, aircraft, bridge, ship, harbor, and car. Each image highlights instances of a specific class using bounding boxes.
are generates using selective search or Region proposal networks. Subsequently, each region is classified and its bounding box refined for accurate object localization. While this approach typically achieves high
detection accuracy, it is gencrally slower than singlo-stage methods. Such as, Faster R-CNN (Ren et al.
2015), Cascade R-CNN (Cai & Vasconcelos 2019), Dynamic R-CNN (Zhang el. al. 2020a), Grid R-CNN
[Lu et al. 2019), Libra R-C R-CNN (Pang et al. 2019). ConvNeXt
(Liu et al. 2022b). ConvNeXtV2 Woo et al.
2023), LSKNet (Li et al.) 2023b),
(iii End2End methods: These methods eliminate hand crafted components and uses direct set prediction.
Such as, DETR (Carion et al. 2020), Deformable DETR(Zhn et al. 2021), DAB-DETR. (Lāu et al.] 2022a)
2021), DenoDet(Dai al.] (024). This will guarantee fair, robust, and diverse comparison of DNOD model for the context of SAR object detection.
Conditional DETR (Meng
A.5 Pseudocodes for MSFM and MADFNO
This section provides a pseudocode of the proposed operators, MSFM (Figure 10 and MADFNO (Figure The pscudocode methodically describes cach step involved, including the fundamental logic, the various inputs, and the specific compulations.
A.6 Hyperaparameters
In Table 5 we present an extensive and thorough compilation of the hyperparameters alongside the training specifications utilized within the construction and application of the DNOD model, providing a comprehen- sive overview for reference.
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Tensor [b, d, h, 8, c]
w_1, W_2 = ComplerTensor [k, c/k, c/k]
b_1, b_2 ComplexTensor [k, c/k]
def BlockMLP (x): MatMul (x, W_1) b_1
= (z)
return MatMul (x, M_2) + b_2
def (x)
bias - x
rfftn (z, dim= (1,2)))
x. r.rehhap d, h, w//2 1, k, c/k)
BlockMLP(x) (x)
x reshapee (b, d, h, w//2 1, c)
(of
irfftn(z, dim (1,2,3))
return bias
Figure 10: Pscudocode for MSFM with multi scale features, adaptive weight sharing and adaptive masking
Table 5: DNOD Hyperparameter
Parameter
Value
Matcher
MateherrraMMthher
One to Many matcher threshold
0.4
One to Many classification loss coefficient
2
One to Many bounding box loss coefficient
5
One to Many GloU loss coefficient.
2
One to One classification loss coefficient
1
One to One bounding box loss coefficient
5
One to One GloU loss coefficient
2
Positional Embedding type
sine
Positional embedding temperature
20
Number of blocks in Fourier mixing
8
Focal Alpha
0.25
Number of classes
7
Weight Decay
0.0001
Learning rate
0.0001
Learning rate drop
0.1
Hidden dimension
256
No of deformable decoder points Non Max Suppression IOU Threshold No of Queries Channel Mixing Dimension Optimizer
6
0.8
1200
2048
AdamW
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def MADFNO (Eo, 9, Rp):
input -> Eo = Tensor [b, n_1, H, W, c]
Tensor [b, n_q, c]
Rp - Tensor [b, n_9, n_l, 4]
output >Qnfinal Tensor [b, n_q, c]
bias
q M(Q)
r (Q)
ressappe (b, n.9, n_s, n_l, n.p, 2)
Rp [p :, None, ;, None, :2]
Rp
Eo = Eo ressapse,e m_1, H, N, n_s, c//n_s)
EoS GridSample (Eo, T) Bilinear interpolation
Eos Eos reshrese (b, n_1+n_p, c)
-> Encoder
z_ in Concat [Eos, Q, (dim=2)1
embeddings
# Shape of z_imp -> (b, n_9, m_l+n_p, c)
Q -> Object queries
2 rfftn (z_ inp, dim=(2))
Rp Reference points
reshape (b, n_9, n_1in_p//2 + 1, k, c/k)
b -> batch size
2 BlockMLP (z)
n_q -> Number of
reshape (b, n_q, n_l+n_p//2 + c)
queries
2 SoftShrink
n_1 Number of levels
irfftn(z. dim (2))
n_s -> Number of slices
2 = z *z_imp
Height
Qfinal meme (dim=2) return Qfinal bias
W -> Width
c -> hidden feature
dimension
W_1, W_2 - ComplexTensor [k, c/k, c/k]
EoS -> Encoder
b_1, b_2 ComplexTensor [k, c/k]
embeddings Sampled
Qfinal -> Final Object
def BlockMLP (x): MatMul (x, W_1) b_1
queries
x Re)U(x)
return MatMul (x, 2_2) + b_2
def {(x);
# input -> x Tensor [b, n_9, c]
output -> Tensor [b, n_9, c] bias = x
rfftn (x, dim=(1))
x. r..rahape (b, n_q//2 1, k, c/k)
x BlockMLP (x)
reshape (b, n_9//2 + 1, c)
I SoftShrink
irfftn (I, dim (1))
return blas
