HR-UMamba++: A High-Resolution Multi-Directional Mamba Framework for Coronary Artery Segmentation in X-Ray Coronary Angiography

Abstract

Coronary artery disease (CAD) remains a leading cause of mortality worldwide, and accurate coronary artery segmentation in X-ray coronary angiography (XCA) is challenged by low contrast, structural ambiguity, and anisotropic vessel trajectories, which hinder quantitative coronary angiography. We propose HR-UMamba++, a U-Mamba-based framework centered on a rotation-aligned multi-directional state-space scan for modeling long-range vessel continuity across multiple orientations. To preserve thin distal branches, the framework is equipped with (i) a persistent high-resolution bypass that injects undownsampled structural details and (ii) a UNet++-style dense decoder topology for cross-scale topological fusion. On an in-house dataset of 739 XCA images from 374 patients, HR-UMamba++ is evaluated using eight segmentation metrics, fractal-geometry descriptors, and multi-view expert scoring. Compared with U-Net, Attention U-Net, HRNet, U-Mamba, DeepLabv3+, and YOLO11-seg, HR-UMamba++ achieves the best performance (Dice 0.8706, IoU 0.7794, HD95 16.99), yielding a relative Dice improvement of 6.0% over U-Mamba and reducing the deviation in fractal dimension by up to 57% relative to U-Net. Expert evaluation across eight angiographic views yields a mean score of 4.24 ± 0.49/5 with high inter-rater agreement. These results indicate that HR-UMamba++ produces anatomically faithful coronary trees and clinically useful segmentations that can serve as robust structural priors for downstream quantitative coronary analysis.

1. Introduction

Coronary artery disease (CAD) is one of the most prevalent cardiovascular disorders and remains a leading cause of death worldwide [1,2]. Epidemiological evidence indicates that CAD accounts for around nine million deaths annually and affects more than 300 million individuals, with its global burden projected to increase further driven by population growth and aging [3,4,5,6,7]. CAD arises primarily from the gradual accumulation of atherosclerotic plaques in the coronary arteries, resulting in luminal narrowing and impaired myocardial perfusion, which in turn leads to varying degrees of myocardial ischemia and functional impairment [8]. Clinically, CAD manifests as stable or unstable angina, silent myocardial ischemia, or acute myocardial infarction [9,10], with acute events frequently triggered by thrombosis following rupture or erosion of thin-cap fibroatheromas [11,12].

In current clinical practice, imaging modalities used for the diagnosis and assessment of CAD include X-ray coronary angiography (XCA) [13], coronary computed tomography angiography (CCTA) [14], optical coherence tomography (OCT) [15], and near-infrared spectroscopy (NIRS) [16]. Among these, XCA is widely regarded as the gold standard for diagnosing CAD [13]. However, visual interpretation of XCA depends heavily on operator experience and exhibits substantial subjectivity and inter-observer variability. Prior studies have reported inter-clinician discrepancies in lesion-boundary identification and stenosis-severity assessment of up to 15–45% [17,18]. Furthermore, manual delineation of coronary vessels is time-consuming, which can compromise diagnostic efficiency and delay clinical decision-making [19,20].

To mitigate the subjectivity and inconsistency inherent in visual assessment, quantitative coronary angiography (QCA) has been developed with the aim of providing objective vascular evaluation in an automated and reproducible manner [21]. QCA enables precise measurement of parameters such as vessel diameter, lesion length, and reference lumen diameter, demonstrating higher consistency compared to traditional visual estimation [21,22,23]. In this workflow, accurately extracting the coronary artery tree from XCA images is a critical prerequisite for QCA implementation [24]. However, this task remains challenging due to inherently low vessel-to-background contrast, overlap caused by 2D projection, and confounders such as noise, motion artifacts, and illumination variability [24,25,26,27]. Figure 1 illustrates the common imaging challenges encountered in clinical XCA images, including insufficient contrast, speckle noise interference, and structural artifacts caused by guidewires and bony structures. Consequently, achieving robust and high-precision automatic segmentation of coronary arteries in XCA images remains a major technical bottleneck in both QCA workflows and medical image analysis.

Despite substantial recent progress, existing coronary-vessel segmentation methods still face several important limitations. First, the structural complexity of coronary vasculature in XCA makes it difficult for traditional unsupervised methods to reliably preserve the full vessel morphology, especially at bifurcations and in distal small-caliber vessels [28,29]. Although deeply supervised methods have achieved state-of-the-art accuracy and robustness, most existing models primarily target the extraction of major epicardial vessels and provide only limited characterization of the full coronary artery tree [30,31]. In addition, many approaches rely on multi-frame sequences or three-dimensional imaging conditions [32,33,34], which constrains their applicability to the clinically more common XCA setting. Finally, most deep learning models remain suboptimal in modeling slender tubular structures and global vascular topology, often yielding prediction masks with broken centerlines, missing details, and discontinuous vascular structures [31,35].

These challenges underscore the need for a segmentation framework that can simultaneously preserve thin distal branches, capture global vascular topology, and remain robust to noise and imaging artifacts in single-frame XCA images. Accordingly, we propose HR-UMamba++, a novel coronary artery segmentation framework built on U-Mamba and equipped with a rotation-aligned multi-directional scanning mechanism that explicitly models long-range vessel continuity in horizontal, vertical, and diagonal directions. To make this long-range directional modeling effective for faint coronary branches, we further introduce two architectural designs:

• Inspired by the segregated processing of magnocellular and parvocellular pathways in the primate retina [36], we construct a persistent high-resolution branch that continuously injects non-downsampled structural details along the entire decoding path, thereby markedly enhancing the spatial discriminability of fine vessels and distal terminal branches.
• We introduce a UNet++-style densely connected decoding topology that enables cross-level and multi-scale semantic feature fusion, strengthening the model’s ability to represent multi-level vascular bifurcations and the complete coronary artery tree.

Through this architectural design, HR-UMamba++ achieves high-resolution detail preservation, cross-scale vascular topology modeling, and robust performance under noise contaminated conditions in XCA images. The proposed model demonstrates clear advantages across multiple quantitative metrics and clinical expert evaluations, providing a reliable segmentation backbone for building automated and clinically applicable QCA workflows. Furthermore, we develop a vascular tree topological evaluation framework based on box-counting fractal dimension, leveraging fractal descriptors as a complementary measure to conventional segmentation metrics and establishing a methodological basis for the exploration of potential fractal-based biomarkers.

2. Related Work

A variety of frameworks have been proposed for automatic segmentation of coronary arteries, which can broadly be categorized into unsupervised and supervised methods [37]. Unsupervised methods can further be divided into three classes: filter-based, tracking-based, and model-based approaches [38]. Filter-based approaches [24,39,40,41,42,43] assume that local vessel cross-sections can be approximated by line- or tube-like intensity profiles. Such methods typically employ spatial- or frequency-domain filters to enhance vessel-like structures and suppress background noise.

Tracking-based approaches [28,44,45] select initial seed points along vessel centerlines or boundaries. Starting from these seeds, the methods predict the propagation direction and radius of the vessels by analyzing local patches of gray-level intensities or gradient features. New center points are then iteratively generated along the inferred direction, gradually tracking and reconstructing the full vascular tree topology. Model-based approaches [46,47,48,49,50,51] likewise require an initial contour or seed region as the starting point for curve evolution. These methods model the dynamic evolution of vessel structures using an energy functional or partial differential equation, and update the contour via energy minimization so that it progressively approaches the true vessel boundary.

Although unsupervised methods avoid labor-intensive manual annotation and often offer advantages in computational efficiency [52,53], their limitations are substantial. Filter-based approaches tend to respond poorly at complex bifurcations and are vulnerable to tubular artifacts such as catheters and the spine, leading to spurious vessel detections [54,55,56]. Tracking-based and model-based methods are highly sensitive to the quality and placement of initial seeds, and suboptimal seed initialization can cause the tracking procedure to deviate from the true lumen or drift into the background [28]. In addition, when neighboring high-intensity structures such as myocardial tissue are present, both tracking- and model-based approaches frequently exhibit boundary leakage into adjacent regions [37,57], further compromising segmentation stability and accuracy.

