Curvature based biomarkers for diabetic retinopathy viaexponential curve fits in SE (2)Citation for published version (APA):Bekkers, E. J., Zhang, J., Duits, R., & ter Haar Romeny, B. M. (2015). Curvature based biomarkers for diabeticretinopathy via exponential curve fits in SE (2). In X. Chen, M. K. Garvin, J. Liu, E. Trucco, & Y. Xu (Eds.),Proceedings of the Ophthalmic Medical Image Analysis : Third International Workshop (OMIA 2016) Held inConjunction with MICCAI 2016, 9-13 October 2016, Munich, Germany (pp. 113-120)https://doi.org/10.17077/omia.1034

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University of IowaIowa Research Online

Proceedings of the Ophthalmic Medical ImageAnalysis International Workshop 2015 Proceedings

Oct 9th, 2015

Curvature Based Biomarkers for DiabeticRetinopathy via Exponential Curve Fits in SE(2)Erik J. BekkersEindhoven University of Technology

Jiong ZhangEindhoven University of Technology

Remco DuitsEindhoven University of Technology

Bart M. ter Haar RomenyEindhoven University of Technology

Follow this and additional works at: http://ir.uiowa.edu/omia

Part of the Ophthalmology Commons

RightsCopyright © 2015 Erik J. Bekkers, Jiong Zhang, Remco Duits, and Bart M. ter Haar Romeny

Hosted by Iowa Research Online. For more information please contact: lib-ir@uiowa.edu.

Recommended CitationBekkers EJ, Zhang J, Duits R, and ter Haar Romeny BM: Curvature Based Biomarkers for Diabetic Retinopathy via Exponential CurveFits in SE(2). In: Chen X, Garvin MK, Liu J, Trucco E, Xu Y, editors. Proceedings of the Ophthalmic Medical Image Analysis SecondInternational Workshop, OMIA 2015, Held in Conjunction with MICCAI 2015, Munich, Germany, October 9, 2015. Iowa City, IA:University of Iowa, 2015, pp. 113-120, available from: http://ir.uiowa.edu/omia/2015_Proceedings/2015/15

Curvature Based Biomarkers for DiabeticRetinopathy via Exponential Curve Fits in SE(2)

Erik J. Bekkers1, Jiong Zhang1, Remco Duits1, and Bart M. ter HaarRomeny2,1

1 Department of Biomedical Engineering and Department of Mathematics andComputer Science, Eindhoven University of Technology, the Netherlands

2 Department of Biomedical and Information Engineering,Northeastern University, Shenyang, China

{e.j.bekkers, j.zhang1, r.duits, b.m.terhaarromeny}@tue.nl

Abstract. We propose a robust and fully automatic method for theanalysis of vessel tortuosity. Our method does not rely on pre-segmentationof vessels, but instead acts directly on retinal image data. The methodis based on theory of best-fit exponential curves in the roto-translationgroup SE(2). We lift 2D images to 3D functions called orientation scoresby including an orientation dimension in the domain. In the extendeddomain of positions and orientations (identified with SE(2)) we studyexponential curves, whose spatial projections have constant curvature.By locally fitting such curves to data in orientation scores, via our newiterative stabilizing refinement method, we are able to assign to eachlocation a curvature and confidence value. These values are then usedto define global tortuosity measures. The method is validated on syn-thetic and retinal images. We show that the tortuosity measures canserve as effective biomarkers for diabetes and different stages of diabeticretinopathy.

Keywords: Retina, vessel tortuosity, biomarkers, diabetes, curvature,orientation scores, roto-translation group

1 Introduction

Systemic diseases, such as diabetes, may cause quantifiable changes to the geom-etry of the retinal microvasculature [1, 2]. One of the most relevant geometricalfeatures of the microvasculature is vessel tortuosity [2–5]. While for some geo-metrical features (such as vessel calibre) there is a general consensus [1] on howthey are associated to several diseases. This is not the case for vessel tortuosity,which makes it still a very relevant topic of research. E.g., in [3] a positive, and in[4] a negative association of vessel tortuosity with progression towards diabeticretinopathy (DR) is found. In this work, we present a novel robust and fullyautomated method for the extraction of tortuosity measures, and show strongpositive associations of the measures with diabetes and progressive stages of DR.