Figure 11: Pseudocode for MADFNO with multi-scale features and deformable attention
A.7 Additional Ablations
A.7.1 DNOD with Different Backbones
To evaluate the effectiveness of DNOD with other backbones, we experimented with ResNet-18 backbone, instead of the ResNet-50 backbone shown in the main paper. The results tabulated in Table 0 Our
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DNOD surpasses DINO and DenoDet, attaining top mAP of 55.92, with significant enhancements in AP50, AP75, and all object sizes. These findings suggest that the enhanced performance of our DNOD architecture is attributable to the model design, and not to the specific pretrained backbone's architecture.
Table 6: Comparison of SAR Object detection methods on the SARDet-100k dataset with ResNet-18 back- bone pretrained on ImageNet.
Method
Backbone
mAP
AP@50
AP@75
APs
APM
APL
DINO DenoDet
ResNet-18
44.76
78.61
46.41
36.40
56.27
55.03
ResNet-18
54.36
84.40
59.12
58.01 49.10
66.61
DNOD (ours)
ResNet-18
55.92
88.92
56.42
51.50
67.72
59.73
A.7.2 DNOD Encoder with Non-NO Spectral Modules
To demonstrate the benefits of neural operators over non-NO frequency or wavelet domain modules, we examined two encoders that utilize frequency or wavelet components along with soft thresholding: the Fourier mixer (Rao et al., 2021) and the wavelet mixer (Patro & Agneeswaran 2023). These were used in the encoder under the same training conditions and data split, incorporating our Decoder (MADFNO). Table shows the results, illustrating the importance of neural operators SAR image objoct detection. Furthermore, our decoder (MADFNO), is the first to use frequency-based neural operator with cross attention mechanism within the DETR framework for object detection.
Table 7: Comparison of SAR Object detection methods on the SARDet-100k dataset with fourier and wavelet mixing modules. Bold indicates it is better than all models. All the models used ResNet-50 backbone pretrained (Pre.) on ImageNet.
Method
Backbone
mAP
AP@50
AP@75
APs
APM M
APL
DNOD with Fourier Mixers ResNet-50
50.83
85.68
52.82
45.98
64.93
56.57
DNOD with Wavelet Mixers
ResNet-50
39.96
72.35
39.50
34.51
50.92
40.31
DNOD
ResNet-50
56.96
90.36
59.69
52.94
71.22
65.43
A.7.3 Sensitivity Analysis of DNOD to Scale and Number of Queries
Table g presents sensitivity analysis of our DNOD model, evaluating the impact of number of scales (2, 3, 4) and queries (300, 600, 1200). Performance remains largely stable when varying the mumber of queries from 300 to 1200, with mAP varies within a narrow band of +1.2 points. The 4-Scale model achieves the overall best results, reaching a peak mAP of 58.11 with 1200 queries, while the 2-Scale model shows slightly lower performance. Increasing the number of queries does not always yield improvements; for example, in the 3-Scale and 2-Scale models, higher queries sometimes lead minimal drops in AP metrics. These results indicate that DNOD is relatively insensitive to query count, and its detection performance is primarily governed by the scale design rather than the query size.
A.8 Computational Cost Analysis
We have performed additional experiments for benchmarking GPU-measured latency (ms per image) in terms of Median latency (50th Percentile Latency) and 95th Percentile Latency, Throughput (FPS), Peak Memory (MB), Avg Power (Watt), Energy/Image (Joule) for various batch sizes as shown in Table 10
Figure 12 presents the trade-off between computational complexity (FLOPs) and mean Average Precision (mAP), with circle size representing the number of parameters. Our DNOD variants achieve balance by de- livering higher accuracy at significantly lower computational costs than many existing detectors. DNOD and DNOD-large are positioned at the top of the accuracy spectrum, outperforming baselines with comparable even higher FLOPPs. A key observation is that DNOD achieves competitive accuracy while maintaining
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Table 8: Accuracy-speed-anee memory tradeoff for DNOD large for different number queries and scales.