In contrast, supervised methods rely on large collections of annotated vessel masks for training. By learning discriminative feature representations from labeled data, these models establish decision boundaries in feature space that separate vascular from non-vascular regions [58,59]. He et al. [60] proposed a segmentation framework based on multi-frame invasive coronary angiography (ICA) sequences, in which temporal information was treated as a third dimension. They employed a 3D encoder–2D decoder architecture to fuse spatiotemporal features and introduced an elastic energy-based loss to encourage vessel connectivity. Han et al. [32] developed a topologically constrained Multiscale Attention Network (MAN) that integrated a Multiscale Squeeze Attention (MSA) mechanism into a U-Net backbone to extract multi-scale features, and employed a topology-aware loss function to preserve the continuity of the coronary tree. Zhao et al. [61] proposed a Hyper Association Graph Matching Network for semantic labeling of coronary arteries. In this framework, coronary semantic annotation was formulated as a hypergraph matching problem, wherein higher-order topological relationships were explicitly exploited to enhance labeling accuracy. He et al. [29] further proposed a coronary segmentation model that incorporated geometric priors and state-space modeling. Their framework integrated a Multiscale Vessel Structure-Aware (MVSA) module and a Kolmogorov–Arnold State Space (KASS) component within a U-shaped architecture to jointly encode vessel morphology and long-range dependencies, and replaced conventional skip connections with cross-stage feature fusion to improve detail preservation. Nannini et al. [33] employed a two-stage U-Net pipeline to segment coronary arteries and quantitatively assess coronary artery calcium and tortuosity from coronary CT angiography (CCTA). A multi-view 2.5D network performed an initial segmentation in the first stage, and a 3D U-Net refined the result in the second stage. Geometric postprocessing was then applied to extract vessel centerlines and morphological descriptors. Iyer et al. [62] introduced an Angiographic Processing Network (APN) module into an end-to-end model, enabling automatic learning of optimal pre-processing filters that sharpen vessel boundaries and enhance local contrast. Xu et al. [34] proposed a hybrid framework that couples spatial attention with a diffusion model for automatic segmentation of CCTA images, effectively addressing the challenges posed by complex vessel morphology, imaging noise, and substantial shape variability.

A synthesis of these recent supervised methods reveals three major limitations: (1) Most supervised coronary segmentation studies focus on delineating the major coronary arteries, with limited ability to model the full coronary artery tree and its fine distal branches. (2) Many frameworks are specifically tailored to CCTA or temporal angiographic sequences, limiting their applicability to the clinically common scenario of X-ray coronary angiography (XCA). (3) Conventional convolutional neural networks struggle to fully capture thin, elongated tubular structures. Their isotropic convolution kernels provide limited ability to encode complex vascular topology and long-range connectivity, which often leads to fragmented or discontinuous vessel predictions in the segmentation masks.

Motivated by these limitations, we propose HR-UMamba++, a novel segmentation framework designed to recover thin distal vessels and maintain topological continuity of the coronary artery tree from XCA images.

3. Materials and Methods

3.1. Dataset Description

The research team collaborated with the Affiliated Hospital of Shandong University of Traditional Chinese Medicine (College Town Branch) and retrospectively assembled a dataset of 739 XCA images from 374 patients who underwent routine clinical examinations between February 2023 and October 2024. The images cover the major epicardial coronary arteries, including the left anterior descending (LAD), left circumflex (LCX), and right coronary artery (RCA), and encompass a broad spectrum of vascular morphologies. To increase diversity and robustness for model training, the dataset includes normal arteries, mild stenosis, moderate-to-severe stenosis, and complete occlusions, reflecting clinically relevant pathophysiological variation.

All subjects underwent standardized X-ray coronary angiography via a transradial approach. Image acquisition was performed using a digital subtraction angiography (DSA) system (Optima IGS 330, GE Healthcare, Beijing, China). Prior to angiography, patients received weight-adjusted heparin for anticoagulation, and during the procedure an iodinated contrast agent (iohexol; $40\pm 10\mathrm{mL}$; $30\mathrm{g}\mathrm{I}/100\mathrm{mL}$) was administered. From each DSA sequence, we extracted representative frames and stored them as single-channel grayscale images with a spatial resolution of $512\times 512$ pixels. Ground-truth vessel masks were annotated by two experienced interventional cardiologists using the LabelMe image annotation platform. Key acquisition and sampling parameters of the dataset are summarized in

Table 1. Imaging and acquisition parameters of the coronary angiography dataset.

Item	Value
Number of images	739
Number of patients	374
Number of male patients	211
Number of female patients	163
Mean patient age (years)	59.7
Image resolution	$512\times 512$
Tube voltage	75–110 kVp
Tube current	4–7 mA

.

To protect patient privacy, all images underwent a rigorous de-identification process before use. Any personal information present in the original images was completely removed through standardized masking and cropping procedures, ensuring that individual subjects could not be re-identified. The study relied exclusively on existing clinical imaging records and did not involve additional interventions or prospective data collection. All procedures were conducted in accordance with the principles of the Declaration of Helsinki and relevant biomedical ethics guidelines. The Academic Ethics and Morality Committee of Shanghai Ocean University reviewed the research protocol and, given the use of fully anonymized retrospective data, granted a waiver of formal ethical approval and informed consent requirements.

Table 2. Comparison of our in-house dataset with published coronary angiography segmentation datasets.

Study	Patients (n)	Annotated Frames	Input Form	Image Size
Lee et al., 2024 [63]	213	500	Single frame	512 × 512
Chang et al., 2024 [64]	100	619	Single frame	512 × 512
Gao et al., 2025 [65]	–	180	2-frame sequence	512 × 512
Ramos-Cortez et al., 2025 [66]	–	750	Single frame	300 × 300 512 × 512
Wei et al., 2025 [67]	110	542	3-frame sequence	800 × 800
Ours	374	739	Single frame	512 × 512

compares our dataset with those used in several representative recent studies. Our in-house dataset includes a larger number of patients than most recent coronary angiography segmentation datasets, and the number of annotated frames is also among the highest reported. In addition, our dataset follows the clinically more common single-frame input setting, and the spatial resolution is consistent with common practice in the literature. Therefore, this dataset provides a solid foundation for validating the effectiveness of the proposed method.

3.2. Overall Architecture of HR-UMamba++

In recent years, vision architectures based on state space models (SSMs) have attracted increasing attention in medical image analysis owing to their ability to efficiently capture long-range dependencies [68]. Building on this paradigm, Ma et al. [69] proposed U-Mamba, which combines CNN-based local feature extraction with Mamba-based sequence modeling in a U-shaped encoder-decoder. Empirical studies have demonstrated that U-Mamba and its derivatives achieve superior performance compared with mainstream CNN- and Transformer-based models across a variety of tasks, including abdominal MRI/CT segmentation, 2D endoscopic imaging, microscopy images, and brain tumor segmentation on MRI [69,70,71,72]. These methods exhibit fewer segmentation artifacts and more stable predictions, suggesting that the U-Mamba paradigm also holds substantial potential for vascular tree segmentation, where both fine-grained local textures and global structural consistency must be preserved.

The proposed HR-UMamba++ architecture is constructed upon the U-Mamba framework and is specifically tailored to the highly filamentous, multi-branching geometry of coronary artery trees. To this end, three key modifications are introduced: (i) a persistent high-resolution feature branch ($H{R}_i$) that continuously injects fine structural details into the decoding pathway; (ii) a UNet++-style dense decoder topology that performs multi-level aggregation across both columns and scales to strengthen semantic fusion; and (iii) a rotation-aligned multi-directional Mamba scanning mechanism designed to capture vessel continuity along multiple spatial orientations. As illustrated in Figure 2, the overall architecture follows an encoder-decoder paradigm while incorporating dense lateral aggregation and a dedicated high-resolution pathway to better accommodate the geometric complexity of coronary vasculature.

3.3. Persistent High-Resolution Branch in HR-UMamba++

In the conventional U-Net architecture, high-resolution features are only transmitted once from encoder to decoder via a single skip connection at each resolution level [73]. As decoding proceeds through repeated concatenation and fusion, fine details from shallow layers can be diluted by deeper semantic features and multi-scale aggregation [74]. This effect is particularly detrimental for thin vessels, small distal branches, and boundary regions, where subtle structural cues are easily overwhelmed, leading to broken or missing segments in the final prediction [75].

To alleviate this limitation, we draw inspiration from the parallel processing mechanisms of the magnocellular and parvocellular pathways in the primate retinal visual system [36] and introduce a persistent high-resolution branch ($H{R}_i$). This branch maintains an undownsampled, high-resolution stream of structural information and continuously injects it into the decoder, thereby providing a stable source of fine-detail compensation throughout the multi-scale integration process.

Let the encoder feature map at the i-th resolution level be denoted as

$$X_{i,0}\in {\mathbb{R}}^{B\times C_i\times H_i\times W_i},$$

(1)

the persistent high-resolution branch refines local structures via a lightweight residual block $\mathcal{D}(·)$ to obtain

$${\mathrm{HR}}_i=\mathcal{D}\left(X_{i,0}\right),i=0,\dots ,S-2.$$

(2)

where S denotes the total number of resolution levels in the encoder.

The refinement block $\mathcal{D}(·)$ is implemented as

$$\mathcal{D}\left(X\right)=\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+X\right).$$

(3)

By maintaining this persistent high-resolution stream, HR-UMamba++ ensures that the decoder consistently receives high-fidelity spatial cues. In contrast to the fully parallel multi-resolution branch design exemplified by HRNet [76], our architecture maintains only a lightweight $H{R}_i$ branch, thereby substantially reducing computational cost. Meanwhile, $H{R}_i$ serves as a stable high-resolution detail bypass, mitigating the degradation of fine vessels caused by repeated fusion of high- and low-resolution features, and thereby making the architecture particularly suitable for the faithful reconstruction of coronary arteries with slender trunks and densely packed distal branches.

