Investigation of a Mathematical Property of the Helix

1
Investigation of a Mathematical Property of the
Helix

• Jed D. Pack, Frédéric Noo
• Medical Imaging Research Laboratory
• University of Utah

2
Motivation
Development of reconstruction algorithms for new
multi-slice CT scanners requires a clear
understanding of geometrical properties of the
helix
3
?-lines

• Definition
• A ?-line is a line conecting two points of a
  helix separated by less than 2? radians

• Theorem
• Every point (x,y,z) inside the cylinder described
  by a helix belongs to exactly one ?-line of that
  helix

4
Proof
(Rcos?o,Rsin?o)
A
B
a
a
(x,y)
ß
2a
5
Proof (continued)
Taking the derivative of z(x,y,?o) with respect
to ?o we get a strictly positive function when
the point is inside the helix This means that the
function z(x,y,?o) is one-to-one in
?o Consequently, there exists a function
?o(x,y,z) which gives a unique ?-line for any
point (x,y,z) Lets look at this graphically!
6
Start with a cylinder
7
Add a helix
8
Parameterization
9
All ?-lines going up from a point
10
Each point generates a surface
11
The two surfaces do not intersect inside the helix
12
The blue one is lower
13
The red is higher
14
Implications of this Property

• Katsevich has shown 2002 that any image voxel
  can be exactly reconstructed if helical CB data
  is available for the segment of the helix
  connecting the extremities of the ?-line passing
  through the voxel
• Consequently, CB data taken along a helix is
  theoretically sufficient to reconstruct every
  point inside the helix
• Research is currently aimed at developing an
  efficient implementation of this algorithm based
  on a careful use of the properties of the helix