Model
Num Queries
mAP
Throughput
Peak Memory
300
56.88
11.76
937.10
DNOD large 4-Scale
600
37.85
11.87
938.27
1200
58.11
11.22
1070.30
300
57.01
14.02
870.96
DNOD large 3-Scale
600
57.70
13.84
872.13
1200
57.51
13.00
981.06
300
56.63
16.04
804.31
DNOD large 2-Scale
600
57.00
16.46
806.45
1200
56.98
16.01
889.76
Table 9: Performance metrics for DNOD large across different number of queries and scales.
Model
Num Queries
mAP
AP@50
AP@75
AP3
AP .M
AP, L
300
56.88
90.54
59.94
53.51
69.63
62.22
DNOD 4-Scale
600
57.85
91.45
60.85
55.24
70.72
63.99
1200
58.11
91.44
61.31
55.31
70.30
64.36
300
57.01
90.83
60.23
53.78
69.52
64.10
DNOD 3-Scale
600
57.70
91.00
60.89
54.97
70.33
64.39
1200
57.51
91.44
60.74
55.25
70.19
64.27
300
56.63
90.27
59.21
53.34
70.02
63.76
DNOD 2-Scale
600
57.00
90.42
59.40
54.07
69.99
64.23
1200
56.98
90.82
59.95
53.57
70.57
63.13
moderate FLOPs, highlighting its efficiency. This analysis demonstrates that DNOD effectively balances per- formance and efficiency, making it suitable for practical deployment. where both accuracy and computational cost are critical.
Table 10: Performance metrics DINO and DNOD across different batch sizes.
Model
Median Latency
P95 Latency
Throughput Peak Memory
Avg. Power
Energy /Image
1
28.61
29.85
34.71
332.73
192.47
5.55
2
32.85
34.02
60.54
472.15
263.91
4.36
DINO
42.55
43.59
93.61
749.63
349.36
3.73
8
75.67
76.16
105.49 49
1309.63
384.55
3.65
16
151.16
152.22
105.74
2425.25
386.17
3.65
25.41
25.84
39.22
413.90
128.10
3.27
2
26.25
26.47
76.03
667.86
219.55
2,89
DNOD
4
33.71
33.84
118.58
1189.54
306.99
2.59
8
60.74
60.88
131.57
2229.51
332.61
2.53
121.78
121.90
131.36
4315.27
338.96
2.58
A.9 Resolution Invariance
To understand the impact of resolution invariance property of neural operators, Figure 13 visualizes repre- sentations extracted from trained models: (a) DNOD (our approach) and (b) DINO. These representations are derived from the same image at different scales. Bilinear interpolation was employed to modily the input scale. Our model demonstrates effective functionality at any resolution, despite having been trained only at a resolution of 512 x 512. contrast, DINO, though specifically designed for multiscale input handling ZZang et al. (2023a),, experiences breakdown beyond certain resolutions. Visualizations also clearly demonstrate that DNOD can distinguish between the background and objects more clearly.
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FLOPs vs mAP (Circle size proportional to #Params)
ONOD large (ours)
DNOD (ours)
Den
GFL
ATSS TOOD
AutoAssign
55
DDOD
enterNet
BING
Cascade R-CNN
ConvNeXtv2
VENet
RCDS
CorNiekt
RepPoints
Libra R-CNN
LSKNet
Grd RCNN
50
Deformable
Dynamic R-CNN
mAp
BETR
PVT-
RetinaNet
45
Conditional DETR
DBB DETR
OOLDF
40
Faster R-CNN
35
YOLOX
25
50
75
100 FLOPs (G)
125
150
175
Figure 12: Comparison of FLOPs versus mAP, with circle size proportional to the number of parameters. DNOD achieves high accuracy with moderate computational cost, outperforming several state-oftthe-at detectors in efficiency.
