Retinal Tortuosity Biomarkers as Early Indicators of Disease: Validation of a Comprehensive Analytical Framework

Abstract

Retinal blood vessel tortuosity is a promising early biomarker for diseases such as diabetic retinopathy. However, the lack of a standardized evaluation method hinders its clinical application. This study presents a framework with 50 features, including 32 developed and refined from our prior unpublished work. All features were tested for sensitivity and scaling to ensure robust performance. To address the influence of blood vessels’ representation on tortuosity estimation, we tested several resampling approaches and proposed the 1-Equidistant Pixel Sampling method (1EPS), which demonstrated accuracy and approximate scale invariance. The framework was evaluated on a public retinal tortuosity dataset, RET-TORT, consisting of 30 arteries and 30 veins ranked in increased tortuosity. Data augmentation expanded the dataset to 330 arteries and veins for improved reliability. Spearman’s rank correlation coefficient analysis revealed resampling variations in tortuosity estimation, with our method outperforming literature and most features favoring arteries. Using the augmented dataset, Gaussian Process Regression achieved near-perfect performance ($R^2$ = 1.0 for arteries; 0.999 for veins). Feature selection analysis identified artery- and vein-specific features. This work highlights the importance of accurate vessel preprocessing and feature sensitivity to scaling on tortuosity estimation and introduces a scalable, robust framework of 50 hand-crafted features for clinical tortuosity assessment.

1. Introduction

Retinal blood vessel tortuosity refers to the twisting and turning of vessels, often resulting from disease progression, aging, or various environmental factors. It is widely recognized as an early indicator of several conditions [1,2], including Diabetic Retinopathy (DR) and Hypertensive Retinopathy (HR) [3,4]. Tortuosity also occurs in other organs such as the heart and kidneys [5,6], but the eye uniquely provides a direct and noninvasive view of blood vessels, providing valuable insight into systemic health. Automated analysis of retinal fundus images therefore holds great potential for early disease detection, quantification, and monitoring, enabling timely diagnosis and management of various systemic diseases. Traditionally, ophthalmologists assess retinal vascular abnormalities by direct visual inspection using slit lamp biomicroscopy, ophthalmoscopes, or fundus images. However, tortuosity estimation remains largely subjective, relying on qualitative scales such as mild, moderate, severe, or extreme. To improve accuracy, experts have introduced specific criteria for vessel segment selection, such as from the optic disc to the periphery, up to the first bifurcation, or between two bifurcations, and emphasized the importance of the structural properties, such as the number and height of curves along segments [7]. However, even with such guidelines, subjective evaluation continues to introduce variability, underlining the need for objective and reproducible methods. Over the years, numerous metrics have been introduced to quantify tortuosity [8,9,10,11,12]. Early approaches relied on manual measurements of vessel length [13], followed by ratios such as arc-over-chord or relative length variation [14]. Despite these advances, no universally accepted standard has emerged [15,16]. This lack of consensus is due to multiple factors: the subjectivity of clinical perception, differences between arteries and veins, variability in image quality and resolution, and the limited availability of large and representative public datasets. Image quality issues such as low resolution or artifacts further complicate segmentation and measurement [16,17]. Given these challenges, there is a clear need for robust and standardized methodologies. This study addresses the gap by reviewing existing approaches and introducing a comprehensive framework that integrates 50 tortuosity features. The framework combines novel and established metrics, grouped according to the tortuosity estimation approach, and incorporates a resampling method designed to ensure invariance to scale. By capturing vessel-type-specific effects and ensuring consistency across imaging conditions, this framework aims to enhance the precision, reliability, and clinical applicability of tortuosity assessment.

2. Methods

2.1. Background on Tortuosity Metrics

The various tortuosity evaluation features proposed in the literature can be broadly categorized into three main groups:

1.

Arc-to-Chord Ratio-Based Features: These features rely on the ratio between the arc length and the chord length of a blood vessel segment, providing a measure of the relative length increase due to tortuosity [18].

2.

Local Curvature-Based Features: These features are derived from curvature measurements at stationary points along blood vessel segments, capturing local variations in curvature [8].

3.

Hybrid Features: These incorporate elements from both arc-to-chord ratio and local curvature approaches, along with structural properties such as the number of sub-curves along a blood vessel segment and their tortuosity values [9,12,19].

While some of these measures have demonstrated promising results, they often capture only specific aspects of tortuosity. For instance, arc-to-chord ratio-based features are effective at quantifying the relative length increase of a vessel arc by comparing it with its chord. However, they yield similar results for segments with smoothly curved vessels and those with visibly more pronounced twists, if the overall segment lengths are the same. This limitation prevents them from adequately capturing the twisting nature of blood vessel segments (see Figure 1). On the other hand, local curvature-based measures can address some of the shortcomings of ratio-based measures by detecting subtle twists and turns in vessel segments. However, they are less effective at capturing structural abnormalities along the vessel walls or boundaries, such as aneurysms [20]. Furthermore, many proposed features tend to perform well for either arteries or veins, but not both, likely due to their anatomical differences [3]. Features that claim similar performance for both vessel types often under perform for arteries compared to those specifically optimized for arteries [4]. This underscores the challenge of designing a single feature or metric capable of accurately capturing the diverse manifestations of tortuosity in both arteries and veins.

2.2. Proposed Methodology

Our proposed method consists of the following steps:

1.

Segment retinal blood vessels, select segments of similar length and caliber, and order vessels by increasing tortuosity or adopt an existing dataset (as in Section 2.3)

2.

Design or adopt a suitable resampling method and evaluate its effectiveness (as described in Section 2.5).

3.

Develop or adopt an ensemble of tortuosity features, then estimate them utilizing the dataset from step 1 (as described in Section 2.4.1 and Section 2.4.2).

4.

Test the invariability of tortuosity features to scaling (as in Section 2.6).

5.

Evaluate tortuosity features using Spearman’s rank correlation coefficient on the dataset from step 1 (as shown in Section 3.1).

6.

Expand dataset from step 1, using data augmentation and repeat the evaluation of tortuosity features and Spearman’s rank correlation coefficient analysis (as illustrated in Section “Augmentation Strategy” and results shown in Appendix C and Appendix D).

7.

Analyze the performance of the tortuosity features ensemble using Gaussian process regression, using dataset from step 6 (as in Section 3.2).

8.

Perform feature selection to identify the most relevant features for accurate tortuosity assessment (as described in Section 3.2).

This structured method ensures a comprehensive evaluation of our tortuosity evaluation framework, incorporating robust statistical validation and machine learning techniques.

2.3. Datasets

This study utilized the Retinal Vessel Tortuosity Dataset (RET-TORT) [9]. The dataset comprises 60 retinal fundus images of individual retinal blood vessel segments, divided equally into 30 artery segments and 30 vein segments. A numerically augmented version of the RET-TORT dataset was also employed to enhance the dataset for robust experimentation. To augment the Retinal Blood Vessels Tortuosity Dataset (RET-TORT), we applied a noise-based duplication strategy that expanded the dataset from 60 to 660 instances, a 30-fold increase. For each feature, we estimated the mean and standard deviation, then added random noise to selected data points to generate synthetic samples while retaining the original class labels. This process, repeated ten times, preserved the dataset’s structure and improved its diversity. By maintaining an equal representation of artery and vein segments (330 instances each), the augmented dataset supports balanced training, enhances model generalization, and reduces the risk of overfitting and class bias. The following section provides a detailed explanation of the augmentation process.

Augmentation Strategy

For each tortuosity feature in our dataset, we computed the empirical mean and standard deviation based on the 30 artery and 30 vein segments available. These statistics reflect the natural variability between vessel segments observed in real retinal measurements. Consequently, augmentation noise was generated for each feature by sampling from a Gaussian distribution:

$${ϵ}_i\sim \mathcal{N}(0,{\sigma }_i)$$

where ${\sigma }_i$ is the feature-specific standard deviation. This ensures that features with inherently low variability receive proportionally smaller perturbations, and features that naturally vary more across vessels receive correspondingly larger perturbations. This is consistent with the assumption that small morphological variations in tortuosity metrics are approximately normally distributed around their local neighborhood. The standard deviation for each feature was derived directly from the observed biological variability in the original dataset, rather than chosen arbitrarily. This approach ensures that the applied perturbations stay within physiologically realistic limits. Consequently, features with low inherent variability receive only minor perturbations, while features that naturally fluctuate more across instances are augmented proportionally to reflect that variation. In addition, the augmentation preserves the dataset’s inherent tortuosity ordering, introducing small, realistic perturbations that reflect natural variability and imaging differences without altering class labels, resulting in biologically plausible augmented samples.

