Visual results from our new algorithm


To test the algorithm, we need to run it on surfaces whose shape is accurately known. Unfortunately, it is difficult to manufacture interesting test cases. For this reason, we have tested the algorithm on both data collected from real objects and data generated synthetically. The synthetic data is generated automatically from various surface definitions by a software simulation of the videokeratograph.

These images show some frames taken from our real-time shape analysis system. The results of the search process are displayed to the user after each iteration. As mentioned before, a good approximation to the final answer is reached in a few seconds, so the user can start analyzing the results immediately. A more accurate picture evolves over the next few minutes.

References for this algorithm (other OPTICAL papers)

Available information

clicking on any picture will bring up a larger one
 * Simulation of the algorithm on an ellipsoid
 * Simulation of the algorithm on a keratoconic surface
 * Results of the algorithm on real data

Simulation of the algorithm on an ellipsoid

Here we have four frames showing the patches converge to a solution for a simple ellipsoid with radii of 8mm, 9mm, and 10mm. For illustration purposes, the exact ellipsoid is also shown. The surface colors indicate the distance between the current solution and the exact ellipsoid. In the final frame, we can see that good convergence has been achieved. We measure the error as the distance in z between the two surfaces, computed at a large number of sample points in the x,y plane. The image data for this example was synthetically generated. The mean error in z for this example was 9.2 x 10^{-6}mm, which is 0.0092 microns. This extremely high accuracy is typical of all synthetic data sets we have tried.

Simulation of the algorithm on a keratoconic surface

These are frames from another run of the algorithm. The data set in this example is also synthetically generated. The aim here is to simulate keratoconus, which is a corneal condition in which there is local region of high curvature. The surface is generated from a sphere with a rotationally symmetric bump grafted onto it. The bump and the sphere meet with curvature continuity. The curvature at the peak of the bump is significantly greater than the curvature of the sphere. This situation has not been handled very well by the existing measurement techniques. Our algorithm, however, has no difficulty in finding an accurate solution. In the images below, the coloring indicates the separation between the surface and a sphere of the same radius as the initial test surface. This illustrates one of our scientific visualization techniques, which is to display the surface separation from a best-fitting ellipsoid. This enhances the deviations so that the bump, which is positionally very close to the sphere, becomes noticeable. In this example, we get extremely high positional accuracy of 0.013 microns.

Results of the algorithm on real data

Finally, these images illustrate the results of the algorithm run on some real data taken from a cornea. In this case, we cannot report accuracy information because the true shape is unknown. However, we are able to render it with our in-house visualization software package. The six images for this figure highlight different representations of the same dataset. For the first image, we have applied a colormap of Gaussian curvature and encoded the height information as the true height of the 3-D surface. The bright red spot in the lower-left represents a local area of high Gaussian curvature. For the second image we have kept the same color representation but changed the encoding of the height to correspond to the Gaussian curvature; the bright red spot in the previous picture now represents a mountain top in this image. We come full circle in the third image and keep the same height encoding of Gaussian curvature but change the color to correspond to the original, true height information. This image highlights the power of the curvature to convey subtle changes in shape; the orange ring of equal altitude travels right through the large local maxima of high curvature.

Images four through six are exactly the same as one through three, with a vector-field applied to the surface. The vectors correspond to the direction of minimum curvature at each point on the surface. For ruled-surface areas of gaussian curvature surrounding high local curvature minima and maxima, these vector field lines appear to track the way water would run off of the surface. This is most apparent in image 5, which has this 'water-flow' effect surrounding the large, steep peak of maximal gaussian curvature.

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