Erik J Bekkers

Assistant professor at University of Amsterdam



The thesis can be downloaded here (48 MB). A compressed version here (8 MB)


Clinical/industrial applications

I did my PhD in mathematical/medical image analysis with the thesis "Retinal Image Analysis using Sub-Riemannian Geometry in SE(2)". The thesis describes algorithms for the automatic and semi-automatic analysis of retinal images. These retinal images are obtained via non-invasive optical cameras that image the inside of the eye. The goal for developing automated retinal image analysis algorithms was twofold:

  1. To facilitate large scale clinical studies aimed at discovering new biomarkers and monitoring disease progression through observations in the eye.
  2. To facilitate computer assisted diagnosis/referral systems.
The developed algorithms involve a mixture of differential geometry, PDEs, and machine learning. The research was conducted in collaboration with industry (part-time appointment at i-Optics - EasyScan BV, the Hague), clinic (Uiversity Eye Hospital Maastricht and He Eye Healthcare, Shenyang, China), and academic partners.

Brain-inspired mathematics

In my thesis I took Inspiration from the findings of Hubel and Wiesel [1] who found that in the first processing layer (V1) of the visual cortex information is organized based on positions and orientations. As such, in my algorithms I analyze image data via densities on position-orientation space ℝ2×S1 that are obtained by filtering the image with filters that model the receptive fields of simple cells in V1 (Fig. 1A-C). In these new objects, the data lives on a non-Euclidean (curved) geometry of coupled positions and orientations. Via a sub-Riemannian geometry on the Lie group SE(2)≡ ℝ2⋊S1 one is able to model the perception of contours via sub-Riemannian geodesics [2,3], and generically deal with crossings as they are neatly disentangled in the higher dimensional representation (Fig.~1D-E).

Theoretical contributions

The academic focus of my past and present research lies within the overlapping fields of mathematics, computer science, and biomedical engineering. Throughout my thesis I employed a coherent mathematical (Lie) group theoretical approach to work on fundamental engineering problems in medical image analysis. The field of group theory studies a wide variety of mathematical entities using their symmetrical properties in a generic uniform manner. The thesis is primarily concerned with the processing and analysis of data that lives on the symmetry group SE(2) of planar rotations and translations. The group theoretical approach allows for algorithm design at a certain level of abstraction, allows us to work with simplified formulas (which would otherwise be intractable) and allows for natural extensions to other groups (e.g. from SE(2) to SO(3) as in [4] or SE(3) as in [5,6].

In the thesis I describe both practical results (fast, robust, and generic algorithms and significant clinical findings on systemic disease related vessel alterations), as well as new theoretical contributions such as:

  1. A new framework for computing data-adaptive sub-Riemannian geodesics in SE(2) [7,8,9,10,6].
  2. Object recognition via densities on SE(2) using a combination of smoothing splines, machine learning and hypo-elliptic smoothing priors [11,12,13,14].
My current research builds upon both main results and has a specific focus on machine learning. My aim is to develop learning algorithms which employ the geometric structure and symmetries in the data using a group theoretical approach (directly extending item 2). Apart from an embedding of my research in the field of machine learning, I aim to place my research in the context of cortical modeling of the primary visual cortex as well (in line with item 1).


The approach of image analysis via a sub-Riemannian geometry on SE(2) enabled the development of the following applications, each of which show state-of-the-art performance in extensive benchmark comparisons:

  1. Anatomical landmark detection via template matching, and template optimization in SE(2).
  2. Crossing preserving vessel enhancement via left-invariant processing of orientation scores.
  3. Vessel tracking and segmentation via local curve optimization.
  4. Vessel tracking via globally optimal sub-Riemannian geodesic extraction in SE(2), where we show clear benefits of our sub-Riemannian framework compared to the Riemannian counterparts in the image/orientation scores.
  5. Vessel geometry analysis and biomarker extraction by direct analysis of orientation scores.

