CVonline: Vision Geometry and Mathematics

  1. Basic Representations
    1. Coordinate Systems
      1. Cartesian: Affine, Rectangular
      2. Cylindrical
      3. Hexagonal
      4. Log-Polar
      5. Polar
      6. Spherical
    2. Digital Topology
    3. Dual Space
    4. Homogeneous Coordinates
    5. Pose/Rotation/Orientation Representations
      1. Axis-angle
      2. Clifford Algebra
      3. Euler Angles
      4. Exponential Map
      5. Quaternion/Dual-Quaternion
      6. Rotation Matrix (See also Homogeneous Coordinates)
      7. Rotation/Slant/Tilt
      8. Yaw/Pitch/Roll
  2. Distance Metrics
    1. Affine
    2. Algebraic
    3. Bhattacharyya
    4. Chi-squared
    5. Earth Mover's/Optimal Mass Transport/Monge-Kantorovich
    6. Euclidean
    7. Fuzzy Intersection
    8. Hausdorff
    9. Jeffrey-Divergence
    10. Kullback-Leibler Divergence (KL)
    11. Mahalanobis
    12. Manhatten or city-block
    13. Minkowski-form
    14. Procrustes
    15. Procrustes Average
    16. Quadric Form
    17. Specific Structure Similarity
      1. Curve Similarity
      2. Region Similarity
      3. Volume Similarity
  3. Elementary Mathematics for Vision
    1. Coordinate Systems, Vectors, Matrices, Derivatives and Gradients, Probability
    2. Derivatives in sampled images
  4. Function Optimization
    1. 1D Function Optimization and Golden Section
    2. Constrained Optimization and Lagrange Multipliers
    3. Multi-Dimensional Optimization
      1. Derivative Free Search
      2. Global Optimization
        1. Ant Colony Optimization
        2. Downhill Simplex
        3. Genetic Algorithms (See Genetic Algorithms/Programming)
        4. Graduated Non-Convexity and Multi-Resolution Methods
        5. Markov Random Field Optimization
        6. Particle Swarm Optimization
        7. Simulated Annealing
      3. Optimization With Derivatives
        1. Levenberg-Marquardt
        2. Newton and Gradient Descent Algorithms
    4. Optimization Model Selection
    5. Variational Methods
  5. Linear Algebra for Computer Vision
    1. Eigenfunctions
    2. Eigenvalues/Eigenvectors
    3. Principal Component and Related Approaches
      1. Dimensionality Reduction
      2. Discriminant Analysis
      3. Factor Analysis
      4. Fisher Linear Discriminant Transformation
      5. Independent Component Analysis
      6. Kernel Linear Discriminant Analysis
      7. Kernel Principal Component Analysis
      8. Non-Negative Matrix Factorization
      9. Optimal Dimension Estimation
      10. Principal Component Analysis/Karhunen-Loeve transformation
      11. Principal Geodesic Analysis (PGA)
      12. Probabilistic Principal Component Analysis
      13. Rao-Blackwell Dimensionality Reduction
    4. Sammon Mapping
    5. Singular Value Decomposition (SVD)
    6. Structure Tensor
  6. Multi-Sensor/Multi-View Geometries
    1. 3D Reconstruction
      1. 2D Projections
      2. Reconstruction from Multiple Images/Orthogonal Views
      3. Slice-based Reconstruction (e.g. PET/CAT/MRI)
    2. Affine and Projective Stereo
    3. Baseline
      1. Narrow Baseline Stereo
      2. Wide Baseline Stereo
    4. Binocular Stereo Algorithms
      1. Cooperative Algorithms
      2. Disparity
      3. Dense Stereo Matching Approaches
      4. Dynamic Programming
      5. Feature Matching Stereo Algorithms
      6. Gradient Matching Stereo Algorithms
      7. Image Rectification
        1. Planar Rectification
        2. Polar Rectification
      8. Log-Polar Stereo
      9. Multi-Scale Stereo Algorithms
      10. Panoramic Image Stereo Algorithms
      11. Phase Matching Stereo Algorithms
      12. Region Matching Stereo Algorithms
      13. Weakly/Uncalibrated Approaches
      14. Spherical Stereo
    5. Epipolar/Multi-View Geometry
      1. Absolute Conic
      2. Absolute Quadric
      3. Epipolar Geometry Definitions
      4. Essential Matrix
      5. Fundamental Matrix
      6. Grassmannian Space/Plucker Embedding
      7. Homography Tensor
