Classes of families of algebraic curves. Local study of a plane algebraic curve. Catalogue of curves: notable examples (cubics, rational quartics and special curves of higher degree).
Hough transform I. Hough transform (HT) for families of algebraic plane curves. Regularity property. Examples. Recognition algorithm.
Hough transform II. Deepening on the notion of Hough regularity: the problem of finite intersections of HT. Examples.
Hough transform III. Deepening on the notion of Hough transform for curves and surfaces in the three-dimensional space.
An approach to the 3D case. The Chinese Remainder Theorem with application to the reconstruction of surfaces equations.
Applications. Algorithmic approach to problems of detection of curves in images by mean of the Hough transform. Lessons (given by the tenured professor of the course Elaborazioni di Immagini) concerning the detection of some profile of interest in either astronomical or medical field.
Possible developments:
1) Adjonts curves, birational models of plane curves, the Riemann--Roch theorem, linear series on curves.
2) Problems on ``almost-vanishing" polynomials.
3) Spline curves, also in relation with the Hough transform technique.
4) Extension to the Hough transform technique to the non-algebraic case: some special cases.