3.4. Encoder Module with Residual Convolutions and Multi-Directional Mamba

As illustrated in Figure 2, the encoder of HR-UMamba++ adopts a multi-stage down-sampling design. At each stage, residual convolutional units are employed to extract multi-scale semantic representations, while rotation-aligned multi-directional Mamba modules are inserted at selected resolutions to enhance the modeling of long-range dependencies within the same spatial scale. An anatomical analysis of common clinical XCA projections [77] indicates that the major coronary arteries predominantly course along the atrioventricular groove and thus appear approximately horizontal in the image; in contrast, vessels such as the LAD extend toward the cardiac apex along the anterior interventricular groove, exhibiting an approximately vertical orientation. Diagonal and lateral branches are often distributed obliquely. To cover these principal orientations while avoiding unnecessary computational overhead, we fix the scanning directions to the four angles $\{0^{\circ },{90}^{\circ },+{45}^{\circ },-{45}^{\circ }\}$. Let the input image be

$$\mathbf{I}\in {\mathbb{R}}^{B\times C_{\mathrm{in}}\times H\times W}.$$

(4)

The encoder consists of a stem block followed by S resolution stages, which sequentially produce feature maps

$$X_{0,0},X_{1,0},\dots ,X_{S-1,0}.$$

(5)

3.4.1. Stem Residual Block

In the stem stage, the input image is first processed by a residual alignment block:

$$X_{0,0}={\mathcal{R}}_0\left(\mathbf{I}\right),$$

(6)

where the residual alignment block ${\mathcal{R}}_0(·)$ is instantiated as

$${\mathcal{R}}_0\left(X\right)=\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+{\mathrm{Conv}}_{1\times 1}\left(X\right)\right),$$

(7)

to perform channel remapping and initial feature compression.

3.4.2. Multi-Stage Residual Down-Sampling

The s-th encoder stage is composed of one strided residual block for down-sampling and multiple non-strided residual refinement blocks for deep feature enrichment. Specifically, we first obtain

$$Z_s^{\left(0\right)}={\mathcal{R}}_s^{\downarrow }\left(X_{s-1,0}\right),$$

(8)

and then recursively refine the features via

$$Z_s^{\left(k\right)}={\mathcal{D}}_s\left(Z_s^{(k-1)}\right),k=1,\dots ,n_s-1.$$

(9)

where $n_s$ denotes the number of residual refinement blocks at stage s. The output of the s-th stage is thus given by

$$X_{s,0}=Z_s^{(n_s-1)}.$$

(10)

The strided residual down-sampling block and the same-scale residual refinement block are defined as

$$\begin{array}{cc}\hfill {\mathcal{R}}_s^{\downarrow }\left(X\right)& =\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+{\mathrm{Conv}}_{1\times 1}\left(X\right)\right),\hfill \\ \hfill {\mathcal{D}}_s\left(X\right)& =\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+X\right).\hfill \end{array}$$

(11)

The spatial resolution of the encoder features is progressively reduced according to

$$(H_s,W_s)=\left({\displaystyle \frac{H}{{\prod }_{i=0}^s{k}_i}},{\displaystyle \frac{W}{{\prod }_{i=0}^s{k}_i}}\right)$$

(12)

where $k_i$ denotes the down-sampling factor of the i-th stage.

3.4.3. Rotation-Aligned Multi-Directional Mamba Scanning Mechanism

In clinical XCA images, coronary vessels often extend over long distances along horizontal, vertical, and oblique directions. To enhance the encoder’s ability to capture long-range dependencies and orientation-specific continuity, HR-UMamba++ integrates a rotation-aligned multi-directional Mamba module at selected encoder stages. The core idea is to rotate 2D feature maps to align different orientations and flatten them into 1D sequences. These sequences are processed by Mamba units for direction-aware state-space modeling, and the resulting features are rotated back and fused in the original coordinate system. As illustrated in Figure 3, four orientations (horizontal, vertical, and two diagonals) are handled by rotating the feature map to $0^{\circ }$, ${90}^{\circ }$, ${45}^{\circ }$, and $-{45}^{\circ }$, respectively, followed by independent Mamba scans to capture vessel structures that predominantly extend along each direction.

Formally, let the input feature at the s-th encoder stage be

$$X_{s,0}\in {\mathbb{R}}^{B\times C\times H\times W}.$$

(13)

To avoid boundary cropping during rotation, $X_{s,0}$ is first reflect-padded to the minimal enclosing square:

$$L=⌈\sqrt{H^2+W^2}⌉,X_{s,0}^{\mathrm{pad}}=\mathcal{P}\left(X_{s,0}\right).$$

(14)

Subsequently, geometric rotations are applied at four orientation angles $\mathrm{\Theta }=\{0^{\circ },{90}^{\circ },{45}^{\circ },-{45}^{\circ }\}$:

$$X_{s,0}^{\left(\theta \right)}={\mathcal{T}}_{\theta }\left(X_{s,0}^{\mathrm{pad}}\right),$$

(15)

and each rotated 2D feature map is flattened into a sequence for state-space modeling:

$$Z_s^{\left(\theta \right)}=\mathrm{Flatten}\left(X_{s,0}^{\left(\theta \right)}\right)\in {\mathbb{R}}^{B\times C\times \left(L^2\right)},{\tilde{Z}}_s^{\left(\theta \right)}=\mathrm{LN}\left(Z_s^{\left(\theta \right)}\right).$$

(16)

The orientation-specific sequences are then fed into a Mamba unit to perform selective state evolution and recurrent convolution, yielding direction-aware responses:

$${\widehat{Z}}_s^{\left(\theta \right)}=\mathrm{Mamba}\left({\tilde{Z}}_s^{\left(\theta \right)}\right).$$

(17)

Each sequence is subsequently reshaped back to a 2D feature map, inverse-rotated to the original coordinate system, and normalized by the corresponding interpolation weights to ensure consistent spatial alignment:

$${\widehat{X}}_s^{\left(\theta \right)}=\mathcal{C}\left({\displaystyle \frac{{\mathcal{T}}_{\theta }^{-1}\left({\mathrm{reshape}}^{-1}\left({\widehat{Z}}_s^{\left(\theta \right)}\right)\right)}{W^{\left(\theta \right)}+\epsilon }}\right),$$

(18)

where $W^{\left(\theta \right)}$ denotes the interpolation weight map associated with rotation angle $\theta$, and $\mathcal{C}(·)$ denotes center cropping used to recover the original spatial resolution $(H,W)$. Finally, the four directional feature maps are concatenated along the channel dimension and projected back to the original channel size C:

$$X_{s,\mathrm{mamba}}=\mathrm{BN}\left({\mathrm{Conv}}_{1\times 1}(\mathrm{BN}([{\widehat{X}}_s^{\left(0^{\circ }\right)},{\widehat{X}}_s^{\left({90}^{\circ }\right)},{\widehat{X}}_s^{\left({45}^{\circ }\right)},{\widehat{X}}_s^{(-{45}^{\circ })}]))\right)$$

(19)

thereby producing orientation-enhanced features at the same spatial scale as the original encoder stage.

3.5. Decoder and Dense Connection Module in HR-UMamba++

The decoder of HR-UMamba++ adopts a UNet++-style dense topology [78] to facilitate lateral aggregation across decoder columns and cross-scale semantic fusion. Let the feature map from the i-th encoder layer be

$$X_{i,0}\in {\mathbb{R}}^{B\times C\times H\times W}.$$

(20)

Each decoder node $X_{i,j}(j\ge 1)$ receives a concatenated input $F_{i,j}$ obtained by fusing all preceding nodes at the same depth, the upsampled feature from the next deeper level, and the corresponding high-resolution branch $H{R}_i$:

$$F_{i,j}=\left[X_{i,0},X_{i,1},\dots ,X_{i,j-1},\mathrm{Up}\left(X_{i+1,j-1}\right),H{R}_i\right],$$

(21)

where the upsampling operator is defined as

$$\mathrm{Up}\left(X\right)={\mathrm{Conv}}_{1\times 1}\left({\mathrm{Interp}}_{\times 2}\left(X\right)\right),$$

(22)

and simultaneously restores the spatial resolution and aligns the channel dimension.

