GEOMETRIC TOMOGRAPHY Richard Gardner 5252021 1 The Xray

GEOMETRIC TOMOGRAPHY Richard Gardner 5/25/2021 1

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? 5/25/2021 2

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. Mc. Mullen, 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 R. J. G. and M. Kiderlen, A solution to from noisy measurements. 5/25/2021 Hammer’s X-ray reconstruction problem, Adv. Math. 214 (2007), 323 -343. 3

Computerized Tomography 5/25/2021 4

1979 Nobel Prize in medicine: Computed Axial Tomography (Work published in 1963 to 1973) Allan Mac. Leod Cormack physicist (1924 - 1998) 5/25/2021 Godfrey Newbold Hounsfield engineer (1919 - ) 5

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. y ĝ(u, v) g(x, y) x f 5/25/2021 v u Xf g ˆ Xf g 6

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 R 3 determined by its parallel X-rays in any set of 7 directions with no three coplanar? • Are there finite sets of directions in R 3 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 R 2 such that a convex body is determined among measurable sets by its X-rays in these directions? 5/25/2021 7

Enter the Brunn-Minkowski theory width function brightness function 5/25/2021 8

Aleksandrov’s Projection Theorem For origin-symmetric convex bodies K and L, Cauchy’s projection formula Surface area measure of K 5/25/2021 9

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 5/25/2021 10

Projection Bodies 5/25/2021 11

Lutwak’s dual Brunn-Minkowski theory section function Funk’s section theorem: For origin-symmetric star-shaped bodies K and L, 5/25/2021 12

Intersection Bodies Erwin Lutwak 5/25/2021 13

Duality in Geometric Tomography Convex bodies Projections Star-shaped bodies 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 5/25/2021 14

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. 5/25/2021 15

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 R 3 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? 5/25/2021 16

Discrete Tomography Discrete X-ray of a finite subset of Zn: v = (1, 0) 1 1 3 4 2 Term “discrete tomography” introduced by Larry Shepp, 1994. Motivated by a new technique in HRTEM (High Resolution Transmission Electron Microscopy), by which discrete Xrays of crystals (point = atom) can effectively be made. 5/25/2021 17

A Comparison of X-rays Geometric tomography Measurable set in Rn Computerized Density tomography function Discrete tomography 5/25/2021 Line integrals Arbitrary directions For convex sets, geometric and Fourier methods Line integrals Arbitrary directions, typically several hundred Mainly Fourier methods 2 to 4 “main” directions, for example, v = (-2, 3) Geometric, algebraic, linear programming, combinatorial Finite subset Line of Zn sums 18

Present Scope of Geometric Tomography Computerized tomography Discrete tomography Convex geometry Integral geometry Point X -rays Parallel X-rays Projections; classical Brunn. Minkowski theory Minkowski geometry Local theory of Banach spaces 5/25/2021 ? Robot vision Imaging Sections through a fixed point; dual Brunn-Minkowski Pattern theory recognition Stereology and local stereology 19

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), 391 -409. 5/25/2021 20

Gauss measure • Artem Zvavitch (2004): For origin-symmetric star bodies K and L, (i) if K is an intersection body and (ii) for convex bodies, if and only if n ≤ 4. Moreover, 5/25/2021 21

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). 5/25/2021 22