      8. Transfer and Novel View Synthesis
      9. Trifocal/Quadrifocal Tensor
    6. Image Based Modelling/Plenoptic Modelling
    7. Image Feature Correspondence Constraints
      1. Active Stereo
      2. Disparity Gradient Limit
      3. Disparity Limit
      4. Epipolar Constraint
      5. Feature Contrast
      6. Feature Orientation
      7. Grey-level Similarity
      8. Lipschitz Continuity
      9. Ordering
      10. Surface Continuity
      11. Surface Smoothness
      12. Uniqueness
      13. Viewpoint Constraint
      14. View Consistency Constraint
    8. Multi-View Matching
    9. Scene Reconstruction/Surface Interpolation
      1. Adaptive Meshing
      2. Constrained Reconstruction
      3. Membrane/Thin Plate Models
      4. Texture Mapping
      5. Triangulation
      6. Volumetric Reconstruction
    10. Trinocular (and more) Stereo
  7. Parameter Estimation
    1. Bayesian Methods
    2. Constrained Least Squares
    3. Linear Least Squares
    4. Optimization (See Functional Optimization)
    5. Robust Techniques (See Robust Estimators)
  8. Probability and Statistics for Computer Vision
    1. Autoregression
    2. Basic Statistics and Bayes Rule in Vision
    3. Bayesian Inference Networks (See generic entry)
    4. Causal Models
    5. Correlation
    6. Covariance and Mahalanobis Distance in Vision
    7. Dempster-Shafer and Evidence Theory
    8. Distribution Mode Analysis
    9. Gaussian / Normal Distribution
    10. Heteroscedastic Noise and HEIV Regression
    11. Homoscedastic Noise
    12. Hidden Markov Models
    13. Honest Probabilities
    14. Hypothesis Testing
    15. Information Theory
    16. Kalman Filters
      1. Unscented Kalman Filter
    17. Kernel Regression
    18. Least Mean Square Estimation and Estimators
    19. Least Median Square Estimation and Estimators
    20. Maximum Likelihood
    21. Model Fitting
    22. Monte Carlo Methods for Vision
    23. Markov Chain Methods for Vision
    24. Markov Random Fields
      1. Applications
      2. Conditional Random Fields
      3. Multi-level MRF
      4. Optimization Methods
        1. Approximate Variational Extremum
        2. Gibbs Sampling
        3. Graduated Nonconvexity
        4. Graph Cuts
        5. Iterated Conditional Modes
        6. "Modern" Graph Cut
        7. Simulated Annealing
      5. Theory
    25. Mixture Models and Expectation Maximization (EM)
      1. Poisson Mixture Model
    26. Normalization
    27. Non-Parametric Methods
    28. Poisson Distribution
    29. Probability Density Estimation
    30. Random Number Generation for Vision
    31. Robust Estimators
    32. Useful Distributions
  9. Projection Geometries and Transformations
    1. Affine
    2. Anamorphic/Catadioptric
    3. Central Projection
    4. Euclidean
    5. Homography
    6. Hierarchy of Geometries
    7. Orthographic
    8. Paraperspective
    9. Perspective
    10. Plane Projection
    11. Projective Space (3D)
    12. Real Camera Projection
    13. Similarity
    14. Weak-Perspective
  10. Properties and Invariants of Projection
    1. Absolute Points
    2. Affine Invariants
    3. Collineations
    4. Conics
    5. Coplanarity Invariants
    6. Cross Ratio
    7. Differential Invariants
    8. Duality
    9. General Projective Invariants
    10. Integral Invariants
    11. Laguerre Formula
    12. Pencil of Lines
    13. Quasi-Invariants
    14. Structural Invariants
  11. Relational Shape Descriptions
    1. Curves
      1. Adjacency/Connectedness
      2. Relative Curvature
      3. Relative Length
      4. Relative Orientation
      5. Separation
    2. Regions
      1. Adjacency/Connectedness
      2. Relative Area/Size
      3. Separation
    3. Surfaces
      1. Adjacency/Connectedness
      2. Relative Area/Size
      3. Relative Orientation
      4. Separation
    4. Volumes
      1. Adjacency/Connectedness
      2. Relative Orientation
      3. Relative Volume/Size
      4. Separation
  12. Shape Properties (See also Geometric Representation of Model Features)
