Definition 1 The Radon Transform [Rf](P) of a function f : R³ → R over the surface of a plane P in space R3 isthe surface integral [4, § 15.7] of f over the plane P[Rf](P) = || f dA.(1)A planethe plane PPe.n may be specified by a unit normal vector n perpendicular to P and the distance p from the origin 0 toin the direction n. A Cartesian equation for the plane Pp,n may then use a dot product with n.P,nSuch integrals arise as measurements in medical diagnostic imaging devices known as Nuclear Magnetic Reso-nance (NMR) or Magnetic Resonance Imaging (MRI) [2], [3].For this problem, the function ƒ has a constant value c > 0 everywhere inside the ball B withradius R > 0 centered at the origin 0, and f has the constant value 0 everywhere outside the ball B. Find [Rf](Ppn):Problem 1x² + y? + z? < R?;0, R2 < x² + y² + z².C,f (x, y, z) = {The plane Pe,n is at a distance p from the origin 0 and perpendicular to the unit normal n =a² + b? + c²(a, b, c), so that1. Finding [Rf](Pp,n) means finding the surface integral of f over the plane Pp,n. The result mayP;ndepend on c, p, and n. Sketching a picture with the plane and the domain of the function may help.

Name: Definition 1 The Radon Transform [Rf](P) of a function f : R³ → R ove
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Description: Definition 1 The Radon Transform [Rf](P) of a function f : R³ → R over the surface of a plane P in space R3 isthe surface integral [4, § 15.7] of f over the plane P[Rf](P) = || f dA.(1)A planethe plane PPe.n may be specified by a unit normal vector

Answered: For this problem, the function f has a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Definition 1 The Radon Transform [Rf](P) of a function f : R³ → R over the surface of a plane P in space R3 is
the surface integral [4, § 15.7] of f over the plane P
[Rf](P) = || f dA.
(1)
A plane
the plane P
Pe.n may be specified by a unit normal vector n perpendicular to P and the distance p from the origin 0 to
in the direction n. A Cartesian equation for the plane Pp,n may then use a dot product with n.
P,n
Such integrals arise as measurements in medical diagnostic imaging devices known as Nuclear Magnetic Reso-
nance (NMR) or Magnetic Resonance Imaging (MRI) [2], [3].
For this problem, the function ƒ has a constant value c > 0 everywhere inside the ball B with
radius R > 0 centered at the origin 0, and f has the constant value 0 everywhere outside the ball B. Find [Rf](Ppn):
Problem 1
x² + y? + z? < R?;
0, R2 < x² + y² + z².
C,
f (x, y, z) = {
The plane Pe,n is at a distance p from the origin 0 and perpendicular to the unit normal n =
a² + b? + c²
(a, b, c), so that
1. Finding [Rf](Pp,n) means finding the surface integral of f over the plane Pp,n. The result may
P;n
depend on c, p, and n. Sketching a picture with the plane and the domain of the function may help.

Expert Solution

Step 1

for this problem the function of has a Constant value
$c > 0$ everywhere beside the ball $B$ with radius
$R > 0$ . Centered at the origin 0 and $f$ has the constant value 0 everywhere outside the ball $B$ .

Then the given function is $f (x, y, z) = \{\begin{cases} c, x^{2} + y^{2} + c^{2} \leq ℝ^{2} \\ 0, ℝ^{2} < x^{2} + y^{2} + z^{2} \end{cases}$

Step 2

The plane $℘_{p, n}$ is at distance $ρ$ from the origin o and perpendicular to the unit normal $n = (a, b, c)$ .

So that

$\begin{array}{rcl} a^{2} + b^{2} + c^{2} & = & 1 \\ = & a \hat{i} + b \hat{j} + c \hat{k} \\ = & a x + b y + c z \end{array}$

If $ℝ$ is the shadow region of a surface defined by the equation of

$f (x, y, z) = c * g$

Where g is an continuous function defined $a < b < c$

on point of s.

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