Geometry and mathematics

A part of the CVonline computer vision resource summarizing the geometric and mathematical methods commonly encountered in computer vision and image processing.

1. Basic Representations

   1. Coordinate systems

      1. Cartesian coordinate system

      2. Cylindrical coordinate system

      3. Hexagonal coordinate system (see external links)

      4. Log-Polar coordinate system

      5. Polar coordinate system

      6. Spherical coordinate system

   2. Digital topology

   3. Dual space

   4. Homogeneous coordinates

   5. Pose/Rotation/Orientation Representations

• • 1. Axis-angle representation

    2. Clifford algebra

    3. Euler angles

    4. Exponential map

    5. Quaternion/Dual quaternion

    6. Rotation matrix

    7. Pitch/Yaw/Roll

2. Distance and similarity metrics

• 1. Affine distance

  2. Algebraic distance

  3. Bregman divergence

  4. Bhattacharyya distance

  5. Chi-square test/metric

  6. Curse of dimensionality

  7. Earth mover's distance

  8. Euclidean distance

  9. Fuzzy intersection

  10. Hausdorff distance

  11. Jaccard Index

  12. Jeffrey divergence

  13. Jensen-Shannon Divergence

  14. Kullback–Leibler divergence

  15. Mahalanobis distance

  16. Manhattan/City block distance

  17. Minkowski distance

  18. Procrustes analysis

  19. Quadratic form

  20. Sørensen-Dice coefficient

  21. Specific structural similarity

      1. Curve similarity

      2. Region similarity

      3. Volume similarity

3. Elementary mathematics for Vision

• 1. Coordinate systems/Vectors/Matrices/Derivatives/Gradients/Probability

  2. Derivatives in sampled images

4. Mathematical optimization

• 1. Golden section search

  2. Lagrange multipliers/Constraint optimization

  3. Multi-dimensional optimization

     1. Random optimization

     2. Global optimization

        1. Ant colony optimization

        2. Downhill simplex

        3. Genetic algorithms

        4. Graduated optimization

        5. Markov random field optimization

        6. Particle swarm optimization

        7. Simulated annealing

     3. Optimization with derivatives

• • • 1. Levenberg–Marquardt

      2. Gradient descent

      3. Quasi-Newton method

• 4. Model selection

  5. Variational methods

5. Linear algebra for computer vision

• 1. Eigenfunction

  2. Eigenvalues and eigenvectors

  3. Norms

     1. Frobenius

     2. Hamming

     3. L or p norms (1, 2, ∞)

     4. Manhatten or taxi

     5. Nuclear

     6. Spectral

  4. Principal component and Related Approaches

• • 1. Dimensionality reduction

    2. Linear discriminant analysis

    3. Factor analysis

    4. Fisher's linear discriminant

    5. Independent component analysis

    6. Kernel linear discriminant analysis

    7. Kernel principal component analysis

    8. Locality preserving projections

    9. Non-negative matrix factorization

    10. Optimal dimension estimation

    11. Sufficient dimension reduction

    12. Principal component analysis/Karhunen–Loève theorem

    13. Principal geodesic analysis

    14. Probabilistic principal component analysis

    15. Rao–Blackwell theorem

• 5. Sammon projection

  6. Singular value decomposition

  7. Structure tensor

6. Multi-sensor/Multi-view geometries

• 1. 3D reconstruction

     1. 3D shape from 2D projections

     2. 3D reconstruction from multiple images

     3. Slice-based reconstruction

  2. Projective reconstruction

  3. Baseline stereo

• • 1. Narrow baseline stereo

    2. Wide baseline stereo

• 4. Binocular stereo algorithms

• • 1. Cooperative stereo algorithms

    2. Binocular disparity

       1. Subpixel disparity

    3. Dense stereo matching approaches

    4. Dynamic programming (stereo)

    5. Feature matching stereo algorithms

    6. Gradient matching stereo algorithms

    7. Image rectification

• • • 1. Planar rectification

      2. Polar rectification

• • 8. Log-polar stereo

    9. Multiresolution analysis

    10. Panoramic image stereo algorithms

    11. Phase matching stereo algorithms

    12. Region matching stereo algorithms

    13. Weakly/Uncalibrated stereo approaches

    14. Spherical stereo

• 5. Epipolar geometry/Multi-view geometry

• • 1. Absolute conic

    2. Absolute quadric

    3. Essential matrix

    4. Fundamental matrix

    5. Grassmannian space/Plücker embedding

    6. Homography tensor

    7. Transfer and novel view synthesis

    8. Trifocal tensor

• 6. Image-based modeling and rendering

  7. Plenoptic modeling

  8. Image feature correspondence

• • 1. Active stereo

    2. Disparity gradient limit (feature correspondence)

    3. Epipolar constraint

    4. Feature contrast

    5. Feature orientation

    6. Grey-level similarity (feature correspondence)

    7. Lipschitz continuity

    8. Surface continuity

    9. Surface smoothness

    10. View consistency constraint

• 9. Scene reconstruction/Surface interpolation

• • 1. Adaptive mesh refinement

    2. Constrained reconstruction

    3. Thin plate models

    4. Texture synthesis/Texture mapping

    5. Triangulation

    6. Volumetric reconstruction

       1. Visual hull

• 10. Trinocular (and more) stereo

7. Parameter Estimation

• 1. Bayesian methods

  2. Constrained least squares

  3. Linear least squares

  4. Optimization

  5. Robust techniques

8. Probability and Statistics for Computer Vision

• 1. Autoregression

  2. Bayes estimator

  3. Bayesian inference networks

  4. Canonical correlation

  5. Causal models

  6. Correlation and dependence

  7. Covariance and Mahalanobis distance in Vision

  8. Dempster–Shafer theory

  9. Density estimation

  10. Gaussian or Normal distribution

  11. Heteroscedastic noise