To ensure that the outputs of all nodes share a consistent dimensionality compatible with the UNet++ grid, each decoder node is implemented as a two-stage convolutional block comprising a channel alignment block ${\mathcal{A}}_{i,j}$ and a feature refinement block ${\mathcal{F}}_{i,j}$, defined as

$$\begin{array}{cc}\hfill {\mathcal{A}}_{i,j}\left(X\right)& =\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+{\mathrm{Conv}}_{1\times 1}\left(X\right)\right),\hfill \\ \hfill {\mathcal{F}}_{i,j}\left(X\right)& =\mathrm{LeakyReLU}\left({\mathrm{Norm}}_2\left({\mathrm{Conv}}_2\left(\mathrm{LeakyReLU}\left({\mathrm{Norm}}_1\left({\mathrm{Conv}}_1\left(X\right)\right)\right)\right)\right)+X\right).\hfill \end{array}$$

(23)

The final output of each decoder node is obtained by sequentially passing the concatenated feature $F_{i,j}$ through these two blocks:

$$X_{i,j}={\mathcal{F}}_{i,j}\left({\mathcal{A}}_{i,j}\left(F_{i,j}\right)\right).$$

(24)

4. Experiments and Results

4.1. Implementation Details

This study is implemented within the nnU-Net v2 framework [79], into which we integrate a custom HR-UMamba++ model while preserving the standardized automated training pipeline. The backbone follows a 2D U-Net architecture with eight resolution stages, using input patches of size $512\times 512$ and a batch size of 3. Each encoder stage contains two $3\times 3$ convolutional layers, with the number of feature channels progressively increasing from 32 at the highest resolution to 512 at the bottleneck. Spatial down-sampling is performed by seven $2\times 2$ pooling operations with stride 2, reducing the feature-map resolution to $4\times 4$. The decoder is composed of seven symmetric up-sampling stages, where $2\times 2$ bilinear interpolation is used at each level to double the spatial resolution while decreasing the channel dimensionality from 512 back to 32. Each decoder stage likewise includes two $3\times 3$ convolutional layers, and after the final up-sampling step the output feature maps are restored to the original spatial resolution of $512\times 512$.

To mitigate the low contrast and noise corruption commonly observed in XCA images, we introduce a lightweight fractional-order Laplacian operator into the preprocessing pipeline. In the frequency domain, the fractional Laplacian of an image I can be expressed as

$$\mathcal{F}\left\{{(-∆)}^{\alpha /2}I\right\}({\omega }_x,{\omega }_y)={\left|\omega \right|}^{\alpha }\mathcal{F}\left\{I\right\}({\omega }_x,{\omega }_y),$$

(25)

where $\mathcal{F}\left\{I\right\}({\omega }_x,{\omega }_y)$ denotes the two-dimensional Fourier transform of I, $\left|\omega \right|=\sqrt{{\omega }_x^2+{\omega }_y^2}$ is the magnitude of the frequency vector, and the parameter $0<\alpha <2$ controls the response to high-frequency components. Based on this operator, the enhanced image is given by

$$I_{\mathrm{enh}}=I+\lambda {(-∆)}^{\alpha /2}I,$$

(26)

where $\lambda$ is an enhancement coefficient. This operation selectively amplifies high-frequency vascular structures, such as vessel boundaries and distal branches, thereby improving the visibility of fine vessels and regions with low contrast. Additional preprocessing and data augmentation strategies are summarized in

Table 3. Preprocessing and data augmentation parameters.

Item	Parameter
Intensity normalization	Z-score
Multiplicative brightness perturbation	$0.7–1.3$
Additive brightness perturbation	$\pm 0.1$
Contrast perturbation	$0.65–1.5$
Gamma correction	$0.7–1.5$
Gaussian noise	$\sigma \in [0,0.1]$
Gaussian blur	kernel size $=3$, $\sigma \in [1.0,3.0]$
Random scaling	$0.85–1.25$
Random rotation	$\pm {180}^{\circ }$
Bidirectional mirroring	axes $=(0,1)$
Low-resolution simulation	down-sampling factor $0.5–1.0$

.

The training objective combines a pixel-wise cross-entropy loss with a soft Dice loss. The cross-entropy term is defined as

$${\mathcal{L}}_{CE}=-{\displaystyle \frac{1}{BHW}}\sum _{b,i,j}\sum _c{y}_{b,c,i,j}log{p}_{b,c,i,j},$$

(27)

where $p_{b,c,i,j}$ and $y_{b,c,i,j}$ denote the predicted probability and one-hot encoded ground-truth label of class c at spatial location $(i,j)$ in sample b. The Dice loss measures the overlap between the predicted segmentation and the ground truth, and is given by

$${\mathcal{L}}_{Dice}=1-{\displaystyle \frac{2\sum py+ϵ}{\sum p+\sum y+ϵ}}.$$

(28)

The final training objective is an equally weighted combination of the two terms:

$${\mathcal{L}}_{Total}={\mathcal{L}}_{CE}+{\mathcal{L}}_{Dice}.$$

(29)

The model is trained for 1000 epochs using a 5-fold cross-validation protocol for data partitioning. We adopt stochastic gradient descent (SGD) with Nesterov momentum of 0.99 as the optimizer, and set the weight decay to $3\times {10}^{-5}$. The initial learning rate is set to $1\times {10}^{-2}$ and progressively decreased according to a polynomial learning rate schedule. Automatic mixed precision (AMP) is enabled throughout training to reduce GPU memory consumption and improve computational efficiency.

All experiments are conducted on a workstation running Ubuntu 22.04.3 LTS equipped with a single NVIDIA GeForce RTX 4090 GPU (24 GB). The CUDA runtime version is 11.8, and both training and inference are implemented in PyTorch 2.1.2+cu118 with Python 3.10. All experiments are performed under this unified software and hardware environment to ensure the comparability and reproducibility of the reported results.

4.2. Evaluation Metrics

To comprehensively evaluate model performance on the vessel segmentation task, we employ eight quantitative metrics: Accuracy, Sensitivity, Specificity, Positive Predictive Value (PPV), Negative Predictive Value (NPV), Dice coefficient, Intersection over Union (IoU), and the 95th-percentile Hausdorff distance (HD95). All metrics are computed from pixel-wise comparisons between the predicted segmentation masks and the manual annotations, where $\mathrm{TP}$, $\mathrm{FP}$, $\mathrm{TN}$, and $\mathrm{FN}$ denote the numbers of true-positive, false-positive, true-negative, and false-negative pixels, respectively.

Accuracy quantifies the overall correctness of the predictions and is defined as

$$\mathrm{Accuracy}={\displaystyle \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{TN}+\mathrm{FP}+\mathrm{FN}}}.$$

(30)

Sensitivity (recall) measures the ability of the model to correctly identify vessel pixels:

$$\mathrm{Sensitivity}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}}.$$

(31)

Specificity measures the capability of the model to correctly recognize background (non-vessel) pixels:

$$\mathrm{Specificity}={\displaystyle \frac{\mathrm{TN}}{\mathrm{TN}+\mathrm{FP}}}.$$

(32)

PPV quantifies the proportion of correctly predicted vessel pixels among all pixels predicted as vessel:

$$\mathrm{PPV}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}}.$$

(33)

NPV quantifies the proportion of correctly predicted background pixels among all pixels predicted as background:

$$\mathrm{NPV}={\displaystyle \frac{\mathrm{TN}}{\mathrm{TN}+\mathrm{FN}}}.$$

(34)

The Dice coefficient measures the overlap between the predicted and ground-truth vessel regions and is one of the most widely used metrics in medical image segmentation:

$$\mathrm{Dice}={\displaystyle \frac{2\mathrm{TP}}{2\mathrm{TP}+\mathrm{FP}+\mathrm{FN}}}.$$

(35)

IoU (Jaccard index) measures the ratio between the intersection and the union of the predicted and ground-truth regions:

$$\mathrm{IoU}={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}+\mathrm{FN}}}.$$

(36)

In addition, HD95 is used to assess the spatial consistency of boundary predictions. The Hausdorff distance measures the maximum deviation between the predicted and ground-truth boundaries in the spatial domain, whereas HD95 takes the 95th percentile of the bidirectional boundary distances to reduce the influence of outliers and annotation noise. Let P and G denote the sets of boundary points extracted from the prediction and the ground truth, respectively; HD95 is defined as

$$\mathrm{HD}95=max\left\{{percentile}_{95}\left(\underset{g\in G}{min}\parallel p-g\parallel \right),{percentile}_{95}\left(\underset{p\in P}{min}\parallel g-p\parallel \right)\right\}.$$

(37)

Among these metrics, Accuracy, Sensitivity, Specificity, PPV, NPV, Dice, and IoU take values within the interval $[0,1]$, with values closer to 1 indicating stronger agreement between the predicted segmentation and the reference annotations. In contrast, HD95 is a distance-based measure, where smaller values approaching 0 indicate more precise localization of the vessel boundaries.

4.3. Ablation Studies

To disentangle the contribution of each core design choice in HR-UMamba++, we conduct ablation studies on three key components: (i) the UNet++-style dense connections, (ii) the persistent high-resolution (HR) branch, and (iii) the rotation-aligned multi-directional Mamba scanning mechanism. We perform both single-module and pairwise ablation experiments. All variants are trained under identical training protocols and data configurations, and the corresponding quantitative results are summarized in

Table 4. Ablation Analysis of the Proposed HR-UMamba++ Framework. The best results for each metric are highlighted in bold.

