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A Multi-Orientation Analysis Approach to Retinal Vessel Tracking

Erik Bekkers, Remco Duits, Tos Berendschot, Bart ter Haar Romeny

arXiv:1212.3530v5cs.CV

TL;DR

Retinal vasculature can be observed non-invasively, but vessel tracking faces complex structures such as crossings and bifurcations. The paper tracks vessels using orientation scores and a geometrical optimization principle, and reports detailed retinal models with clean topology and strong comparison to state-of-the-art algorithms.

  • Problem

    Vessel-tracking methods have reported limitations in handling crossings, bifurcations, and other complex situations, despite the value of retinal vasculature for studying disease.

    Source: §1 · pp. 1–2

  • Method

    The paper tracks vessels directly in orientation scores using geometrical curve optimization, introducing ETOS and a faster multi-scale CTOS alternative.

    Source: §1 · p. 2, §3 · p. 8, §5 · p. 20

  • Results

    The method constructs detailed hierarchical retinal models, provides reliable width measurements, compares excellently with state-of-the-art algorithms, and yields clean topological models with few false-positive vessels.

    Source: §5 · p. 21

  • Takeaways & Limitations

    The resulting retinal vasculature models provide a basis for quantitative analysis of vessel geometry and topology.

    Source: §5 · p. 21

    CTOS is very fast, but its combined scale and orientation detection is slightly less stable because each depends on correct detection of the other.

    Source: §3.1.2 · p. 12

Abstract

from arXiv · show

This paper presents a method for retinal vasculature extraction based on biologically inspired multi-orientation analysis. We apply multi-orientation analysis via so-called invertible orientation scores, modeling the cortical columns in the visual system of higher mammals. This allows us to generically deal with many hitherto complex problems inherent to vessel tracking, such as crossings, bifurcations, parallel vessels, vessels of varying widths and vessels with high curvature. Our approach applies tracking in invertible orientation scores via a novel geometrical principle for curve optimization in the Euclidean motion group SE(2). The method runs fully automatically and provides a detailed model of the retinal vasculature, which is crucial as a sound basis for further quantitative analysis of the retina, especially in screening applications.

1 Introduction

Retinal images offer a non-invasive view of vasculature affected by ocular and systemic diseases. The paper introduces orientation-score tracking to address complex vessel structures and construct retinal vasculature models.

1 Introduction

  • Vessel tracking recursively expands a vascular model from seed points, preserving connected segments and providing geometric and topological information for analysis.Pixel classification instead labels pixels as vessel or background and does not necessarily preserve connectedness.Source: §1 · p. 1
  • Orientation-score tracking addresses crossings, bifurcations, and other complex structures with two new algorithms that operate directly on the score domain.The authors extend one algorithm into a method for constructing models of the complete retinal vasculature.Source: §1 · p. 2
  • Orientation scores map image positions and orientations to complex values; invertible scores retain image information and disentangle local orientations.This representation allows multiple orientations at a position, supporting analysis of crossings.Source: §1 · p. 2
  • Both tracking approaches use a geometrical principle for curve optimization in the Euclidean motion group.ETOS is an all-scale approach associated with cake wavelets; CTOS uses multi-scale orientation decomposition and classical Gabor wavelets.Source: §1 · pp. 2–3

2 Orientation scores

Orientation scores represent image content across position and orientation, allowing multiple orientations at one position to be preserved. The section describes their invertible construction, SE(2) geometry, and cake and Gabor wavelet implementations.

2 Orientation scores

  • Orientation scores map each image position-orientation combination to a value, preserving multiple local orientations for tracking through crossings.Invertibility supports automatic disentanglement of local orientations involved in crossings.Source: §2 · pp. 3–4

2.1 Construction of orientation scores

  • The transform convolves an image with rotated anisotropic wavelets, and invertibility permits exact reconstruction from the orientation score.The frequency function Mψ indicates how well image spatial frequencies are preserved and thus the stability of the inverse transform.Source: §2 · p. 3, §2.1 · p. 4

2.2 The domain of orientation scores: SE(2)

  • SE(2) models the orientation-score domain as rigid-body motions, reflecting that its spatial directions vary with orientation.The group action combines translation and rotation.Source: §2.2 · p. 5