Vessel tortuosity descriptors are typically computed via an extensive pipeline(including manual interventions) of image pre-processing, segmentation, thinning

X. Chen, M. K. Garvin, J. Liu, E. Trucco, Y. Xu (Eds.): OMIA 2015, Held in Conjunctionwith MICCAI 2015, Munich, Germany, Iowa Research Online, pp. 113–120, 2015. Availablefrom: http://ir.uiowa.edu/omia/2015_Proceedings/2015/

and splitting of the vascular network, after which tortuosity values are computedfrom the extracted vessel centerlines [2, 3, 5, 6]. In such pipelines, errors intro-duced in each processing step may accumulate, and information might get lostalong the way. As an alternative, we propose a reduced pipeline that does not relyon explicit segmentation of the blood vessels, but instead computes tortuosityfeatures directly from retinal image data.

The proposed method is based on theory of best exponential curve fits inthe roto-translation group SE(2), developed by Duits, Franken and Janssen [7–9]. To this end, we lift 2D images to 3D functions called orientation scores byincluding an orientation dimension [10]. In the extended domain of positionsand orientations (identified with SE(2)) we study so-called exponential curves,whose curvatures are constant. By locally fitting exponential curves [8] to datain orientation scores, we are able to assign to each location a curvature andmeasurement-confidence value, which we use to define global tortuosity mea-sures. Additionally, we improve the accuracy of best-exponential curve fits via anovel refinement procedure, resulting in more accurate curvature estimations.

This article is structured as follows: Sec. 2 provides necessary prerequisites:the notion of curves in the space of positions and orientations R2 o S1 (Sub-sec. 2.1), theory on orientation scores (functions on this space) and on the com-putation of best-exponential curve fits from these functions (Subsec. 2.2). InSubsec. 2.3 we then introduce our tortuosity measures based on the aforemen-tioned theory. In Sec. 3 we validate the accuracy of the curvature extractionand associated confidence, and demonstrate the potential of using our tortuositymeasures as biomarkers in diabetes research. The article is concluded in Sec. 4.

Fig. 1: A: Two exponential curves γch and γcl in SE(2), with high and lowcurvature respectively. The coefficients c = (cξ, cη, cθ)T of tangent vectors γc(t),expressed in the left invariant basis at location γc(t) are constant along theexponential curves γc. This is emphasized in B and C, where one also observesa steeper slope in θ-direction of the tangent vector γch compared to γcl .

114 E. J. Bekkers et al.

2 Theory

2.1 Exponential Curves in SE(2)

The domain R2oS1 ≡ SE(2). The joint space of positions and orientations, withelements (x, θ) ∈ R2oS1, is essentially the roto-translation group SE(2) of planartranslations and rotations, equipped with group product g ·g′ = (x, θ) · (x′, θ′) =(Rθx

′ + x, θ + θ′) [9, ch. 2.1].Curves and tangent vectors in SE(2). Planar curves γ2D(t) = (x(t), y(t))T ∈

R2 can be naturally lifted to curves γSE(2)(t) = (x(t), y(t), θ(t))T ∈ SE(2) in thespace of positions and orientations by considering the direction of the tangentvector γ2D(t) as the third coordinate (θ(t) = arg(x(t) + i y(t))). Tangent vectorsof planar curves γ(t) = (x(t), y(t)) ∈ T (R2) are usually spanned by a globalbasis {ex, ey}, with ex = (1, 0) and ey = (0, 1), i.e., T (R2) = span{ex, ey}. InSE(2) we must work with a rotating frame of reference {eξ(g), eη(g), eθ(g)} ={cos θex+sin θey,− sin θex+cos θey, eθ}, aligned with the orientation at each g ∈SE(2), rather than with a global frame {ex, ey, eθ}. The tangent space at eachg is spanned by the left-invariant frame Tg(SE(2)) = span{eξ(g), eη(g), eθ(g)}.

