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Stochastic Geometry for Image Analysis

Book Description

This book develops the stochastic geometry framework for image analysis purpose. Two main frameworks are described: marked point process and random closed sets models. We derive the main issues for defining an appropriate model. The algorithms for sampling and optimizing the models as well as for estimating parameters are reviewed. Numerous applications, covering remote sensing images, biological and medical imaging, are detailed. This book provides all the necessary tools for developing an image analysis application based on modern stochastic modeling.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Chapter 1: Introduction
  5. Chapter 2: Marked Point Processes for Object Detection
    1. 2.1. Principal definitions
    2. 2.2. Density of a point process
    3. 2.3. Marked point processes
    4. 2.4. Point processes and image analysis
      1. 2.4.1. Bayesian versus non-Bayesian
      2. 2.4.2. A priori versus reference measure
  6. Chapter 3: Random Sets for Texture Analysis
    1. 3.1. Introduction
    2. 3.2. Random sets
      1. 3.2.1. Insufficiency of the spatial law
      2. 3.2.2. Introduction of a topological context
      3. 3.2.3. The theory of random closed sets (RACS)
      4. 3.2.4. Some examples
      5. 3.2.5. Stationarity and isotropy
    3. 3.3. Some geostatistical aspects
      1. 3.3.1. The ergodicity assumption
      2. 3.3.2. Inference of the DF of a stationary ergodic RACS
        1. 3.3.2.1. Construction of the estimator
        2. 3.3.2.2. On sampling
      3. 3.3.3. Individual analysis of objects
    4. 3.4. Some morphological aspects
      1. 3.4.1. Geometric interpretation
        1. 3.4.1.1. Point
        2. 3.4.1.2. Pair of points
        3. 3.4.1.3. Segment
        4. 3.4.1.4. Ball
      2. 3.4.2. Filtering
        1. 3.4.2.1. Opening and closing
        2. 3.4.2.2. Sequential alternate filtering
    5. 3.5. Appendix: demonstration of Miles’ formulae for the Boolean model
  7. Chapter 4: Simulation and Optimization
    1. 4.1. Discrete simulations: Markov chain Monte Carlo algorithms
      1. 4.1.1. Irreducibility, recurrence, and ergodicity
        1. 4.1.1.1. Definitions
        2. 4.1.1.2. Stationarity
        3. 4.1.1.3. Convergence
        4. 4.1.1.4. Irreducibility
        5. 4.1.1.5. Aperiodicity
        6. 4.1.1.6. Harris recurrence
        7. 4.1.1.7. Ergodicity
        8. 4.1.1.8. Geometric ergodicity
        9. 4.1.1.9. Central limit theorem
      2. 4.1.2. Metropolis–Hastings algorithm
      3. 4.1.3. Dimensional jumps
        1. 4.1.3.1. Mixture of kernels
        2. 4.1.3.2. π-reversibility
      4. 4.1.4. Standard proposition kernels
        1. 4.1.4.1. Simple perturbations
        2. 4.1.4.2. Model switch
        3. 4.1.4.3. Birth and death
      5. 4.1.5. Specific proposition kernels
        1. 4.1.5.1. Creating complex transitions from standard transitions
        2. 4.1.5.2. Data-driven perturbations
        3. 4.1.5.3. Perturbations directed by the current state
        4. 4.1.5.4. Composition of kernels
    2. 4.2. Continuous simulations
      1. 4.2.1. Diffusion algorithm
      2. 4.2.2. Birth and death algorithm
      3. 4.2.3. Muliple births and deaths algorithm
        1. 4.2.3.1. Convergence of the distributions
        2. 4.2.3.2. Birth and death process
      4. 4.2.4. Discrete approximation
        1. 4.2.4.1. Acceleration of the multiple births and deaths algorithm
    3. 4.3. Mixed simulations
      1. 4.3.1. Jump process
      2. 4.3.2. Diffusion process
      3. 4.3.3. Coordination of jumps and diffusions
    4. 4.4. Simulated annealing
      1. 4.4.1. Cooling schedule
      2. 4.4.2. Initial temperature T0
      3. 4.4.3. Logarithmic decrease
      4. 4.4.4. Geometric decrease
      5. 4.4.5. Adaptive reduction
      6. 4.4.6. Stopping criterion/final temperature
  8. Chapter 5: Parametric Inference for Marked Point Processes in Image Analysis
    1. 5.1. Introduction
    2. 5.2. First question: what and where are the objects in the image?
    3. 5.3. Second question: what are the parameters of the point process that models the objects observed in the image?
      1. 5.3.1. Complete data
        1. 5.3.1.1. Maximum likelihood
        2. 5.3.1.2. Maximum pseudolikelihood
      2. 5.3.2. Incomplete data: EM algorithm
    4. 5.4. Conclusion and perspectives
    5. 5.5. Acknowledgments
  9. Chapter 6: How to Set Up a Point Process?
    1. 6.1. From disks to polygons, via a discussion of segments
    2. 6.2. From no overlap to alignment
    3. 6.3. From the likelihood to a hypothesis test
    4. 6.4. From Metropolis–Hastings to multiple births and deaths
  10. Chapter 7: Population Counting
    1. 7.1. Detection of Virchow–Robin spaces
      1. 7.1.1. Data modeling
      2. 7.1.2. Marked point process
      3. 7.1.3. Reversible jump MCMC algorithm
      4. 7.1.4. Results
    2. 7.2. Evaluation of forestry resources
      1. 7.2.1 2D model
        1. 7.2.1.1. Prior
        2. 7.2.1.2. Data term
        3. 7.2.1.3. Optimization
        4. 7.2.1.4. Results
      2. 7.2.2. 3D model
        1. 7.2.2.1. Results
    3. 7.3. Counting a population of famingos
      1. 7.3.1. Estimation of the famingo color
      2. 7.3.2. Simulation and optimization by multiple births and deaths
      3. 7.3.3. Results
    4. 7.4. Counting the boats at a port
      1. 7.4.1. Initialization of the optimization algorithm
        1. 7.4.1.1. Parameter γd
        2. 7.4.1.2. Calibration of the d0 parameter
      2. 7.4.2. Initial results
      3. 7.4.3. Modification of the data energy
        1. 7.4.3.1. First modification of the prior energy
        2. 7.4.3.2. Second modification of the prior energy
  11. Chapter 8: Structure Extraction
    1. 8.1. Detection of the road network
    2. 8.2. Extraction of building footprints
    3. 8.3. Representation of natural textures
      1. 8.3.1 Simple model
        1. 8.3.1.1. Data term
        2. 8.3.1.2. Sampling by jump diffusion
        3. 8.3.1.3. Results
      2. 8.3.2. Models with complex interactions
  12. Chapter 9: Shape Recognition
    1. 9.1. Modeling of a LIDAR signal
      1. 9.1.1. Motivation
      2. 9.1.2. Model library
        1. 9.1.2.1. Energy formulation
      3. 9.1.3. Sampling
      4. 9.1.4. Results
        1. 9.1.4.1. Simulated data
        2. 9.1.4.2. Satellite data: large footprint waveforms
        3. 9.1.4.3. Airborne data: small footprint waveforms
        4. 9.1.4.4. Application to the classification of 3D point clouds
    2. 9.2. 3D reconstruction of buildings
      1. 9.2.1. Library of 3D models
      2. 9.2.2. Bayesian formulation
        1. 9.2.2.1. Likelihood
        2. 9.2.2.2. A priori
      3. 9.2.3. Optimization
      4. 9.2.4. Results and discussion
  13. Bibliography
  14. List of Authors
  15. Index