A new notion of discrete tangent, called order d discrete tangent,
adapted to noisy curves, is proposed. It is based on the definition of
discrete tangents given by A. Vialard in 1996, on the definition of
fuzzy segments and on the linear algorithm of fuzzy segments
recognition. The algorithm calculating the order d discrete tangent at a point of a curve relies on simple calculations and is linear according to the number of points of the obtained tangent. From the definition of an order d discrete tangent, we deduced an estimation of the normal vector and of the curvature at a point of a discrete curve for a given order d.
Let a 26-connected, bounded subset of Z3, such that the projection in the plane Oxy is one to one. We present an incremental algorithm which determines if this subset is or is not a piece of digital plane. This problem is solved thanks to the diophantine definition of digital planes and by using only 2D geometrical constructions.
The aim of this work is to increase the precision of the computations involved in brain tumor stereotactic radiosurgery. It is proposed to apply the newest algorithm of contour segmentation to the contours of the MR/CT slices. It is shown that this algorithm improves significantly the accuracy and speed of the classical contour segmentation algorithms and produces a one-to-one transformation between the Euclidean space and the discrete pixel space. For the distance computations involved in dose distribution computation a subpixel precision is obtained. The segmentation algorithm computes the exact equation of the discrete lines forming a pixel contour of an object (the skull, a tumor, etc.). Since it actually computes the equation of the lines forming the contour and not an approximated line like most algorithms, the algorithm is reversible. From the segmented contour the pixel contour can be exactly reconstructed without loss or displacement of any pixel. The segmentation algorithm is quick since it works in linear time. The present application involves the computation of dose distribution for stereotactic radiosurgery. The intersection between the ray and the skull, the sinus cavities and the tumor can be computed with subpixel accuracy. This obviously improves the dose distribution computation. There are many other applications for this algorithm, for example segmentation, slice reconstruction or simply better rendering of anatomical information, etc. The advantage of this segmentation algorithm over classical approaches lies not only in the results presented here, but also in the fact that a 3D extension should be available soon. This strongly suggests the building of a real 3D planning system, simplifies the inverse planning problem and increases the precision of the computations involved in stereotactic radiosurgery.
The difficulties met in the study of digital planes come from a bad definition. We introduce a diophantine point of view, which, despite its formal appearance, is much more convenient and fits perfectly with the analogous notion of digital lines. We give algorithms for the drawing of digital lines and planes and solve the problem of their intersection.