Exponential curves in SE(2). An exponential curve is a curve whose tangentvector components c = (cξ, cη, cθ)T expressed in the local left-invariant basis{eξ, eη, eθ}|γc are constant, i.e., γc(t) = cξeξ(γc(t)) + cηeη(γc(t)) + cθeθ(γc(t)),

for all t ∈ R. Exponential curves in SE(2) can be regarded as ”straight lines”with respect to the curved geometry of SE(2). The exponential curve through

g with tangent c is given by γgc (t) = g · exp(t(∑3i=1 c

iAi)) with {A1, A2, A3} ={eξ(0, 0), eη(0, 0), eθ(0, 0)} the basis for the Lie algebra. By direct computationit follows that γgc is a helix with constant curvature and torsion in SE(2). Fordetails see [11, 8]. For intuition see Fig. 1. The explicit formulas for these expo-nential curves are well-known (see e.g. [12, 8]). Here however, we do not needthese formulas as we directly deduce curvature of spatially projected curvesPR2γc (cf. Fig. 1) from vector c via

κ =cθ sign(cξ)√|cξ|2 + |cη|2

. (1)

See also [9, ch. 2.9] for more details.

2.2 Image Data Analyzed as Function on SE(2)

Orientation scores. We analyse image data in the form of orientation scores,which are functions on SE(2). An orientation score U can be constructed froman image f by means of correlation with some anisotropic wavelet ψ via

U(x, θ) = (Wψf)(x, θ) = (ψθ ? f)(x) =

∫

R2

ψ(R−1θ (x− x))f(x)dx, (2)

where ψ ∈ L2(R2) is the correlation kernel, aligned with the x-axis, where Wψ

denotes the transformation between image f and orientation score U , and ? de-notes correlation. The overline denotes complex conjugation, ψθ(x) = ψ(R−1

θ x)

Curvature Based Biomarkers for Diabetic Retinopathy 115

Fig. 2: Construction of an orientation score (OS) (middle panel) from an image(left panel) via the OS-transformWψ. In the score a derivative frame {∂ξ, ∂η, ∂θ},aligned with group elements (x, y, θ), is used for tangent vector (c∗) estimation ofexponential curves γc∗ . Using c∗, curvature and measurement confidence valuescan be computed, which are encoded resp. in color and opacity in the right panel.

and Rθ is a counter clockwise rotation over angle θ. We choose cake wavelets[10] for ψ. The Fourier transforms of cake wavelets uniformly cover the frequencydomain, and have thereby the advantage over other oriented wavelets (e.g. Ga-bor wavelets) that they allow for a stable inverse transformation W∗ψ from theorientation score back to the image. As such, they ensure that no data-evidenceis lost in the transformation. The left two panels of Fig. 2 show an image withdifferent curvature circles and the corresponding orientation score.

Best exponential curve fits. We compute curvature values directly from tan-gent vectors (see Eq. (1)) of exponential curves that locally best fit the data.In medical image analysis applications the direction of minimal principal curva-ture, obtained via eigensystem analysis of the Hessian matrix, is often used in thecomputation of vectors tangent to oriented (tubular) structures. This concept isfor example used in the Frangi vesselness filter [13]. Here we exploit a similarapproach, however, when considering the curved domain R2 o S1 we must payattention to the following:

1. Rather than using a global {∂x, ∂y, ∂θ} derivative frame (we use short handnotation ∂i = ∂

∂i ) we must take into consideration the curved geometry ofthe domain, and compute the Hessian matrix via left-invariant derivatives:

HU = Mµ−2

∂2ξU ∂ξ∂ηU ∂θ∂ξU

∂ξ∂ηU ∂2ηU ∂θ∂ηU

∂ξ∂θU ∂η∂θU ∂2θU

Mµ−2 , (3)

with {∂ξ, ∂η, ∂θ} = {cos θ∂x + sin θ∂y,− sin θ∂x + cos θ∂y, ∂θ} , and withMµ = diag(µ, µ, 1). Here parameter µ, with unit 1/length, is introduced todeal with the different physical dimensions in domain R2 o S1 [8, 9].

2. Since left-invariant derivatives are non-commutative, e.g. ∂θ∂ξU 6= ∂ξ∂θU ,the Hessian matrix HU is not symmetric. In order to obtain real-valuedeigenvalues of a dimensionless matrix, we symmetrize the Hessian matrixvia HµU = Mµ(HU)TMµ2(HU)Mµ, and perform eigenanalysis on HµU .