  12. Hidden Markov models

  13. Homoscedastic noise

  14. Information theory

  15. Kalman filters

      1. Unscented Kalman filters

  16. Kernel regression

  17. Least mean squares estimation

  18. Least median square estimation and estimators

  19. Log-normal distribution

  20. Logistic regression

  21. Markov chain/Markov chain Monte Carlo methods

  22. Markov random field

• • 1. Applications

    2. Conditional random fields

    3. Multi-level Markov random fields

    4. Optimization methods

       1. Gibbs sampling

       2. Graduated optimization

       3. Graph cuts in computer vision

       4. Iterated conditional modes

       5. Simulated annealing

• 23. Maximum likelihood

  24. Mixture models and expectation-maximization (EM)

• • 1. Gaussian mixture model

    2. Categorical mixture model

• 25. Model/Curve fitting

  26. Monte Carlo method

  27. Multimodal distribution

  28. Normalization

  29. Non-parametric statistics

• • 1. Non-parametric regression

    2. Kernel density estimation

• 30. Point process

  31. Poisson distribution

  32. Probability axioms

  33. Random number generation

  34. Robust estimators

  35. Statistical hypothesis testing/Analysis of variance

  36. von-Mises-Fisher and other directional statistics

9. Projective geometry/Projective transformations

• 1. Affine projection model/Affine transformation

  2. Anamorphic projection/Catadioptric system

  3. Central cylindrical projection

  4. Orthographic projection

  5. Map projection

  6. Homography

  7. Hierarchy of geometries

  8. Perspective projection

  9. Projective plane

  10. Projective space

  11. Real camera projection

  12. Similarity matrix

  13. Weak-perspective

      1. Tomasi-Kanade factorization

10. Projective invariants/cross-ratio

• 1. Absolute points (points at infinity)

  2. Affine invariants

     1. Affine geometry of curves

  3. Collineation

  4. Conics/Quadrics

  5. Coplanarity

  6. Differential invariants

  7. Duality

  8. Integral invariants

  9. Laguerre formula

  10. Pencils

  11. Quasi-invariants

  12. Structural invariants

• • 1. Cartan's equivalence method

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

• 1. Geometric Morphometrics

  2. Kendall´s Shape Space

  3. Points and local invariants

     1. Scale-invariant feature transform

  4. Curves and Curve Invariants

• • 1. Affine curvature

    2. Arc length

    3. Bending energy

    4. Chord distribution

    5. Curvature, Torsion of a curve, Radius of curvature

    6. Differential geometry, Frenet–Serret formulas

    7. Invariant Points: Inflections/Bitangents

• 5. Image regions and region invariants

• • 1. Compactness measure of a shape

    2. Area

    3. Perimeter

    4. Center of mass, Centroid

    5. Eccentricity, Elongatedness

    6. Euler number/Genus

    7. Extremal points

    8. Feret's diameter

    9. Fourier descriptors

    10. Minimum bounding rectangle

    11. Image moments

        1. Affine moments

        2. Bessel-Fourier moments

        3. Binary moments

        4. Color moments

        5. Central moments

        6. Eigenmoments

        7. Fourier-Mellin moment invariants

        8. Gaussian-Hermite moments

        9. Texture moments

        10. Hahn moments

        11. Krawtchouk moments

        12. Legendre moments

        13. Orthogonal moments

        14. Racah moments

        15. Chebyshev moments

        16. Zernike and velocity moments

    12. Orientation

    13. Sphericity

    14. Rectangularity

    15. Rectilinearity

    16. Roundness

    17. Topological invariants

• • • 1. Euler characteristic

• 6. Differential geometry of surfaces

• • 1. Parametric surfaces

    2. Common shape classes and representations

       1. Cone representations

       2. Cyclide

       3. Cylinder representations

       4. Ellipsoid/Sphere Representations

       5. Thin plate splines

       6. Plane representations

       7. Polyhedra representations

       8. Quadric representations

       9. Torus representations

    3. Fundamental surface forms

• • • 1. First fundamental form

      2. Second fundamental form

• • 4. Gauge coordinates

    5. Hessian

    6. Laplace–Beltrami operator

    7. Metric derivative

    8. Principal curvature 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. Minimal surface

      7. Parabolic points

      8. Ridges

      9. Umbilics

• • 9. Quadratic variation

    10. Ricci flow

    11. Surface area

    12. Surface normals and tangent planes

    13. Orientability

• 7. Symmetry

• • 1. Affine symmetry

    2. Bilateral symmetry

    3. Rotational symmetry

    4. Skew symmetry

• 8. Volumes

• • 1. Elongatedness

    2. 3D moments and moment invariants

• 9. Volume

13. Transformations (geometric), registration and pose estimation methods

• 1. Poste estimation

  2. 2D to 2D pose estimation

     1. Methods

  3. 2D to 3D pose estimation

• • 1. Methods

• 4. 3D to 3D pose estimation

• • 1. Methods

• 5. Affine transformation

• • 1. Minimal data estimation

• 6. Bundle adjustment

  7. Euclidean transformation

• • 1. Least-square euclidean transformation estimates

    2. Minimal data euclidean transformation estimation

    3. Robust euclidean transformation estimates

• 8. Homographic transformation

• • 1. Least-square homography transformation estimates

    2. Robust homography transformation estimates

• 9. Kalman filter pose estimation methods

  10. Partially constrained pose

• • 1. Incomplete information

    2. Intrinsic degrees of freedom

• 11. Projective transformation

• • 1. Direct linear transformation

    2. Robust estimates

• 12. Similarity transformation

  13. Articulated body pose estimation