Model Variant	Accuracy	Sensitivity	Specificity	PPV	NPV	Dice	IoU	HD95
Proposed Model	0.9877	0.8179	0.9975	0.9461	0.9895	0.8706	0.7794	16.9943
Single-module ablations
w/o Dense conn.	0.9843	0.7746	0.9968	0.9293	0.9867	0.8364	0.7300	20.3191
w/o HR branch	0.9844	0.8003	0.9955	0.9002	0.9880	0.8413	0.7375	18.7250
w/o MD SSM	0.9852	0.7954	0.9969	0.9259	0.9876	0.8478	0.7470	17.2209
Double-module ablations
w/o Dense conn. & HR Branch	0.9842	0.7677	0.9963	0.9156	0.9871	0.8294	0.7164	20.3458
w/o Dense conn. & MD SSM	0.9837	0.7672	0.9964	0.9174	0.9865	0.8275	0.7161	19.5036
w/o HR branch & MD SSM	0.9842	0.7852	0.9960	0.9139	0.9873	0.8379	0.7303	18.7881
U-Mamba (baseline)	0.9833	0.7768	0.9952	0.8852	0.9872	0.8215	0.7005	20.3530

.

As shown in Table 4, the complete HR-UMamba++ model achieves the best performance across all eight evaluation metrics. Removing any individual module consistently leads to a deterioration in segmentation performance, indicating that these components collectively contribute to the overall performance improvements in coronary vessel segmentation.

4.3.1. Effect of Dense Connections

As reported in Table 4, removing the dense connections yields a pronounced decline in the model’s ability to capture vessel connectivity and thin filamentous structures. Specifically, Sensitivity and IoU decrease by 5.3% and 6.3%, respectively, while HD95 increases by approximately 19.6% compared with the full model. These changes highlight the critical role of dense connections in maintaining the continuity and spatial coherence of coronary vasculature.

Figure 4 illustrates three representative cases. In Case 1, faint vessel boundaries in a low-perfusion region are misclassified as background by the ablated model, resulting in local discontinuities. In Case 2, the continuity of small-caliber vessels is substantially compromised, and gaps emerge at the junctions between the main trunk and its side branches. In Case 3, the ablated model fails to preserve the complex bifurcation topology, with several thin branches completely missing. In contrast, the full HR-UMamba++ model accurately reconstructs the global vascular topology in all cases, confirming that dense connections are essential for effective multi-scale feature aggregation and fine-grained vessel structure recovery.

4.3.2. Effect of the Persistent High-Resolution Branch

The persistent high-resolution (HR) branch maintains a dedicated high-resolution feature stream throughout the feature extraction process, enabling the network to preserve fine spatial details and vessel boundaries and thereby providing more discriminative cues for thin, filamentous coronary structures. As reported in Table 4, removing the HR branch reduces the Dice coefficient and IoU by approximately 3.4% and 5.4%, respectively, while increasing HD95 by about 10.2%, underscoring its importance for accurate and anatomically coherent vessel delineation. To further quantify the role of the persistent HR branch in preserving multi-scale vascular structures, we perform a frequency-domain analysis of the outputs produced by the full HR-UMamba++ model and its ablated counterpart. Specifically, the predicted segmentation masks from both models are first apodized with a Hanning window and have their DC components removed, after which a 2D Fourier transform is applied to obtain the corresponding power spectra. We then compute the proportion of mid-to-high frequency energy within the normalized radial frequency range of 0.25–0.9.

The experimental results show that the complete model exhibits, on average, a 4.10% increase in mid-to-high frequency energy, indicating that the HR branch substantially enhances the preservation of fine vascular filaments and directional continuity. As illustrated in Figure 5, the log-scaled power spectra reveal that the model without the HR branch suffers from a marked loss of high-frequency energy, leading to degraded texture representation and reduced connectivity of thin vessel segments. In contrast, the additional energy in the full model is primarily concentrated in radial streaks aligned with the dominant coronary directions, suggesting that the HR branch selectively amplifies structure-related high-frequency components rather than noise, thereby improving the continuity and completeness of the reconstructed coronary artery tree.

4.3.3. Effect of the Rotation-Aligned Multi-Directional Mamba Module

Removing the rotation-aligned multi-directional Mamba scanning mechanism reduces the Dice coefficient from 0.8706 to 0.8478 and the IoU from 0.7794 to 0.7470, indicating that this component contributes meaningfully to the overall segmentation performance. To further assess its role in maintaining directional coherence and global connectivity of the coronary vasculature, we additionally include 200 angiograms with visually normal coronary anatomy and define a binary structural-integrity indicator based on whether the predicted mask contains exactly one foreground connected component. Ideally, all 200 predictions should exhibit a single connected vascular tree.

Experimental results show that the complete model preserves single-component connectivity in 187 out of 200 images, whereas the ablated variant achieves this in only 175 cases. Because both models generate segmentation masks on the same image set, these outcomes constitute a paired binary dataset, for which we employ McNemar’s test to assess the statistical significance of the difference [80]; the corresponding paired outcomes are summarized in the contingency table in

Table 5. McNemar contingency table for single-component connectivity preservation.

	Ablation Correct	Ablation Wrong
Full Model Correct	172	15
Full Model Wrong	3	10

. A two-sided exact binomial test yields a p-value of 0.0075, demonstrating that removal of the rotation-aligned multi-directional Mamba module significantly degrades the model’s ability to preserve global vascular connectivity. Consistently, the paired risk difference increases by 6.0 percentage points, indicating that this mechanism provides a stable improvement in the structural consistency of the predicted vessel trees.

4.4. Performance Comparison with Canonical Baselines

To further assess the overall performance of HR-UMamba++ on the coronary vessel segmentation task, we compare it against five representative semantic segmentation baselines: U-Net [81], Attention U-Net [82], HRNet [76], U-Mamba [69], and DeepLabv3+ [83]. Given the competitive performance of the YOLO family reported in coronary vessel segmentation [84,85,86], we additionally include YOLO11-seg as an instance segmentation baseline. As YOLO targets instance segmentation [87], we convert each binary vessel mask into instance labels via connected-component decomposition, and merge the predicted instances back into a semantic vessel map for metric evaluation. YOLO11-seg is initialized from pretrained YOLO11m-seg weights and trained with the official default hyperparameters. All models are trained and evaluated under identical data configurations and training protocols to ensure the fairness and reproducibility of the comparison.

Table 6. Quantitative comparison between the proposed HR-UMamba++ model and six representative vessel-segmentation baselines. The best results for each metric are highlighted in bold.

Model Variant	Accuracy	Sensitivity	Specificity	PPV	NPV	Dice	IoU	HD95
U-Net	0.9804	0.7189	0.9938	0.8530	0.9855	0.7758	0.6362	27.6258
Attention U-Net	0.9802	0.7153	0.9941	0.8616	0.9851	0.7777	0.6389	23.0382
HRNet	0.9827	0.7684	0.9942	0.8790	0.9874	0.8184	0.6962	23.1386
U-Mamba	0.9833	0.7768	0.9952	0.8852	0.9872	0.8215	0.7005	20.3530
DeepLabv3+	0.9804	0.7692	0.9920	0.8290	0.9872	0.7937	0.6624	17.2217
YOLO11-seg	0.9802	0.7490	0.9916	0.8364	0.9874	0.7882	0.6519	18.7155
Proposed Model	0.9877	0.8179	0.9975	0.9461	0.9895	0.8706	0.7794	16.9943

summarizes the quantitative results across eight evaluation metrics, while Figure 6 provides a visual comparison of the predicted segmentation masks.

The CNN-based U-Net and Attention U-Net exhibit stable performance on large and moderately sized structures; however, their limited capacity for rich high-resolution feature representation leads to suboptimal detection of fine vessels. Consequently, their Sensitivity scores are restricted to 0.7189 and 0.7153, with Dice coefficients of 0.7758 and 0.7777, respectively. In the visual results, both models frequently produce interruptions in slender vessels and miss small peripheral branches. This issue is particularly evident in the complex tree-like region of Case 2, where multiple low-contrast bifurcations are incorrectly collapsed into discontinuous fragments, highlighting the limitations of purely convolutional encoders in modeling delicate vascular details.

By maintaining high-resolution representations via parallel multi-resolution streams with repeated feature fusion, HRNet provides a stronger baseline for vessel delineation. It improves Sensitivity to 0.7684 and Dice to 0.8184, outperforming U-Net and Attention U-Net. Qualitatively, HRNet recovers more secondary branches in Case 2 and produces fewer breaks along medium-caliber segments. However, its boundary accuracy and topological completeness remain limited: distal vessels with rapidly changing orientations can still appear locally discontinuous, and small peripheral branches are prone to truncation. These observations suggest that preserving high-resolution features alone is insufficient to maintain directional coherence and long-range continuity across the full coronary tree, and that explicitly modeling long-range dependencies remains necessary.

Leveraging state space modeling, U-Mamba substantially enhances the capture of long-range dependencies and achieves noticeable improvements over conventional CNN-based architectures, with Sensitivity increasing to 0.7768 and the Dice coefficient to 0.8215. The visual comparison indicates that U-Mamba maintains more consistent continuity along the main trunk and medium-caliber segments. Nevertheless, connection gaps remain in distal branches with rapidly changing orientations. For instance, U-Mamba fails to correctly recover small vessels such as the acute marginal branch and right ventricular branch in Case 1, indicating that its modeling of directional consistency is still insufficient to fully preserve the complete topology of thin coronary vessels.

