# Mathematics for Ophtalmology

by Julia Shatokhina

Optical coherence tomography (OCT) is a powerful interferometric technique providing non-invasive 3D imaging of sub-surface structures with resolutions on the micrometer scale. Significant developments of the OCT techniques in the recent decades were observed especially in biomedical applications. Thus, in ophthalmology the OCT imaging technique has revolutionized the possibilities of in-vivo examination of the eye. Nowadays it is widely applied for high-resolution 3D in-vivo imaging of the retina cells and supports early diagnostic and treatment decisions in the eye health care.

In order to be able to image the retina cells with the highest possible resolution, one needs to focus the imaging light in the retina at best. Before the imaging light reaches the retina plane, it actually propagates through the whole eye medium where the wavefront gets distorted. So, if we send a plane wavefront to the eye and hope to get a perfect tight focus in the retina plane, we simply fail. What one has to do instead, is reshape the incoming wavefront in such a way that after propagation through the eye the beam converges in a nice focal point. This is the aim of Adaptive Optics (AO) – a hardware based technique allowing one to reshape the wavefronts in real time applications.

A part of light returning from the eye is directed to a wavefront sensing device. This provides intensity data that serve as an indirect measurement of the wavefront. Restoration of the wavefront itself requires development of a forward mathematical model of the wavefront sensor as well as its subsequent inversion. These steps involve knowledge from various fields like optical theory, digital signal processing, inverse problems and regularization theory as well as computational complexity theory. Typically, in real-world application, the problem of wavefront reconstruction has to be solved continuously with the sensor data flow (in a loop), each step within a few milliseconds. Algorithm optimization is crucial here. Finally, the reconstructed wavefronts are applied to a deformable mirror in the imaging path, and the pre-corrected light enters the eye. This is the concept of the AO-OCT in ophthalmology.

In parallel to software optimization, overall performance of AO-OCT setup can also be boosted by the hardware improvements. For instance, instead of the commonly applied Shack-Hartmann sensor (SH-WFS), we go for a relatively newer pyramid sensor (P-WFS), which possesses a great benefit of being significantly more sensitive. The limitations imposed in ophthalmological AO-OCT on the laser powers that are allowed to be sent to the eye require longer exposure time for reasonable signal-to-noise ratios. Here the increased sensitivity of the pyramid sensor compared to SH enables an impressive reduction of the exposure time by the order of magnitude. Application of the pyramid sensor together with the mathematical software optimization result in a boosted in-vivo AO-OCT performance with unprecedented correction bandwidth.

More on the exciting results can be found in preprint: Elisabeth Brunner, Julia Shatokhina, et al: Adaptive optics imaging with a pyramid wavefront sensor for visual science