2.3 Cake wavelets

  • Cake wavelets uniformly sample the Fourier domain, supporting stable reconstruction and capture of image scales without bias to a specific scale.Their quadrature property lets tracking detect vessel edges from the imaginary part of orientation scores without first computing perpendicular derivatives.Source: §2 · p. 3, §2.3 · pp. 5, 7

2.4 Gabor wavelets

  • Gabor wavelets target specific spatial frequencies, matching differently sized vessels, but a single-scale transform loses information and is non-invertible.A multiscale approach is needed for invertibility and is more computationally expensive.Source: §2 · p. 3, §2.4 · p. 7
  • Decomposing double-sided wavelet scores into forward and backward directions distinguishes π from 2π symmetries and supports bifurcation handling.A single-sided wavelet can construct a directional score while retaining access to the original double-sided score.Source: §2.5 · p. 8

3 Vessel tracking in orientation scores via optimization in transversal tangent planes V

This section presents ETOS and CTOS for tracking vessel edges or centerlines in orientation scores through geometric optimization. It also evaluates ETOS on complex vessel junctions and retinal vessel-width measurements against established methods and REVIEW ground truth.

3 Vessel tracking in orientation scores via optimization in transversal tangent planes V

  • Orientation scores organize image information by orientation to disentangle crossings, and the proposed trackers use local path optimization in transversal tangent planes.The score’s quadrature property represents edge information in its imaginary part.Source: §3 · pp. 8–9

3.1.1 ETOS: Edge tracking in orientation scores

  • ETOS tracks paired vessel edges simultaneously through two one-dimensional optimizations, using an edge-probability envelope to locate likely edges and their orientation.Paired tracking can maintain both edges at crossings, parallel vessels, and bifurcations, but abrupt width changes may be missed or less detailed.Source: §3.1.1 · pp. 9–10

3.1.2 CTOS: Multi-scale vessel center-line tracking in orientation scores

  • CTOS detects vessel centerlines in the real part of multi-scale orientation scores, using center, orientation, and scale detection steps.Scale-selective Gabor wavelets can filter central light reflexes; CTOS is fast but less stable because orientation and scale detection depend on each other.Source: §3.1.2 · p. 12

3.2 Validation

  • The algorithms were tested on green-channel fundus images after low-frequency luminosity drifts were detected by Gaussian blurring and subtracted.The typical Gaussian-blur standard deviation was σ = 32.Source: §3.2 · p. 12

3.2.1 Algorithm behavior at complex vessel junctions

  • The complex-junction evaluation used four HRFI sub-images with 27 manually marked seed points covering crossings, bifurcations, parallel vessels, and curved vessels.Tracking used a 2-pixel step size, a 40-pixel scan-line width, 36 orientations, and an edge-envelope standard deviation of 3.Source: §3.2.1 · p. 12
  • In complex situations, ETOS performed best with invertible cake-kernel orientation scores, failing to track only blood vessel 5 from image 3 while tracking all other vessels, even at very low edge contrast.Source: §3.2.1 · p. 14

3.2.2 Validation of width measurements

  • ETOS vessel widths were compared with REVIEW ground truth, formed by averaging three observers’ measurements across 16 images and 5,066 available width measurements.The plotted 5,059 ground-truth and ETOS width pairs have regression y = 0.85 + 0.88x; the caption describes an offset of less than a pixel and a strong positive relation.Source: §3.2.2 · pp. 14–15, Figure 12 · p. 15
  • Width-measurement success rates fall below 100% when measurements fail to converge, such as when measured edges are too far from ground truth.The table’s standard-deviation measure reflects error consistency, with lower values favorable.Source: §3.2.2 · pp. 15–16
  • ETOS outperforms classic width-measurement techniques and performs well against state-of-the-art methods, with lower error standard deviation than observers on CLRIS and comparable performance on other datasets.The paper concludes that ETOS reliably measures widths while tracking vessels through complex situations.Source: §3.2.2 · p. 16
  • The regression y = 0.85 + 0.88x indicates strong positive agreement, but ETOS slightly overestimates vessels up to 7 pixels wide and underestimates larger vessels.Few outliers in the scatter plot are consistent with the low error standard deviation reported in Table 1.Source: §3.2.2 · p. 16

4 Vasculature tracking

The vasculature-tracking algorithm builds on ETOS and adds steps for constructing complete retinal vasculature models. The complete algorithm is validated in Section 4.2.

