DuFal: Dual-Frequency-Aware Learning for High-Fidelity Extremely Sparse-view CBCT Reconstruction
Authors: Cuong Tran Van
Trong-Thang Pham
Ngoc-Son Nguyen
Duy Minh Ho Nguyen
Ngan Le
Venue: Accepted by TMLR ()

Abstract: Sparse-view Cone-Beam Computed Tomography reconstruction from limited X-ray projections remains a challenging problem in medical imaging due to the inherent undersampling of fine-grained anatomical details, which correspond to high-frequency components. Conventional CNN-based methods often struggle to recover these fine structures, as they are typically biased toward learning low-frequency information. To address this challenge, this paper presents DuFal (Dual-Frequency-Aware Learning), a novel framework that integrates frequency-domain and spatial-domain processing via a dual-path architecture. The core innovation lies in our High-Local Factorized Fourier Neural Operator, which comprises two complementary branches: a Global High-Frequency Enhanced Fourier Neural Operator that captures global frequency patterns and a Local High-Frequency Enhanced Fourier Neural Operator that processes spatially partitioned patches to preserve spatial locality that might be lost in global frequency analysis. To improve efficiency, we design a Spectral-Channel Factorization scheme that reduces the Fourier Neural Operator parameter count.  We also design a Cross-Attention Frequency Fusion module to integrate spatial and frequency features effectively. The fused features are then decoded through a Feature Decoder to produce projection representations, which are subsequently processed through an Intensity Field Decoding pipeline to reconstruct a final Computed Tomography volume. Experimental results on the LUNA16 and ToothFairy datasets demonstrate that DuFal significantly outperforms existing state-of-the-art methods in preserving high-frequency anatomical features, particularly under extremely sparse-view settings.

Published in Transactions on Machine Learning Research (12/2025)
DuFal: Dual-Frequency-Awar Learning for High-Fidelity Extremely Sparse-view CBCT Reconstruction
Cuong Tran Van*
cuong. tuong.anduy@ggail.com
FPT Software AI Center, Viet Nam
Trong-Thame Pham
tp0300vark.edi edu
AICV Lab, EECS Department, University of Arkansas
Ngoc Son Nguyen FPT Softwore AI Center, Viet Nam
sonnn450fpt.com com
Duy Minh Ho Nguyen
ho_ minh duy. Sugggnggyenndfii... de Mas Planck Research School for Intelligent Systems (IMPRS-IS), University of Stutigart, and DFKI, Germany
Ngan Le ATCV Lab, EECS Department, University of Arkansas
thile@uork.ed edu
*Co-first authors
Reviewed on OpenReview:
Abstract
Sparse-view Cone-Beam Computed Tomography reconstruction from limited X-ray projec- tions remains challenging problem in medical imaging due to the inherent undersampling of fine-grained anatomical details, which correspond to high-frequency components. Con- ventional CNN-based methods often struggle to recover these fine structures, as they are typically biased toward learning low-frequency information. To address this challenge, this paper presents DuFal (Duul--rreueency-Awii Aware Learning), a novel framework that integrates frequency-doman and spatial-domain processing via dual-path architecture. The core innovation lies in our High-Local Factorized Fourier Neural Operator, which comprises two complementary branches: Global High-Frequency Enhanced Fourier Neural Operator that captures global frequency patterns and Local High-Frequency Enhanced Fourier Neural Operator that processes spatially partitioned patches to preserve spatial locality that might be lost in global frequency analysis. To improve efficiency, we design Spectral-Channel Factorization scheme that reduces the Fourior Neural Operator parameter count. We also design a Pross-Attention Frequency Fusion module integrate spatial and frequency features effectively. The fused features are then decoded through a Feature Decoder to produce projection representations, which are subsequently processed through Intensity Field Decoding pipeline to reconstruct final Computed Tomography volume. Experimental results on the LUNA16 and ToothFairy datasets demonstrate that DuFal significantly out- performs existing state-of-the-art methods in preserving high-frequency anatomical features, particularly under extremely sparse-view settings.
1 Introduction
Computed Tomography (CT) plays a vital role in medical imaging, providing clinicians with detailed, non-invasive visualization of internal anatomical structures to support accurate diagnosis and trcatment planning, Among CT modalities, Cone-Bean Computed Tomography (CBCT) (Alamri et al. (2012) has gained prominence due to its ability to produce high-resolution 3D images with faster scanning times compared to

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traditional Fan-Beam CT Decabooter et al. 2024h. However, achieving high image quality often necessitates higher radiation doses, which raise safety concerns for patients. To mitigate this issue, substantial research has focused on developing low-dose CT imaging protocols, particularly through sparse-view CBCT reconstruction, which reduccs radiation exposure by acquiring fower projection views during scanning.
Traditional reconstruction methods such as Filtered Back Projection (FBP) [Turbell 2001), Feldkamp Davis Kress (FDK) (Turbeli 2001), and iterative techniques like Algebraic Reconstruction Technique (ART) and Simultancous Algebraic Reconstruction Technique (SART) (Andersen & Kak 1984) perform well when dense-view data (2 100 projections) is available. However, their performance deteriorates significantly as the number of views decreases. In sparse-view (20-50 projections) and especially extremely sparse-view (S 10 projections) settings, these methods struggle due to the ill-posed nature of the inverse problem, limiting their applicability in low-dose imaging scenarios.
In pursuing extremely sparsc-view (5 10) CBCT reconstruction, which aims to achieve the lowest possible radiation dose, deep learning has emerged a powerful approach to address the challenges posed by limited projection data. Current sparse-view CBCT reconstruction methods can be categorized into three main approaches. First, neural attenuation field models such Neural Attenuation Fields (NAF) (Zha et al.) 2022b) and SAX-NeRF (Cai et al. 2024) directly model the underlying anatomical structures from X-ray projections. Second, post-processing enhancement methods apply CNN-based architectures to improve low-quality CT reconstructions, including DuDoNet (Lin et al. 2019 for artifact reduction, FBPConvNet (Jin et al. 2017) for filtered pnok-projection enhancement, and various denoising techniques using encoder-decoder frameworks with skip connections. However, CNN-based models are biased toward low-frequency information, as demonstrated in previous studies (Xu et al., 2020 Magid et al.; 2021 Lin et al. 2022 Khayatkhoei &
D Elgamnal 2022), which hinders their ability to capture high-frequency features essential for preserving fine anatomical details in medical imaging. Third, end-to-end implicit neural representation methods such as DIF-Net (Lin et al. 2023), C2RV-Net (Lin et al. 2024c), and Gau-Sian (Lin et al. 2024a) attempt to reconstruct CT volumes directly from X-ray projections without intermediate low-quality CT generation. Despite this direct approach, these methods remain constrained by spatial domain processing limitations that similarly struggle to preserve the figh-frreuency details essential for medical imaging precision.
Recognizing the importance of capturing high-froquency information, FreeSeed (Ma et al. 2023) attempts hard fix with a band-pass Fourier module to learn frequency representations, but this approach lacks flexibility because it is hard-coded into each convolutional layer, making this mechanism difficult to seamlessly integrate into other models. Moreover, FreeSeed only operates as a post-processing denoising method on already-recoostrueeed CT images rather than performing direct reconstruction from raw X-ray data. While global high-frequency processing may effectively capture edges and structural boundaries, small objects such as nodules in medical images require localized patch-wise high-frequency analysis to preserve fine details that are important for detection tasks.
This work introduces the DuFal Frail-Frequenc--Aware Learning) framework for sparse-view CBCT recon- struction. DuFal follows a two-stage design: (i) the Frequency-enhanced Dual-encoding stage includes a dual-encoder that processes each projection with a standard spatial encoder in parallel with a proposed Frequency Encoder, then passes through a Feature Decoder to obtain the necessary features for the next stage. The Frequency Encoder captures complementary global and local spectral cues using multiple High-Local Factorized Fowrier Newral Operator (HiLocPPNO) layers, while the spatial encoder models structural context. in the image domain. And (fi) the Intensity Field Decoding stage, which is standard decoding pipeline from Deep Intensity Field (DIF) that takes the decoded features and reconstructs them into 3D CT volume.
Since Fourier Neural Operators (FNO) exhibit rapid parameter growth as the number of retained frequency modes increases, we introduce Speetral-Channel Channel Factorization (SCF), a weight factorization that substantially reduces parameters while maintaining comparable accuracy. To preserve important spectral information when merging the two pathways, we use Crtes-Attention Frequency Fusion (CAFF), which operates in the frequency domain to combine spatial queries with frequency keys/values at each resolution level. The fused multi-scale features are decoded and subsequently passed through the DIF pipeline for volumetric reconstruction. Both the Frequency Encoder and CAFF are designed as modular units that can be seamlessly integrated into
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existing frameworks. As shown in Table 1 DuFal is, to our knowledge, the first end-to-end method for sparse-view CBCT reconstruction that jointly models spatial features and global-local frequency information.
