## Ridges in Image and Data AnalysisThe concept of ridges has appeared numerous times in the image processing liter ature. Sometimes the term is used in an intuitive sense. Other times a concrete definition is provided. In almost all cases the concept is used for very specific ap plications. When analyzing images or data sets, it is very natural for a scientist to measure critical behavior by considering maxima or minima of the data. These critical points are relatively easy to compute. Numerical packages always provide support for root finding or optimization, whether it be through bisection, Newton's method, conjugate gradient method, or other standard methods. It has not been natural for scientists to consider critical behavior in a higher-order sense. The con cept of ridge as a manifold of critical points is a natural extension of the concept of local maximum as an isolated critical point. However, almost no attention has been given to formalizing the concept. There is a need for a formal development. There is a need for understanding the computation issues that arise in the imple mentations. The purpose of this book is to address both needs by providing a formal mathematical foundation and a computational framework for ridges. The intended audience for this book includes anyone interested in exploring the use fulness of ridges in data analysis. |

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### Contents

II | 1 |

IV | 6 |

V | 9 |

VI | 10 |

VII | 13 |

VIII | 14 |

IX | 16 |

X | 17 |

XLVII | 75 |

XLVIII | 82 |

XLIX | 86 |

LI | 91 |

LII | 94 |

LIII | 97 |

LIV | 99 |

LV | 102 |

XII | 18 |

XIII | 19 |

XIV | 20 |

XV | 22 |

XVII | 23 |

XVIII | 25 |

XIX | 29 |

XX | 30 |

XXI | 35 |

XXII | 39 |

XXIII | 42 |

XXIV | 45 |

XXV | 46 |

XXVI | 47 |

XXVII | 48 |

XXVIII | 49 |

XXX | 52 |

XXXII | 53 |

XXXIII | 56 |

XXXIV | 57 |

XXXVI | 58 |

XXXVII | 60 |

XXXVIII | 61 |

XXXIX | 65 |

XL | 66 |

XLII | 68 |

XLIII | 69 |

XLIV | 70 |

XLV | 71 |

XLVI | 72 |

### Common terms and phrases

1-dimensional ridges algorithm applied atoms B-spline basis BASOC bisection boundaries calculations Christoffel symbols columns component compute contours contravariant coordinates corresponding covariant derivatives cube curvature tensor data set defined degree denote diagonal Differentiating dimensional dimensions directional derivatives edge eigensystem eigenvalues eigenvectors electron density equation Euclidean evaluation example Figure fluid gradient graph height ridge definition implementation indices intermediate tensor intersection kx++ lattice level curve level surface linear linearly independent local maximum loop manifold matrix maxima maximum medialness function method metric tensor minimization minimum molecule normal vectors notation obtain occur orthogonal orthonormal system parameterized partial derivatives permutation tensor polynomial principal curvatures protein quadratic form quantities requires ridge construction ridge point Ridge Tangents ridges and courses ridges are sought Riemann curvature tensor Riemannian scalar scale space Section solving sought in regions spline structure subregion summation symmetric tangent vectors tridiagonal vertex vertices volume rendering yields

### Popular passages

Page 207 - T. Lindeberg. Scale-Space Theory in Computer Vision. The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, 1994.

Page 204 - Object representation by cores: Identifying and representing primitive spatial regions," Vision Research 35, 1917-1930 (1995).