The augmentation was repeated ten times, expanding the dataset tenfold to increase training samples while maintaining a balance between data diversity and computational efficiency. An implicit sensitivity analysis, based on the model’s downstream performance, showed that using empirically derived noise levels preserved stable feature rankings and consistent classification accuracy, whereas artificially increased noise degraded performance. This confirms that the chosen perturbations appropriately reflect natural biological variability, and preliminary tests on an independent dataset further validated the robustness of the feature selection.

2.4. Proposed Framework

This section outlines the structure of our proposed framework. We begin by introducing our ensemble of tortuosity features designed for disease detection, which includes both our refined features (denoted by the group’s initial letter followed by “P” for “Proposed”) and established features adopted from the literature (marked similarly with an “L” for “Literature”). Subsequently, we present the mathematical equations used to estimate each feature.

2.4.1. Tortuosity Feature Ensemble

The proposed tortuosity ensemble included 48 features and 2 length measurements, 16 adopted from the existing literature, and 32 features originally developed in prior unpublished work [7]. While initially introduced in that work, these features have been refined and extended here with detailed characterization. The motivation for constructing an ensemble arises from the fact that no single tortuosity metric can robustly capture both arterial and venous tortuosity with high accuracy, as tortuosity manifests differently across the retinal vasculature. This distinction is well established in clinical observations, ref. [3] notes that retinal arterial attenuation is a major indicator of hypertensive retinopathy, whereas retinal venous dilation or increased venous tortuosity are common signs of diabetic retinopathy. Furthermore, as discussed above, individual approaches often capture only partial aspects of tortuosity; for example, distance-based features may overlook localised sub-curves, and curvature-based features may fail to reflect vessel-wall irregularities such as aneurysms. Therefore, building an ensemble of tortuosity features enables the capture of multiple, complementary manifestations of tortuosity, relevant not only to hypertensive and diabetic retinopathy but also to other ocular diseases. Importantly, this ensemble also enhances clinical interpretability, allowing specific tortuosity biomarkers to be identified, especially in longitudinal studies of progressive disease. The features were grouped based on their respective tortuosity metric properties as follows:

Structural Approach Features (S)

Structural approach features capture changes in the structure of blood vessel segments. Examples include the number of minima and maxima points (local extrema), the number of sub-curves, and their average heights. These metrics are derived by calculating the gradient at each point along a vessel segment and analyzing variations in the gradient to identify critical points. These critical points help reveal prominent structural changes and sub structures within the blood vessel.

Distance Approach Features (D)

Distance approach features quantify tortuosity by calculating the ratio of the arc length to the chord length of a blood vessel segment. This method, first introduced by [18], is widely recognized and adopted in the literature.

Curvature Approach Features (C)

These features focus on the local curvature at each point along a blood vessel segment. Most of these features were initially proposed by [18].

Combined Approach Features (Co)

These features merge structural, curvature, or distance-based properties to provide a more comprehensive representation of tortuosity along the vessel segment.

Signal Approach Features (Si)

Signal-based features leverage Fourier Transform analysis to evaluate tortuosity. For this purpose, four spatial functions derived from the blood vessel segments are used as inputs for the analysis. These functions include:

1.

The displacement points: the distances between each point in the vessel segment and its chord.

2.

The first derivative of the x-coordinates along the centerline of the vessel segment.

3.

The second derivative of the x-coordinates along the centerline of the vessel segment.

4.

The signed curvature at each point along the vessel segment.

The features derived from these inputs serve as tortuosity estimators. Originally proposed in unpublished work by [7], these metrics have been refined and extended in the present study to enhance their effectiveness.

2.4.2. Framework’s Features Estimation

Centerline Points Resampling

Before conducting the mathematical estimation, a resampling method is necessary. This process redefines or redistributes points along blood vessel segments to ensure uniform spacing and optimize the resolution for accurate feature estimation. Further details can be found in Section 2.5.

Length Measurement

Two length measurements were implemented within our framework:

Chord Length ($L_x$): This represents the straight-line distance between the endpoints of a vessel segment, calculated using the Euclidean distance formula:

$$L_x=\sqrt{{(x_n-x_1)}^2+{(y_n-y_1)}^2}$$

(1)

Arc Length ($L_c$): The actual curve length of the blood vessel segment is computed using two methods: Discrete Sum Method ($L_{c1}$): Arc length is determined as the sum of Euclidean distances between consecutive points [18]:

$$L_{c1}=\sum _{i=1}^{n-1}\sqrt{{(x_i-x_{i+1})}^2+{(y_i-y_{i+1})}^2}$$

(2)

Integral Method ($L_{c2}$): Arc length is calculated by integrating the curve’s gradient over a parameterized interval [18]:

$$L_{c2}={\int }_{t_0}^{t_1}\sqrt{{\left(x^{\prime }\left(t\right)\right)}^2+{\left(y^{\prime }\left(t\right)\right)}^2}dt$$

(3)

Feature Groups

Table 1. Overview of the proposed tortuosity feature framework, including feature groups, sub-groups, feature codes, descriptions, corresponding mathematical formulations, original sources, and their assigned invariability categories.

Group	Sub-Group	Index	Code	Formula	Source	Category
Distance-Approach		1	DL1	Equation (A1)	[8]	Invariant
		2	Lx	Equation (1)
		3	Lc2	Equation (3)
		4	Lc1	Equation (2)
Structural-Approach		5	SP1	Description in Appendix A	Proposed	Sensitive
		6	SP2		Proposed	Sensitive
		7	SP3		Proposed	Sensitive
		8	SP4		Proposed	Extra-Sensitive
		9	SP5		Proposed	Sensitive
Combined-Approach		10	CoL1	Equation (A2)	[9]	Invariant
		11	CoL2	Equation (A3)	[21]	Sensitive
		12	CoP1	Equation (A4)	Proposed	Invariant
		13	CoP2	Equation (A5)	Proposed	Extra-Sensitive
		14	CoP3	Equation (A6)	Proposed	Invariant
Curvature-Approach	Group 1	15	CL1	Equation (A8)	[8]	Sensitive
		16	CL2	Equation (A9)	[8]	Sensitive
		17	CL3	Equation (A10)	[8]	Sensitive
		18	CL4	Equation (A11)	[8]	Sensitive
	Group 2	19	CL5	Equation (A12)	[8]	Invariant
		20	CL6	Equation (A13)	[8]	Invariant
		21	CL7	Equation (A14)	[8]	Invariant
		22	CL8	Equation (A15)	[8]	Invariant
		23	CL9	Equation (A16)	[8]	Invariant
		24	CL10	Equation (A17)	[8]	Invariant
	Group 3	25	CL11	Equation (A22)	[22]	Extremely-Sensitive
		26	CL12	Equation (A23)	[23]	Invariant
		27	CL13	Equation (A24)	[24]	Invariant
	Group 4	28	CP1	Equation (A18)	Proposed	Sensitive
		29	CP2	Equation (A19)	Proposed	Invariant
		30	CP3	Equation (A20)	Proposed	Invariant
		31	CP4	Equation (A21)	Proposed	Invariant
Signal-Approach	Group 1	32	SiP1	Equation (A25)	Proposed	Sensitive
		33	SiP2		Proposed	Extra-Sensitive
		34	SiP3		Proposed	Sensitive
		35	SiP4		Proposed	Invariant
		36	SiP5		Proposed	Sensitive
		37	SiP6		Proposed	Invariant
		38	SiP7		Proposed	Invariant
		39	SiP8		Proposed	Sensitive
	Group 2	40	SiP9		Proposed	Sensitive
		41	SiP10		Proposed	Sensitive
		42	SiP11		Proposed	Invariant
		43	SiP12		Proposed	Invariant
		44	SiP13		Proposed	Sensitive
		45	SiP14		Proposed	Sensitive
	Group 3	46	SiP15		Proposed	Sensitive
		47	SiP 16		Proposed	Sensitive
		48	SiP 17		Proposed	Sensitive
	Group 4	49	SiP18		Proposed	Invariant
		50	SiP19		Proposed	Invariant
		51	SiP20		Proposed	Invariant

highlights the diverse set of metrics included in the framework, combining both newly proposed features and established measures from the literature. The proposed features form the majority of the framework, reflecting the key advancements and contributions introduced in this work. In addition, the inclusion of invariability classification allows the identification of metrics best suited for different analytical purposes. A detailed explanation and full mathematical formulation of all features are provided in Appendix “Features Estimation”, with extended feature descriptions presented in

Table A1. Framework’s Features: Groups, Codes, Descriptions, Sources, and Invariability Categories.

