Question

Definition 1 Denote by Lra the straight line that is perpendicular to the direction [cos(a), sin(a)] and at distance r from the origin 0 (0, 0). Thus (r, y) is on the line Lr.a if and only if r cos (a) y sin(a) = Common choices are r E R and 0 a<T. Another potential choice might be r 2 0 and -T < a < T. Remark 2 The line Lra 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 Lra 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 Lra may be parametrized by its arc length with r. X(s) Y(8) 8.sin(a) cos (a) (r, a) is the integral of f along Lr, a cos(a) - sin(a) r (1) r S. Definition 3 The Radon Transform of a function f at (RS(r, a) fds Lra (2) Computed Tomography (CT) scanners measure Example 4 Denote by D the dise with radius S centered at the origin: where 2 S2 Define a function x ("chi") by x(r, y) = 1 if (x, y) is in D, whereas x(r, y) = 0 if (x,y) is not in D If Ir S, then the line La is too far from 0 to intersect D; thus, [Rx(r, a) = 0 for all r >S and If IrS, then the line Lra intersects D along a segment of length 2vS2- r2. Therefore, Rf(r, a). Hardware and software reconstruct f(r, y) O. 2/S2-2, r< S; 0, IrS. { (Rx](r, a) = 1 xds Lra (3) Choose either theoretical Problem 1 or computational Problem 2 ellipse Problem 1 For all items, c is a constant. Each item specifies a disc or For each item, Xe (xr, y) = c if (x, y) is in E, whereas xe (x, y) = 0 if (x,y) is not in E For each item, find algebraic formula for [Rxe](r,a) in terms of r, a, c, and parameters defining E (ro, yo): where (r -ro)(y-yo) S2 an (1.1) For this item, E is the disc with radius S centered at (1.2) For this item, E is the ellipse centered at 0 where (2/a2) + (y?/b?) < 1. (1.3) For this item, E is the ellipse centered at 0 with principal semi-axes of lengths a in the direction [COS(7), sin(y) and b in the direction - sin(), cos(y)) (1.4) For this item, E is the ellipse centered at (ro, yo) where [(x - ro)2/a2] + [(y - yo/b2 1. (1.5) For this item, is the llipse centered at (ro,Yo) with principal semi-axes of lengths a in the direction (cos(), sin(7)] and b in the direction sin(), cos(y)].

Answer 1

Given thal chak JUJW Abere algebic an defng Paramders C1,) For his , E is he disc with Ceniered at Cyo Yacius S C1-1Cy lere is in E Wheve Xg Cy not Scanned with CS CamScanner Ever