In summary, our key contributions are as follows:
We propose DuFal, complete end-to-end framework that integrates a dual-cncoding architecture with Frequency-enhanced Dual-encoding for sparse-viet CBCT reconstruction. Our framework processes X-ray projections through parallel spatial and frequency pathways, cnabling simultancous capture of structural context and high-frequency anatomical details that are often lost in conventional CNN- approaches.
To complement the global receptive field of FNO, we introduce HiLocFFNO blocks that combine global high-frequency enhancement with local patch-based processing through dual FNO branches. This design preserves fine-grained details at both global and local scales.
We propose novel weight decomposition method called SCF that factorizes complex FNO weight tensors into separate channcl-mixing and spectral-weighttng components, achioving 82.20% total parameter reduction compared to using vanilla FNO while maintaining reconstruction quality for high-resolution medical imaging.
We design a CAFF mechanism that operates directly in the frequency domain to integrate spatial and spectral features through cross-attention on real and imaginary components separately, This fusion strategy preserves complementary information from both encoding pathways, especially the frequency characteristics essential for high-quality CT reconstruction.
2 Related Work
2.1
Sparse-view CBCT Reconstruction
Traditional CT reconstruction methods such as Filtered Back Projection (FDK) (Feldkamp et al. 1984
and Algebraic Reconstruction Techniques (ART) (Andersen Kak, 1984) require hundreds of projections acquired over a full rotation. When limited sparse views (< 100 views), these methods often produce severe artifacts and degraded image quality. To mitigate this issue, deep learning methods have become popular
in sparsc-view reconstruction. Post-hoc denoising methods (Han et al. 2016 Jin et al., 2017 Ma et al.
2023 Wu et al. 2022] have been intrude for the reconstruction of conventional han/parallel-best example, FreeSeed (Ma et al. 2023) applies filtered back projection (FBP) to reconstruct artifact-laden CT volumos and designs a specialized architecture to capture frequency information for denoising and refining missing details. However, FreeSeed only captures frequency information at the global level to expand the overall receptive field. Moreover, when adapted to CBCT through slice- wise processing, these methods often fail to maintain the spatial consistency of reconstructed 3D volumes (Lin et al. 2024a
CT. For
Advancing to multi-view CBCT reconstruction. several representative approaches utilize only one two perpendicular X-ray projections (Ying et al. 2019 Corona-Figueroa et al. 2022 Kyung et al.) 2023 Liu
et al. 2024b Jeong et al. 20255; Liu & Bai 2024). Although these techniques demonstrate potential, they are specifically engineered for manlle/otthogonalweeo sparse-view rection scenarios.
reconstruction and are difficult to extend to general
For more realistic multi-view configurations, i.e., CBCT reconstruction, promising approach is to exploit implicit neural representations (INRs), which represent the CT volume continuous attenuation field. Self-supervised INR methods such NAF (Zha et al. 2022b and NeRP (Shen et al. 2022b) minimize the discrepancy between real and rendered X-ray images without requiring full 3D supervision. However, these self self-supervised methods require good priors before per-sample optimizalion, which makes them unsuitable for extremely sparse views with very limited prior data. Supervised INR methods like DIF-Net (Lin et al. 2023 encode multiple X-ray views with 2D encoder and use these map 3D coordinates to latent features projection space. Gaussaann (Lin al.] (2024a) further improves on DIF-Net by introducing 3D Gaussians spatial feature carriers, allowing more expressive and geometry-aware reconstructions. However, these
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methods heavily depend on 2D encoders, which ignore the fundamental problem of CNNss their preference for low-frequency information.
Our framework DuFal is a supervised INR method (like DIF-Net and DIF-Gaussian), where we view CT volume as a continuous attenuation field. However, we address the fundamental limitations of naive spatial encoders by introducing a dedicated Frequency Encoder that can be flexibly integrated with standard spatial encoders and an intuitive cross-attention-aasee based fusion layer. Moreover, our Frequency Encoder captures both global and local frequency information through two different branches that specialize in global and local frequency information, addressing the lack of local frequency information in FreeSeed. Together, DuFal becomes a dual-domain architecture that enables comprchensive feature encoding by simultancously processing both spatial structural context and frequency domain characteristics of sparse X-ray projections.
Table 1: Capability comparison of existing methods on extreme sparse-view CBCT reconstruction. DuFal captures several key aspects: spatial modeling, end-to-end reconstruction capability, and comprehensive froquency modeling.
Frequency Modelling
Methods
Spatial Modelling
End-to-end Reconstruction
Global
Local
FDK
Feldkamp et 1984
X
SART
Andersen & Kak 1984
x
NAF
Zba ct al.
2022a 2a
X
X
NERP
Shen ot al 2022
/
X
X
FBPCony Net Jin et al
2017
X
X
X
Freeseed
(Ma el a.. 2123
X
x
/
DIF-net
[in et al.
2123
V
x
X
DIF DIF-Gaussian [Lin et al.
2024a
x
X
DuFal (Ours)
2.2 Fourier Transform in Medical Image Analysis
The Fourier transform plays a crucial role medical image analysis due to its ability to represent frequency components, making it valuable for tasks requiring fine detail and structure preservation. In segmentation, spectral methods have been used enhance boundary sensitivity and perform denoising, as seen in works like Speetformer (Patro al. 2025) and FourierFormer (Nguyen et al. 2022). In diffusion models, spectral-domain formulations accelerate sampling and improve robustness to noise (Li et al.
2023b Si et al. 2024). For 3D
medical reconstruction, Fourier-basee positional encodings are widely adopted in neural implicit models to
capture high-frequency variations and spatial smoothness (Shen et al. 2022b Zha et al. 2022b Tancik et al.
2020
Pedraza Ortega et al. 2007 Liu ct al. 2024a
Despite these advances, Fourier-based techniques sparse-view CBCT reconstruction have not been well explored. Our work introduces principled use of FNO tailored for this domain, with a focus on preserving high-frequency details through the HiLocFFNO layer. Unlike prior approaches designed for 3D reconstruction tasks that incorporate frequency-awarr attention only at later stages of the network or as post-processing [Mal
et al. 2023 Pedraza Ortega et al. 2007 Liu et al. 2024a), our model integrates frequency-domain reasoning into the encoder.
Additionally, we choose the supervised INR framework to utilize its high performance and speed. Note that naively applying FNO significantly reduces our inference speed. We propose the SCF mochanism improve efficiency through low-rank Fourier weight factorization, which significantly reduces both parameter count and computational overhead. This compact design makes model well-suited for sparse-view CBCT scenarios where only a limited number of projections are available and computational resources are constrained.
3 Preliminary
3.1 Problem Definition
Given a set of sparse 2D X-ray projections I= (4, Iz...., Ig]. where each I3 HI is a grayscale image of height H, and width W,, and the corresponding gantry angles {0i, 02...., ax), the objective
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is to learn function:
((T=) (ox) E ERHexWeeD.
(1)
that maps the sparse projection data and their angles to a volumetric CT image Y. Here, Hos We, and De represent the height, width, and depth of the reconstructed volume, respectively.
3.2 Supervised Implicit Neural Representation Framework
Among the existing supervised INR. frameworks, we borrow from the DIF framework (Lin et al. 2023 which serves as a straightforward baseline that converts sparse X-ray projections into continuous intensity field by relying solely on spatial features. In DIF framework, the process involves two major modules: (a) Feature Encoding and (b) Intensity Field Decoding.
Feature Encoding. A UNet encoder + processes each projection image Ta and yields the feature tensor Eg = ((+),, constructing a feature set & = [Ei Ei,,.. E where Eg E RC Here, C' denotes the number of feat ure channels, while H and W are the height and width of the spatial resolution in the encoded feature space.
To fundamentally improve the projection representations, we introduce a novel dual-encoding network, Crqquency-Enhancee Dual-encoding, which injects frequency cues inlo & prior to decoding. More deteils are presented in Section
Intensity Field Decoding. This module is implemented in two steps as follows:
Step 1: Mulli-view Fusion. This step aims to encode multi-view features & into single feature in 3D space. For any given 3D spatial point x E R3, we transform it to the 2D coordinate system of each projection view k using the geometric projection function ş0(z, ap,,8), 3), where represents the angle of view k and 3 encompasscs the imaging geometry paramcters (such as source-to-detcctor distance, pixel spacing, and detector posilioning). This yields the 2D projection coordinates o/ E IR² on the detector panel for view k. The corresponding feature representation vector then extracted through bilinear interpolation: Tы =f) RCC, where Eg is the feature map and a provides the sampling coordinates. These K view-specific representations {ri...., TK) are fused using set function 5 to obtain the aggregated embedding F = (((ri...., rxx] E RC.