Index	Type	Code	Description	Source	Category
1	Combined	CoP2	Path-over-chord: Adds sub-curve heights (VP1).	Proposed	EXTRA-SENSITIVE
2	Combined	CoP1	Arc-over-chord combined with the number of maxima points along a blood vessel segment.	Proposed	INVARIANT
3	Combined	CoP3	Path-over-chord: Adds sub-curve under-spaces (VP2).	Proposed	SENSITIVE
4	Curvature G4	CP2	Sum of the absolute first derivatives along a blood vessel segment, normalized by the sample interval.	Proposed	INVARIANT
5	Curvature G4	CP3	Sum of the absolute second derivatives along a blood vessel segment, normalized by the sample interval.	Proposed	INVARIANT
6	Curvature G4	CP4	Sum of the absolute third derivatives along a blood vessel segment, normalized by the sample interval.	Proposed	INVARIANT
7	Curvature G4	CP1	Sum of combined unsigned curvature and absolute differences of gradients along a segment.	Proposed	SENSITIVE
8	Signal G1	SiP2	FFT measure using the displacement points and the curvature.	Proposed	EXTRA-SENSITIVE
9	Signal G1	SiP4	FFT measure using the second derivatives and path over the chord.	Proposed	INVARIANT
10	Signal G1	SiP6	FFT measure using curvature and path over the chord.	Proposed	INVARIANT
11	Signal G1	SiP7	FFT measure using slopes and paths over the chord.	Proposed	INVARIANT
12	Signal G1	SiP1	FFT measure using displacement points and path over the chord.	Proposed	SENSITIVE
13	Signal G1	SiP3	FFT using the first derivatives and curvature.	Proposed	SENSITIVE
14	Signal G1	SiP5	FFT measure using the second derivatives and curvature.	Proposed	SENSITIVE
15	Signal G1	SiP8	FFT measure using slope and curvature.	Proposed	SENSITIVE
16	Signal G2	SiP11	Sum of the magnitude using second derivatives.	Proposed	INVARIANT
17	Signal G2	SiP12	Sum of the magnitude using curvature.	Proposed	INVARIANT
18	Signal G2	SiP9	Sum of the magnitude using displacement points.	Proposed	SENSITIVE
19	Signal G2	SiP10	Sum of magnitude using first derivatives.	Proposed	SENSITIVE
20	Signal G2	SiP13	Sum of the magnitude norm by path length using slope.	Proposed	SENSITIVE
21	Signal G2	SiP14	Sum of magnitude using slope.	Proposed	SENSITIVE
22	Signal G3	SiP15	Sum of the abs angles of first derivatives norm by path length.	Proposed	SENSITIVE
23	Signal G3	SiP16	Sum of the abs angles of curvature points norm by path length.	Proposed	SENSITIVE
24	Signal G3	SiP17	Sum of the abs angles of slopes along segment norm by path length.	Proposed	SENSITIVE
25	Signal G4	SiP18	Magnitude variance using first derivatives.	Proposed	INVARIANT
26	Signal G4	SiP19	Magnitude variance using first derivatives and norm by path length.	Proposed	INVARIANT
27	Signal G4	SiP20	Magnitude variance using curvature.	Proposed	INVARIANT
28	Structural	SP4	Sum of sub-curves spaces.	Proposed	EXTRA-SENSITIVE
29	Structural	SP1	Sub-curves number.	Proposed	SENSITIVE
30	Structural	SP2	Maxima points.	Proposed	SENSITIVE
31	Structural	SP3	Minima points.	Proposed	SENSITIVE
32	Structural	SP5	Curves heights.	Proposed	SENSITIVE
33	Combined	CoL1	Tortuosity measure by Grisan.	Literature	INVARIANT
34	Combined	CoL2	Inflection count metric 1.	Literature	SENSITIVE
35	Curvature G1	CL4	Total squared unsigned curvature.	Literature	SENSITIVE
36	Curvature G1	CL1	Total signed curvature.	Literature	SENSITIVE
37	Curvature G1	CL2	Total squared signed curvature.	Literature	SENSITIVE
38	Curvature G1	CL3	Total unsigned curvature.	Literature	SENSITIVE
39	Curvature G2	CL10	Total squared unsigned curvature over chord length.	Literature	INVARIANT
40	Curvature G2	CL6	Total squared signed curvature over chord length.	Literature	INVARIANT
41	Curvature G2	CL7	Total unsigned curvature over arc length.	Literature	INVARIANT
42	Curvature G2	CL8	Total unsigned curvature over chord length.	Literature	INVARIANT
43	Curvature G2	CL9	Total squared unsigned curvature over arc length.	Literature	INVARIANT
44	Curvature G2	CL5	Total squared signed curvature norm by arc length.	Literature	INVARIANT
45	Curvature G3	CL11	Tortuosity measure by Rashmi.	Literature	EXTREMELY-SENSITIVE
46	Curvature G3	CL12	Tortuosity coefficient.	Literature	INVARIANT
47	Curvature G3	CL13	Mean direction angle change.	Literature	INVARIANT
48	Distance	DL1	Arc over chord.	Literature	INVARIANT
49	Distance	Lc	Arc Length measured using differentiation.	-	-
50	Distance	Lc2	Path length using the sum of the geodesic distances between each two consecutive points along the segment.	-	-
51	Distance	Lx	Chord length measured using the Euclidean distance.	-	-

.

2.5. Resampling Approach

Resampling refers to modifying the spatial distribution of data points along a curve to achieve a more consistent and accurate representation. The manually traced centerline points of the vessel segments described in Section 2.3, represented as $\{(x_i,y_i)\mid i=1,2,\dots ,n\}$, were found to lack sufficient precision in numerically capturing the true vessel paths visible in the fundus images (see Figure 2). Therefore, a resampling mechanism is required to resample, and, if necessary, upsample, the centerline points using linear (or higher-order) interpolation, followed by smoothing with an appropriate smoothing filter or curve-fitting method.

As noted above, resampling is essential in studies of direct vascular morphology analysis to preserve biologically meaningful representations of vessel geometry. However, existing resampling and upsampling strategies vary widely, and a lack of standardization in the literature makes it difficult to reproduce tortuosity-related findings. To address these limitations, we propose an improved resampling strategy informed by the evaluation of two distinct approaches. Resample Method 1 fits a smoothing spline to the original centerline points and resamples the curve at a fixed, user-specified spatial interval. The method relies on a piecewise linear approximation of the spline, generated at sufficiently dense resolution, to reconstruct the vessel path while enforcing uniform point spacing, resulting in a comparatively coarser geometric representation (see Figure 3, left). In contrast, Method 2, our proposed (1EPS) approach, uses the MATLAB interparc function by D’Errico [25] to generate a refined centerline in which consecutive points are uniformly spaced at exactly one pixel. Through spline-based parametric interpolation, this method produces a high-resolution, smooth, and evenly sampled representation of the vessel geometry, making it better suited for precise tortuosity analysis (see Figure 3, right).

Shape-Fidelity Evaluation of the Resampling Approaches

To assess the geometric accuracy of Resample Method 1 and Method 2, we evaluated both approaches using the Relative Arc Length Error (RALE) and the Hausdorff Distance (HD), computed separately for the 30 artery and 30 vein segments. and the arc length for the (RALE) assesment perfomed using both length metrics ($L_{C_1}$ and $L_{C_2}$). The RALE was computed as:

$$\mathrm{RALE}={\displaystyle \frac{\left|{\ell }_{\mathrm{orig}}-{\ell }_{\mathrm{resampled}}\right|}{{\ell }_{\mathrm{orig}}}},$$

where ${\ell }_{\mathrm{orig}}$ and ${\ell }_{\mathrm{resampled}}$ denote the arc lengths of the original and resampled centerlines, respectively.

The Hausdorff Distance was defined as:

$$d_H(A,B)=max\left\{\underset{a\in A}{sup}\underset{b\in B}{inf}\parallel a-b\parallel ,\underset{b\in B}{sup}\underset{a\in A}{inf}\parallel a-b\parallel \right\}$$

where A and B represent the sets of points belonging to the original and resampled centerline.

