Conceptually, we can consider modern videokeratoscopes as a derivative of ophthalmometers , so much so that the first method developed in this regard is based on the same principles of the ophthalmometer [5]. Indeed, we can consider a topographic measure as many keratometries with ever larger fixed mires, centered on the same axis. The corneal curvature so measured is called the sagittal (or axial) curvature and can be defined as the radius of an arc of a circle, centered on the optical axis of videokeratoscope (or the keratometer) when this is well aligned to the corneal vertex , which has the same tangent as the cornea to the point concerned, as shown in Fig. 6.2 on the left. In the branches of topography this measure was carried out essentially for comparison: once the device has been calibrated by acquiring spheres (calibration spheres ) with known curvature, the curvature of a sample is estimated by comparing the position of the mires reflected by its surface and the ones obtained from the calibration spheres [6].

The curvature as defined earlier suffers from a low sensitivity for determining shape variations localized in the peripheral cornea. A map of sagittal curvature expressed in diopters, while not being the direct expression of the real refractive power, can represent quite well the optical characteristics of the corneal surface, but it is not able to accurately describe the morphological aspects of the cornea. For this reason, it is felt necessary to support the sagittal curvature with magnitudes which best show the details of the shape of the corneal surface.

The tangential curvature (sometimes called instantaneous or local) is the geometric unit that describes the steepening of the cornea, for a given point and in a given direction: it is therefore defined as the radius of the osculating circle for each point belonging to the meridian in question, as shown in Fig. 6.2 on the right. Unlike the sagittal curvature, the center of tangential curvature is not constrained to lie on a reference axis: this condition does not occur unless the cornea is not exactly spherical, and the tangential and the sagittal curvatures are different at every point and coincide only at the vertex. The tangential curvature of a surface is independent of the position of the reference axis along the meridian considered: its sagittal curvature is instead a function of this axis.

Topographic maps that represent elevation data are probably more understandable, since they are similar to those used in geographic cartography to describe the elevations of the Earth’s surface or the depth of the seabed. The corneal elevation maps are expressed in microns [μm] the position of each point of the cornea, in terms of the elevations along the z axis, with respect to a reference surface. As in geographic maps, where the reference surface is the mean sea level (the geoid), the elevation of the corneal surface needs to be expressed in respect to some reference surface: usually spherical, aspherical, or asphero-toric. The reference surfaces, with respect to which the height is measured, are almost always the best-fit surfaces (best approximating surfaces in the sense of the least square error) and have some parameters that characterize them: the spheres are defined by their radius of curvature, the aspherical surfaces by the apical radius of curvature and eccentricity, the asphero-toric surfaces by the apical radius of curvature, eccentricity, and by toricity, that is, the difference of curvature of the flattest apical meridian and the steepest one. If even one of these parameters is changed, it alters the reference surface and consequently the relative heights of the cornea under examination with respect to it. For this reason, the parameters defining the reference surface are critical to the interpretation of the height map.

In recent generation reflection-based topographers, tangential curvature, sagittal curvature, elevations, and normals to the surface are calculated simultaneously by a process called arc step [7–10]. For videokeratoscopies, the definition of this algorithm marked a real turning point, expanding the range of measurements possible, from only the sagittal curvature to all the morphological or refractive measurements arising from the shape of the cornea.