DeepLabv3+ benefits from its strong multi-scale feature extraction capability, achieving an HD95 of 17.22, which reflects more accurate boundary localization. Visual comparison shows that its vessel contours are relatively smooth. However, in the fine distal branches of Case 1, it still exhibits terminal branch truncation and local structural adhesion, implying that atrous convolutions alone are not fully adequate for capturing high-frequency vascular texture details. Although YOLO11-seg demonstrates strong performance in object detection and region-level segmentation, its multi-task design introduces trade-offs when modeling subtle vessel patterns, and its overall performance does not surpass that of more specialized semantic segmentation architectures.

In contrast, the proposed HR-UMamba++ achieves the best performance across all eight evaluation metrics. It attains an Accuracy of 0.9877, while Sensitivity is further improved to 0.8179, and the Dice coefficient and IoU reach 0.8706 and 0.7794, respectively. In the visual comparison of the three representative cases in Figure 6, the proposed model consistently reconstructs the most complete coronary artery tree topology, preserving the continuity and directional coherence of thin bifurcations. It effectively avoids the common issues observed in other models, such as small-branch breakage, trajectory deviation, and local omissions, thereby demonstrating its clear overall advantage in modeling complex coronary vessel structures.

4.5. Fractal-Geometry-Based Topological Assessment of Coronary Arterial Trees

The coronary arterial tree exhibits a characteristic branching morphology with pronounced multiscale and self-similar patterns [88,89]. Its geometric complexity and hierarchical branching architecture can be described in a scale-invariant manner by the fractal dimension. To further assess the ability of different models to preserve the topological structure of the coronary arterial tree, we complement conventional overlap- and distance-based metrics with a box-counting-based fractal-dimension analysis.

Specifically, we first apply skeletonization to the binary vessel masks obtained from the manual annotations and model predictions to extract vessel centerlines, so that the fractal dimension primarily reflects the branching hierarchy and spatial spread of the vascular tree. Let $M\in {\{0,1\}}^{H\times W}$ denote a binary vessel mask, where $M(x,y)=1$ indicates that pixel $(x,y)$ belongs to the vessel foreground. We apply the Zhang-Suen thinning algorithm to iteratively remove boundary pixels that do not affect connectivity, thereby shrinking the vessels into a one-pixel-wide centerline. Let ${\left\{M^{\left(t\right)}\right\}}_{t\ge 0}$ be the thinning sequence, defined by

$$M^{\left(0\right)}=M,M^{(t+1)}=M^{\left(t\right)}\setminus \mathcal{R}\left(M^{\left(t\right)}\right),$$

(38)

where $\mathcal{R}\left(M^{\left(t\right)}\right)$ denotes the set of pixels removable at iteration t. We first define the foreground set and its boundary at iteration t as

$$F^{\left(t\right)}=\{p\in \mathrm{\Omega }\mid M^{\left(t\right)}\left(p\right)=1\},\partial F^{\left(t\right)}=\{p\in F^{\left(t\right)}|\exists q\in N_8\left(p\right),M^{\left(t\right)}\left(q\right)=0\},$$

(39)

where $\mathrm{\Omega }$ is the image lattice and $N_8\left(p\right)$ denotes the 8-neighborhood of pixel p. For any candidate pixel $p_1$ and its clockwise-ordered neighbors $\{p_2,\dots ,p_9\}$, we define the number of foreground neighbors $B\left(p_1\right)$ and the number of $0\to 1$ transitions $A\left(p_1\right)$ as

$$B\left(p_1\right)=\sum _{i=2}^9{M}^{\left(t\right)}\left(p_i\right),$$

(40)

$$A\left(p_1\right)=\sum _{i=2}^9\left(1-M^{\left(t\right)}\left(p_i\right)\right)M^{\left(t\right)}\left(p_{i+1}\right),(p_{10}\equiv p_2).$$

(41)

To avoid directional bias that may cause skeleton breakage, we adopt a two-subiteration scheme with parallel deletions. The removable sets for the two subiterations are given by

$${\mathcal{R}}^{(t,1)}=\left\{p_1\in \partial F^{\left(t\right)}|\begin{array}{c}2\le B\left(p_1\right)\le 6,\hfill \\ A\left(p_1\right)=1,\hfill \\ M^{\left(t\right)}\left(p_2\right)M^{\left(t\right)}\left(p_4\right)M^{\left(t\right)}\left(p_6\right)=0,\hfill \\ M^{\left(t\right)}\left(p_4\right)M^{\left(t\right)}\left(p_6\right)M^{\left(t\right)}\left(p_8\right)=0\hfill \end{array}\right\},$$

(42)

$${\tilde{M}}^{\left(t\right)}=M^{\left(t\right)}\setminus {\mathcal{R}}^{(t,1)},$$

(43)

$${\mathcal{R}}^{(t,2)}=\left\{p_1\in \partial {\tilde{F}}^{\left(t\right)}|\begin{array}{c}2\le B_{\tilde{M}}\left(p_1\right)\le 6,\hfill \\ A_{\tilde{M}}\left(p_1\right)=1,\hfill \\ {\tilde{M}}^{\left(t\right)}\left(p_2\right){\tilde{M}}^{\left(t\right)}\left(p_4\right){\tilde{M}}^{\left(t\right)}\left(p_8\right)=0,\hfill \\ {\tilde{M}}^{\left(t\right)}\left(p_2\right){\tilde{M}}^{\left(t\right)}\left(p_6\right){\tilde{M}}^{\left(t\right)}\left(p_8\right)=0\hfill \end{array}\right\},$$

(44)

$$M^{(t+1)}={\tilde{M}}^{\left(t\right)}\setminus {\mathcal{R}}^{(t,2)}.$$

(45)

$\partial {\tilde{F}}^{\left(t\right)}$ is the boundary recomputed from ${\tilde{M}}^{\left(t\right)}$, and $B_{\tilde{M}},A_{\tilde{M}}$ denote the corresponding quantities B and A evaluated on ${\tilde{M}}^{\left(t\right)}$.

The iterations terminate when ${\mathcal{R}}^{(T,1)}=⌀$ and ${\mathcal{R}}^{(T,2)}=⌀$, and the final skeleton is defined as

$$S=\mathcal{S}\left(M\right)=M^{\left(T\right)}\in {\{0,1\}}^{H\times W}.$$

(46)

The resulting S is a one-pixel-wide centerline skeleton that preserves the hierarchical branching structure of the vascular tree, enabling subsequent fractal-dimension analysis to focus on topological complexity rather than variations in vessel caliber.

For a set of spatial scales $\left\{{ϵ}_k\right\}$, we perform box-counting on S by tiling the image with square boxes of side length ${ϵ}_k$ and counting the number of boxes $N\left({ϵ}_k\right)$ that contain at least one foreground pixel. Figure 7 illustrates the box-counting procedure applied to the ground truth (GT) and the DeepLabv3+ prediction for the same image. Repeating this counting across multiple scales yields the paired observations $\left(log(1/{ϵ}_k),logN\left({ϵ}_k\right)\right)$. We then fit the following linear model in the log-log domain:

$$logN\left({ϵ}_k\right)=-Dlog{ϵ}_k+C,$$

(47)

where the slope D provides an estimate of the fractal dimension of the vessel-tree skeleton. For each image, we compute the fractal dimensions of the ground-truth skeleton $D_{\mathrm{GT}}$ and the predicted skeleton $D_{\mathrm{pred}}$, and define

$$∆D=\left|D_{\mathrm{pred}}-D_{\mathrm{GT}}\right|$$

(48)

as a measure of the preservation of fractal topological complexity. Smaller values of $∆D$ indicate that the model more faithfully preserves the geometric characteristics of the true coronary arterial tree. The final score for each model is obtained by averaging $∆D$ over all test images.

Table 7. Fractal-geometry-based evaluation of coronary arterial tree topology. $D_{\mathrm{pred}}$ denotes the mean fractal dimension of the predicted vessel skeleton, and $∆D=|D_{\mathrm{pred}}-D_{\mathrm{GT}}|$ denotes the deviation in fractal complexity. The best results for each metric are highlighted in bold.

Method	${\mathit{D}}_{\mathbf{pred}}$	$\mathbf{Mean}\left(\mathit{∆}\mathit{D}\right)\pm \mathbf{SD}$
U-Net	1.0258	0.0818 ± 0.0390
Attention U-Net	1.0327	0.0811 ± 0.0367
HRNet	1.0697	0.0468 ± 0.0245
U-Mamba	1.0712	0.0377 ± 0.0193
DeepLabv3+	1.0714	0.0531 ± 0.0301
YOLO11-seg	1.0506	0.0614 ± 0.0346
Proposed Model	1.0721	0.0355 ± 0.0178

summarizes the fractal-geometry-based evaluation results for all representative methods. The mean fractal dimension of the ground-truth coronary arterial trees is approximately $D_{\mathrm{GT}}=1.1076$, which is comparable in magnitude to values reported for other two-dimensional vascular networks [90]. This indicates that the test set exhibits a realistic level of topological complexity.