Quantitative shape-fidelity analysis using RALE and HD highlights the performance differences between the two resampling strategies. As shown in

Table 2. Quantitative shape-fidelity assessment of Resampling Method 1 and Method 2 (1EPS) on the RET-TORT dataset. The table reports the mean Relative Arc Length Error, evaluated using two length metrics ($L_{C_1}$ and $L_{C_2}$), and the mean Hausdorff Distance for arterial and venous segments, providing a comparative measure of each method’s ability to preserve the geometry of the original vessel centerline.

Resample Approach	Length Metric	Mean Relative Error in Arc Length		Mean Hausdorff Distance
		A	V	A	V
Resample 1	$L_{C_1}$	0.0035	0.0023	27.9947	27.2125
	$L_{C_2}$	0.2520	0.2835
Resample 2 (1EPS)	$L_{C_1}$	0.0037	0.0037	28.6641	28.0586
	$L_{C_2}$	0.0270	0.0323

, Resampling Method 1 introduced considerable distortions in arc length, particularly for the $L_{C_2}$ metric, with RALE values of 0.2520 for arteries and 0.2835 for veins, indicating an overestimation of curve-length. In contrast, our proposed Method 2 (1EPS) produced consistently low RALE values across both metrics, reducing $L_{C_2}$ errors to 0.0270 for arteries and 0.0323 for veins. Although the Hausdorff Distance values for Method 1 and Method 2 were very similar, indicating comparable overall geometric fidelity, Method 2 maintained slightly higher agreement while avoiding the large shape distortions observed in Method 1. This is further supported by the observed measurements of an artery and a vein arc lengths in

Table 3. Comparison of length measurements for an artery and a vein segments across three representations: the original polyline, the polyline resampled with Method 1, and the polyline resampled with Method 2 (1EPS).

Feature	Metric	Original		Resample 1		Resample 2 (1EPS)
		A	V	A	V	A	V
Chord length (Lx)	(1)	886.2037	730.1342	886.2037	730.1342	886.2037	730.1342
Arc Length (Lc1)	(2)	999.4621	795.0340	1001.7	799.03	1002.3	799.163
Arc Length (Lc2)	(3)	1059.6	813.2823	1294.8	1047.8	1003.3	800.1698

, where Method 2 yielded stable and nearly identical length estimates for both $L_{C_1}$ and $L_{C_2}$. Collectively, the RALE, HD, and length-metric results demonstrate that Method 2 provides superior shape preservation and high-fidelity geometric reconstruction during resampling.

These results highlight the critical role of resampling in the tortuosity estimation process. It has been observed that the application of an imprecise resampling can introduce variability in fundamental measurements, such as the arc length, as illustrated in Table 3. This variability has the potential to introduce inaccuracies in the subsequent tortuosity estimation. In contrast, a precise resampling approach produces a more accurate numerical and geometrical representation of the original blood vessel segments. This approach is expected to produce reliable and consistent tortuosity measures, facilitating more accurate assessments across different datasets.

2.6. Scale Variation

A tortuosity feature should produce consistent values for the same segment, irrespective of variations in the resolution of the image from which the segment is extracted. To validate this criterion, the feature ensemble was tested for invariance using two identical images at different resolutions. A fundus image was resized to generate two identical versions: Image 1 with a resolution of 2032 pixels × 1934 pixels and Image 2 with a resolution of 1016 pixels × 967 pixels. Both images were segmented using the Al-Diri segmentation method [26], and the resulting segments were manually classified as “artery” and “vein.” Ten matching arterial and venous segments from both images were selected, as illustrated in Figure 4.

The tortuosity of these segments was measured using the feature ensemble to evaluate their scaling invariance. The absolute differences between the feature values of the same segments from both images were computed. The mean of these absolute differences was calculated for each feature across the 10 segments. Ideally, invariant features should exhibit a mean value close to zero, signifying minimal variability under a uniform similarity transform, as defined in (4).

$$\left|\right(\begin{array}{c}Img{1}_{\left(F_{1-n}\right)}\end{array}-\begin{array}{c}Img{2}_{\left(F_{1-n}\right)}\end{array}\left)\right|\approx 0$$

(4)

The Elbow Method was utilized to identify and categorize the framework features and to determine the optimal number of clusters. This widely used heuristic approach analyzes the trade-off between the number of clusters and the within-cluster variance. Based on the findings of the Elbow Method, the K-Means clustering algorithm was applied to group the features effectively.

Following the clustering process, redundant and non-invariant features were systematically removed to streamline the feature set. The remaining features were then classified into four distinct categories based on their stability and sensitivity:

• Invariant/Stable features
• Sensitive features
• Extra-sensitive features
• Extremely-sensitive features

Detailed descriptions and the corresponding feature classifications for each category are provided in

Table A2. Descriptive statistics of the absolute differences between Image 1 and Image 2 features for invariant features.

Feature Sensitivity	Feature Index	Min	Max	Range	Mean	Std
Invariant	SiP4	0.0000	0.0000	0.0000	0.0000	0.0000
	SiP20	0.0000	0.0005	0.0005	0.0001	0.0002
	SiP11	0.0000	0.0011	0.0011	0.0003	0.0003
	CoL1	0.0000	0.0020	0.0020	0.0005	0.0007
	CL12	0.0000	0.0014	0.0014	0.0005	0.0005
	CP4	0.0000	0.0015	0.0015	0.0006	0.0005
	SiP12	0.0000	0.0054	0.0054	0.0008	0.0016
	SiP19	0.0002	0.0018	0.0016	0.0008	0.0004
	SiP7	0.0000	0.0075	0.0075	0.0012	0.0023
	CP3	0.0004	0.0071	0.0067	0.0020	0.0027
	CoP1	0.0002	0.0062	0.0060	0.0020	0.0021
	CL7	0.0001	0.0092	0.0092	0.0028	0.0032
	CL8	0.0002	0.0104	0.0102	0.0031	0.0036
	CL5	0.0002	0.0161	0.0159	0.0033	0.0048
	CL6	0.0002	0.0181	0.0179	0.0037	0.0054
	CL13	0.0004	0.0136	0.0132	0.0052	0.0053
	DL1	0.0009	0.0212	0.0203	0.0067	0.0073
	CP2	0.0001	0.0183	0.0182	0.0083	0.0055
	SiP6	0.0012	0.0230	0.0217	0.0084	0.0072
	SiP18	0.0005	0.0248	0.0243	0.0097	0.0079
	CL9	0.0018	0.0434	0.0417	0.0135	0.0133
	CL10	0.0022	0.0484	0.0461	0.0148	0.0147

and

Table A3. Descriptive statistics of the absolute differences between Image 1 and Image 2 features for sensitive features.

Feature Sensitivity	Feature Index	Min	Max	Range	Mean	Std
Sensitive	CP1	0.0002	0.0945	0.0944	0.0192	0.0288
	SiP1	0.0001	0.0964	0.0963	0.0232	0.0387
	SiP10	0.0100	0.0865	0.0765	0.0370	0.0276
	SP5	0.0000	0.3888	0.3888	0.0477	0.1209
	CoP3	0.0013	0.3866	0.3853	0.0549	0.1182
	SiP15	0.0019	0.6899	0.6880	0.0964	0.2096
	SiP13	0.0009	0.9550	0.9541	0.2269	0.3391
	CoL2	0.0018	1.7117	1.7100	0.2707	0.5841
	SiP14	0.0049	1.0886	1.0837	0.2826	0.3948
	SP1	0.0000	1.0000	1.0000	0.3000	0.4830
	CL2	0.0899	0.8216	0.7318	0.3969	0.2589
	SP3	0.0000	2.0000	2.0000	0.4000	0.8433
	SiP17	0.0082	1.6611	1.6529	0.4985	0.5462
	SP2	0.0000	2.0000	2.0000	0.5000	0.8498
	SiP16	0.0147	1.3279	1.3132	0.5625	0.4673
	CL1	0.0341	3.7711	3.7370	0.6257	1.1236
	SiP3	0.7370	3.3020	2.5650	1.4851	0.8409
	CL3	0.7840	3.3770	2.5930	1.5090	0.8632
	SiP5	0.7949	3.3890	2.5941	1.5172	0.8644
	SiP8	0.0449	3.3935	3.3486	1.5919	0.9456
	CL4	2.4414	32.5369	30.0955	12.2729	9.7811
	SiP9	0.0035	56.9894	56.9859	13.2256	20.3982
Extra-sensitive	SiP2	1.2150	423.095	421.8799	134.13	138.67
	CoP2	0.001	1320.095	1320.094	217.030	408.17
	SP4	0.000	1320.077	1320.0772	217.031	408.17
Extremely-sensitive	CL11	1.61	1433.52	1431.9	382.32	562.79

.

