1.

Show that the orientation ? of an object f( x, y) in the field of view of an image can be determined from the expression

where m _pq are the ordinary image moments:

2.

Given the Radon transform F _R( x ?, ?) of an object where x ? is related to x by the rotation matrix R, derive an expression for the moments of F _R( x ?, ?) given by

where m _{n ? i, i} are the Cartesian moments of the original function f( x, y). Invert this result to express the Cartesian moments in terms of the Radon transform moments and show

From these results it is clear that only four projections are needed (at ? = 0, ?/4, ?2, 3 ?/4) to compute the Cartesian moments. Discuss the relative computational requirements for calculating Cartesian moments directly versus by Radon moments.

3.

Derive the expression for the angular autocorrelation of a rectangular pillbox and show that it is given by

where w is width, l is length, and x ? = R x. What happens at l = w? Plot this function for l = 4 w, 2 w, and w. Show how the values R(0) and R( ?/2) can be used to calculate the area, aspect ratio, length, and...