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2003 | Buch

Geometric Curve Evolution and Image Processing

verfasst von: Frédéric Cao

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

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Über dieses Buch

In image processing, "motions by curvature" provide an efficient way to smooth curves representing the boundaries of objects. In such a motion, each point of the curve moves, at any instant, with a normal velocity equal to a function of the curvature at this point. This book is a rigorous and self-contained exposition of the techniques of "motion by curvature". The approach is axiomatic and formulated in terms of geometric invariance with respect to the position of the observer. This is translated into mathematical terms, and the author develops the approach of Olver, Sapiro and Tannenbaum, which classifies all curve evolution equations. He then draws a complete parallel with another axiomatic approach using level-set methods: this leads to generalized curvature motions. Finally, novel, and very accurate, numerical schemes are proposed allowing one to compute the solution of highly degenerate evolution equations in a completely invariant way. The convergence of this scheme is also proved.

Inhaltsverzeichnis

Frontmatter
1. Curve evolution and image processing
Contents.
  • 1.1 Shape recognition
    • 1.1.1 Axioms for shape recognition...
    • 1.1.2 ... and their consequences
  • 1.2 Curve smoothing
    • 1.2.1 The linear curve scale space
    • 1.2.2 Towards an intrinsic heat equation
  • 1.3 An axiomatic approach of curve evolution
    • 1.3.1 Basic requirements
    • 1.3.2 First conclusions and first models
  • 1.4 Image and contour smoothing
  • 1.5 Applications
    • 1.5.1 Active contours
    • 1.5.2 Principles of a shape recognition algorithm
    • 1.5.3 Optical character recognition
  • 1.6 Organization of the volume
  • 1.7 Bibliographical notes
Frédéric Cao
2. Rudimentary bases of curve geometry
Contents.
  • 2.1 Jordan curves
  • 2.2 Length of a curve
  • 2.3 Euclidean parameterization
  • 2.4 Motion of graphs
Frédéric Cao
3. Geometric curve shortening flow
Contents.
  • 3.1 What kind of equations for curve smoothing?
    • 3.1.1 Invariant flows
    • 3.1.2 Symmetry group of flow
  • 3.2 Differential invariants
    • 3.2.1 General form of invariant flows
    • 3.2.2 The mean curvature flow is the Euclidean intrinsic heat flow
    • 3.2.3 The affine invariant flow: the simplest affine invariant curve flow
  • 3.3 General properties of second order parabolic flows: a digest
    • 3.3.1 Existence and uniqueness for mean curvature and affine flows
    • 3.3.2 Short-time existence in the general case
    • 3.3.3 Evolution of convex curves
    • 3.3.4 Evolution of the length
    • 3.3.5 Evolution of the area
  • 3.4 Smoothing staircases
  • 3.5 Bibliographical notes
Frédéric Cao
4. Curve evolution and level sets
Contents.
  • 4.1 From curve operators to function operators and vice versa
    • 4.1.1 Signed distance function and supporting function
    • 4.1.2 Monotone and translation invariant operators
    • 4.1.3 Level sets and their properties
    • 4.1.4 Extension of sets operators to functions operators
    • 4.1.5 Characterization of monotone, contrast invariant operator
    • 4.1.6 Asymptotic behavior of morphological operators
    • 4.1.7 Morphological operators yield PDEs
  • 4.2 Curve evolution and Scale Space theory
    • 4.2.1 Multiscale analysis are given by PDEs
    • 4.2.2 Morphological scale space
  • 4.3 Viscosity solutions
    • 4.3.1 Definition of viscosity solution
    • 4.3.2 Proof of uniqueness: the maximum principle
    • 4.3.3 Existence of solution by Perron’s Method
    • 4.3.4 Contrast invariance of level sets flow
    • 4.3.5 Viscosity solutions shorten level lines
  • 4.4 Morphological operators and viscosity solution
    • 4.4.1 Median filter and mean curvature motion
    • 4.4.2 Affine invariant schemes
  • 4.5 Conclusions
  • 4.6 Curvature thresholding
    • 4.6.1 Viscosity approach
    • 4.6.2 Opening and closing
  • 4.7 Alternative weak solutions of curve evolution
    • 4.7.1 Brakke’s varifold solution
    • 4.7.2 Reaction diffusion approximation
    • 4.7.3 Minimal barriers
  • 4.8 Bibliographical notes
Frédéric Cao
5. Classical numerical methods for curve evolution
Contents.
  • 5.1 Parametric methods
    • 5.1.1 Finite difference methods
    • 5.1.2 Finite element schemes
  • 5.2 Non parametric methods
    • 5.2.1 Sethian’s level sets methods
    • 5.2.2 Alvarez and Guichard’s finite differences scheme
    • 5.2.3 A monotone and convergent finite difference schemes
    • 5.2.4 Bence, Merriman and Osher scheme for mean curvature motion
    • 5.2.5 Elliptic regularization
Frédéric Cao
6. A geometrical scheme for curve evolution
Contents.
  • 6.1 Preliminary definitions
  • 6.2 Erosion
  • 6.3 Properties of the erosion
  • 6.4 Erosion and level sets
    • 6.4.1 Consistency
    • 6.4.2 Convergence
  • 6.5 Evolution by a power of the curvature
    • 6.5.1 Consistency
    • 6.5.2 Convergence
  • 6.6 A numerical implementation of the erosion
    • 6.6.1 Scale covariance
    • 6.6.2 General algorithm
    • 6.6.3 Eroded set and envelope
    • 6.6.4 Swallow tails
  • 6.7 Numerical experiments
    • 6.7.1 Evolving circles
    • 6.7.2 Convex polygons
    • 6.7.3 Unclosed curve
    • 6.7.4 Evolving non convex curves
    • 6.7.5 Invariance
    • 6.7.6 Numerical maximum principle
    • 6.7.7 Image filtering
  • 6.8 Bibliographical notes
Frédéric Cao
Conclusion and perspectives
Contents.
  • Discussion
  • Open problems
Frédéric Cao
A. Proof of Thm. 4.34
 
Abstract not available
Frédéric Cao
References
 
Abstract not available
Frédéric Cao
Index
 
Abstract not available
Frédéric Cao
Metadaten
Titel
Geometric Curve Evolution and Image Processing
verfasst von
Frédéric Cao
Copyright-Jahr
2003
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-36392-7
Print ISBN
978-3-540-00402-8
DOI
https://doi.org/10.1007/b10404