As shown in the preceding analysis, several features, most notably CL11, CP3, and the SiP* spectral family, are intrinsically sensitive to high-frequency zig-zag noise arising from centerline discretizations. CL11 is computed using a local slope-difference (turning-angle) operator, effectively a 3-point stencil that approximates curvature through differences in successive slopes. This formulation responds strongly to small point-to-point oscillations, so even a ±0.5-pixel jitter in the centerline can generate disproportionately large curvature estimates. CP3, defined as the mean absolute second derivative of the centerline, is even more vulnerable: numerical differentiation inherently amplifies high-frequency noise, and the second derivative scales with ${\displaystyle \frac{1}{{(\Delta s)}^2}}$, where s denotes the arc-length parameter of the vessel centerline, and $\Delta s$ is the sampling interval (i.e., the distance between successive resampled points along the curve). meaning that tiny digitization artifacts can dominate the true anatomical signal. The SiP* features, which rely on FFT-based spectral decomposition, are similarly affected because zig-zag jitter injects excess energy into high-frequency bands, biasing the spectral power distribution and reducing feature reliability. Our proposed Resample Method 2, which enforces uniform 1-pixel spacing along the centerline, mitigates these effects to some extent by removing irregular sampling and improving numerical stability of the derivative operators. However, because resampling alone cannot fully suppress high-frequency digitization noise, an explicit smoothing step is likely necessary. We therefore recommend incorporating a light smoothing or denoising stage, supported by an ablation study, to materially reduce zig-zag sensitivity and further strengthen the robustness of these curvature- and spectrum-based features.

2.6.1. Evaluation of the Absolute Differences Between Image 1 and Image 2 Features

The study revealed that certain features were particularly sensitive to minor variations in the segmentation process. Notably, several features displayed erratic behavior in segments 3, 6, 9, and 10. A detailed analysis highlighted specific discrepancies. For example, the segmentation of segment 9 in Image 2 did not originate from the bifurcation point observed in Image 1, and segment 9 in Image 1 was slightly longer. These differences are illustrated in Figure 5. Furthermore, segment 3 in Image 1 exhibited, visually, a greater curvature compared to its counterpart in Image 2, a characteristic also observed in segment 6.

After excluding these segments, the clustering analysis was repeated, leading to a marked reduction in sensitivity across all feature groups, particularly in the extra-sensitive and extremely sensitive categories, see

Table A5. Descriptive statistics of absolute difference analysis for sensitive features between Image 1 and Image 2 values after excluding segments 3, 6, 9, and 10.

Feature Sensitivity	Feature Index	Min	Max	Range	Mean	Std
Sensitive	SiP18	0.0049	0.0248	0.0199	0.0134	0.0083
	SiP1	0.0001	0.0964	0.0963	0.0244	0.0480
	SiP15	0.0019	0.0533	0.0514	0.0289	0.0258
	SiP14	0.0077	0.0948	0.0870	0.0392	0.0382
	SiP10	0.0102	0.0614	0.0513	0.0406	0.0240
	SP5	0.0000	0.3888	0.3888	0.1089	0.1874
	CoP3	0.0013	0.3866	0.3853	0.1152	0.1826
	CL2	0.1038	0.8216	0.7178	0.3514	0.3360
	SiP16	0.0147	1.1476	1.1329	0.4184	0.4993
	CoL2	0.0018	1.7117	1.7100	0.4313	0.8536
	SiP17	0.0082	1.6611	1.6529	0.4925	0.7883
	SP1	0.0000	1.0000	1.0000	0.5000	0.5774
	SP3	0.0000	2.0000	2.0000	0.5000	1.0000
	SP2	0.0000	2.0000	2.0000	0.7500	0.9574
	CL1	0.0341	3.7711	3.7370	1.1565	1.7683
	SiP3	0.7370	2.1459	1.4089	1.4338	0.6781
	SiP8	0.7691	2.2151	1.4459	1.4391	0.6755
	CL3	0.7938	2.1892	1.3954	1.4670	0.6703
	SiP5	0.7949	2.2074	1.4124	1.4743	0.6751
	CL11	1.6062	19.2774	17.6712	6.5067	8.5260
	CL4	2.4414	25.3618	22.9204	13.3971	9.5436
	SiP9	0.0035	56.9894	56.9859	19.5890	26.8732
Extra-sensitive	SiP2	46.67	423.095	376.42	162.42	176.23
Extremely-sensitive	SP4	0	1320.077	1320.08	460.7	594.25
	CoP2	0.0013	1320.1	1320.094	460.70	594.26

. While these discrepancies in sensitivity did not significantly impact the current experiment, they highlight critical considerations for studies focused on disease progression or longitudinal analysis of images. Careful attention to the inherent sensitivity of certain features is essential to ensure reliability in such applications, as discussed previously. Figure 6 illustrates the stability of invariant features when applied to two identical segments, labeled Segment 8, from Image 1 and Image 2.

2.6.2. Validation of Scale-Invariant Features

To ensure that the vessel features used in our analysis are robust to changes in image scale, we conducted both theoretical and empirical assessments of their scale invariance. Key features, including DL1 and chord-length–normalized metrics (CL8/$L_c$ and CL10/${\mathrm{L}}_c$), were examined to confirm that their values remain consistent under uniform scaling transformations. In addition, we investigated the behavior of these features under non-uniform (anisotropic) scaling to determine the limits of invariance and assess potential deviations. The theoretical scale invariance of these features is formalized in Lemma 1, which provides a proof for uniform scaling and highlights expected deviations under anisotropic transformations. This lemma is complemented by empirical results shown in Figure 7 and Figure 8, demonstrating feature stability across multiple scaling factors and confirming the robustness of these features for vessel morphology analysis.

Lemma 1

(Scale Invariance of Selected Vessel Features). The features DL1, CL8/$L_c$, and CL10/$L_c$ are invariant under uniform similarity transforms (i.e., uniform scaling of image coordinates).

Proof. 

• DL1: Defined as a ratio of distances within the vessel, DL1 = $d_1/d_2$. Under uniform scaling by factor s, $d_1^{\prime }=s·d_1$ and $d_2^{\prime }=s·d_2$, yielding ${\mathrm{DL}1}^{\prime }$ = $d_1^{\prime }/d_2^{\prime }$ = $d_1/d_2$ = DL1.
• CL8/L_c and CL10/L_c: Normalized by chord length $L_c$, these features satisfy CL8_n = CL8/$L_c$ and CL10_n = CL10/$L_c$. Uniform scaling by s gives ${\mathrm{CL}8}_n^{\prime }=(s·CL8)/(s·L_c)=CL8/L_c$ and similarly for CL10_n.

Empirical Test of Non-Uniform Scaling: When applying anisotropic scaling (e.g., x scaled by $s_x$ and y by $s_y$, $s_x\ne s_y$), DL1, which depends on ratios of distances, may remain approximately invariant if distances are dominated by one axis, but CL8/L_c and CL10/L_c show measurable deviations. This demonstrates that these features are strictly invariant only under uniform similarity transforms, and anisotropic scaling introduces predictable changes in their values.    □

The empirical validation of Lemma 1 is presented in Figure 7. Feature values were computed across a range of uniform scaling factors to demonstrate their invariance, and anisotropic scaling scenarios were also included to highlight deviations that occur when the x- and y-dimensions are scaled differently. As expected, DL1 and the chord-length–normalized features CL8/L_c and CL10/L_c remain approximately constant under uniform scaling, closely matching the theoretical predictions. In contrast, anisotropic scaling introduces measurable changes in CL8/L_c and CL10/L_c, illustrating the practical limits of scale invariance. This combined theoretical and empirical assessment confirms that these features are robust for vessel morphology analysis while clarifying the conditions under which deviations may occur.

2.7. Spearman’s Correlation and Gaussian Process Regression Analysis

All experiments in this study were conducted using MATLAB (2023a) and Python (3.9.13). The MATLAB environment was used for image processing and feature extraction, while both MATLAB and Python were employed for statistical analyses to comprehensively evaluate the proposed metrics and methods.

Spearman’s rank correlation coefficient was employed to analyze the ranked dataset, which was ordered by increasing tortuosity. As outlined by [27], correlation analysis assesses the degree of association between variables, quantifying the relative strength of their relationship. Unlike Pearson’s correlation, Spearman’s coefficient uses the ranks of the data rather than their raw values, making it particularly suitable for non-parametric and monotonic relationships. In this study, the ranks were computed for all features, and the correlation coefficients were derived based on these rankings.