The arc-step algorithm is summarized as follows: for each point corresponding to the edge of a ring of each meridian, a ray-tracing procedure is carried out, which, from the known positions of the rings on the Placido disk and the detected positions of the keratoscopy rings, allows to derive normals of the surface to be measured. Such information depends on the correct positioning of the surface to be measured and the exact knowledge of the location of the rings of the Placido disk : the first condition is obtained by different methods (maximum sharpness algorithms on a focusing video stream, triangulations, or interruptions of light beams), while the second is known from construction data and is refined by means of instrument calibration. Once the normals to the corneal surface are known (and therefore its derivatives) an iterative algorithm is started. At step 0, the condition of zero derivative in correspondence of the corneal vertex is imposed and, setting the validity of the law of reflection for the first ring, an arc that meets the slope conditions is identified. The algorithm of arc step, as the name suggests, is iterative, and therefore cannot proceed without the knowledge of data obtained in the previous step (step i – 1) to derive data for the current step (step i). Knowing the coordinates and the normal of (i – 1)th point, and the normal to the ith point, the coordinates of the new point are determined, by setting conditions of differentiability (surface continuity and continuity of tangency). The reconstructed curve, repeating the pattern shown earlier for all the rings is continuous with continuous first derivative. The connecting functions are circular arcs in the traditional algorithm but can also be polynomial splines or conic arcs (by virtue of their greater flexibility in pursuing the corneal slope). There are essentially two limits in these reconstructions methods and are negligible compared to the advantages they offer:

The optical scanning instruments directly measure the sagittal height of the front surface and the back of the cornea (i.e., the corneal elevation with respect to a reference plane). Usually, the transverse field of view goes from limbus to limbus and the vertical is such as to include the entire anterior chamber limited by the iris and the visible part of the lens. The big advantage of this class of instruments is therefore, in addition to the high transverse coverage, the ability to view and measure the whole anterior chamber and all its surfaces (front cornea, posterior cornea, iris, and lens).

The optical scanning devices base their operation on the projection of beams or light slits which, due to scattering effects, diffuse in every direction part of the energy of the incident light. Therefore, thanks to the scattering, the tissue crossed behaves like a real extended emitter: part of the back-diffused light radiation is captured by a suitable optics able to form the image on the CCD.

The measurement of the two front and rear corneal surfaces allows agile deduction of pachymetry information: it is the measurement in microns of the distance between the front surface and posterior surface in the direction normal to the anterior corneal surface. It is very useful information from the clinical point of view: it can allow, for example, to assess the evolution of keratoconus or to decide whether a patient may undergo refractive surgery and what can be the maximum amount of the bearable correction of the cornea. In contact lens practice, differential pachymetry can also highlight possible edematous conditions due to the wearing of contact lenses.

The simplest and most intuitive way to obtain the image of a corneal section is to project a luminous slit along a meridional plane of the cornea and to shoot the backscattered light through suitable optics, having its axis of at 90° with respect to this plane and its focus on the plane of the slit (Fig. 6.3 left). This configuration, however, in practice does not work because of obstacles to the path of the shot such as the nose or eyelids. These obstacles force the choice of a nonorthogonal axis of shot to lighting plane and consequently to have a good part of the image out of focus. The Scheimpflug method (see Fig. 6.4) is useful in this case (or Carpentier, or should we say by Scheimpflug’s [12] own admission) that shows how to tilt the CCD in order to focus a plane inclined to the axis of the optical system shoot, as shown in Fig. 6.3 on the right. Once scanned the first image, the light blade and the Scheimpflug system can be rotated in order to acquire images of a succession of equidistant angular sections that allow to completely map the anterior segment of the eye. The anterior corneal surface is determined by identifying its edge on the image in section and correcting the distortion effect due to the optical configuration. The internal structures (posterior corneal surface, iris, angles) are determined by identifying the corresponding edges on images in section, correcting the effect of distortion due to the optics in Scheimpflug configuration and correcting the distorting effect given by the fact that they are seen through the overlying corneal surfaces.

Until now, optical tomography coherence has been applied with great success in ophthalmology for the study of the retina and for the measurement of the ocular distances, firstly of the axial length. Only recently industries have turned their attention to this for topography and tomography of the anterior segment of the eye. The reasons for this delay are to be found mainly in the complexity of this technology that is reflected at least for now in high prices of products, in their robustness, in the difficulty of obtaining images that contain the anterior segment in its entirety, and in the difficulty of obtaining accurate measurements comparable to those typical of the simplest techniques in spite of the promising quality of the images.