Step 2: Intensily Reconstruction. This step aims to reconstruet the intensity of each 3D point to produce CT volume. A four-layer MLP to RC + R maps the fused vector to a continuous intensity prediction ((x) (F).. To generate a full CT volume, sample uniform 3D grid with coordinates Tisk = (i.j. and evaluate ((Z1zk) over this grid to yield a volumetric output Y.
By integrating our proposed Frepuency-Enbamcedd Enhanced Dual-encoding into the DIF frumework, call it the DuPal frumework. In the proposed DuFal framework, we focus on improving feature encoding within the Feature Encoding module by incorporating fine-grained information while keeping the Intensity Field Decoding module unchanged from the original design.
Objectives. Assume the original CT volume has resolution H. x We x De with physical spacing (sh, Swy 8a) in millimeters. During training, instead of supervising the full voxel grid, a set of N 3D points randomly sampled X {x1, .2,...,,y in world coordinates, where each I, E R is sampled uniformly within the bounding box from (0,0,0) to (s, Way saDc).
The network estimates intensity values Ý = {Ū1, ......, } for the sampled points through the following process: for each 3D point Φ,, it projects the point to all K projection views to obtain 2D coordinates, extracts view-specific feature representations from È using bilinear interpolation, fuses these multi-view features using the aggregation function 8, and finally decodes the fused reprosentation into an intensity value using the MLP decoder v. Ground-truth intensities Y (y1, (y1, .....,.,,,,,..... ,yN are obtained from the CT volume using trilinear interpolation at each Zn.
The objective function is the mean squared error between predicted and ground-truth intensitics: C(Y E (yn }.
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a. Frequency-enhanced Dual-encoder U-net
E
b. Intensity Field Decoding
Frequency Encoder
E1
I
Multi-view Fusion
f1
fi
Intensity Reconstruction
Frequency Feature Fusioon
$
Sep Connectin
projections
Ix
$1
$L
d
d1
Spatial Encoder
Decoder
Output Ý
Figure 1: Illustration of DuFal for CT image roconstruction from multiple projection views. Our proposod Frequency-enhanced Dual-encoding (part a) processes a projection Ig from K input projections through two parallel encoders: Frequency Eacoder that analyzes frequency components of T to produce frequency features
f&} and Spatial Encoder that produces spatial features St,.,,...., 3L). .L). The model then fuses both domains through CAFF module before decoding through series of decoder layers at the Feature Decoder with skip connections to produce the feature EĄ of the input image T By repeating this process for all projections, we obtaín the extracted features {Ei...., Ex}- The Intensity Field Decoding (part b) operates as in Section 3.2; it first performs Multi-view Fusion by projecting 3D query points to extract siew-specific features through bilinear interpolation and then aggregates them using set function to create unified 3D representations. Subsequently, Intensity Reconstruction employs multi-layer MLP to decode these fused features into continuous intensity values, which are sampled over uniform 3D grid to generale the final volunetric CT output Y.
4 Proposed Method: DuFal
The quality of the encoded X-ray features contributes greatly to the quality of the reconstructed CT images. Yet, the encoder, which consists of CNN layers, is inherently biased toward low-frequency information and fails to capture fine details. To this end, we propose Frequency-enhancel Dual-encoding As shown in Figure [ for a given projection 4& E " xW,, it is processed in parallel by two encoders: a Frequency Encoder and Spatial Encoder. The spatial encoder produces set of spatial feature maps { where Si X capturing structural context from the image domain, where C the hidden channel layer 1. Simultaneously, the frequency encoder produces complementary set of high-frequency feature maps (f where ft E C enhancing the network's 's ability to preserve fine anatomical details.
At each resolution layer, the two representations and fi are fused via Cross-Attentia Frequency Fusion (CAFF) to obtain a unified feature sf E x H that integrates both spatial and spectral information. The fused features { are then passed to a Feature Decoder, which reconstructs a consolidated feature map Es E xC for each projection Finally, the fused multi-view features & {Ei, Eg,..., Ex} are passed through Intensity Field Decoding to reconstruct the CT intensity volume Y X..DD.
4.1 Frequency Encoder
Since Frequency Encoder processes a single projection Iş at a time, we omit the view index & from the image and feature symbols in this section for notational simplicity.

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HiLoc Fourier Factorized Neural Operator
Frequency Encoder
M
SCF
AAAA KUSTDTOR
W
GaLU
fua
BANA ATLIET
VUV
VAV CAMA
C
Figure 2: Overview of the proposed HiLocFFNO block within the Frequency Encoder. Each HiLocFFNO block comprises two major branches: The gHiF is illustrated at the top using solid blue line (-) and arrows (-), and the IHiF is depicted at the bottom using dashed blue line (---) and arrows (--+). Both branches employ SCF to replace the full complex weight with separate channel-mixing and spectral-weighhtnng kernels. After inverse Fourier transform, the resulting frequency features zglob and g/ocal are fused with a
linear projection W, activated by GeLU, and refined by a Calerkin-Attention (GA) (Cao 2021 layer to yield the next-level representation fi+1-
Our Frequency Encoder is inspired by the FNO architecture (Li al, 2020 (Kalimuthu al. 2025 Despite the advances of FNO in capturing frequency domain features, which help them excel in solving partial differential equation tasks dominated by low-frequency components, they exhibit limitations in natural and medical imaging where high-frequency features are equally essential, such as edges and abnormalities. To address this challenge, we propose a novel FNO arehitecture that enhances high-frequency information capture at both the local detail level and the global context level. Particularly, our Frequency Encoder comprises multiple E2NO-aseed modules, collectively referred to as HiLocFFNO()), forming an iterative architecture:
I HILocFFNO, Si HiLocFFNO, HiLocFFNO,
HiLocFFNO,
ft.
(2)
where each fi E RiXH for / = ..., L, f(1 = HiLocFFNO(f)), and fi = HiLocFFNO(I), While I is grayscale image, we can view 7 as a feature map with shape H, W,. Each HiLocFFNO(() contains two branches: a Global High-frequency Enhanced FNO (gHiF) and Local High-frequency Enhanced FNO (IHiF). As shown in Figure 2 the gHiF is illustrated at the top using solid arrows (-), while the IHiF is depicted at the bottom using dashed arrows (-++). Let F denote the Fourier transform and F-1 its inverse, HiLocFFNO is implemented as follows:
Global High-frequency Enhanced FNO (gHiF). In our implementation, the input feature map ft e CiXHiXW is first transformed via a 2D Fourier transform F applied along the spatial dimensions:
21 F(f) [C: Xmodes; Xmodes;
where modes] and modes2 denote the number of retained frequency components along the height and width dimensions, respectively. In contrast to vanilla FNO which typically retains low-frequency modes, gHiF selectively prescrves high-frequency components, enabling better capture of fino-grained features.
In the vanilla FNO, each output coefficient is produced with full complex weight RS e& xnodes, xnodees resulting in total of 2 C C modes; modes2 parameters. The output frequency component is given by:
& ul
Σ u, u.
(4)
where & {1...., Oi} indexes the output channel dimension, a {l, ..., Ci} indexes the input channel dimension, and (u, v) index the frequency coordinates. This can grow rapidly with larger values of C₁ modes),
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and modes2. In response, inspired by the depthwise separable convolution strategy in MobileNet Howard et al. (2017) we propose SCF that decomposes the weight tensor into two smaller, independent components: R CCixC for channel mixing and R E [CCixodea xnodeaz for spectral weighting. This results in the following computation:
C¹
& [,, u] =
Σ
Ro al Ro,e u, u, a[,, u..
(5)
q=l
Our proposed factorized form requires only 2 (Cr Ct + C2 modes) modes2) parameters, yielding saving ratio of
2 * C * C++ C * modes * modes
(6)
2 Ci C modes] modes2
modes] modes2 Ci
relative to the original count. For instance, in the final layer LTh given C = 512, Cj = 1024, and modes; = modes2 = 16, the factorized weight retains only 0.49% of the original parameters, representing about a 99.51% reduction.
Finally, an inverse Fourier transform yields the frequency feature
glob
(B)) E RIXHI.W
(7)
Local High-frequency Enhanced FNO (IHiF). Prioritizing high-frequency modes alone is insufficient to fully capture important fine-grained details. To address this issue, we propose a parallel FNO branch that operates on spatially partitioned patches while retaining all frequency modes to capture local information. Using the same imput feature fi as the global branch, we partition it into N, x N, non-overlapping patches, where N₂ = H./Hp = Wi/Wp: resulting in N² local patches fn.. E RCx where Πp = 0......y N2. Each patch fnp. is independently processed using the same frequency-domaain operations with factorized weights IBOxx HpxWp, We aggregate (Equations 3 and 7 in the global branch to produce the local feature 2ny.! E IR
all patches to obtain the final local feature z) z/ocal by reshaping the patches from N2 x T x Hp x W, back to the full-resolution feature map z/ocal TR H
w; in the same order the partitioning step.