To further validate the strength and predictive relevance of the extracted tortuosity features beyond the Spearman’s rank correlation analysis, Gaussian Process Regression (GPR) was employed in this study. GPR is a non-parametric Bayesian regression approach that models complex nonlinear relationships and provides uncertainty estimates for predictions, making it particularly suitable for biomedical analysis where data size may be limited and interpretability is essential. Its ability to quantify confidence in feature contributions enables a robust assessment of each feature’s predictive power, while also facilitating effective feature selection by highlighting the most informative descriptors. Given the initial dataset comprised only 30 ranked vessel segments, for each vessel type, numerical data augmentation was applied to expand the dataset to 330 instances per vessel category (see Section “Augmentation Strategy”), allowing the GPR model to be trained more effectively and improving its capacity to generalize tortuosity patterns across varying vessel morphologies.

3. Results

3.1. Spearman’s Rank Correlation Coefficient Analysis

The results, detailed in

Table 4. The comparison of Spearman’s Rank correlation coefficient analysis between some of our framework metrics/features and those from previous studies, as well as the effect of resampling methods on metrics’ performances.

Author	Framework	Literature (Grisan 2008) [9]		Literature (Chuang 2023 [12])		Resample Method 1		Resample Method 2
	Index	Artery	Vein	Artery	Vein	Artery	Vein	Artery	Vein
[8]	DL1	0.792	0.656	0.829	0.632	0.843	0.638	0.855	0.644
[8]	CL3	0.922	0.837	0.818	0.881	0.822	0.775	0.807	0.792
[8]	CL4	0.925	0.826	0.844	0.876	0.822	0.775	0.807	0.792
[8]	CL7	0.919	0.814	0.785	0.868	0.800	0.772	0.750	0.797
[8]	CL9	0.917	0.773	0.823	0.871	0.804	0.773	0.773	0.802
[8]	CL8	0.939	0.842	0.803	0.886	0.806	0.778	0.818	0.797
[8]	CL10	0.928	0.804	0.838	0.876	0.814	0.776	0.815	0.799
[24]	CL13	0.920	0.814	0.675	0.842	−0.708	−0.692	−0.138	0.126
[21]	CoL2	0.648	0.575	0.009	0.493	0.440	0.345	0.361	0.015
[9]	CoL1	0.949	0.853	0.723	0.834	0.917	0.753	0.863	0.541
[22]	CL11	-	-	-	-	0.927	0.809	0.896	0.766
[7]	CoP1	-	-	-	-	0.756	0.485	0.733	0.489
[7]	CoP3	-	-	-	-	0.864	0.566	0.729	0.481
[7]	SiP6	-	-	-	-	0.779	0.794	0.855	0.644
[7]	SiP1	-	-	-	-	0.919	0.703	0.710	0.711

, summarize the Spearman’s rank correlation coefficients between selected features from the framework and the clinically assigned order in the RET-TORT dataset. The study analyzed retinal vascular tortuosity using various feature approaches, with key results summarized as follows:

• Structural Features: Arterial correlations were significant for SP1 and SP2 ($\rho$ = 0.810 and $\rho$ = 0.806), while SP4 showed a strong negative correlation ($\rho$ = −0.842). Veins exhibited weaker correlations overall, with SP5 achieving the highest ($\rho$ = 0.353).
• Distance-Based Features: DL1 demonstrated strong performance for both arteries ($\rho$ = 0.855) and veins ($\rho$ = 0.644).
• Combined Features: CoL1 achieved the highest correlations for arteries and veins ($\rho$ = 0.863 and $\rho$ = 0.541, respectively). Other features such as CoP1 and CoP3 also performed well in the arteries, with moderate correlations ($\rho$ = 0.733 and $\rho$ = 0.729).
• Curvature-Based Features: Group 1: CL3 and CL4 showed the strongest correlations for both arteries and veins ($\rho$ = 0.807 and $\rho$ = 0.792). Group 2: CL8 and CL9 stood out, with $\rho$ values of 0.818 (arteries) and 0.802 (veins). Group 3: CL11 achieved the highest correlation overall ($\rho$ = 0.896 for arteries, $\rho$ = 0.766 for veins). Group 4: CP3 performed best among the group, with ($\rho$ = 0.895) for arteries and ($\rho$ = 0.773) for veins.
• Signal-Based Features: Group 1: SiP3, SiP5, and SiP8 exhibited strong performance, with all equally reporting ($\rho$ = 0.794) for veins and SiP6 reporting ($\rho$ = 0.855) the highest for arteries. Group 2: SiP14 had the highest correlation for arteries ($\rho$ = 0.833), while SiP11 was most effective for veins ($\rho$ = 0.711). Group 4: SiP20 performed well across both vessel types ($\rho$ = 0.725 for arteries, $\rho$ = 0.711 for veins).

Arteries generally showed stronger correlations with structural and distance-based features, while veins responded better to curvature and signal-based features. Features with high correlations, such as DL1, CL11, CP3, and SiP6, demonstrate robustness for predictive modelling. Some features exhibited negative correlations (e.g., SiP18), necessitating further analysis. This comprehensive analysis highlights the complementary strengths of different feature groups in capturing the distinct arterial and venous tortuosity characteristics.

Table 4 compares Spearman’s rank correlation coefficients for selected metrics from the proposed framework with those from previous works, highlighting the impact of resampling techniques on performance. Resample Method 1 showed strong performance, especially for arteries, with higher correlations observed for features previously reported in the literature, such as Distance (DL1), as well as Combined and Curvature-based features (CL3, CL11, CoP3). For DL1, Resample Method 2 achieved a correlation of ($\rho =0.855$) for arteries, surpassing previous results reported in [9] ($\rho =0.792$) and [12] ($\rho =0.829$), reinforcing its effectiveness as the superior resampling approach. CL11 also showed high correlation for arteries, achieving ($\rho =0.927$) with Resample Method 1, while Resample Method 2 slightly reduced this to ($\rho =0.896$). Although Resample Method 1 often provided higher correlation values, Resample Method 2, as highlighted in this study, is considered more accurate due to its improved consistency and reduced bias. CL3 and CL4 remained competitive compared to the literature, with CL3 achieving ($\rho =0.922$) in [9] and showing strong performance with Resample Method 1 ($\rho =0.822$). However, a slight drop was observed with Resample Method 2 ($\rho =0.807$), reinforcing the trend that Method 1 tended to yield higher correlations. Resampling results were mixed across signal-based and combined features. Signal based features, particularly SiP6, demonstrated improved performance with Resample Method 2 ($\rho =0.855$ for arteries), reinforcing its reliability. Meanwhile, SiP1 performed exceptionally well for arteries under Resample Method 1 ($\rho =0.919$), outperforming previous works, though its performance declined under Resample Method 2 ($\rho =0.710$ for arteries). For veins, both resampling methods showed varying performance, with Method 1 occasionally yielding higher correlation values. However, combined features (CoP1, CoP3) demonstrated only moderate to low performance across both methods, highlighting challenges in capturing the complexity of vein tortuosity. Overall, Method 2 offered improved precision and lower error, supporting its suitability for accurate resampling. See Appendix C and Appendix D for all framework’s features Spearman’s rank coefficient correlation analysis results.

3.2. Gaussian Process Regression Analysis Feature Selection Analysis

Gaussian Process Regression (GPR) models were employed with various kernels, demonstrating strong compatibility with augmented datasets. These models excel when sufficient data is provided, enabling accurate estimation of relationships and associated uncertainties. Applying the Minimum Redundancy Maximum Relevance (MRMR) method to the augmented dataset ensured that only the most relevant features were selected, effectively reducing noise introduced by the augmentation process. Following this, MATLAB’s Regression Learner application was utilized to conduct GPR analysis to predict the augmented dataset’s clinical ordering. GPR, a non-parametric supervised learning method, is commonly employed for solving regression and probabilistic classification problems. The dataset was divided into two groups, arteries and veins, each containing 330 instances. Both the complete feature set and a selected subset of features (using MRMR) were evaluated using multiple GPR models with Rational Quadratic, Squared Exponential, Matern, and Exponential kernels. A 70–30% training–testing split was applied, followed by stratified 5-fold cross-validation to ensure a robust and reliable evaluation. Performance metrics, including the Mean Absolute Error (MAE), Root Mean Square Error (RMSE), and the $R^2$ were reported for each group. These results are detailed in

Table 5. Gaussian Process Regression (GPR) analysis of the arteries dataset with augmented data. The table reports training and testing metrics across models using all framework features.