4 Vasculature tracking

  • The tracking pipeline adds optic-disk detection, seed-point detection, robust edge initialization, automatic stopping, junction handling, and junction resolving.These additions support construction of complete retinal vasculature models.Source: §4 · p. 16
  • The complete algorithm is validated in Section 4.2.Source: §4 · p. 16

4.1 Methods

The methods initialize tracking near the optic disk, then extend ETOS from selected seed points and edges. They stop tracks automatically and detect, classify, number, and resolve junctions to build vasculature models.

4.1.1 Optic disk detection

  • The optic disk is first roughly located by variance filtering, then refined by edge focusing and a weighted circular Hough transform.The refinement assumes an approximately circular optic disk and removes vessels with a closing operator before boundary detection.Source: §4.1.1 · p. 16

4.1.2 Vessel likelihood map and seed point detection

  • Seed points are selected from circular profiles in an invertible-orientation-score vessel-likelihood map, then assigned the orientation with the highest score modulus.Profiles use radii R = {ROD, 1.5ROD}; candidates below the circle's average likelihood are discarded.Source: §4.1.2 · pp. 16–17

4.1.3 Initial edge detection

  • Initial vessel edges are chosen by scoring candidate edge pairs around the seed, then tracing neighboring patch edges in scale and selecting the strongest at the toppoint scale.The edge-pair score favors elongated vessel structures and penalizes distance from the initialized center.Source: §4.1.3 · pp. 17–18

4.1.4 Stopping criteria

  • Tracking stops when a vessel leaves the region of interest, overlaps an already tracked segment for the specified iterations, or falls below the vessel-value threshold.The threshold is based on the average vessel value in the optic-disk region.Source: §4.1.4 · p. 18

4.1.5 Junction detection, classification and numbering

  • Junction candidates are detected from additional orientation maxima, clustered by position and orientation, then classified using local-orientation alignment in SE(2).Each seed point receives a segment ID and a parent-vessel ID, preserving the hierarchical relations among segments.Source: §4.1.5 · pp. 18–19

4.1.6 Junction resolving

  • Unresolved junctions are handled according to their position on the established track, overlap width, and similarity between source and target tracks.The detailed resolving scheme is cited in.Source: §4.1.6 · p. 19

4.2 Validation

The authors assessed the topology of generated vasculature models on nine HRFI images. Junction detection was more precise for crossings than bifurcations, while false-positive vessel segments were uncommon.

4.2.1 Validation of topological correctness

  • The evaluation covered 495 junctions from nine generated models, comprising 381 bifurcations and 114 crossings across healthy, diabetes, and glaucoma HRFI datasets.Bifurcation errors included junctions that were actually part of crossings and branches not matching ground-truth vessels.Source: §4.2.1 · p. 20
  • 96.03% of crossings and 76.54% of bifurcations were correctly detected by precision rate.These rates correspond to 109/114 crossings and 290/381 bifurcations, respectively.Source: §4.2.1 · p. 20, Table 2 · p. 20
  • Only 5.25% of bifurcations were false vessels, and the authors report that almost all modeled segments represented true blood vessels.Misclassified crossings affect topology; false-vessel branches also add false-positive segments to the model.Source: §4.2.1 · p. 20

5 Conclusion

The conclusion presents ETOS on invertible orientation scores as the basis for hierarchical retinal vasculature models and reports clean topology. It also identifies a geometric tracking principle in SE(2).