116 E. J. Bekkers et al.

Eigenvector Mµc∗ with lowest eigenvalue of the dimensionless (µ-scaled) matrixHµU does not give the minimal principal curvature direction, but rather providesthe solution to the following optimization problem [8]:

c∗(g) = arg minc∈R3,‖c‖µ=1

{∥∥∥∥d

dt(∇U(γgc (t)))

∣∣∣∣t=0

∥∥∥∥2

µ

}, (4)

with left-invariant gradient ∇U = Mµ−2(∂ξU, ∂ηU, ∂θU)T and ‖c‖2µ = µ2|cξ|2 +

µ2|cη|2 + |cθ|2. Intuitively speaking, c∗ gives the tangent vector components ofthe exponential curve γgc∗(t), starting at position g, along which the left-invariantgradient has fewest variations (Fig. 2). Alternatively, in work by van Ginkel [14],tangent estimation was done based on the structure tensor. A full overview ofexponential curve fit models, with 3D extensions, can be found in [8].

2.3 Curvature, Confidence and Global Tortuosity Measures

In our implementation we use Gaussian derivatives to compute the Hessian ma-trix, i.e., in Eq. (3) we substitute U ← Gσs,σo ∗U , with Gσs,σo(x, θ) a Gaussiankernel with spatial isotropic scale 1

2σ2s , and orientation scale 1

2σ2o . For each tan-

gent component c∗(g) of the best exponential curve fit at location g, we computea curvature value κ(g) directly via Eq. (1). For each location we also determinea confidence measure s(g) based on blob detection via the Laplacian computedin the plane orthogonal to the tangent direction c∗(g) via

s(g) = (S(U))(g) = (−∆oU(g))+ =(−(eo1(g))TMµ2HU(g)Mµ2eo1(g)− (eo2(g))TMµ2HU(g)Mµ2eo2(g)

)+, (5)

where (v)+ = max{v, 0}, and Mµeo1(g) and Mµeo2(g) are two eigenvectors ofHµU orthonormal to Mµc∗(g) [9, ch. 5.3]. Improved accuracy of the confidenceand curvature measurements is achieved via the following stabilizing refinementscheme:

sn+1 = S(sn), with s1 = S(U), κn+1 = K(sn), with κ1 = K(U), (6)

where we denote the computation of the confidence map s from input volume U(using Equations (3)-(5)) with S(U), and the computation of curvature κ (using

Equations (3),(4) and (1)) with (K(U))(g) = c(g)θ sign(c(g)ξ)√|c(g)ξ|2+|c(g)η|2

. From the cur-

vature and measurement confidence functions we compute the following globaltortuosity features:

µ|κ| =1

stotal

∫ ∞

−∞

∫ π

0

|κ(x, θ)|s(x, θ)dxdθ (7a)

σ|κ| =

√1

stotal

∫ ∞

−∞

∫ π

0

(|κ(x, θ)| − µ|κ|)2s(x, θ)dxdθ, (7b)

with stotal =∫∞−∞

∫ π0s(x, θ)dxdθ. Features µ|κ| and σ|κ| give respectively the

weighted mean and standard deviation of absolute curvatures.

Curvature Based Biomarkers for Diabetic Retinopathy 117

3 Validation and Application to Clinical Data

The scales σs = 3 and σo = π18 are fixed in all experiments, and are chosen as

to best match the cross-sectional scales of vessels in the orientation score (Sub-sec. 2.3). We set µ = σo

σs, and sampled the orientation score with 18 orientations.

Validation. Our method was validated on two synthetic images (201px by201px) with Gaussian white noise (SNR=1): One image composed of three cir-cles with radii of 50px, 70px and 90px; One image composed of three crossingEuler spirals. The curvatures computed with our method (third and fourth col-umn Fig. 3), with n = 1 and n = 10 refinement iterations (Eq. (6)), werecompared against the ground truth (second column Fig. 3). In the curvaturemaps, curvature is encoded with color and confidence with opacity (see e.g. alsoFig. 2). Visual comparison shows a remarkable agreement between our methodand the ground truth, and we observe improved precision of both the confidenceand curvature measurements with an increasing number of refinement iterationsn. This is also confirmed by the comparison of curvature measurements againstthe ground truth via scatter plots (most right two figures in Fig. 3). The rootmean squared error of |κ| was reduced from 0.0138 for n = 1 to 0.0024 for n = 10.