Finally, fuse the outputs of two branches and pass the result to Calerkin-Attention (GA) Cao 2021
layer. After the GA layer, we obtain fi+1. Formally, we have
f++1 GA(GeLU(W fi + Joral + 21 )loh))
where W E CCxx is a linear transformation.
4.2 Spatial Encoder
At each depth 1, it performs two unpadded 3 x 3 convolutions, each followed by a ReLU, and then applies a x 2 max-pooling with stride 2. The max-pool halves the spatial resolution (H H2//2, Wi-1 = W2/2), and the subsequent convolutional stage doubles the channel width (Ct+1 = 2C)). Repeating this pattern for L levels yields the hierarchy of feature maps {s))/ where each st RCxxX Wi ready for fusion with the frequency features.
4.3 Cross-Attention Frequency Fusion (CAFF)
3 illustrates the overall flow of our CAFF module. Given two input feature maps s,, fi E Ri our goal is to fuse them in the frequency domain. Each feature is first projected via convolution and transformed into the frequency domain:
Figure
8, = =(Coo((s))) = R, +i. F(Co(C()()) = =R, +i
(9)
where R. 3,, By E x are the real and imaginary components, respectively.
To enable spatially-awer feature reweighting, we apply a cross-attention operator CA v, v. separately the real and imaginary parts, where k, v, and g denote the key, value, and query.
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CAFF
K.V
fi
Conv
F
CA
3
K.V
$1
F
8
R
Conv
F
J
CA
J,
81
Figure The CAFF flowchart. This diagram shows the fusion process between spatial features 8t and frequency features in the frequency domain. Both input features are first convolved and transformed into the frequency domain using Fourier transform F, decompasing into real (Ry, Rs) and imaginary (3 3x) components. Cross-attention (CA) operations are applied separately to real and imaginary parts, where frequency components serve as keys/valuee and spatial components as qucries. The attended frequency components are combined with original spatial frequency components (R,, 3.) through residual addition @ then transformed back to spatial domain via inverse Fourier transform F-1 to produce the fused feature s/c- + indicates the frequency processing path, - shows the spatial processing path, and --+ represents the residual connections.
R = CA R.), = CA Sy, 3,)
(10)
The attended frequency components ( and 3 are then combined with the original spatial features via residual addition:
R. (R, R.). $ = (3, + Sz).
(11)
Finally, the fused complex-valued representation is transformed back into the spatial domain via an inverse Fourier transform:
= +o..) E CixHiw..
(12)
4.4 Feature Decoder
Starting from the deepest fused feature s it performs upsampling stages. At stage (1 = ..... . 2), a 2 transposed convolution doubles the spatial size, concatenates the result with the skip connection $/-1 and refines the concatenated tensor using two successive x Conv-ReLU blocks d,-1 CAT (Up(i),, s(-1); where d2 = s/ Up(-) denotes transposed convolution, and CAT(-, denotes the two-block concatenation. This sequence yields decoder fcatures {d1/21 with di R C x H1 W; Finally, the top-level feature to the projection representation Es = Convix (di) E BC At the end of our Fourier-aut augmenee Dual-encoding, we obtain the encoded features of K projections È = {Ei .,,,,. Ex},
x convolution projects
which are then decoded via Intensity Field Decoding described in Scction 3.2
5 Experiments
5.1
Experiment setup
Datascts. We evaluate our method on two CT datascts. The LUNA16 datasct (Sctio et al. 2017 contains 888 chest CT scans split into 738 training, 50 validation, and 100 testing samples. Scans are resampled and standardized to 256 vaxels with [1.6, 1.6, 1.6) mm spacing, following Lin et al. 2024ch. The ToothFairy dataset (Cipriano et al.] 2022) consists of 443 dental CBCT scans with varying resolutions split into 343 training, 25 validation, and 75 testing scans. Scans are resampled and standardized 2563 voxels with spacing (0.54, 0.54, 0.21] mm.

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Table 2: Performance evaluation on the LUNA16 dataset. 1 indicates that higher values are better. Best rosults are bolded and second-best results underlined.
6- View
V-View
10-View VieV
Methods
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
FDK
Fellkamp et al.
1984
15.3G
31.41
15.95
31.00
16.25
31.79
SART
Andersen & Kak 1984
18.94
49.47
20.60
54.63
21.75
58.94
NAF
Zha
2022a
18.76
51.16
20.51
60.81
22.17
62.22
NeRP
S Shen et nl.
2022a
23.55
74.46
25.53
80.67
26.12
81.30
Freeseed
Ma sl.
2023
25.59
77.36
26.86
78.92
27.23
79.25
DiF-Net
2023
24.68
71.46
25.67
76.08
DIF Gaussian
Lin el al.
2024a
26.48
81.78
26.93
82.09
29.29
87.55
DuFal (ours)
28.20 + 0.38
85.83
+ 0.27
28.77 + 0.42
85.50
1.65
29.41 + 0.18
87.67
0.22
Multi-view 2D projections of size 256 x 256 are simulated using open-source toolbox with viewing angles uniformly sampled over 180 We experiment with 6, 8, and 10 views to assess the effect of projection count.
Implementation Details. We implement the HiLocFFNO as a multi-layer encoder with L = 4 iterative blocks. Each block contains a dual-branch FNO module: a global branch preserving the top modes; = modes2 = 16 Fourier components and a local branch operating on 16 x 16 x 16 non-overlapping patches. The GA layer fuses global, local, and projected input features using spatial-domain linear projection. The hidden dimension C varies across different stages of the network. The fusion block uses complex-valued cross-attention in the frequency domain with 4 attention heads. All models are trained using the Adam optimizer with learning rate of 2 10-4, batch size 4, and cosine annealing scheduler.
Baselines. We compare our model with several existing baselines, including traditional methods FDK Feldkamp et al. 1984) and SART (Andersen & Kak 1984), which require no training and are directly inferred using implementations from a public GitHub repository. For NAF (Zha et al.) 2022a). NeRP (Shen et al.) 2022a), and FreeSeed (a et al. 2023), we report the results as provided in (Lin al. 2024a) Additionally, we reproduce DIF-Net (Lin al. [2023] and DlF-Gaussian (Lin al. 2024a) using their official codebases, training for 400 epochs the LUNA16 dataset and 600 epochs the ToothFairy dataset following their
original paper settings. We note that spatial-domain methods such C2RV-Net (Lin et al.) 2024b were not publicly available at submission time, precluding comparison on our datascts; however, we provide a comparison using our own implementation in Appendix A.4
Evaluation Metrics. We follow previous works ZZha et al. 2022a Lin et al. 2023 to evaluate the reconstructed CT using the following metrics: Peak Signal-to-Noise Ratio (PSNR) decibels (dB) and Structural Similarity Index Measurement (SSIM) in percentage (%). Higher PSNR/SSIM values indicate better reconstruction quality. We also report weighted scores CEPSNN//WSSTNN in the ablation study to provide an additional perspective the metrics when focusing only regions of interest.
5.2 Quantitative Results
Results on LUNA16 Dataset. As shown in Table 2. DuFal consistently achieves the highest average metrics across all view settings (6, 8, and 10 views). This advantage is particularly pronouneed in very sparse view configurations such as and views, indicating that our enhanced frequency feature capture capability provides proven benefits even in ill-posed sparse view settings. Among neural methods, NeRP typically underperforms compared to recent approaches, while Daa-Gaussian ranks as the second-best. method in most settings, though approach consistently surpasses both across all view configurations.
https://github. www.:huww..cccceE//.TEEEI /ERENETGETT
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Table Performance evaluation on the ToothFairy dataset. indicates that higher values are better. Best rosults are bolded and second-best results underlined.
6- View
S-View
View
Methods
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
FDK
Feldkamp et
1984
17.07
39.90
18.42
43.29
19.58
47.21
SART
Andersen & Kak 1984
20.04
64,98
21.92
67.86
22.82
71,53
NAF
Zha
2022a
20.38
63.52
22.39
67.21
23.81
72.52
NeRP
Shen et nl.
2022a
21.27
72.06
24.18
78.83
25.99
82.08
FreeSeed
MR et al.
2023
26.35
78.98
27.08
81.38
27.63
81.40
DiF-Net
2023
25.35
26.09
86.09
DIF Gaussian
Lin el al.