DenoDet Dai et al. (004]] is resolution-specific due to the "Deformable Group FC" component in its ar- chitecture, which relies on parameters that are resolution dependent. As such, using a different resolution requires retraining with new parameters. Thus, we did not include DenoDet in this experiment. Further- more, although our framework employs encoder and decoder neural operators, the backbone is limited to ResNet-50, which is not neural operator. As such, we demonstrate the resolution invariance property solely via visualizations. In future research, we plan to substitute our backbone with a neural operator framework, which allows us to present metrics for various resolution inputs.
A.10
Additional Results
A.10.1
Quantitative Analysis Across Object Categories
Table II presents an evaluation of our model for each of the six distinct categories (ship, aircraft, car, tank, bridge, harbor) that comprise the SARDet-100k dataset. To determine the model that performs the best in all categories, we implemented ranking methodology follows. The models were individually ranked for each category, and then the mean rank was calculated for every model. In particular, our model achieved the lowest mean rank compared to all other models, highlighting its superior performance across all classes.
A.10.2 Qualitative Analysis Low Quality Image Context
To augment the qualitative analysis shown in the main paper, we introduced additional comparative studies in low-quality image contexts. Figures 14 and [5]dcpict a scenario involving the detection of multiple classes (bridge and harbor) in noisy Low-resolution SAR images. These figures show that our DNOD accurately detected the bounding boxes for both classes, including multiple instances of the same class.
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S12
800 800
800 800
1026 x 1024
x 1024
2048
a
Figure 13: Comparision of (a) DNOD and (b) DINO for resolution invariance by performing inference of pretrained model at multiple scales (256, 512, 800, 1024, 2048)
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Table 11: Per-class average precision comparison with SoTA methods on the SARDet-100K dataset.
Method
Pre.
Ship
Aircraft
Car
Tank
Bridge
Harbor
Avg Rank
One-stage
FCOS
IN
59.79(21)
(594
(0.75
41.78(11)
34.17
63.44
17
GFL
IN
(3.92(6)
(7)
(2)
(4
36)41(9)
(5.04(6)
6.3
RepPoints
(7.
(2.)
(191
40.69
35.12
(6.71(22)
17.6
ATSS
IN (1153(11))
(5.94(12)
(1.77
(6)20
37)22
(7)48(3)
8.0
CenterNet
IN
61.24
56.35(9)
(1.74
45.31
35.91
63.29
11.8
PAA
IN
60. 60.16(20)
56.17(10)
60.09
41.07(12)
35.96(14)
60.12
15.8
PVT-T
IN
53.30(25)
52.91
59.03
30.20
59.11 22.51
(19)
24.0
RetinaNet
IN 55.36(23)
(2.0
(0.88
32.72(23)
51.12 24.81(26)
(28)
23.6
TOOD
IN 62.28(s)
(1661
(2.53
45.96(4)
36.64())
65.24(6)
8.0
DDOD
IN (2)
56.08(11)
62.48(8)
43.98(9)
36.34(11)
62.87
9.6
VFNet
IN 62.14(9)
55.84(13)
61.97
(2008
(4.11(21)
62.28
13.0
AutoAssign
IN 62.03
55.70(16)
(16)
48.55(1)
38.25(4)
57.45
11.6
YOLOF
IN 52.62
52.71(27) (2.64(26)
22.
23.74(27)
52.42
27.1
YOLOX
IN
46.08(30)
(3083(30)
53.43
26. 26.26(25)
(3.14
18.95
28.5
Two-stage
Faster R-CNN
IN
50.45
50.36
(258
(4.