Conventional encoder-decoder architectures exhibit notable limitations in preserving fractal complexity. The predicted fractal dimensions of U-Net and Attention U-Net are $D_{\mathrm{pred}}=1.0258$ and $1.0327$, respectively, with $\mathrm{Mean}(∆D)$ values around $0.08$. This suggests that, despite achieving relatively high overlap metrics, these models tend to over-simplify fine distal branches and bifurcation patterns, leading to a substantial underestimation of the global fractal complexity. By maintaining high-resolution representations via parallel multi-resolution streams, HRNet increases the predicted fractal dimension to $D_{\mathrm{pred}}=1.0697$ and reduces the deviation to $\mathrm{Mean}(∆D)=0.0468$, corresponding to an approximate 43% reduction in fractal-dimension deviation relative to U-Net. Nevertheless, its predicted topology remains simplified compared with the ground truth, suggesting that preserving high-resolution features alone may be insufficient to fully recover the finest distal arborization.

DeepLabv3+, which incorporates multiscale atrous convolutions, partially alleviates this issue. Its predicted fractal dimension increases to $D_{\mathrm{pred}}=1.0714$, and $\mathrm{Mean}(∆D)$ decreases to $0.0531$, corresponding to an approximate 35% reduction in fractal-dimension deviation compared with U-Net. YOLO11-seg yields intermediate performance, with $D_{\mathrm{pred}}=1.0506$ and $\mathrm{Mean}(∆D)=0.0614$, indicating that a noticeable loss of fractal complexity persists.

The state-space-based U-Mamba and the proposed HR-UMamba++ model achieve the closest agreement with the ground-truth fractal geometry. U-Mamba attains $D_{\mathrm{pred}}=1.0712$ with $\mathrm{Mean}(∆D)=0.0377$. Building on this, HR-UMamba++ further improves the preservation of vascular topology by incorporating a persistent high-resolution branch and a rotation-aligned multi-directional Mamba scanning mechanism, yielding an average predicted fractal dimension of $D_{\mathrm{pred}}=1.0721$ and the lowest $\mathrm{Mean}(∆D)=0.0355$ among all methods. Relative to U-Net, HR-UMamba++ reduces the fractal-dimension deviation by approximately 57%, and it also achieves reductions of about 42% and 33% compared with YOLO11-seg and DeepLabv3+, respectively, supporting improved preservation of the coronary arterial tree’s topological complexity.

To assess the statistical reliability of the observed improvements, we compared per-image fractal-dimension deviations ($∆D$) using paired nonparametric tests. A Friedman test across all methods indicated a significant overall difference (${\chi }_F^2\left(6\right)=63.11$, $p=1.05\times {10}^{-11}$; $N=50$, $k=7$). Post-hoc comparisons between HR-UMamba++ and each baseline were then performed using two-sided Wilcoxon signed-rank tests with Holm–Bonferroni adjustment to control the family-wise error rate at $\alpha =0.05$. The proposed method achieved significantly lower $∆D$ than U-Net ($p_{\mathrm{adj}}=3.75\times {10}^{-5}$), Attention U-Net ($p_{\mathrm{adj}}=3.75\times {10}^{-5}$), HRNet ($p_{\mathrm{adj}}=2.08\times {10}^{-4}$), U-Mamba ($p_{\mathrm{adj}}=0.041$), DeepLabv3+ ($p_{\mathrm{adj}}=1.05\times {10}^{-5}$), and YOLO11-seg ($p_{\mathrm{adj}}=4.18\times {10}^{-9}$), supporting a statistically reliable improvement in fractal-topology consistency.

4.6. Clinical Usability Evaluation

To further evaluate the clinical usability of the model across different coronary angiographic projections, we randomly selected 20 representative frames from each of eight commonly used views: LAO view for RCA, AP caudal (${20}^{\circ }$), LAO ${45}^{\circ }$ + CAU ${20}^{\circ }$ (spider view), RAO ${30}^{\circ }$ + CAU ${20}^{\circ }$, RCA cranial view, RAO ${30}^{\circ }$ + CRA ${20}^{\circ }$, AP cranial (${30}^{\circ }$), and LAO ${45}^{\circ }$ + CRA ${20}^{\circ }$. These frames were processed using the HR-UMamba++ model and independently evaluated by three cardiovascular specialists with more than five years of clinical experience, none of whom were involved in model development.

A five-point Likert scale was employed to rate the segmentation quality. Each physician assigned a single composite score from 1 to 5 after holistically considering the delineation completeness of the main trunk and major branches, the preservation of small vessels and bifurcations, and the overall diagnostic usability of the segmentation. The scoring criteria and their corresponding clinical interpretations are summarized in

Table 8. Five-point Likert scale for expert ratings and corresponding clinical interpretations.

Score	Clinical Interpretation
1	Completely unusable; segmentation is severely erroneous and cannot be used for any diagnostic reference.
2	Partially usable, but with substantial errors that may mislead clinical judgment.
3	Basically usable; segmentation contains noticeable errors but can still assist decision-making when interpreted with caution.
4	Good; segmentation is reliable in most regions and can serve as stable auxiliary diagnostic information.
5	Excellent; segmentation closely matches visual assessment and can be directly used as a reference for clinical decision-making.

.

In total, 160 angiographic frames (20 per view) were evaluated by three physicians, yielding 480 individual ratings. The overall mean score across all views was $4.24\pm 0.49$, and the three raters exhibited highly similar means and standard deviations (Doctor 1: $4.24\pm 0.53$; Doctor 2: $4.24\pm 0.52$; Doctor 3: $4.24\pm 0.52$), suggesting no evident systematic rater bias. As summarized in

Table 9. Expert rating statistics and inter-rater reliability for different coronary angiographic projections. Bold values indicate the overall summary across all views.

View	Mean Score ± SD	ICC (2,1)	ICC (2,3)
LAO view for RCA	4.48 ± 0.48	0.808	0.927
AP caudal (${20}^{\circ }$)	4.18 ± 0.48	0.808	0.927
LAO ${45}^{\circ }$ + CAU ${20}^{\circ }$ (spider view)	4.22 ± 0.39	0.812	0.928
RAO ${30}^{\circ }$ + CAU ${20}^{\circ }$	4.08 ± 0.51	0.827	0.935
RCA cranial view	4.52 ± 0.48	0.809	0.927
RAO ${30}^{\circ }$ + CRA ${20}^{\circ }$	4.30 ± 0.47	0.803	0.925
AP cranial (${30}^{\circ }$)	4.23 ± 0.47	0.806	0.926
LAO ${45}^{\circ }$ + CRA ${20}^{\circ }$	3.92 ± 0.47	0.804	0.925
Overall	4.24 ± 0.49	0.828	0.935

, the mean scores for individual views ranged from 3.92 to 4.52. Among them, RCA cranial view and LAO view for RCA achieved the highest ratings ($4.52\pm 0.48$ and $4.48\pm 0.48$, respectively), whereas LAO ${45}^{\circ }$ + CRA ${20}^{\circ }$ and RAO ${30}^{\circ }$ + CAU ${20}^{\circ }$ received slightly lower, yet still favorable, scores ($3.92\pm 0.47$ and $4.08\pm 0.51$, respectively). The remaining projections consistently maintained mean ratings around 4.2.

With respect to inter-rater reliability, the overall intraclass correlation coefficients were ICC(2,1) = 0.83 and ICC(2,3) = 0.94. Further view-specific analysis showed that the single-rater consistency ICC(2,1) ranged from 0.80 to 0.83, whereas the average-measures consistency ICC(2,3) ranged from 0.92 to 0.94 across the eight views (Table 9). According to established benchmarks [91], these values indicate good-to-excellent agreement among the three experts in their subjective assessment of the model outputs.

To further visualize the distribution of scores across different projections, we plotted violin diagrams of the 5-point Likert ratings for all three raters and eight views (Figure 8). The distributions exhibit a strong concentration of scores at 4 and 5 for all views, with only approximately 4.6% of ratings equal to 3 and no ratings of 1 or 2. These results indicate that HR-UMamba++ provides robust and clinically useful coronary vessel segmentations across diverse angiographic angles, supporting its potential for practical deployment in routine clinical workflows.

4.7. Computational Complexity Analysis

To quantify the computational cost of each design component, we report the number of trainable parameters (Params), floating-point operations per forward pass (FLOPs), peak GPU memory footprint during inference (Memory), and inference latency (Inference) under a single-image setting with an input size of $512\times 512$ (batch size = 1). Each variant adds one component to the baseline U-Mamba, whereas HR-UMamba++ combines all components.

As shown in

Table 10. Complexity comparison of HR-UMamba++ and variants.

Method	Params (M)	FLOPs (G)	Memory (GB)	Inference (ms/img)
Baseline U-Mamba	64.01	411.13	0.87	12.96
+ Multi-directional SSM	67.17	518.68	1.21	24.45
+ HR Branch	88.88	639.63	1.16	16.19
+ Dense Connections	131.87	2117.77	1.78	32.04
HR-UMamba++	184.02	2557.98	2.33

, introducing the multi-directional SSM increases the parameter count by only about 4.9%. However, the redundant computation incurred by multi-branch directional modeling leads to a 26.2% increase in FLOPs and a 39.1% increase in GPU memory, with the inference latency rising to 24.45 ms/img. The HR branch remains computationally efficient while continuously injecting high-resolution details, increasing the per-image inference time by only 3.23 ms.