    1. Geometric Morphometrics
    2. Kendall's Shape Space
    3. Points and Local Invariants
    4. Curves and Curve Invariants (See also Curves)
      1. Affine Arc Length and Affine Curvature
      2. Arc Length
      3. Bending Energy
      4. Chord Distribution
      5. Curvature, Torsion, Curvature Radius
      6. Differential Geometry, Frenet Frame, Frenet-Serret Formulas
      7. Invariant Points: Inflections/Bitangents
    5. Image Regions and Region Invariants
      1. Angularity ratio
      2. Area, Perimeter
      3. Boundary Properties
      4. Center-of-Mass
      5. Convexity Ratio
      6. Eccentricity, Circularity, Compactness, Elongatedness
      7. Elongation Factor
      8. Euler number/Genus
      9. Extremal Points
      10. Feret's Diameter, Martin's Diameter
      11. Fourier Descriptors
      12. Minimum Bounding Rectangle
      13. 2D Moments and their Invariants
        1. Affine Moments
        2. Binary Moments
        3. Color Moments
        4. Eigenmoments
        5. Fourier-Mellin Moment Invariants
        6. Grey-Level or Texture Moments
        7. Hahn Moments
        8. Krawtchouk Moments
        9. Legendre Moments
        10. Orthogonal Moments: Legendre, Zernike
        11. Racah Moments
        12. Tchebichef/Chebichev Moments
        13. Velocity Moments
        14. Zernike Moments
      14. Orientation
      15. Sphericity ratio
      16. Rectangularity
      17. Rectilinearity
      18. Roundness ratio
      19. Topological Descriptors
      20. Wadell's circularity shape ratio
    6. Surfaces
      1. Apparent Contour and Local Geometry
      2. Common Shape Classes and Representations
        1. Cone
        2. Cyclide
        3. Cylinder
        4. Ellipsoid/Sphere
        5. Membrane/Thin Plate Spline (See here)
        6. Plane
        7. Polyhedra
        8. Quadric
        9. Torus
      3. Fundamental Forms
      4. Gauge Coordinates
      5. Hessian
      6. Metric Determinant
      7. Principal Curvatures and Directions and other Local Shape Representations
        1. Deviation from Flatness
        2. Gauss-Bonnet Surface Description
        3. Gaussian Curvature
        4. Koenderink's Shape Classification
        5. Mean Curvature
        6. Mean and Gaussian Curvature Shape Classification
        7. Minimal Points
        8. Parabolic Points
        9. Ridges and Valleys
        10. Umbilics
      8. Quadratic Variation
      9. Ricci Flow
      10. Surface Area
      11. Surface Normals and Tangent Planes/Tangent Spaces
      12. Surface Orientation and Gradient Space
    7. Symmetry (See also Symmetry Detection)
      1. Affine
      2. Bilateral
      3. Rotation
      4. Skew
    8. Volumes
      1. Elongatedness
      2. 3D Moments and Moment Invariants
      3. Volume
  13. Transformations (Geometric), Registration and Pose Estimation Methods
    1. 2D to 2D Pose Estimation Methods
      1. Line-Based Methods
      2. Point-Based Methods
    2. 2D to 3D Pose Estimation Methods
      1. Line-Based Methods
      2. Point-Based Methods
    3. 3D to 3D Pose Estimation Methods
      1. Line-Based Methods
      2. Point-Based Methods
    4. Affine Transformation
      1. Minimal Data Estimation
      2. Least-square Estimates
      3. Robust Estimates
    5. Bundle Adjustment
    6. Euclidean Transformation
      1. Least-square Estimates
      2. Minimal Data Estimation
      3. Robust Estimates
    7. Homography Transformation
      1. Least-square Estimates
      2. Minimal Data Estimation
      3. Robust Estimates
    8. Kalman Filter Pose Estimation Methods
    9. Partially Constrained Pose
      1. Incomplete Information
      2. Intrinsic Degrees of Freedom
    10. Projective Transformation
      1. Least-square Estimates
      2. Minimal Data Estimation
      3. Robust Estimates
    11. Similarity Transformation
      1. Least-square Estimates
      2. Minimal Data Estimation
      3. Robust Estimates

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