Generic Grey Value Functions and the Line of Extremal Slope

Joshua Stough

MATH 210, Jim Damon

May 5, 2003

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.

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Outline

•Definitions

•Generic properties of the edge line

•Generic properties of the evolving under linear diffusion

•The hypersurface of extremal slope, and selected results

Canny edges

H(g), g = 0, g 0

, P, g

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 A3 circles of curvature

Outline

•Definitions

•Generic properties of the edge line

•Generic properties of the evolving under linear diffusion

•The hypersurface of extremal slope, and selected results

Generic Evolutions of on families of diffused greyvalue surfaces

If f is analytic on a domain U, then a point z0 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 z0 (Krantz 1999, p. 119). Basically, z0 fits (is consistent with) its surroundings.

Edge line evolution for coronal CT scanSigma = sqrt(2*t)

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 = 8x2 – 8y2 = 0 x = y

•P, w/ h.o.t, det(H) = 0 x = (4 + 9*t2y4) / (12*ty)

•P not on pure Morse saddle

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

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 Ak, has A3k-2 points ( has non-simple critical points at Dk4, E6,7,8 points of g.)

•Generically (codim 0) has only isolated A1 points at A1 points of g.

• In codim 1, can have A1 points at regular points of g, and can also have A4 points at A2 points of g.