1. GEOMETRIC TOMOGRAPHY
Richard Gardner
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2. The X-ray Problems (P. C. Hammer)
Proc. Symp. Pure Math. Vol VII: Convexity
(Providence, RI), AMS, 1963, pp
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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3. 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. Volčič, 1986)
In these situations a viable algorithm exists for reconstruction, even from noisy measurements.
R.J.G. and M. Kiderlen, A solution to Hammer’s X-ray reconstruction problem, Adv. Math. 214 (2007),
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4. Computerized Tomography
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5. 1979 Nobel Prize in medicine: Computed Axial Tomography
(Work published in 1963 to 1973)
A. McLeod Cormack and G. Newbold Hounsfield in Medicine for computed axial tomography.
1979
The prize was awarded jointly to:
Alan M. Cormack (*1924 in Johannesburg, South Africa)
USA, Tufts University, Medford, MA,
and
Sir Godfrey N. Hounsfield (*1919)
Great Britain, Central Research Laboratories, EMI, London,
"for the development of computer assisted tomography".
Allan MacLeod Cormack physicist ( )
Godfrey Newbold Hounsfield engineer (1919- )
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6. 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
ĝ(u,v)
g(x,y) x u ˆ
Xf g
Xf g f
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7. 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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8. Enter the Brunn-Minkowski theory
width function
brightness function
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9. Aleksandrov’s Projection Theorem
For origin-symmetric convex bodies K and L,
Cauchy’s projection formula
Surface area measure of K
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10. Shephard’s 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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11. Projection Bodies
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12. Lutwak’s dual Brunn-Minkowski theory
section function
Funk’s section theorem:
For origin-symmetric star-shaped
bodies K and L,
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13. Intersection Bodies
Erwin Lutwak
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14. 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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15. 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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16. 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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17. Discrete Tomography Discrete X-ray of a finite subset of Zn:1
Term “discrete tomography” introduced by Larry Shepp, 1994. 1
v = (1,0) 3 4 2
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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18. 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
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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19. Present Scope of Geometric Tomography
Computerized tomography
Discrete tomography
Robot vision
Convex geometry
Point X-rays
Parallel X-rays
Imaging
Projections; classical Brunn-Minkowski theory
Sections through a fixed point; dual Brunn-Minkowski theory ?
Integral geometry
Pattern recognition
Minkowski geometry
Stereology and local stereology
Local theory of Banach spaces
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20. 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 K=dim L=n 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),
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21. Gauss measure if K is an intersection body and (ii) for convex bodies,
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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22. 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).
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