33.11 18.69
27.6
(29)
Cascade R-CNN
IN
(1. 56.43(8)
63.25(2)
44.35(8)
36.89(6)
53.81
(26)
8.5
Dynamic R-CNN
IN
61.32(13)
53.86(23)
(0.00(23)
33.68(22)
55.25 34.40(19)
20.5
Grid R-CNN
(0.43
55.61(16)
61.94(13)
36.03(21)
31.16(23)
55.13(24)
19.3
Libra R-CNN
IN
61.32(1a)
54.03(21)
61.56
38.12(18)
35.97
61.50
15.8
NeXt
IN
60.55
57.35(3)
62.13(11)
38.12(18)
36.81
63.95
11.0
ConvNeXtV2
IN
61.48
55.83
63.23
39.65
39.16(3)
64.09
9.3
LSKNet
IN
(223
56.76(6)
62.74())
36.09
35.01
64.38
12.6
End-lo-Fru End
DETR
IN
54.94(24)
51.17
50.11
(29)
(29) (6.06
32.80
59.31 (18)
24.1
Deformable DETR
IN
(0.94
54.16(20)
61.22
39.14(17)
36.09
60.46
16.5
DAB-DETR
IN
53. (3)1))2))
50.32
49.47
24.06(28)
28.47(25)
55.07 (25)
27.0
Conditional DETR
IN
(2.77(27)
49.58(29)
51.00
22.73(30)
29.98(24)
58.16
26.3
DINO 4-Scale
IN
64.87(4)
56.78(5)
62.72(6)
(9.80
34.97
61.26
10.3
DenoDel
IN
64.91 57.36(2)
63.66
45.79
36.39
67.17
4.1
* DNOD (Ours)
IN
64.65(5)
56.48(7)
62.60
40.73
43.27
74.05
5.5
DNOD large (Ours)
66.31
57.18
62.28
48.40
43.12
71.38
3.6
A.10.3 Additional Visual Insights
Figure 16 displays additional visual representations of DNOD's performance efficacy. Significantly, DNOD accurately localizes bounding boxes compared to DINO, distinguishing from DETR-based models for object detection in SAR imagery.
28

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Published in Transactions on Machine Learning Research (11/2025)
harbor: 631
DARORLO.77 2773
Deidgee: 0.56
(a) DenoDet
(b) DINO
brigge. 0.73
EIRgOI 25
Tridgw:
Crisgo:
Bridpe
(c) DNOD (ours)
(d) Ground Truth
Figure 14: Qualitative comparison of DNOD predictions with those of DenoDet Dai et al. 2024 and DINO (Zhang et al. 2023a) shows that while DenoDet successfully identified only the harbor, and DINO managed detect only the bridge, our DNOD demonstrated superior performance by accurately identifying both the harbor and the bridge.
29

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Published in Transactions on Machine Learning Research (11/2025)
Drioge:
HaP S
(a) DenoDet
(b) DINO
(c) DNOD (ours)
(d) Ground Truth
Figure 15: Qualitative comparison of DNOD predictions with DenoDet Dai et al. 2024), and DINO (Zhang
et al. 2023a Both of the basclines only detected single harbor, but our DNOD detected both the harbors.
30

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Published in Transactions on Machine Learning Research (11/2025)
Clank Lank
BAN
harboo
tank
harbor
harber
CNI
(a) Ground Truth
(b) DINO
(d) DNOD (Ours)
Figure 16: Exploring the Detection Models (a) DINO and (b) DNOD; This is an extension to the analysis from main paper.
31


Reviews:
[{"review_id": "NpzJM8708E", "reviewer": "Reviewer_VMU3", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose two spectral-type neural network architecture components that are specialized for SAR; these build off the existing DETR architecture. They train/evaluate it on the SARDET-100k dataset, and compare that result with metrics published in the literature.", "claims_and_evidence_yn": "No", "claims_explanation": "1. The main argument for the paper's novelty is: \"this is the first work to introduce neural operators for object detection\"\nRefutation: \"Enhancing Object Detection in Layout Analysis: Leveraging Vision Transformer and Fourier Neural Operator\". I don't currently have access but maybe this as well: \"Location Invariant Flood Prediction using Fourier Neural Operator\"\n\n2. New architecture components are indeed novel\n\n3. Integrating with DETR: this isn't a novel contribution right? It's necessary for your components to function / not a separate thing from #2.\n\n4. You evaluated ONLY the new algorithm; the comparison is copied from the DenoDet paper. That is not a \"comprehensive empirical evaluation\". Please tell me if I am mis-reading this.", "audience_yn": "No", "audience_explanation": "Maybe I am being too harsh: of course there must be someone in the TMLR audience who want to see the latest SAR numbers. But it seems more like an incremental SAR conference result-- not a machine learning journal result.", "requested_changes": "- Can you justify the new architectural components using principles not specific to SAR? A more detailed results section might be able to speak toward this, if you can characterize what your model is doing differently than the others (what is the nature of the success it found that other models did not)\n\n- Why are the two papers I mentioned above not neural operators applied to object detection?\n\n- \"a Fourier transform is executed across scale, height, and width\" -- Does this mean Fourier across the scale axis? I've never heard of that: what is the dimension, resolution, and units of this axis?  