Skip to content

Geometric Tomography

Spend $50 to get a free DVD!

ISBN-10: 0521684935

ISBN-13: 9780521684934

Edition: 2nd 2006 (Revised)

Authors: Richard J. Gardner, G. C. Rota, B. Doran, M. Ismail, T. Y. Lam

List price: $80.95
Blue ribbon 30 day, 100% satisfaction guarantee!
Out of stock
what's this?
Rush Rewards U
Members Receive:
Carrot Coin icon
XP icon
You have reached 400 XP and carrot coins. That is the daily max!

Description:

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions,…    
Customers also bought

Book details

List price: $80.95
Edition: 2nd
Copyright year: 2006
Publisher: Cambridge University Press
Publication date: 6/19/2006
Binding: Paperback
Pages: 516
Size: 6.00" wide x 9.00" long x 1.25" tall
Weight: 1.606
Language: English

Richard Gardner has been Professor of Mathematics at Western Washington University since 1991. He is the author of 70 papers and founded geometric tomography as a subject in its own right with the publication of the first edition of this book in 1995.

Preface to the second edition
Preface
Background material
Basic concepts and terminology
Transformations
Basic convexity
The Hausdorff metric
Measure and integration
The support function
Star sets and the radial function
Polar duality
Differentiability properties
Parallel X-rays of planar convex bodies
What is an X-ray?
X-rays and Steiner symmetrals of planar convex bodies
Open problems
Notes
Computerized tomography
Parallel X-rays and Steiner symmetrals of convex bodies
Exact reconstruction
Well-posedness and stability
Reconstruction of convex bodies from possibly noisy data
Geometric probing
Jakob Steiner (1796-1863)
Parallel X-rays in n dimensions
Parallel X-rays and k-symmetrals
X-rays of convex bodies in E[superscript n]
X-rays of bounded measurable sets
Open problems
Notes
Parallel X-rays and k-symmetrals of convex bodies
Switching components and discrete tomography
Parallel X-rays and k-symmetrals of measurable sets
Blaschke shaking
Reconstruction of polygons and polyhedra from possibly noisy X-rays
Ridge functions and the additivity conjecture
X-rays of bounded density functions
Johann Radon (1887-1956)
Projections and projection functions
Homothetic and similar projections
The width function and central symmetral
Projection functions and the Blaschke body
Open problems
Notes
Homothetic and similar projections
Bodies with congruent or affinely equivalent projections
Sets of constant width and brightness
Blaschke bodies and Blaschke sums
Determination by one projection function
Determination by more than one projection function
Determination by directed projection functions, etc
Reconstruction
Mean projection bodies
Projections of convex polytopes
Critical projections
Almost-spherical or almost-ellipsoidal projections, and related results
Aleksander Danilovich Aleksandrov (1912-1999)
Projection bodies and volume inequalities
Projection bodies and related concepts
Smaller bodies with larger projections
Stability
Reconstruction from brightness functions
Open problems
Notes
Projection bodies and zonoids
The Fourier transform approach I: The brightness function and projection bodies
The Minkowski map and Minkowski linear combinations of projection bodies
Generalized zonoids
Bodies whose projections are zonoids
Projection bodies of order i
The L[superscript p]-Brunn-Minkowski theory and L[superscript p]-projection bodies
Characterizations in terms of mixed volumes
Results related to Aleksandrov's projection theorem
Smaller bodies with larger projections
Stability results
Reconstruction from brightness functions
Hermann Minkowski (1864-1909)
Point X-rays
Point X-rays and chordal symmetrals
The X-ray of order i
Point X-rays of planar convex bodies
X-rays in the projective plane
Open problems
Notes
Point X-rays and chordal symmetrals
Point X-rays of planar convex bodies
Reconstruction
Well-posedness
Point X-rays in higher dimensions
Discrete point X-rays
Point projections
Wilhelm Suss (1895-1958) and the Japanese school
Chord functions and equichordal problems
i-chord functions and i-chordal symmetrals
Chord functions of star sets
Equichordal problems
Open problems
Notes
Chord functions, i-chordal symmetrals, and ith radial sums
Chord functions of star sets
Equichordal problems
Wilhelm Blaschke (1885-1962)
Sections, section functions, and point X-rays
Homothetic and similar sections
Section functions and point X-rays
Point X-rays of measurable sets
Open problems
Notes
Homothetic and similar sections
Bodies with congruent or affinely equivalent sections
Sets of constant section
Determination by section functions
Determination by half-volumes
Point X-rays of measurable sets
Sections of convex polytopes
Critical sections
Almost-spherical or almost-ellipsoidal sections
A characterization of star-shaped sets
Sections by other sets of planes
Integral geometry
Mean section bodies
Stereology
Local stereology
Paul Funk (1886-1969)
Intersection bodies and volume inequalities
Intersection bodies of star bodies
Larger bodies with smaller sections
Cross-section bodies
Open problems
Notes
Intersection bodies
The Fourier transform approach II: The section function and intersection bodies
The map I
Generalized intersection bodies
Bodies whose central sections are intersection bodies
Intersection bodies of order i
k-intersection bodies and related notions
Characterizations in terms of dual mixed volumes
Larger bodies with smaller sections I: The Busemann-Petty problem
Larger bodies with smaller sections II: Generalizations and variants of the Busemann-Petty problem
Stability results
Cross-section bodies
Problems involving both projections and sections
Herbert Busemann (1905-1994)
Estimates from projection and section functions
Centroid bodies
Some affine isoperimetric inequalities
Volume estimates from projection functions
Volume estimates from section functions
Open problems
Notes
Centroid bodies and polar projection bodies
The floating body problem
Affine surface area, the covariogram, and convolution and sectional bodies
Affine isoperimetric inequalities
The L[superscript p]-Brunn-Minkowski theory: centroid bodies, ellipsoids, and inequalities
Volume estimates from projection functions
Volume estimates from section functions
The slicing problem
Central limit theorems for convex bodies
Estimates concerning both projections and sections
Estimates for inradius and circumradius
Hugo Hadwiger (1908-1981)
Appendixes
Mixed volumes and dual mixed volumes
An example
Area measures
Mixed volumes and mixed area measures
Reconstruction from surface area measures
Quermassintegrals and intrinsic volumes
Projection formulas
Dual mixed volumes
Inequalities
Inequalities involving means and sums
The Brunn-Minkowski inequality
The Aleksandrov-Fenchel inequality
The dual Aleksandrov-Fenchel inequality
Other inequalities
Integral transforms
X-ray transforms
The cosine and spherical Radon transforms
Open problem
References
Notation
Author index
Subject index