2024a
25.74
74.64
26.27
75.23
28.00
83.11
DuFal (ours)
28.79 + 0.27
86.08
1.20
29.45 + 0.58
87.56
1.14
30.51 + 0.47 89.49 0.51
FDK
SART
DIF-Net
DIF-Gaussian
DuFal
Ground-Truth
Axial
Coronal
Sagittal
Figure 4: Visualization of 10-view reconstructed chest CT (from top to bottom: axial, coronal, and sagittal slice). This figure highlights the key advantage of our method in preserving fine details. While DeF-Net. and DIF-Gaussian successfully recover the general anatomy, they produce overly smoothed images that lose high-frequency information, as evidenced by the blurry details in the zoomed-in views. In contrast, our method yields visibly sharper reconstructions across the axial, coronal, and sagittal planes, achieving a fidelity much closer to the ground-truth image.
Results on ToothFairy Dataset. As reported in Table our method consistently achieves the highest average metrics across all view settings (6, 8, and 10 views), with SSIM showing notable improvements over baseline methods. Similar to the LUNA16 dataset, particularly in very sparse view configurations such 6 and views, confirming its robustness under limited projection scenarios. FreeSced, post-processing model that emphasizes the ability to capture high-frequency details by incorporating FFT layers, outperforms DIF-Gaussian in sparse view settings like 6 or 8 views but generally underperforms compared to our approach across all view configurations. Among other INR-based mothods, NeRP and DIF-Net typically underperform compared to more recent approaches, while our method consistently surpasses all basclines, demonstrating the effectiveness of our enhanced frequency feature capture capability for dental CT reconstruction.
model demonstrates enhanced performance,
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FDK
SART
DIF-Net
DIF-Gaussian
DuFal
Ground-Trune
Axial
Coronal
Sagittal
6
6
Figure 5: Visualization of 10-view reconstructed dental CT (from top to bottom: axial, coronal, and sagittal slice). While the competing DIF-Gaussian method reconstructs the general bone structure, it oversmooths the image, completely obscuring fine details like the mandibular canals. Our method yields a significantly sharper result that clearly defines the cortical bone and successfully resolves the mandibular canals in the coronal and sagittal views.
5.3 Qualitative Analysis
We present visual comparisons to further demonstrate the differences between our model and bascline methods. We focus on 6-view reconstructions as this represents the most challenging sparse-view scenario and best showcases reconstructton ion quality differences (Figures 4 and 5 Traditional methods such as FDK (Feldkamp)
et al. 1984 and SART (Andersen & Kak 1984) exhibit noticeable artifacts and fail to preserve fine anatomical structures, while DIF-Net (Lin al.] 2023) and DIF-Gaussian (Lin al. 2024a] show improvements but still struggle to capture critical high-frequency details and lissue boundaries. Our model produces reconstructions with enhanced clarity and accurate contrast representation, effectively preserving high-frequencs features while minimizing artifacts. On the ToothFairy dataset, similar patterns emerge: traditional methods fail to recover fine dental structures. Our approach demonstrates improved capability in reconstructing sharp boundaries and complex anatomical features while maintaining high fidelity under sparse projection conditions across different medical imaging domains.
6 Ablation Study
6.1
The Effect of IHiF in HiLocFFNO
Table 4 evalluates the contribution of the IHiF branch within the HiLocFFNO block. This component captures local spatial relationships through batch-based processing, complementing the global frequency analysis in gHiF. The results demonstrate that including the IHiF branch provides performance improvements across both datasets, with particularly notable gains on the ToothFairy dataset. This validates the importance of combining both gHiF and IHiF for optimal roconstruction quality.
6.2 Comparison between CAFF and Naive
Table presents a comparative analysis between our proposed CAFF and conventional spatial-domain cross-attention mechanisms. The baseline spatial cross-attention approach processes both spatial feature
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Table 4: Ablation study for IHiF in HiLocFFNO.
LUNA16
ToothFairy
IHiF
PSNR
SSIM f
PSNR
SSIM 1
29.56
87.14
29.45
86.02
29.63
87.91
30.79
89.67
81 and frequency feature fi through convolution operations before applying cross-attention in the spatial domain, where serves as queries and fi functions as keys and values. In contrast, our CAFF operates cross attention on the frequency domain to integrate multi-modal feature representations. The experimental results demonstrate that CAFF achieves superior reconstruction performance compared to spatial cross-attention (CA), confirming the effectiveness of frequency-domain feature fusion for sparse-view CBCT reconstruction.
Table Ablation study comparing different feature fusson strategies on the LUNA16 and ToothFairy datasets.
LUNA16
ToothFairy
Fusion Method
PSNR t
SSIM
PSNR t SSIM
Add
29.28
87.40
29.77
88.52
Concat
29.21
87.33
29.70
88.43
Spatial CA
29.59
87.25
30.75
89.49
CAFF (ours)
29.63
87.91
30.79
89.67
Table 6: Impact of Factorization PSNR, SSIM, and Model Parameters.
Factorization
LUNA16
ToothFairy
Model Size
PSNR
SSIM
PSNR
SSIM
Params (M)
Reduction
No Factorization
29.54 29.63 (+0.09)
87.94 87.91 (-0.03)
30.94 30.79 (-0.15)
90.27 89.67 (-0.6)
445M
Factorized
79M
82.2%
6.3
SCF's Factorization Analysis
Table evaluates the impact of factorization in SCF performance and computational efficiency across both asets. Importantly, the factorized variant maintains comparable reconstruction quality, with only minimal variations in PSNR and slight decreases in SSIM scores, demonstrating that reconstruction quality is largely preserved despite the significant parameter reduction. Most significantly, factorization achieves a substantial 82.2% reduction model parameters, decreasing from 445M to 79M parameters. This dramatic computational efficiency gain demonstrates that factorization maintains comparable performance while substantially reducing parameter complexity, making the approach more practical for deployment in resource-constrained environments.
6.4 HiLocFFNO Plug- and-Play Integration
Table 7 presents an ablation study on the plug-and-play integration of the proposed HiLocFPNO into various reconstruction models across the LUNA16 and ToothFairy datasets. The inclusion FENO consistently improves performance, with particularly notable gains on the OTF-Gaussian model, where PSNR and SSIM show improvements on both datasets. This suggests that FFNO effectively enhances both image fidelity and structural quality.
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6.5 Efficiency Analysis
As shown in Table 8 and Figure G DuFal demonstrates a favorable compulational Irade-off, particularly in inference speed. Most notably, our method achieves significantly faster inference than DiF-Net (1.938 3.778 s/sample), and the training time of DuFal does not scale diroctly with paramcter count. In medical Al applications where precision the primary concern, this computational trade-off is highly acceptable given the quality improvements.
While Dulalls training time longer due to the dual-path architecture and Fourier operator computations, this overhead is incurred only once during offiine training. The model's faster inference speed provides clear advantages in real-world clinical deployment. Furthermore, training efficiency can be substantially improved through distributed data-paralle training with multiple GPUs and larger batch sizes. Future work could explore initialization from pre-trained medical foundation models to accelerate convergence while maintaining robust performance.
Table 7: Ablation study of plug-and-play of the HiLocFFNO modular on different models.
LUNA16
Toith
Methods
HiLocFFNO
PSNR f SSIM 1
PSNR
SSIM
X
25.35
75.01
27.01
76.12
FBPConvNet Lin et al.
2023
/
25.55
75.68
27.35
76.86
X
26.53
76.08
26.67
86.09
DiF-Net Lin et al. 2023
V
26.85
76.62
27.79
86.49
X
29.29
87.55
28.60
83.11
DiF- -aa--sssian Lin et al.
2024a
V
29.63
87.91
30.79
89.67
Table 8: Computational efficiency comparison on the LUNA16 dataset. + indicates that lower values are
better for speed metrics.
Methods
Params (M)
Inference Speed (s/sample)
Training Time (h) &
Freesoed
48.41
1.274
29.6
DiF-Net
31.11
3.778
24.5
DiF-Gaussian
31.71
0.695
26.5
DuFal (ours)
78.92
1.938
34.1
6.6 Ablation on High-Fidelity Reconstruction
6.6.1 ROI-Weighted Metrics.
Conventional PSNR and SSIM can misleading when degradation occurs in clinically important regions.
As shown Figure 7 adding Gaussian noise or Laplace noise only inside the lung ROT (c) yields a higher PSNR (10.59 and 12.12 dB) than adding the same noise to the background (d, 10.37 and 11.78 dB), even though radiologists prefer (d) over (c). To address this issue, we introduce ROI-weighted metrics (W-PSNR, W-SSIM), which focus evaluation diagnostically important regions. For example, W-PSNR assigns score of 38.15 dB to the image after background masking, correctly reflecting its perceptual quality.
