GEOMETRIC TOMOGRAPHY

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GEOMETRIC TOMOGRAPHY
Richard Gardner
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The X-ray Problems (P. C. Hammer)

• Proc. Symp. Pure Math. Vol VII Convexity
• (Providence, RI), AMS, 1963, pp. 498-9

Suppose there is a convex hole in an
otherwise homogeneous solid and that X-ray
pictures taken are so sharp that the darkness
at each point determines the length of a chord
along an X-ray line. (No diffusion, please.) How
many pictures must be taken to permit exact
reconstruction of the body if a. The
X-rays issue from a finite point source?
b. The X-rays are assumed parallel?
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Parallel and Point X-rays
There are sets of 4 directions so that the
parallel X-rays of a planar convex body in those
directions determine it uniquely. (R.J.G. and
P. McMullen, 1980)
A planar convex body is determined by its
X-rays taken from any set of 4 points with no
three collinear. (A. Volcic, 1986)
In these situations a viable algorithm
exists for reconstruction, even from noisy
measurements.
R.J.G. and M. Kiderlen, A solution to Hammers
X-ray reconstruction problem, Adv. Math. 214
(2007), 323-343.
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Computerized Tomography
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1979 Nobel Prize in medicine Computed Axial
Tomography
(Work published in 1963 to 1973)
Allan MacLeod Cormack physicist(1924 - 1998)
Godfrey Newbold Hounsfield engineer(1919- )
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Projection-Slice Theorem

• The (one-dimensional) Fourier transform of the
  X-ray of a density function g(x,y) at a given
  angle equals the slice of the (two-dimensional)
  Fourier transform of g at the same angle.

v
y
g(u,v)
g(x,y)
x
u
ˆ
Xf g
Xf g
f
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Unsolved Problems I
Geometric Tomography, second edition,
poses 66 open problems.

• Some examples of problems on X-rays still
  open
• Is a convex body in R3 determined by its parallel
  X-rays in any set of 7 directions with no three
  coplanar?
• Are there finite sets of directions in R3 such
  that a convex body is determined by its
  2-dimensional X-rays orthogonal to these
  directions?
• Is there a finite set of directions in R2 such
  that a convex body is determined among measurable
  sets by its X-rays in these directions?

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Enter the Brunn-Minkowski theory
width function
brightness function
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Aleksandrovs Projection Theorem
For origin-symmetric convex bodies K and L,
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Shephards Problem

• Petty and Schneider (1967)
• For origin-symmetric convex bodies K and L,

(i) if L is a projection body and (ii) if and
only if n 2.
A counterexample
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Projection Bodies
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Lutwaks dual Brunn-Minkowski theory
section function
Funks section theorem For origin-symmetric
star-shaped bodies K and L,
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Intersection Bodies

• Erwin Lutwak

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Duality in Geometric Tomography
Convex bodies Star-shaped bodies
Projections Sections through o
Support function Radial function
Brightness function Section function
Projection body Intersection body
Cosine transform Spherical Radon transform
Mixed volumes Dual mixed volumes
Brunn-Minkowski ineq. Dual B-M inequality
Aleksandrov-Fenchel Dual A-F inequality
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Geometric Tomography
The area of mathematics dealing with the
retrieval of information about a geometric object
from data about its sections, or projections, or
both. The term geometric object is
deliberately vague a convex polytope or body
would certainly qualify, as would a star-shaped
body, or even, when appropriate, a compact or
measurable set.
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Unsolved Problems II

• If K and L are convex bodies in Rn (n 3) whose
  projections on every hyperplane are congruent, is
  K a translate of L or L?
• If K and L are origin-symmetric star-shaped
  (w.r.t. o) bodies in R3 whose intersections with
  every hyperplane through o have equal perimeters,
  is K L?
• Let n 3. (i) Is a convex body K in Rn
  determined, up to translation and reflection in
  o, by its inner section function (i.e., is K
  determined by its cross-section body CK)? (ii) If
  CK is a ball, is K a ball?

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Discrete Tomography
Discrete X-ray of a finite subset of Zn
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Term discrete tomography introduced by Larry
Shepp, 1994.
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Motivated by a new technique in HRTEM (High
Resolution Transmission Electron Microscopy), by
which discrete X-rays of crystals (point atom)
can effectively be made.
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A Comparison of X-rays
Geometric tomography Measurable set in Rn Line integrals Arbitrary directions For convex sets, geometric and Fourier methods
Computerized tomography Density function Line integrals Arbitrary directions, typically several hundred Mainly Fourier methods
Discrete tomography Finite subset of Zn Line sums 2 to 4 main directions, for example, v (-2,3) Geometric, algebraic, linear programming, combinatorial
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Present Scope of Geometric Tomography
Computerized tomography
Discrete tomography
Robot vision
Convex geometry
Point X-rays
Parallel X-rays
Imaging
Sections through a fixed point dual
Brunn-Minkowski theory
Projections classical Brunn-Minkowski theory
?
Integral geometry
Pattern recognition
Minkowski geometry
Stereology and local stereology
Local theory of Banach spaces
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Unsolved Problems III

• (Discrete Aleksandrov projection theorem.) Let n
  3, and let K and L be centrally symmetric
  convex lattice sets in Zn with
• dim Kdim Ln such that for each u in Zn we
  have
• Is K a translate of L?

R.J.G., P. Gronchi and C. Zong, Sums,
projections, and sections of lattice sets, and
the discrete covariogram, Discrete Comput. Geom.
34 (2005), 391-409.
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Gauss measure

• Artem Zvavitch (2004)
• For origin-symmetric star bodies K and L,

• if K is an intersection body and (ii) for convex
  bodies,
• if and only if n 4.

Moreover,
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Unsolved Problems IV

• (Variations of Aleksandrov projection theorem.)
  Let n 3, and let K and L be origin-symmetric
  convex bodies in Rn. Does K L if any of
  the following conditions holds for each u in Sn-1

or for any of several other set functions
that satisfy a Brunn-Minkowski-type inequality
(for example, the first eigenvalue of the
Laplacian and torsional rigidity).