Figure 1. A: The V1 region of the visual cortex contains orientation sensitive cells whose receptive fields (RFs) resemble line detectors. B: Their orientation preference varies smoothly over the brain's surface; pinwheel regions locally encode for all orientations (see color-coding). C: We similarly represent images by lifting the data (using a transformation 𝒲ψ to densities on ℝ2⋊S1 filtering with simple cell-like RF filters >ψ; now crossings are disentangled. D: In a Euclidean geometry on ℝ2×S1 the distance between the green and red arrow to the origin is the same; in a sub-Riemannian (SR) geometry on SE(2)≡ ℝ2⋊S1 the green arrow is closer; the solid curves are SR geodesics. E: With an SR geometry one can quantify the perceptual notions of alignment and robustly model/extract blood vessel trajectories.


  1. D. H. Hubel and T. N. Wiesel. "Receptive fields of single neurones in the cat's striate cortex". In: The Journal of Physiology 148.3 (1959), pp. 574-591.
  2. J. Petitot. "The neurogeometry of pinwheels as a sub-Riemannian contact structure". In: Journal of Physiology-Paris 97.2 (2003), pp. 265-309.
  3. G. Citti and A. Sarti. "A cortical based model of perceptual completion in the roto-translation space". In: Journal of Mathematical Imaging and Vision 24.3 (2006), pp. 307-326.
  4. A. P. Mashtakov, R. Duits, Y. L. Sachkov, E. J. Bekkers, and I. Beschastnyi. "Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in SO(3)". In: Journal of Mathematical Imaging and Vision 58.2 (2017), pp. 239-264.
  5. E. J. Bekkers, D. Chen, and J. M. Portegies. "Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D". In: Journal of Mathematical Imaging and Vision (2018).
  6. R. Duits, S. P. L. Meesters, J.-M. Mirebeau, and J. M. Portegies. "Optimal Paths for Variants of the 2D and 3D Reeds{Shepp Car with Applications in Image Analysis". In: Journal of Mathematical Imaging and Vision (Feb. 2018).
  7. E. J. Bekkers, R. Duits, A. P. Mashtakov, and G. R. Sanguinetti. "A PDE Approach to Data-Driven Sub-Riemannian Geodesics in SE(2)". In: SIAM Journal on Imaging Sciences 8.4 (2015), pp. 2740-2770.
  8. E. J. Bekkers, R. Duits, A. P. Mashtakov, and G. R. Sanguinetti. "Data-Driven Sub-Riemannian Geodesics". In: Scale Space and Variational Methods in Computer Vision. Ed. by J.-F. Aujol, M. Nikolova, and N. Papakadis. Lecture Notes in Computer Science. Springer, 2015, pp. 613-625.
  9. E. J. Bekkers, R. Duits, A. P. Mashtakov, and Y. L. Sachkov. "Vessel Tracking via Sub-Riemannian Geodesics on the Projective Line Bundle". In: International Conference on Geometric Science of Information. Springer, 2017, pp. 773-781.
  10. G. R. Sanguinetti, E. J. Bekkers, R. Duits, M. H. J. Janssen, A. P. Mashtakov, and J.-M. Mirebeau. "Sub-Riemannian Fast Marching in SE(2)". In: Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications. Springer International Publishing, 2015, pp. 366-374.
  11. E. J. Bekkers, M. Loog, B. M. ter Haar Romeny, and R. Duits. "Template matching via densities on the roto-translation group". In: IEEE transactions on pattern analysis and machine intelligence 40.2 (2018), pp. 452-466.
  12. E. J. Bekkers, M. W. Lafarge, M. Veta, K. A. J. Eppenhof, J. P. W. Pluim, and R. Duits. "Roto-translation covariant convolutional networks for medical image analysis". In: arXiv preprint arXiv:1804.03393 (accepted at MICCAI 2018). 2018.
  13. E. J. Bekkers, R. Duits, and M. Loog. "Training of Templates for Object Recognition in Invertible Orientation Scores: Application to Optic Nerve Head Detection in Retinal Images". In: Energy Minimization Methods in Computer Vision and Pattern Recognition. Ed. by X.-C. Tai, E. Bae, T. Chan, and M. Lysaker. Vol. 8932. Lecture Notes in Computer Science. Springer International Publishing, 2015, pp. 464-477.
  14. E. J. Bekkers, R. Duits, and B. M. ter Haar Romeny. "Optic Nerve Head Detection via Group Correlations in Multi-orientation Transforms". In: Image Analysis and Recognition. Ed. by A. Campilho and M. Kamel. LNCS. Springer, 2014, pp. 293-302.