1. The static formulation of the original snake model is given by the minimization of the energy functional

E : Ad im, E (c) = Ex(c (s)) + E2(c(s)), defined in Section 7.2.1. Supposing that c e C4, show that the Euler-Lagrange equations become:

- (wlc'(s))' + (w2c"(s))" + VP(c(s)) = 0.

2. Discuss the effect of the parameters w\ and w2 over the original snake model in exercise 1 by using the following equations for a curve c:

where a is the arc length, K is the curvature, and ~T and are the unitary tangent and normal vectors, respectively.

3. Show that the original snake model is not invariant under affine transformations given by the general form:

4. Discuss the role of the characteristic function for the T-surfaces model.

5. Let us consider a characteristic function in f : ft2 i{0,1} defined over a CF triangulation of ft2. In this case, given a triangle, it can be verified (do it as an exercise) that it has exactly two transverse edges or it does not have transverse edges. Based on this property, write a pseudocode for an algorithm to generate the polygonal curves, after computing the intersections with the triangulation (see Section 7.2.3).

6. Would it be possible to design a T-surfaces model based on a cellular decomposition of the image domain? What would be the advantages over the traditional T-surfaces?

7. Choose a gray scale image, binarize it applying several values of thresholds. Later, with the same initial image, apply the following sequence of operations and compare the results: Canny's edge detector of thresholds 30 and 80; invert the result; apply over the result the erosion operation with a cross structuring element. Observe the isolated regions with other values of thresholds of your choice.

8. Choose a binary image, apply the following sequence of operations and describe the net effect (B is the structuring element of your choice):

(d) XB = (X © Bob )/(X © Bbk), where Bob is the set formed from pixels in B that should belong to the object, and Bbk is the set formed from pixels in B that should belong to the background.

9. Considering the implicit representation of a curve, G(x, y) = 0, show that the normal it and the curvature K can be computed by:

respectively, where the gradient and the divergent (V-) are computed with respect to the spatial coordinates (x, y).

10. Take the anisotropic diffusion scheme (see Section 7.8):

Show that if ||V11| < T, the edges are blurring and if ||V11| > T they become sharper.

11. Let us suppose hand gas constants in the GVF model given by the equation:

Consider the stationary solution and take the Fourier transform of the corresponding stationary equation to analyze the GVF in the frequency space.

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