Problem 1 For all items, c is a constant. Each item specifies a disc or ellipse E. For each item, Xe(x, y) = c if (x, y) is in Ɛ, whereas xe(x,y) = 0 if (x, y) is not in E. For each item, find an algebraic formula for [Rxe] (r, a) in terms of r, a, c, and parameters defining E. (1.1) For this item, & is the disc with radius S centered at (20, 30): where (x-xo)2 + (y-yo)2 <S². (1.2) For this item, & is the ellipse centered at 0 where (r2/a2) + (y2/62) ≤1. (1.3) For this item, & is the ellipse centered at 0 with principal semi-axes of lengths a in the direction [cos(7), sin(7)] and b in the direction [-sin(7), cos(7)]. (1.4) For this item, & is the ellipse centered at (zo, yo) where [(x-xo)2/a2] + [(y-yo)2/62] ≤1. (1.5) For this item, & is the ellipse centered at (xo, yo) with principal semi-axes of lengths a in the direction (cos(7), sin(y)] and b in the direction [-sin(y), cos(7)]. Definition 1 Denote by Lr,a the straight line that is perpendicular to the direction [cos(a), sin(a)] and at distance r from the origin 0=(0,0). Thus (x, y) is on the line Lr,a if and only if x cos(a)+y.sin(a) = r. Common choices are re R and 0 < a <. Another potential choice might be r≥ 0 and - <a≤x. Remark 2 The line Lr,a is a distance r from (0,0) in the direction perpendicular to [cos(a),sin(a)]. Consequently, the point r (cos(a), sin(a)] is on Lr,a The line Lr,a is perpendicular to [cos(a), sin(a)] and therefore parallel to the unit vector [-sin(a), cos(a)]. Therefore, the line Lr,a may be parametrized by its arc length with X(s) Y(s) = T. · cos(a) T. sin(a) + -8 sin(a)]. 8. cos(a) Definition 3 The Radon Transform of a function f at (r, a) is the integral of f along Lr.a Rf)(r,a)=fds. (1) (2) Computed Tomography (CT) scanners measure [R](r, a). Hardware and software reconstruct f(x, y). Example 4 Denote by D the disc with radius S centered at the origin: where x2 + y2 S². Define a function x ("chi") by x(x, y) = 1 if (x, y) is in D, whereas x(x, y) = 0 if (x, y) is not in D. If r>S, then the line Lr,a is too far from 0 to intersect D; thus, [Rx](r, a) = 0 for all |r|> S and a. If r S, then the line Lr,a intersects D along a segment of length 2√√S2-2. Therefore, Rx](r, a) = xds = { 2√2- 0, r> S. Choose either theoretical Problem 1 or computational Problem 2. (3)