Model	Training				Testing
	RMSE	MSE	${\mathit{R}}^{\mathbf{2}}$	MAE	MAE	MSE	RMSE	${\mathit{R}}^{\mathbf{2}}$
Rational quadratic GPR	0.00036	0.00000	1	0.00025	0.00125	0.00000	0.00157	0.99997
Squared exponential GPR	0.00022	0.00000	1	0.00016	0.00011	0.0000000197	0.00014	1
Exponential GPR	0.07606	0.00579	0.93702	0.06071	0.04802	0.00349	0.05910	0.95931

,

Table 6. Gaussian Process Regression (GPR) analysis of the veins dataset with augmented data. The table presents training and testing results across models using all framework features.

Model	Training				Testing
	RMSE	MSE	${\mathit{R}}^{\mathbf{2}}$	MAE	MAE	MSE	RMSE	${\mathit{R}}^{\mathbf{2}}$
Rational quadratic GPR	0.0003414	0.0000001	0.9999987	0.0002495	0.0001345	0.0000000	0.0001741	0.9999997
Squared exponential GPR	0.0005840	0.0000003	0.9999961	0.0002643	0.0002441	0.0000001	0.0002550	0.9999993
Exponential GPR	0.1578929	0.0249302	0.7180713	0.1156539	0.1077298	0.0222542	0.1491786	0.7609155

,

Table 7. Gaussian Process Regression (GPR) analysis for arteries using the augmented dataset with MRMR-selected 25 features. The table presents training and testing results for different GPR models, evaluated using RMSE, MSE, $R^2$, and MAE.

Model	Training				Testing
	RMSE	MSE	${\mathit{R}}^{\mathbf{2}}$	MAE	MAE	MSE	RMSE	${\mathit{R}}^{\mathbf{2}}$
Rational quadratic GPR	0.0762	0.0058	0.9371	0.0624	0.0694	0.0048	0.0597	0.9417
Squared exponential GPR	0.0775	0.0060	0.9350	0.0630	0.0694	0.0048	0.0597	0.9417
Matern 5/7 GPR	0.0738	0.0054	0.9410	0.0602	0.0694	0.0048	0.0596	0.9418
Exponential GPR	0.0889	0.0079	0.9144	0.0742	0.0899	0.0081	0.0745	0.9024

and

Table 8. Gaussian Process Regression (GPR) analysis for veins using the augmented dataset with MRMR-selected 25 features. The table presents training and testing results for different GPR models, evaluated using RMSE, MSE, $R^2$, and MAE.

Model	Training				Testing
	RMSE	MSE	${\mathit{R}}^{\mathbf{2}}$	MAE	MAE	MSE	RMSE	${\mathit{R}}^{\mathbf{2}}$
Rational quadratic GPR	0.1422	0.0202	0.7715	0.1031	0.1335	0.0178	0.1030	0.8086
Squared exponential GPR	0.1422	0.0202	0.7715	0.1031	0.1335	0.0178	0.1030	0.8086
Matern 5/7 GPR	0.1422	0.0202	0.7715	0.1030	0.1341	0.0180	0.1035	0.8068
Exponential GPR	0.1677	0.0281	0.6819	0.1208	0.1586	0.0252	0.1185	0.7297

. Additionally,

Table 9. Analysis of selected features using the Minimum Redundancy Maximum Relevance (MRMR) method. The table highlights common, unique arterial (A), and unique venous (V) features for each feature group.

Feature Group	Common Features Between A and V	Unique for A	Unique for V
Structural approach	-	SP4, SP3, SP1	-
Distance approach	DL1	-	-
Combined approach	CoL2, CoL1, CP1, CP2	CoP1, CoP3	CoP2
Curvature approach	CL12, CL8, CL4, CL10	CP3, CL4, CL9	CL3, CP2, CL11, CL13, CL1, CL3, CL5, CL6
Signal approach	SiP12, SiP18, SiP20, SiP11, SiP5, SiP19	SiP6, SiP2	SiP3

outlines the common and unique feature sets resulting from the MRMR application, along with the feature groups they belong to.

Squared Exponential consistently showed superior performance with the highest $R^2$ values and minimal errors in both arterial and venous datasets. Arterial data models tend to exhibit higher $R^2$ values and lower errors compared to venous data models, suggesting more predictable patterns in arteries. Models using MRMR selected features maintained high predictive accuracy while reducing dimensionality and computational complexity. This confirms the effectiveness of MRMR in eliminating noise from the augmented dataset. Common features such as DL1 and CoL2 highlight over lapping predictive markers for arteries and veins. However, unique features, such as SP4 for arteries and SiP3 for veins, emphasize their distinct functional or structural differences. Signal and curvature-based approaches yielded the most significant shared features across both vessel types, with signal-based features such as SiP5, SiP20, and SiP11 demonstrating consistently high and comparable Spearman’s correlations for both arteries and veins in previous analyses. Augmented data enhances the robustness of GPR models by improving generalization and minimizing over-fitting. This is evidenced by consistently high $R^2$ values and low errors across all models. Performance metrics (MAE, RMSE, and $R^2$ reveal that Rational Quadratic and Squared Exponential kernels outperform other models, particularly in predicting arterial data with augmented datasets. Exponential kernels show relatively lower accuracy, indicating less suitability for this dataset.

While the GPR models demonstrate strong predictive performance, it is important to note that the “clinical order” used for supervision is based on a single clinician’s ranking, which introduces a degree of subjectivity. Additionally, the way the ground truth is defined, such as using grouped ordinal categories instead of strict rankings, may affect model performance. To enhance the reliability and clinical relevance of these findings, further evaluation on larger and specialized retinal datasets with annotations from multiple clinicians is recommended.

4. Discussion

4.1. Spearman’s Rank Correlation Coefficient Analysis on RET-TORT Dataset and Augmented RET-TORT Dataset

Our findings demonstrate that tortuosity affects retinal arteries and veins differently, which is expected due to their distinct anatomical structures. Arteries are more elastic and tend to constrict passively to help maintain blood pressure, whereas veins are more prone to stretching, becoming elongated and convoluted. Spearman’s correlation coefficient analysis on the RET-TORT dataset revealed that our framework’s metrics exhibited varying performance across vessel types, as detailed in the following sections. Structural Features Arterial correlations were highly significant for structural features SP1 and SP2 ($\rho =0.810$ and $\rho =0.806$), while SP4 showed a strong negative correlation ($\rho =-0.842$). Venous correlations were weaker, with SP5 achieving the highest value ($\rho =0.353$). This suggests that structural features are more effective for arteries, capturing their distinct geometrical characteristics. Distance-Based Feature DL1 showed strong correlations for both arteries ($\rho =0.855$) and veins ($\rho =0.644$), highlighting its reliability in capturing vascular tortuosity across different vessel types. Combined Features demonstrated strong performance in some cases. CoL1 achieved the strongest correlations for both arteries ($\rho =0.863$) and demonstrated moderate performance for veins ($\rho =0.541$). While other combined features, such as CoP1 and CoP3, performed moderately for arteries ($\rho =0.733$ and $\rho =0.729$), their performance for veins remained relatively low, emphasizing the need for optimization to capture venous tortuosity patterns better. Curvature-Based Features exhibited robust performance. CL3 and CL4 (Group 1) achieved strong correlations for arteries and veins ($\rho =0.807$ and $\rho =0.792$). CL11 (Group 3) demonstrated the highest correlation overall for arteries ($\rho =0.896$) and veins ($\rho =0.766$). CP3 (Group 4) was also a top performer for arteries ($\rho =0.895$) and veins ($\rho =0.773$). Signal-based features excelled across both vessel types. SiP3, SiP5, and SiP8 (Group 1) performed well, with SiP5 achieving ($\rho =0.824$) for arteries and ($\rho =0.794$) for veins. SiP14 (Group 2) was the most effective arterial feature ($\rho =0.833$), while SiP11 performed best for veins ($\rho =0.711$). Arterial correlations were stronger for structural and distance-based features, while venous correlations were more robust for curvature and signal-based features. Top performing features, such as DL1, CL11, CP3, and SiP6, exhibited robustness, making them strong candidates for predictive modeling. Negative correlations observed for features like CP2 and SiP18 highlight areas for further investigation. Although the Spearman’s rank correlation coefficient analysis on the augmented RET-TORT dataset showed moderate to weak overall correlations, it reaffirmed the distinct performance trends observed between arteries and veins. Structural Features: SP1 showed the highest cor relation for arteries ($\rho =0.614$), indicating a moderate association, while SP2 performed best for veins ($\rho =0.315$), reflecting a weak correlation. Distance-Based Features: DL1 remained a reliable feature for both arteries ($\rho =0.626$), indicating a moderate correlation, and veins ($\rho =0.400$), reflecting a weak to moderate association. Curvature-Based Features:CL3, CL8, and CP3 consistently showed moderate correlations across vessel types, with CP1 achieving a relatively strong moderate correlation for veins ($\rho =0.577$). Signal-Based Features: SiP5 and SiP8 were top performers, capturing both large-scale and localized variations. Structural features excel for arteries, while curvature and signal-based features better capture venous tortuosity. Spearman’s rank correlation is sensitive to changes in ranking order, and augmentation often introduces small inconsistencies that scramble the original ranks, leading to lower correlations; therefore, we do not recommend the use of Spearman’s rank analysis with such augmented datasets.