It looks like you had only \"2-4 scales\"\n\n- table 2: your bold/underline scheme is inconsistent (dnod regular and large both bold, no underline for AP@50, parameters and flops),\n\n- \"The baseline results were obtained from DenoDet (Dai et al., 2024)\" -- you should cite Table 2 as a table taken from another paper. It's 93% from that other paper.", "broader_impact_concerns": "N/A", "additional_comments": "I apologize for the tardiness of my response.", "extra_scores": {}, "extra_text": {}}, {"review_id": "FhIUuoeMTH", "reviewer": "Reviewer_6B9R", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors propose a new deep learning method, DNOD, for object detection in SAR images, addressing SAR‑specific challenges such as speckle noise and the detection of small objects. In particular, they leverage neural operators by introducing MSFM and MADFNO as the encoder and decoder, respectively, on top of DETR. As a result, they achieve state‑of‑the‑art performance on SARDet‑100K. The contributions of this paper can be summarized as follows:\n- First application of neural operators to object detection: This is the first study to introduce neural operator techniques into the object detection domain, especially for SAR imagery, thereby opening a new research direction.\n- A novel architecture tailored to SAR: The authors develop two new modules specialized for SAR (the MSFM encoder and the MADFNO decoder) to reduce speckle noise and improve small‑object detection.\n- Integration into the DETR framework: They integrate the proposed neural operator modules into a DETR‑based detector and build DNOD, an end‑to‑end trainable and deployable object detection model.\n- Demonstration of strong detection performance: On the SARDet‑100K dataset, the method surpasses existing approaches in SOTA accuracy, notably improving mAP and the detection of small and medium objects, while also showing favorable efficiency in terms of accuracy per parameter.", "claims_and_evidence_yn": "No", "claims_explanation": "The empirical claim of performance gains on SARDet‑100K is relatively well supported from both quantitative and qualitative perspectives. The paper provides comparison tables using the same dataset and evaluation metrics against a large set of baselines (one‑stage, two‑stage, and DETR‑style detectors), showing that DNOD and DNOD‑Large outperform prior best methods (such as DenoDet) in mAP, with especially clear improvements for small and medium objects.\n\nBy contrast, the paper’s more central claims, namely \"why it has to be a neural operator\" and \"how discretization invariance alleviates the challenges posed by small objects and scale diversity'', are not directly substantiated by the experiments. The rationale that frequency‑domain mixing and soft‑thresholding for noise suppression are effective is reasonable, but these are not advantages unique to neural operators. Similar effects can be achieved with AFNO‑style token mixers, Fourier/FFT blocks, wavelets, or frequency attention (e.g., DenoDet). There is no head‑to‑head, equal‑recipe comparison under counterfactual conditions (\"being an NO\" versus \"not being an NO but using frequency modules\"), nor are NO‑specific consequences directly visualized and quantified (e.g., extrapolation across training/evaluation resolutions, cross‑sensor or cross‑band generalization, superiority under noise‑strength sweeps). Consequently, the necessity claim runs ahead of the presented evidence. Furthermore, because the current evaluation is limited to COCO‑style, amplitude‑only (intensity) SARDet‑100K images and does not incorporate SAR‑intrinsic physical quantities such as phase, coherence, or polarization, the implied advantage of \"NOs specifically for SAR\" is less convincing.", "audience_yn": "Yes", "audience_explanation": "Object detection in SAR images has been studied for a long time, but because the data are satellite images, most targets are small, making the task persistently challenging. How to detect small targets is of high interest to the machine learning community. In addition, unlike general RGB images, SAR has distinctive frequency characteristics and a special sensing modality, so designing methods that handle such data in a principled way is a highly interesting problem.", "requested_changes": "- The core claim of this paper is that neural operators (NOs) bring an essential advantage to SAR object detection. To directly support this claim experimentally, please conduct head‑to‑head comparisons against non‑NO frequency‑domain modules (AFNO‑like token mixers, Fourier/FFT blocks, wavelets, DenoDet‑style frequency attention) under the exact same training recipe and data splits. (Critical)\n\n- If you wish to claim ''generalizability to standard object detection,'' please either provide initial experiments on optical datasets such as COCO or, alternatively, tone down the claim by explicitly limiting it to SAR intensity imagery. (Strengthening)\n\n- You state that ``the discretization invariance property of the neural operator will reduce challenges related to small targets.'' However, I did not find experiments that directly demonstrate this effect in the paper. For example, do you have concrete results showing that detection performance is maintained when image resolution is reduced? (Critical)\n\n- Operational metrics for inference speed, memory, and energy: beyond FLOPs and parameter counts, please report GPU‑measured latency (ms per image), throughput (FPS), peak memory usage, and batch‑size constraints. Since your approach makes heavy use of FFTs, provide a clear outlook on real‑time feasibility and include measured comparisons against