Dense connections require cross-scale feature concatenation and fusion convolutions, resulting in approximately doubled parameters and GPU memory compared with the baseline, while the FLOPs surge to 2117.77G. Although dense connections constitute the dominant source of complexity growth, they also deliver the most pronounced gain among single-module variants, as adding dense connections improves IoU by 4.3% and reduces HD95 by 7.7% (Table 4).

Overall, the proposed HR-UMamba++ reaches 184.02M parameters (+187.5%), 2557.98G FLOPs (+522.2%), and a GPU memory footprint of 2.33 GB (+167.8%), with an inference latency of 48.86 ms/img (+277.0%). Despite the substantial increase in computational cost, it is noteworthy that clinical XCA sequences are typically acquired at 15 fps (i.e., a frame interval of approximately 66.7 ms) [63]. Since the proposed model’s per-frame latency is below this interval, it is able to complete inference before the next frame arrives, thereby achieving clinically meaningful real-time performance. Meanwhile, the 2.33 GB memory footprint is far below the capacity limits of mainstream GPUs, indicating that the model retains strong feasibility for clinical deployment.

5. Discussion

5.1. Summary of Main Findings

This study addresses long-standing challenges in single-frame coronary angiography-based vessel segmentation, including low contrast, structural ambiguity, and pronounced directional variability. Building upon the U-Mamba paradigm, we propose HR-UMamba++, centered on a rotation-aligned multi-directional Mamba module and incorporating a persistent high-resolution branch and a UNet++-style dense decoder topology.

Compared with existing approaches, traditional unsupervised methods struggle to preserve vessel connectivity and distal branches under low-contrast conditions and overlapping artifacts, and they are easily confounded by guidewires and bony structures [28,54,55,56,57]. Although recent supervised methods achieve substantial improvements in overall segmentation accuracy, most of them primarily focus on delineating major coronary trunks and remain limited in capturing the complete coronary arterial tree and higher-order bifurcation patterns [30,92,93,94]. State space model-based architectures such as Mamba and its variants have shown promising performance across various medical imaging tasks, yet they are still suboptimal when dealing with highly filamentous, strongly anisotropic vascular networks [95,96]. To address these limitations, we adapt the U-Mamba framework to the specific characteristics of XCA imaging and develop the HR-UMamba++ architecture. In a systematic comparison with representative methods, HR-UMamba++ achieves the best performance across all eight evaluation metrics (Table 6).

The ablation experiments clarify the functional contributions of each architectural component. Removing the UNet++-style dense decoding module leads to a decrease in Sensitivity and IoU by approximately 5.3% and 6.3%, respectively, and an increase in HD95 by nearly 20%. Qualitative results further reveal broken vessel continuity in low-perfusion regions and disruptions of complex bifurcation topology (Figure 4).

The ablation of the persistent high-resolution branch primarily results in deterioration in fine structural details. After removing this branch, Dice and IoU decrease by about 3.4% and 5.4%, respectively, whereas HD95 increases by approximately 10.2%. Frequency-domain power-spectrum analysis shows that the complete model exhibits, on average, a 4.10% increase in mid-to-high frequency energy within the normalized radial frequency range 0.25–0.9. This energy enhancement is concentrated in radial frequency bands aligned with the dominant vessel orientations (Figure 5), indicating that the high-resolution branch selectively preserves and amplifies high-frequency components associated with the geometric structure of the coronary vasculature.

The rotation-aligned multi-directional Mamba scanning mechanism mainly strengthens directional coherence and global connectivity of the vessel tree. When this module is removed, the Dice coefficient and IoU drop from 0.8706 and 0.7794 to 0.8478 and 0.7470, respectively. Furthermore, a single-component connectivity analysis on 200 angiograms with visually normal coronary anatomy (Table 5) shows that the complete model maintains a single foreground connected component in 93.5% of cases, whereas the ablated variant achieves this in only 87.5% of images. This difference in topological consistency is statistically significant, suggesting that rotating the original projection images into orientation-aligned views and feeding them into separate Mamba units for state-space evolution facilitates continuous vessel tracking along multiple directions.

In addition, we incorporate fractal-geometry-based descriptors to quantitatively analyze the fractal dimension of skeletonized coronary trees (Table 7). The fractal-dimension analysis demonstrates that the predictions of HR-UMamba++ exhibit the closest agreement with the ground-truth coronary trees, achieving the lowest $\mathrm{Mean}(∆D)$ among all evaluated methods. This finding indicates that HR-UMamba++ better preserves the self-similar geometric complexity of the coronary arterial tree in terms of fractal geometry.

Importantly, beyond algorithmic metrics, we also conduct a systematic evaluation of the clinical usability of HR-UMamba++ using expert ratings. The overall clinical score of the model is $4.24\pm 0.49$, with mean scores across different angiographic views ranging from 3.92 to 4.52. No ratings of 1 or 2 (completely unusable or potentially misleading) are observed, and only approximately 4.6% of the ratings correspond to a score of 3 (Table 9, Figure 8). These results indicate that HR-UMamba++ yields clinically usable vessel maps across diverse angiographic views, which is critical for downstream quantitative coronary analysis (QCA). In standard QCA workflows, centerlines and lumen contours are typically extracted along the entire coronary tree to compute key indices such as minimal lumen diameter (MLD), reference vessel diameter (RVD), diameter stenosis (DS), and lesion length (LL) [22,97]. Topological distortion in the segmented vessel tree can propagate to unstable centerline extraction and contour tracking, thereby biasing stenosis quantification and lesion-length measurements. By better preserving the global continuity of the vessel tree, as well as the connectivity of distal branches and bifurcation regions, HR-UMamba++ can provide a reliable structural prior for automated or semi-automated QCA measurements, reducing downstream manual editing and correction.

Overall, the proposed framework demonstrates stable and consistent advantages across quantitative evaluation metrics, structural integrity analyses, and subjective clinical assessments, confirming its suitability for complex coronary scenarios and its promising potential clinical value.

5.2. Limitations and Future Work

Despite the substantial performance gains achieved by HR-UMamba++, several limitations remain. First, the dataset used in this study is collected from a single medical center, where imaging equipment, exposure parameters, and operational protocols are relatively fixed. For algorithms intended to be deployed in multi-center and cross-device settings, it is necessary to incorporate data from different hospitals, vendors, and patient populations to more comprehensively assess the cross-domain robustness and generalizability of the model.

Second, in a small minority of cases, as illustrated in Figure 9, the model may still misclassify interventional devices as vascular structures in images containing artifacts such as catheters, guidewires, balloons, and pacing electrodes. Although many existing studies on XCA image analysis choose to exclude such abnormal frames during pre-processing [98,99], from a clinical perspective it is more desirable to improve the robustness of the algorithm while retaining these images. Enhancing artifact robustness is crucial for improving segmentation quality and generalization under realistic operating conditions. Therefore, future work will focus on constructing an extended dataset that includes a broader spectrum of artifact types and on incorporating a dedicated artifact-detection branch to locally suppress these regions. This strategy is expected to further improve the overall stability and reliability of HR-UMamba++ in real clinical environments.

6. Conclusions

This study targets the pervasive challenges of low contrast, structural ambiguity, and pronounced directional variability that hinder coronary arterial tree segmentation in single-frame X-ray coronary angiography. Building on the U-Mamba framework, we propose HR-UMamba++, a model specifically tailored for coronary vessel-tree segmentation. HR-UMamba++ is centered on a rotation-aligned multi-directional Mamba scan, complemented by a persistent high-resolution branch and a UNet++-style dense decoder. Together, these components enhance the representation of complex vascular networks from three complementary perspectives: fine-grained detail preservation, cross-scale topological modeling, and multi-directional long-range dependency capture.

Comprehensive baseline comparisons and ablation studies demonstrate that HR-UMamba++ achieves superior performance across all quantitative metrics, while exhibiting more reliable preservation of thin vessels, distal branch continuity, and global topological integrity. A box-counting-based fractal-dimension analysis of the coronary arterial tree further shows that HR-UMamba++ achieves closer agreement between the predicted and reference fractal dimensions and yields smaller deviations $∆D$ than competing models, reflecting a better preservation of the self-similar geometric characteristics of the coronary arterial tree. In addition, expert subjective assessments across eight commonly used angiographic projections show that HR-UMamba++ attains high reliability in terms of clinical interpretability and usability, suggesting its potential to provide high-quality structural priors for quantitative coronary angiography, stenosis severity assessment, and interventional decision support.

Future work will extend the validation of HR-UMamba++ to multi-center datasets to further assess its cross-domain generalization capability, and will focus on designing more robust structural modeling strategies tailored to challenging clinical scenarios. These efforts are expected to facilitate the integration of HR-UMamba++ into routine clinical workflows.