Under the ROI-weighted metrics, Tables 9 and 10 demonstrate that our method also achieves the highest W-PSNR and W-SSIM across all sparse-vitee view settings on both LUNA16 and ToothFairy datasets. For example, LUNA16, DuFal achieves 28.52 dB in W-PSNR and 88.96 in W-SSIM, outperforming the runner-up method Gaussans by +1.15 dB and +2.05 points, respectively. Similar improvements are observed on the ToithFairy dataset (Table 10 where method consistently surpasses DiF-Gaussian across all sparse-view
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PSNR vs Speed
SSIM vs Speed
Freeseed
36.0
DiF-Net
DiF GFFusaussian
81.8
DuFal (ours)
PSNR (dB)
SSIM (%)
26.5
25.6
4.7
15 20 25 30 35
1.2
25 25 30
35
Inference Speed (s/sample)
Inference Speed (s/sample)
Figure 6: Performance vs. Inference Speed Trade-off on LUNA16 Dataset (6-View). Scatter plots comparing reconstruction quality against computational efficiency for different methods. Left: PSNR Inference speed. Right: SSIM vs. Inference speed. DuFal achieves the highest reconstruction quality in both metrics while maintaining good inference speed.
PSNR 10.37
W-PSNR: 10.59
W-FSNR 38
PSNR: 12.12
PSNR 11.78
W-PSNR: 12.12
W-PSNR: 38 15
a)
d)
Figure 7: (a) Original image and region interest in mask format; (b) Two types of noise: Gaussian noise (top) and Laplace noise (bottom); (c) Noise added only within the lung region of interest; (d) Noise added only to the background. Global PSNR favors (c), whereas ROI-weighted PSNR (W-PSNR) correctly ranks (d) higher, demonstrating the need for ROI-weighted metrics.
configurations.
The consistent superiority across both datasets under conventional and ROT-weighted metrics establishes the robustness of our proposed approach.
6.6.2 Downstream task: Segmentation
To evaluate the clinical utility of our roconstructed CT volumes, we conduct downstream segmentation tasks that assess whether the preserved anatomical details enable accurate automated analysis. High-fidelity reconstruction is crucial for the segmentation task because segmentation algorithms rely on clear anatomical boundaries and structural details that can be compromised by reconstruction artifacts or blurring.
For lung segmentation evaluation on LUNA16, we employed the pre-trained LungMask model (Hofmanninger et al. 2020 for automated lung region extraction. Traditional CBCT reconstruction methods (FDK and
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Table Performance ROI-weighted evaluation on the LUNA16 dataset.
6- View
8- V-ieew
10- View
Methods
W-PSNR
W-SSIM
W-PSNR
W-SSIM
W-PSNR
W-SSIM
FDK Feldkamp et al.
1984
15.82
72.68
74.21
16.83
75.32
SART Andersen & Kak
1984
21.21
81.43
22.57
82.37
23.52
83.32
DiF-Net LLn et sl.
2023
26.16
85.59
26.69
86.51
27.66
87.19
DiF-Gaussian
2024a
27.37
86.91
27.84
87.26
30.06
90.60
DuFal (ours)
28.52
88.96
29.83
90.29
30.59
91.03
Table 10: Performance ROI-weighled evaluation the ToothFairy dataset.
6- View
8- ViView
10-View
Methods
W-PSNR
W-SSIM
W-PSNR
W-SSIM
W-PSNR
W-SSIM
DiF-Gaussian [Lin al. 2024a
80.73
27.47
83.11
28.08
83.97
DuFal (ours)
27.11
81.90
27.92
83.35
28.52
84.62
Full view
10-view
Axial
Coronal
Sagittal
Figure 8: Comparison of LungMask segmentation performance on full-view versus 10-view reconstructed CT images across three anatomical planes. Left panel shows results on full-view (ground truth) CTss right panel shows results 10-view reconstructed CTs. For cach view (axial, coronal, sagittal from top to bottom), three columns display: (1) original CT slice, (2) segmentation mask overlaid on CT slice in red, and (3) isolated predicted lung mask.
SART) exhibit poor segmentation performance, with Dice scores shown in Table II indicating severe degradation of analomical boundaries in extremely sparse-view scenarios. In contrast, deep learning-basee reconstruction methods achieve substantially better segmentation performance, with Dice scores above 96. Figure & illustrates the qualitative segmentation results and further confirms DuFal's strong performance. DuFal demonstrates robust preservation of anatomical boundaries with the highest precision and recall values, achieving the best overall segmentation performance.
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Table 11: Lung Segmentation Comparison on LUNA16 Dataset.
Model
Dicet
Jaccardf
Precisionf
Recallf
Views
97.73
95.60
97.10
98.37
FDK
8.89
4.82
6.96
12.31
SART
54.45
39.22
58.94
50.99
DIF-Net
96.68
92.12
95.41
97.98
DuFal (ours)
97.05
94.29
95.69
98.45
7 Conclusion
This work presents the DuFal framework, a novel approach address the extremely sparse-view CBCT reconstruction challenge. Our proposed framework fundamentally addresses the limitations of conventional CNN-based methods that struggle to capture the high-frequency anatomical details essential for accurate medical imaging. Through the introduction of Frequency-Fahancee Dual-encoding architecture, we demonstrate that simultaneous processing of spatial structural context and frequency domain characteristics significantly improves reconstruction quality in sparse X-ray projection scenarios. The key innovation lies in our HiLocFFNN blocks, which effectively combine global high-frequency enhancement (gHiF) with local patch- based processing (IHiF), ensuring that both major anatomical structures (such as organ boundaries and bone structures) and fine-grained details such pulmonary nodules are preserved. Our SCF technique achieves remarkable parameter efficiency while maintaining reconstruction quality. Additionally, our CAFF mechanism enables effective integration of spatial and spectral fcatures in the frequency domain. Therefore, DuFal sets a new standard for sparse-view CBCT reconstruction that prioritizes both computational efficiency and medical imaging reconstruction performance.
Discussion Future Work.
Modality generalization: The proposed DuFal has been extentively evaluated on CT datasets, demonstrating its effectiveness in preserving figh-freeuency anatomical details under extremely sparse-view conditions. While current experiments focus CT, many other medical imaging modalities, such as MRI, PET, and ultrasound could potentially benefit from this approach, offering promising avenue for future research.
Enhanced interpretability: While many prior approaches implicitly learn frequency information, DuFal explicitly models it, making the learned representations more amenable to interpretation. Future work could extend this concept by disentangling additional semantic or structural features, polentially enabling explanations that align more closely with clinical reasoning.
Artifact mitigation: DuFal's froquency-spatial dual architceture offers promising potential for handling motion artifacts. The key insight that motion artifacts affect spatial and frequency representations differently, creating detectable inconsistencies between domains, whereas anatomical features should exhibit consistent patterns across both representations. However, current work focuses exclusively on static sparse-view reconstruction, as publicly available datasets with well- annotated motion artifacts are lacking. As future work, we plan to collaborate with clinical partners to evaluate DuFal's robustness on real dynamic CBCT datasets.
Broader Impact. The DuFal framework represents a significant advancement. in sparse-view CBCT reconstruction with direct clinical implications for routine screening, emergency imaging, and pediatric applications where radiation reduction is critical. The integration of frequency-aware processing addresses the fundamental challenge of preserving high-frequency anatomical details crucial for accurate diagnosis, while our modular design enables integration into existing deep learning pipelines.
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Acknowledgments This material is based upon work supported by the National Science Foundation (NSF) under Award No NSF 2443877, 1946391, 2223793 EFRI BRAID, National Institutes of Health (NTH) IR01CA27773-01.. The authors thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Duy M. H. Nguyen.
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A
Appendix
A.1 Analysis of Query-Key- Value Configuration in CAFF
To examine the rationale behind our cross-domain attention design, conduct additional ablation study that. interchanges the roles of spatial and frequency features in the attention mechanism. As shown in Table 12] reversing the roles results in a consistent performance drop across both datasets. This outcome supports our design choice of assigning spatial features as queries and frequency features as keys and valucs. Spatial features capture the primary structural context of anatomical regions, while frequency features encode complementary high-frequency details. Allowing spatial tokens to query frequency representations enables more coherent integration between global frequency cues and localized spatial patterns. Conversely, when frequency tokens query spatial ones, the model loses spatial anchoring, leading to less effective feature fusion and degraded reconstruction quality.
Table 12: Ablation study query-key-value domain configuration in the CAFF module.
LUNA16
Toith
K-V
PSNR t
SSIM f
PSNR f
SSIM 1
Frequency
Spatial
29.24 29.63
87.32
29.67
88.41
Spatial
Frequency
87.91
30.79
89.67
A.2 Sensitivity Analysis of Frequency Modes
To assess the sensitivity of the global High-Frequency (gHiF) branch to the number of retained Fourier modes, we conducted experiments by halving and doubling the default number of modes (16). Table 13 shows the performance on both datasets with 10 views.