DenoDet/DINO. (Critical)\n\n- Your evaluation is conducted on intensity (amplitude‑only) images. If feasible, please add small‑scale experiments using SLC/complex phase, coherence, or multi‑polarization (HH/HV/VH/VV) data to demonstrate the value of NOs as complex Fourier operators and as multi‑input operators. If this is not currently feasible, at least include concrete plans and technical prospects for such future extensions in the paper. (Strengthening)\n\n- The number of detectable objects depends on the number of queries. Please present accuracy–speed–memory trade‑off curves when varying the number of queries (e.g., 300/600/900/1200) and the number of scales used (2/3/4). (Critical)\n\n- Please verify in at least one or two settings that the same advantages hold with lighter backbones beyond ResNet‑50 (e.g., ResNet‑18 or Swin‑T). It is important to show that the effect of NOs is not backbone‑dependent. (Critical)\n\n- Augment the exposition with intuitive diagrams and analogies that clarify the relationship between the NO mathematics (kernel integrals, lifting/projection, discretization invariance) and the attention structure in DETR, so that readers outside the immediate subarea can follow the ideas. (Strengthening)", "broader_impact_concerns": "N/A", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}, {"review_id": "xUGWyJWELO", "reviewer": "Reviewer_HfNX", "summary": "", "questions": "", "limitations": "", "rating": null, "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": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper integrates two FFT-based operations into a DETR-based model in order to improve on object detection in SAR.  The first (Multi-scale Fourier Mixing \"MSFM\") applies elementwise multiplication with a kernel learned in the 3d fourier transform space of (scale, x, y), which I believe is basically a 3d conv in the spatial domain with possibly large kernel.  The second (Multi-Scale Adaptive Deformable Fourier Neural Operator \"MADFNO\") applies a similar learned fourier-space kernel to features sampled from predicted deformable locations (similar to deformable detr, but using fourier mixing of concatenated Q,E features as the mixing operator instead of attention).  The overall system is evaluated on SAR-100K with good improvements compared to existing detectors.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The performance comparisons and ablation with progressive enabling of the components is enough to show improvement over baselines.  Although that may be just enough to answer \"yes\" here, I think there could be clearer connections between the issues identified (noise vs small objects), and how the mechanics of this method help address them.  See below questions/changes requests.", "audience_yn": "Yes", "audience_explanation": "This appears to be an effective method for SAR object detection.", "requested_changes": "While there are mechanical descriptions of the operations the new components perform, there isn't much that illustrates what they do or their behaviors beyond this, or how these particular operations actually address the issues identified with noise.\n\nFor example, following the very general description of a fourier kernel on p5, p.6 describes that the component performs \"spatial mixing across multiple scales via the Fourier transform ... followed by channel mixing ... The final embeddings are obtained after multiple encoder blocks\".  While this states the steps the module does, I would also have liked to see more illustrative examples of what these operations look like, how they behave, and how this is linked to the issues of noise identified before.\n\nStepping through an example image or region, showing a slice of feature maps and how they behave in the fourier domain vs spatial domain, could help here.  So might measurements on the frequency of the noise somehow compared to the transformations or mixing kernels.  I'm also not sure how this compares to large-kernel depthwise convolutions across space and scale, applied in the spatial domain (does this also help?).\n\nFor the second component with deformations, this is even more the case --- I believe the deformable part is done in the spatial domain (though I'm not entirely sure about that) --- but what is the impact of adding fourier mixing to deformations?  Simply showing an improvement is not enough to really see what this does.", "broader_impact_concerns": "Since SAR object detection is still somewhat newer than RGB and exclusively applied to satellite data, I think at least mentioning some of the current applications of SAR detection (including surveillance) and the impacts of improved detection on them would be appropriate to help ground the larger context here.  I don't think a separate section is needed for this --- adding this to the Introduction section would be sufficient.", "additional_comments": "* what is the input representation to the model?  how is SAR encoded when fed to the model, and is this encoding the same for all comparison baselines?\n\n* it would be helpful to plot a FLOPS vs AP curve for table 2\n\n* eq 6: what is lowercase \"m\" ?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)