1. Generic Grey Value Functions and the Line of Extremal Slope Joshua Stough MATH 210, Jim Damon May 5, 2003

2. Motivation: determine properties of the edge line for generic grey value surfaces G(x,y) = (x, y, g(x,y)); image graph g(x,y) = Previous work on describing and detecting edge lines uses idealized/degenerate models of g. Mathematical approach: determine properties of graph of generic smooth g. Lens distortion, noise conceivably lead to generic g.

3. Outline Definitions Generic properties of the edge line  Generic properties of the evolving  under linear diffusion The hypersurface of extremal slope, and selected results

4. Canny edges  H(  g),  g  = 0,  g  0

5. , P,  g

6. Generic properties of   contains points only of negative Gaussian curvature  meets P tangentially at isolated points and  and P are smooth at these points. The level curves at   P are tranverse to both. The only singular points of  are tansverse double points corresponding to Morse saddle points of G  g does not meet P and intersects  transversely at isolated points  has isolated curvature extrema corresponding to A 3 circles of curvature

7. Outline Definitions Generic properties of the edge line  Generic properties of the evolving  under linear diffusion The hypersurface of extremal slope, and selected results

8. Generic Evolutions of  on families of diffused greyvalue surfaces If f is analytic on a domain U, then a point z 0 on the boundary  U is called regular if f extends to be a analytic function on an open set containing U and also the point z 0 (Krantz 1999, p. 119). Basically, z 0 fits (is consistent with) its surroundings.

9. Edge line evolution for coronal CT scan Sigma = sqrt(2*t)

10. Morse saddle stability implies  stability g(x,y) = x^2 - y^2 + t*x*y^3 (H.O.T) , w/o h.o.t,  H(  g),  g  = 8x 2 – 8y 2 = 0  x =  y P, w/ h.o.t, det(H) = 0  x = (4 + 9*t 2 y 4 ) / (12*ty) P not on pure Morse saddle

11. Evolution of the Edge Line in Forming a rhamphoid cusp: g t = x^2 + 6*t*y + y^3 + 2*t

12. The hypersurface of extremal slope    is a hypersurface with isolated singular points. The generic geometry of   (  \ {x   :  g = 0}) (punctured set) is the same as a general hypersurface (  without the closure?). At singular points of g of type A k,  has A 3k-2 points (  has non-simple critical points at D k  4, E 6,7,8 points of g.) Generically (codim 0)  has only isolated A 1 points at A 1 points of g. In codim 1,  can have A 1 points at regular points of g, and  can also have A 4 points at A 2 points of g.