Table 13: Sensitivity analysis for frequency modes in gHiF (10 views).
Modes
LUNA16
Toirh
(gHiF)
PSNR 1
SSIM 1
PSNR
SSIM
8
29.26
87.23
30.11
88.97
16
29.63
87.91
30.79
89.67
32
29.22
87.35
30.15
89.05
Mean/Std 29.37 / 0.03 87.50 / 0.09
30.35 / 0.10 89.23
0.10
The results exhibit low standard deviations (< 0.1) across both metrics and datasets, indicating that DuFal maintains stable reconstruction quality despite variations in the number of frequency modes. Increasing and decreasing the number of modes slightly degrade reconstruction quality, confirming that the chosen configuration achieves a good balance between capturing high-frequency components and maintaining stable optimization. These findings further demonstrate that Duffll is robust moderate changes in the frequency resolution of the gHiF branch, validating the general stability of our frequency modeling strategy.
A.3 Impact of Local Patch Size
We evaluate the influence of patch size in the local High-Frequency (IHiF) branch on preserving fine spatial structures. Table 14 compares different non-overlapping patch sizes (83, 163, 32*).
DuFal demonstrates stable performance with low standard deviations across all patch sizes, confirming that the model is robust to changes in local partitioning. Both smaller (89) and larger (323) patch sizes lead to only
22

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Published in Transactions on Machine Learning Research (12/2025)
Table 14: Ablation study local patch size in IHiF.
Patch Size
LUNA16
ToothFairy
(IHiF)
PSNR 1
SSIM
PSNR 1
SSIM t
83
29.31
87.58
29.70
88.40
16
29.63
87.91
30.79
89.67
323
29.26
87.35
29.71
88.36
Mean/Suy
29.4 0.03
87.61 / 0.05
30.1
0.26
88.81 / 0.37
Table 15: Comparison with C2RV-Net under different initialization settings on the LUNA16 da dataset.
6- View
8- V-eeww
10- View
Methods
Init.
Scale
PSNR1
SSIMT
PSNR
SSIM1
PSNR
SSIM
FDK
Scratch
Single
15.36
31.41
15.95
31.00
16.25
31.79
SART
Scratch
Single
18.94
49.47
20.60
54.63
21.75
58.94
NAF
Scratch
Single
18.76
54.16
20.51
60.84
22.17
62.22
NeRP
Scratch
Single
23.55
74.46
25.53
80.67
26.12
81.30
Freeseed
Scratch
Single
25.59
77.36
26.86
78.92
27.23
79.25
DiF-Net
Scratch
Single
24.68
71.46
25.67
76.47
26.53
76.08
DiF-Gaussian
Scratch
Single
26.48
81.78
26.93
82.09
29.29
87.55
DiF-Gaussian
Pro-trained
Single
27.50
84.73
28.48
85.69
29.65
88.70
C2RV-Net
Multiple
29.23
87.47
29.95
88.46
30.70
89.16
DuFal (Ours)
Scratch
Single
28.31
86.00
29.13
87.41
29.63
87.91
DuFal (Ours)
Pre-trained
Single
29.58
88.14
30.74
89.61
31.49
90.59
slight performance degradation compared to the default configuration. Smaller patches may overemphasize fine texture variations, while larger patches reduce spatial locality and weaken boundary precision. The default (163) configuration achieves the best overall balance between local detail preservation and contextual representation, as reflected by the highest mean PSNR/SSIM. These findings validate that DuFal maintains consistent reconstruction quality even under varying local frequency resolutions.
A.4 Comparison with C2RV-Net
We further compare DuFal against the recent C2RV-Net Lin et al. (2024c) (CVPR 2024). It is important to note that DuFal is a modular frequency enhancement framework rather than standalone architecture; its frequency-aware components can be integrated into existing spatial-domain reconstruction networks to improve figh-freeuency recovery. To validate this property, we implemented DuFal based on the spatial DiF-Gaussian model using pre-trained ResNet-50 Nguyen et al. (2023). As shown in Table 15 DuFal surpasses both C2RV-Net and Gaussian under comparable sparse-view configurations on the LUNA16 dataset, demonstrating that the observed improvements primarily stem from proposed frequency-domain integration.
23


Reviews:
[{"review_id": "jkReiXHZe0", "reviewer": "Reviewer_7wfB", "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 proposes DuFal, a dual-frequency-aware learning framework tailored for high-fidelity extremely sparse-view Cone-Beam Computed Tomography (CBCT) reconstruction. Its core contributions can be summarized as follows:\n\nDual-encoding Architecture: It designs a Frequency-Enhanced Dual-encoding structure that processes X-ray projections through parallel Spatial and Frequency Encoders. This addresses the inherent bias of conventional CNN-based methods toward low-frequency information, enabling simultaneous capture of spatial structural context and high-frequency anatomical details (e.g., pulmonary nodules, mandibular canals) critical for medical imaging.\n\nHiLocFFNO Blocks: To complement the global receptive field of Fourier Neural Operators (FNO), the paper introduces High-Local Factorized Fourier Neural Operator (HiLocFFNO) blocks. These blocks integrate two complementary branches—Global High-frequency Enhanced FNO (gHiF) for capturing global frequency patterns and Local High-frequency Enhanced FNO (lHiF) for processing spatially partitioned patches—preserving fine-grained details at both global and local scales.\n\nSpectral-Channel Factorization (SCF): To improve the efficiency of FNO (which suffers from rapid parameter growth with more frequency modes), the paper proposes SCF. This weight decomposition technique splits complex FNO weight tensors into channel-mixing and spectral-weighting components, achieving an 82.2% parameter reduction (from 445M to 79M) while maintaining high-resolution medical image reconstruction quality.\n\nCross-Attention Frequency Fusion (CAFF): A CAFF module is designed to fuse spatial and frequency features directly in the frequency domain. By applying cross-attention to the real and imaginary components of Fourier-transformed features separately, it preserves complementary information from both encoders, which is essential for high-quality CT reconstruction.\n\nClinical Utility Validation: The paper verifies DuFal’s practical value through downstream tasks (e.g., lung segmentation on the LUNA16 dataset) and ROI-weighted metrics (W-PSNR, W-SSIM). It demonstrates that DuFal not only outperforms state-of-the-art methods in standard metrics (PSNR, SSIM) but also preserves anatomical boundaries critical for clinical diagnosis, even under extremely sparse-view (≤10 projections) conditions.\n\nKey Strengths\n\nTargeted Solution to a Critical Clinical Problem: The framework directly addresses the core challenge of sparse-view CBCT—reducing radiation dose while preserving high-frequency anatomical details. This aligns with a key clinical need (e.g., pediatric imaging, routine screening) where radiation reduction is non-negotiable.\n\nInnovative Integration of Frequency and Spatial Domains: Unlike prior methods (e.g., FreeSeed, which only uses global frequency post-processing), DuFal embeds frequency-aware learning into the encoder stage and combines global-local frequency analysis. This design effectively mitigates the low-frequency bias of CNNs and FNO’s limitations in local detail capture.\n\nEfficiency and Modularity: SCF significantly reduces model parameters without sacrificing performance, making DuFal suitable for resource-constrained clinical environments. Additionally, HiLocFFNO and CAFF are modular, allowing seamless integration into existing reconstruction pipelines (e.g., DIF-Net, DIF-Gaussian), as validated by plug-and-play ablation studies.\n\nComprehensive Evaluation: The paper uses two diverse datasets (LUNA16 for chest CT, ToothFairy for dental CBCT) and evaluates performance across multiple sparse-view settings (6, 8, 10 views). It also includes qualitative analysis (visual comparisons of axial/coronal/sagittal slices) and clinical downstream tasks, ensuring results are robust and clinically relevant.\n\nKey Weaknesses\n\nLimited Generalization to Other Modalities: While the paper mentions potential extension to MRI, PET, and ultrasound, it only validates DuFal on CT datasets. The unique characteristics of other modalities (e.g., MRI’s low signal-to-noise ratio, ultrasound’s speckle artifacts) may require adjustments to the frequency-processing modules, which are not explored here.\n\nLack of Ablation on Frequency Mode Selection: The gHiF branch retains the \"top 16 Fourier components\" (Section 5.1) for global high-frequency processing, but the paper does not justify why 16 modes are optimal. A sensitivity analysis (e.g., testing 8, 16, 32 modes) would strengthen the rationale for this hyperparameter choice.\n\nInsufficient Discussion of Motion Artifacts: The \"Future Work\" section proposes using cross-domain inconsistencies (spatial vs. frequency) to detect motion artifacts, but the current work does not evaluate DuFal’s performance on datasets with real motion artifacts. Given that motion is a common issue in clinical CBCT (e.g., patient breathing), this limits the framework’s practical applicability.\n\nAmbiguity in Local Patch Partitioning: The lHiF branch uses 16×16×16 non-overlapping patches (Section 5.1), but the paper does not explain how patch size is determined. Different patch sizes (e.g., 8×8×8, 32×32×32) may impact local detail preservation, and a comparative analysis is missing.