5 Conclusion

  • The authors report that orientation-score disentanglement supports tracking vessels through crossings, and that invertible scores produced the best results among the score types tested.CTOS is described as very fast but less stable than ETOS at crossings and parallel vessels.Source: §5 · p. 20
  • The ETOS-based vasculature tracker produced detailed hierarchical models, with reliable width measurements and few false-positive vessels reported in validation.Source: §5 · p. 21
  • Tracking in orientation scores uses V-plane optimization, a geometric principle within a sub-Riemannian manifold in SE(2).The paper relates this principle to probabilistic approaches to optimal curves and identifies geometric control in enhanced orientation scores as future work.Source: §5 · p. 21

A A mathematical underpinning of optimization in the eη −eθ tangent planes V

The section grounds ETOS’s optimization in the curved geometry of SE(2), using local searches in transversal (η, θ) planes and a cost-based principle for horizontal curves. For completion fields, the principle connects to elastica and sub-Riemannian geodesics in specific settings, including cubic B-splines in the Heisenberg approximation.

A. The geometrical principle behind the ETOS algorithm

  • The approach optimizes within transversal planes V = span{∂θ, ∂η}, while horizontal curves have tangents in H = span{∂ξ, ∂θ}.Here ∂ξ is the spatial propagation direction and ∂η is the transverse spatial direction in the moving frame.Source: §A.1 · p. 21, A A mathematical underpinning of… · p. 21
  • The proposed principle selects horizontal curves satisfying ∂θC = 0 and ∂ηC = 0, which the authors expect to be optimal under the stated local cost conditions.The curve optimization uses a sub-Riemannian metric that penalizes bending and stretching.Source: §A.1 · p. 22

A.2 Application of the geometrical principle to completion fields

  • Completion fields are products of resolvent Green’s functions for contour completion and enhancement, and their relation to Brownian bridges connects them to geodesics.As λ tends to zero, the completion field for contour enhancement concentrates on sub-Riemannian geodesics; the geodesic relation is stated for endpoints sufficiently close.Source: §A.2 · pp. 22–23
  • The direction-process modes coincide with elastica, while the corresponding optimization curves numerically seem to coincide with elastica for the completion distribution.For contour enhancement, production of sub-Riemannian geodesics remains a point for future investigation.Source: §A.2 · p. 23, §A.4 · p. 24

A.3 The Heisenberg approximation of completion fields produces B-splines

  • In the Heisenberg approximation, the solution curves are cubic B-splines lifted as (x, y(x), y′(x)), with y(x) a third-order polynomial matching boundary conditions.The figure illustrates the lifted spline for specified endpoints and orientations.Source: §A.3 · p. 23, Figure 21 · p. 24

A.2 Application of the geometrical principle to completion fields

  • For exact SE(2) completion fields, numerical computations seem to confirm that the optimization curves coincide with elastica.The text presents this as a conjecture supported by numerical computations, not as a proven identity.Source: §A.4 · p. 24, Figure 22 · p. 24

B The closure of the distributional orientation score transform is an L2-isometry on the whole L2(R2) space.

The section addresses conditions that obstruct an ordinary square-integrable wavelet formulation by extending the orientation score transform through distributional wavelets. Under a stated condition, the transform’s closure is an L2-isometry, and the framework does not require bandlimited images.

B. The closure of the distributional orientation score transform is an L2-isometry on the whole L2(R2) space

  • The usual condition violates ψ ∈ L1(R2) ∩ L2(R2), motivating distributional wavelets and a distributional orientation score transform.The paper uses a dual Sobolev space and a Gelfand triple to formulate this extension.Source: B The closure of the distributional… · p. 24
  • The orientation score framework remains valid without restricting images to be bandlimited, regardless of sampling size or rate.The text notes bandlimiting can be imposed to avoid technicalities, but says it is not crucial.Source: B The closure of the distributional… · pp. 24–25
  • Under a condition on ψ, the orientation score transform W is closable and its closure is an L2-isometry from the image space into L2(SE(2)).The stated condition includes βψ = β^-2; the isometric closure is DβWDβ^-1.Source: B The closure of the distributional… · p. 25
  • For an L2-isometric mapping, the paper requires proper distributional wavelets; cake-wavelet limits typically become oriented δ-distributions as ϱ tends to infinity.For proper wavelets of class II, the limits exist pointwise and in H−I-sense, but not in L2-sense.Source: B The closure of the distributional… · p. 25
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