The following four statements are true. Advance an argument that establishes the reason(s)...

The following four statements are true. Advance an argument that establishes the reason(s) for their validity. Part (a) is true in general. Parts (b) through (d) are true only for digital sets. To show the validity of (b) through (d), draw a discrete, square grid (as shown in Problem 9.1) and give an example for each case using sets composed of points on this grid. (Hint: Keep the number of points in each case as small as possible while still establishing the validity of the statements.)

(a) The erosion of a convex set by a convex structuring element is a convex set.

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(b) The dilation of a convex set by a convex structuring element is not necessarily always convex.

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(c) The points in a convex digital set are not always connected.

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(d) It is possible to have a set of points in which a line joining every pair of points in the set lies within the set but the set is not convex.

9.1 Digital images in this book are embedded in square grid arrangements and pixels are allowed to be 4-, 8-, or m-connected. However, other grid arrangements are possible. Specifically, a hexagonal grid arrangement that leads to 6-connectivity, is used sometimes (see the following figure).

(a) How would you convert an image from a square grid to a hexagonal grid?

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(b) Discuss the shape invariance to rotation of objects represented in a square grid as opposed to a hexagonal grid.

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(c) Is it possible to have ambiguous diagonal configurations in a hexagonal grid, as is the case with 8-connectivity? (See Section 2.5.2.)