\n\nTraining Time Trade-off: While DuFal’s inference speed is faster than DIF-Net (1.938 vs. 3.778 s/sample), its training time (34.1 hours) is longer than baselines like DIF-Net (24.5 hours) and DIF-Gaussian (26.5 hours). The paper does not discuss strategies to optimize training efficiency, which could be a barrier for large-scale clinical deployment.", "claims_and_evidence_yn": "Yes", "claims_explanation": "As specified in the above evaluation of contributions.", "audience_yn": "Yes", "audience_explanation": "As specified in the above evaluation of contributions.", "requested_changes": "1. Add sensitivity analysis for gHiF’s frequency mode count.\n\n2. Evaluate performance on datasets with motion artifacts.\n\n3. Discuss strategies to optimize training time.\n\n4.Clarify the implementation of the Galerkin-Attention (GA) layer.\n\n5. Provide more details on the “set function δ” in Multi-view Fusion.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "KFVk2MImKl", "reviewer": "Reviewer_d4eY", "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": "This paper introduces DuFal, a new method to create 3D CT scans from very few 2D X-ray images. Their key insight is that standard AI models miss fine details because they focus too much on \"blurry\" low-frequency info. To fix this, they built (1) Dual-Encoding: One path (spatial encoder) handles the basic structure of the image, while a second, new path (frequency encoder) specifically captures the fine details and edges (2) HiLocFFNO Blocks: Within the Frequency Encoder, they design a new module called HiLocFFNO that combines two specialized branches: a global branch for capturing overarching high-frequency patterns and a local, patch-based branch for preserving fine-grained details that might be lost in a global analysis.(3) Spectral-Channel Factorization (SCF): To make the complex Frequency Encoder practical, they introduce a weight factorization technique that dramatically reduces its number of parameters (by over 82%) without significantly compromising performance, addressing a major computational bottleneck of similar methods.\n\nKey Strengths\n- The dual-path architecture is a well-motivated and principled approach to a known problem (CNN frequency bias)\n- Strong Empirical Validation\n- The authors show that their core HiLocFFNO component can be integrated into other existing models as a \"plug-and-play\" module to boost their performance, which increases the impact and potential adoption of their work.\n\nKey Weakness\n- Missing a Critical Baseline Comparison: The most significant weakness is the omission of C2RV-Net (Lin et al., 2024b) from all quantitative comparisons (e.g., Table 2 & 3) , even though the authors explicitly cite it in the introduction. **Based on C2RV-Net's own published results on the LUNA16 benchmark, it appears to achieve a higher PSNR/SSIM score than DuFal.**\n### LUNA16 Performance Comparison: C2RV-Net vs. DuFal\n\n| Method | 6-View (PSNR / SSIM) | 8-View (PSNR / SSIM) | 10-View (PSNR / SSIM) | Source |\n| :--- | :--- | :--- | :--- | :--- |\n| **C2RV-Net** | **29.23 / 87.47** | **29.95 / 88.46** | **30.70 / 89.16** | C2RV Paper (Table 1) |\n| **DuFal** | 28.31 / 86.00 | 29.13 / 87.41 | 29.63 / 87.91 | DuFal Paper (Table 2) |\n\n- Model Complexity: Despite the impressive parameter reduction via SCF, the final DuFal model remains significantly larger (78.9M parameters) and requires longer training times than several key baselines. This could be a barrier for deployment in resource-constrained environments.\n- Reliance on Qualitative Claims: In light of the missing C2RV-Net comparison, the argument for superiority leans more heavily on qualitative results (e.g., \"sharper\" images). While these visuals are compelling, a head-to-head quantitative comparison with the strongest known baseline would have provided a more definitive conclusion.", "claims_and_evidence_yn": "No", "claims_explanation": "While the paper presents substantial evidence, a critical claim is not fully supported, making the overall case less convincing.\n\n## Supported claims:\n\nClaim: The dual-encoding architecture and HiLocFFNO blocks improve feature capture.\n- Tables 4, 5, 7 demonstrate the performance gain from each component (IHiF branch, CAFF fusion, plug-and-play use).\n- Figures 4, 5 offer clear visual proof of details.\n\nClaim: The SCF factorization drastically reduces parameters.\n- Table 6 shows an 82.2% parameter reduction with minimal performance loss. \n\nClaim: DuFal outperforms a specific set of baseline methods.\n- Tables 2 and 3 show consistent quantitative superiority over listed baselines like FDK, SART, NAF, and DIF-Net\n\n\n## Unsupported / Weakened Claim (Lack of Critical Evidence)\n\nThe paper's overarching claim of state-of-the-art (SOTA) performance is not convincingly supported due to a critical omission.\n- Lacking Evidence: The paper completely omits a comparison with C2RV-Net (Lin et al., 2024b), a directly relevant and contemporary SOTA method.\n- While the authors provide strong qualitative evidence (Figures 4 & 5) and use alternative metrics (W-PSNR, Segmentation Dice)  to argue for better high-frequency detail, this does not excuse the failure to compare against the strongest known baseline on the field's standard metrics.", "audience_yn": "Yes", "audience_explanation": "Yes, the paper's introduction of a frequency-aware neural operator to overcome CNN spectral bias presents a novel and impactful contribution that will interest researchers in medical imaging.", "requested_changes": "1. Perform a direct, quantitative and qualitative comparison with C2RV-Net (Lin et al., 2024b)\n2. Applying metric W-PSNR to the Tooth Fairy dataset as well. \n3. The current table shows DuFal's Recall (98.45) as slightly higher than the GT's (98.37), which is confusing. Can the author explain?", "broader_impact_concerns": "N/A. The authors' Broader Impact discussion is sufficient.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "FvF4sHdXED", "reviewer": "Reviewer_S3D9", "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": "The authors propose a novel implicit neural representation (INR) framework for extremely sparse-view cone-beam CT (CBCT) reconstruction. Their approach employs two complementary modules: one operating in the frequency domain and the other in the spatial domain. The outputs from these two branches are merged through a cross-attention-based fusion module that integrates features from both the frequency and spatial domains. The proposed framework demonstrates superior reconstruction performance compared to existing baselines on two widely used datasets. The authors also conduct ablation studies to assess the contributions of individual components of their framework.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The paper provides clear empirical evidence through comparisons with baselines and ablation experiments.", "audience_yn": "Yes", "audience_explanation": "The paper addresses a relevant problem in CBCT reconstruction. Its methodological novelty and demonstrated improvements make it of interest to researchers working on CBCT reconstruction and neural operator methods.", "requested_changes": "- **All Tables:** Please include the standard deviation of reported metrics and specify over how many model runs or random seeds the averages are computed. This applies to all tables in the paper for better statistical confidence.\n- **Page 3, second-to-last line:** The phrase \"*2D and map 3D*\" is unclear.\n- **Page 7:** The proposed **Spectral-Channel Factorization (SCF)** appears conceptually similar to depthwise separable convolution. Are these two operations mathematically equivalent, or is there a key distinction that should be emphasised? I understand that SCF works in the frequency domain while depthwise separable convolution works in the spatial domain but the basic idea seems to be the same and it could be that the two operations are even mathematically equivalent in that if the input to SCF is converted to spatial-domain then the operation that gives the same output as SCF would be a deptpthwise separable conv operation (followed by conversion to frequency domain to match SCF's output). Please elaborate on the similarities and the differences.\n- **Page 8:** The paper introduces *Galerkin-Attention* but does not include a reference for it. Please provide a citation.\n- **Page 8, Section 4.3 (CAFF):** Does the choice of which branch provides the query (Q) and which provides the key/value (K,V) affect the results?\n- **Page 13, Section 6.2:** The differences shown in Table 5 seem small. Are they statistically significant? **Suggestion:** I think an ablation that might better isolate the impact of the cross-attention mechanism itself would be to compare it against a simple concatenation or summation of the two branches’ features without attention, rather than only studying the impact of converting the features to the frequency domain in CAFF. Perhaps the differences would be much larger if the ablation were whether having a module to fuse features is better than no module at all (i.e., simple concatenation or summation).\n- **Page 15, Section 6.6.2:** Please clarify whether the segmentation evaluation is performed on a single slice or multiple slices, and if multiple, how those slices are selected.", "broader_impact_concerns": "I have no concerns regarding the ethical implications of the work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)