Application to clinical data. Tortuosity measures µ|κ| and σ|κ| were computedon images of two publicly available databases: 1) the high resolution fundus(HRF) database [15], consisting of 15 images of healthy controls, and 15 images ofdiabetes patients; 2) the MESSIDOR database [16], consisting of 1200 images ofdiabetes patients which are graded for diabetic retinopathy: R0 (no retinopathy),R1, R2 and R3 (severe retinopathy). All images are made with 45 degree fieldof view (FOV) cameras, however with varying image resolutions. In order tohave approximately the same physical pixel size in all images, they are croppedand resized such that the FOV area spans a width of 1024px. Curvature andconfidence measures were computed with n = 3 refinement iterations.

Fig. 4 shows a selection of results. Fig. 5 and Tab. 1 show the distributionof feature values for different subgroups of the HRF and MESSIDOR database.Based on a Mann-Whitney U test (p-values reported in Tab. 1) we conclude thatall subgroups show a significant increase in µ|κ| and σ|κ| in comparison to thecorresponding base groups (healthy for HRF, and R0 for MESSIDOR).

We also observe that our method detects microbleeds and hemorrhages ashigh curvature regions (Fig. 4). While this was not our intention, it is a very wel-come property when using features µ|κ| and σ|κ| as biomarkers for DR. However,for research dedicated to the retinal vasculature one only wants to analyse bloodvessels. We plan to address this feature in future work via the construction ofvessel specific confidence measures (e.g. vesselness in orientation scores [13, 8]).

4 Conclusion

We developed new vessel tortuosity descriptors based on curvature estimationsfrom best exponential curve fits in orientation scores. Furthermore, a novel re-finement scheme was presented for more accurate curvature and confidence mea-

118 E. J. Bekkers et al.

Fig. 3: From left to right: input image (SNR=1), ground truth color-coded cur-vature map, measured curvature map with resp. n = 1 and n = 10 refinements,scatter plot of ground truth vs measured curvatures for resp. n = 1 and n = 10.

Fig. 4: Results on three images of the MESSIDOR database. Measured absolutecurvature |κ| encoded in color, and confidence s encoded with opacity, overlainon the original image, together with the histogram of measured |κ| values.

Fig. 5: Box-and-whisker plots oftortuosity measures µ|κ| andσ|κ| in subgroups of the HRFand MESSIDOR database.

Table 1: Tortuosity measures µ|κ| and σ|κ| inthe HRF and MESSIDOR database.

Subgroup Mean ± (STD)µ|κ| (10−2) σ|κ| (10−2)

———————— HRF ————————Healthy 1.372 ± (0.069) 1.796 ± (0.072)Diabetic 1.521 ± (0.130) 2.073 ± (0.185)

p-valuea < 0.001 < 0.001——————– MESSIDOR ——————

R0 1.624 ± (0.120) 2.333 ± (0.134)R1 1.657 ± (0.124) 2.365 ± (0.131)

p-valueb 0.007 0.020R2 1.698 ± (0.122) 2.436 ± (0.144)

p-valueb < 0.001 < 0.001R3 1.795 ± (0.160) 2.674 ± (0.235)

p-valueb < 0.001 < 0.001a Compared to Healthy.b Compared to R0.

Curvature Based Biomarkers for Diabetic Retinopathy 119

sures. Validation on synthetic images showed high accuracy of our curvature ex-traction approach. Application to clinical retinal image datasets showed strongpositive associations of the proposed tortuosity descriptors with diabetes anddifferent stages of diabetic retinopathy (DR). As such, we see high potential ofthe method to be used in a screening setting for diabetes and DR detection.

Acknowledgements: This work is part of He Programme of Innovation, which is (partly) fi-

nanced by the Netherlands Organisation for Scientific Research (NWO). The research leading to

these results has also received funding from the ERC council under the EC’s 7th Framework Pro-

gramme (FP7/2007–2013) / ERC grant agr. No. 335555.

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