4.2. Resampling Approaches

The study highlighted the critical importance of employing robust resampling techniques in the evaluation of retinal vessel tortuosity, as curvature, distance, and frequency-based tortuosity metrics are highly sensitive to the number of sample points. At higher sampling rates, these metrics can produce values that differ significantly from those obtained at lower rates [28], (see Table 4), highlighting the necessity for a standardized and reliable sampling strategy. To address this, two resampling methods were evaluated, with Resample Method 2 (1EPS) demonstrating superior geometric fidelity and generating vessel segments that more accurately represent the original morphology. Although Method 1 occasionally achieved higher Spearman’s correlation coefficients with the clinical ordering, Method 2 exhibited greater consistency and accuracy, based on accurate length measurements, particularly for artery specific metrics such as DL1. The effectiveness of this method was further validated by testing tortuosity value accuracy across varying vessel resolutions, and by conducting both theoretical and empirical assessments of feature scale invariance, together confirming the robustness of Resample Method 2. Based on these findings, our framework features were classified according to their sensitivity to scaling, with over 20 features identified as invariant, under uniform similarity transform, and suitable for longitudinal image analysis to monitor temporal progression of retinal diseases. Ultimately, Resample Method 2 is recommended for its ability to preserve vessel morphology, maintain approximately scale invariance, and ensure reliable tortuosity assessment across both arterial and venous segments. Furthermore, the absence of clearly defined preprocessing steps in previous studies complicates direct comparisons and reinforces the need for standardized methodologies.

4.3. Gaussian Process Regression (GPR) Performance

Gaussian Process Regression (GPR) models demonstrated high predictive accuracy: To enhance the performance observed with Spearman’s correlation and fully utilize the framework’s feature set, Gaussian Process Regression (GPR) was applied to the augmented dataset. This allowed for effective feature selection, enabling the identification of vessel-specific tortuosity characteristics. The Squared Exponential kernel consistently yielded the highest $R^2$ values and lowest errors, particularly for arteries. Feature selection using the Minimum Redundancy Maximum Relevance (MRMR) method reduced computational complexity while maintaining accuracy. Key features identified included L1 and CoL2 for arteries, and SiP3 for veins. The augmentation strategy improved generalization and minimized over fitting, enhancing the overall robustness of the framework.

4.4. Innovation and Clinical Relevance

Our study presents a comprehensive framework for retinal blood vessel tortuosity estimation, offering a validated, step-by-step approach for analyzing tortuosity and supporting early disease detection. The framework accounts for anatomical differences between arteries and veins by introducing an ensemble of features designed to capture diverse tortuosity patterns, which are also approximately invariant to image scaling. Additionally, it emphasizes the importance of resampling approaches and introduces a robust resampling strategy tested in this work. This framework holds promise for integration into clinical settings, such as diabetic eye screening programs, where it could automate and enhance the accuracy of early diabetic retinopathy detection, reducing reliance on manual assessment and clinician expertise. The findings support the development of objective, automated screening approaches that could guide diabetic eye-care policy and provide a scalable foundation for research into vascular biomarkers across systemic diseases. The long-term goal is to integrate this framework into routine clinical workflows for early ocular disease detection and longitudinal monitoring. The system is currently being tested in diverse populations, including hypertensive and diabetic retinopathy patients, healthy controls, and age related macular degeneration, within a semi-automatic pipeline that segments retinal images and extracts geometric features at both image and vessel levels. A key remaining challenge is improving artery–vein classification to correctly resolve vessel crossings and ensure reliable image-level tortuosity estimates. Addressing this limitation is essential for advancing the framework toward seamless clinical deployment.

4.5. Limitations

While the proposed framework includes features invariant to image scaling and accounts for anatomical differences between arteries and veins, it may still be sensitive to other sources of variability such as image quality, noise, or inconsistencies in manual vessel segmentation, potentially limiting generalizability across different datasets or imaging devices. Despite the use of data augmentation, the dataset may not fully represent the diversity of real-world clinical scenarios, which could impact model performance when applied to new patient demographics, imaging conditions, or disease stages. Furthermore, reliance on augmented data introduces the risk of overfitting to synthetic patterns that do not exist in actual clinical images, especially in feature-driven models. The Spearman’s rank correlation analysis conducted on the augmented dataset showed reduced performance compared to the original data, suggesting that this metric should be used with caution when rank-scrambling augmentations are applied. Additionally, while vessel ranking for tortuosity severity was performed by a clinician, it was inherently subjective and based on clinical experience rather than standardized diagnostic criteria. This introduces variability and may affect the consistency and clinical interpretability of the results, highlighting the need for objective ground truth benchmarks in future studies. Finally, the framework was developed and validated primarily on fundus photography and may not generalize to other imaging modalities such as OCTA or fluorescein angiography without further validation.

5. Conclusions

This study presents a robust, well-built framework for retinal vascular tortuosity analysis, designed around a reliable resampling method and features that remain invariant to scaling. It offers a comprehensive approach by integrating structural, distance, combined, curvature, and signal-based features with advanced modeling techniques. The framework accurately captures all forms of tortuosity in both arterial and venous blood vessels, potentially supporting the early detection and prevention of conditions such as diabetic and hypertensive retinopathy. By combining a large number of estimation features with a solid resampling methodology, it ensures precise tortuosity detection while maintaining scale invariance, making it suitable for images of varying resolutions. The key findings of this study include the following:

• Feature Effectiveness: Structural and distance-based features excelled in detecting arterial tortuosity, while curvature and signal-based features were particularly effective for veins. Top-performing features included DL1, CL11, CP3, and SiP6, with the latter two features proposed in this study.
• Resampling Benefits: The study demonstrated that the proposed resampling method 2 (1EPS) produces more representative vessel segments and therefore produces more accurate tortuosity evaluations compared to approaches such as method 1. Spearman’s correlation analysis confirmed that the choice of resampling method significantly impacts vessel measurements and tortuosity estimation. While Method 1 showed higher performance for specific features (e.g., CL11 and SiP1), Method 2 outperformed Method 1 in terms of performance and, presumably, greater accuracy, especially for distance-based features such as DL1.
• Gaussian Process Regression (GPR) Modelling: The use of MRMR-selected features and Squared Exponential kernels achieved high predictive accuracy of the clinical order of both arteries and veins, demonstrating practical applicability for detecting specific early structural changes along the course of blood vessels.

6. Recommendations and Future Work

The proposed framework shows strong potential for clinical application, particularly in enabling fully automated screening programs for diabetic and hypertensive retinopathy. To further validate its applicability for disease detection, it should be tested on a large-scale retinal pathology dataset. Additionally, incorporating consecutive retinal image analysis using the framework’s scale-invariant features could enhance its capacity to monitor disease progression and identify specific retinal biomarkers. Embedding this framework into real-time image analysis systems would provide a more objective and quantitative approach to retinopathy detection, reducing reliance on subjective clinical assessments. These recommendations aim to strengthen the framework’s clinical utility and facilitate its integration into routine diagnostic workflows.

Future developments should focus on ensuring that all extracted features remain invariant to scaling, particularly for use in longitudinal studies that monitor disease progression. Expanding the framework’s application to larger and more diverse datasets will be essential to confirm its generalizability in detecting hypertensive and diabetic retinal characteristics. To support widespread clinical adoption, integrating an automatic artery–vein classification algorithm could streamline analysis and enable end-to-end automation. Further investigations should also explore the effects of axial length and eye curvature on measurement accuracy, compare different imaging modalities (such as standard fundus photography versus wide-field scanning laser ophthalmoscopy), and extend the framework to systemic and ocular diseases beyond retinopathy. Collectively, these directions will strengthen the framework’s robustness and establish it as a valuable tool for early disease detection, clinical research, and patient monitoring.