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Morel J.-M., Solimini S. — Variational Models for Image Segmentation: with seven image processing experiments (Progress in Nonlinear Differential Equations and Their Applications)
Morel J.-M., Solimini S. — Variational Models for Image Segmentation: with seven image processing experiments (Progress in Nonlinear Differential Equations and Their Applications)



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Название: Variational Models for Image Segmentation: with seven image processing experiments (Progress in Nonlinear Differential Equations and Their Applications)

Автор: Morel J.-M., Solimini S.

Аннотация:

This text contains a synthesis and a mathematical analysis of a wide set of algorithms and theories whose aim is the automatic segmentation of digital images as well as the understanding of visual perception. A common formalism for these theories and algorithms is obtained in variational form. Thanks to the formalization, mathematical questions about the soundness can be raised and answered. Perception theory has to deal with the complex interaction between regions and \"edges\" (or boundaries) in an image; in the variational segmentation energies, \"edge\" terms compete with \"region\" terms in a way which is intended to impose regularity on both regions and boundaries. The first part of the book presents a unified presentation of the evidence in favour of the conjecture. It is proven that the competition of one-dimensional and two-dimensional energy terms in a variational formulation cannot create fractal-like behaviour for the edges. The proof of regularity for the edges of a segmentation constantly involves concepts from geometric measure theory, which proves to be central in image processing theory. The second part of the book provides a fast and self-contained presentation of the classical theory of rectifiable sets (the \"edges\") and unrectifiable sets (\"fractals\").


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1994

Количество страниц: 264

Добавлена в каталог: 14.10.2011

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$A^{(k)}$, $A_{(m)}$, reflected sets of A      9.1
$C(x, W, \varphi)$, angular neighbourhood of W      (9.68)
$C(x, W, \varphi, r)$, angular neighbourhood of V at x with angle $\varphi$ and radius r      12.1
$K_{m, n}$      (8.9)
$P_{\theta}$, orthogonal projection onto the hyperplane orthogonal to the direction $\theta$      12.3
$\alpha$-set      8 preliminary
$\bar{d}^{\alpha}_{K}(x)$, upper spherical density of K at x      8.2
$\frac{\partial u}{\partial n}$, derivative of u in the direction normal to K      13.1
$\mathcal{D}^{\alpha}_{K}(A)$, $\mathcal{H}^{\alpha}$-mean density of K on A      8.1
$\omega^{s}_{j}(u)$, oscillation of u      (14.22)
$\sigma$-algebra      6.1
$\underline{d}^{\alpha}_{K}(x)$, lower spherical density of K at x      8.2
$\varepsilon$-net      10.1
$|\mathcal{A}|_{\alpha}$      6.1
1-normal segmentation      5.2
2-normal segmentation      5.2
2.1 sketch      4.1 3.3
2.5 sketch      4.1
Affine invariant (segmentation)      4.5
Ahlfors set      15.5 (15.30)
Almost covering      (7.7)
Almost everywhere (a.e.)      8 preliminary
Almost isolated (point of a segmentation)      (14.17)
Alvarez — Guichard      2.4
Alvarez — Lions — Morel      2.4
Ambrosio — Tortorelli      4.4
Angular neighbourhood      (9.66)
Anisotropic diffusion      2.3
Approximated tangent space      (9.20)
Ascoli — Arzela theorem      5.2
B(x, y), cylinder      12.1
Ballester — Caselles — Gonzalez      4.4
Beaulieu — Goldberg      3.3
Besl — Jain      3.1
Blake — Zisserman      4.4
Borel set      6.2
Brice — Fennema      3.3
Brouwer fixed point theorem      (11.16)
C(x, y), double cone      12.1
Caratheodory criterion      6.23
Caselles — Catte — Coll — Dibos      3.3
Catte — Coll      2.2
Causality      1.2
Chambolle      4.4
Channel      3.1
Coarea formula      16.2
Cohen — Vinet — Sander — Gagalowicz      3.3
Common boundary of two regions      5.2
Compact metric space      10.1
Complet metric space      10.1
Connected, path connected, connected set with finite Hausdorff measure      6.3
Constrained optimization      3.3
Convergence of 2-normal segmentations      5.6
Convergence of compact sets      (10.1)
Convergence of piecewise constant segmentations      5.2
Convergence of rectifiable curves      5.2
Convergence of tangent affine spaces      (9.59)
Corners      4.1
Covering      6.1 see
Covering lemma      7 (13.15)
Covering, almost covering      (7.7) (11.23)
Covering, disjoint covering      (7.1) (7.8) (7.10) (7.12) (7.14)
Curvature of edges      3.2
Curvature of smooth curves      5.2
Curvature, curvature estimate for 2-normal segmentations      (5.8)
d(X, Y), Hausdorff distance of two sets      10.1
Denoising      2.4
Densities, $\mathcal{H}^{\alpha}$-mean, upper and lower density      (8.1)
Densities, conic density      (9.68)
Densities, density of the range of a Lipschitz map      (11.13)
Densities, lower conic density (of unrectifiable sets)      12 (12.4)
Densities, spherical densities      8.2
Density properties, atomization condition, first and second projection property      15.4
Density properties, essentiality property      15.1
Density properties, uniform concentration property      (10.9) 15.4
Density properties, uniform density property      15.2
Differentiability of Lipschitz maps      (11.10)
Differentiability of optimal 2-normal segmentations      (5.8)
Dual curve lemmas      16.2
EDGE      1.1
Edge (of a 1-normal segmentation)      5.2
Edge growing      3.2
Edge map      4.2
Elliptic equation      13
Enhancement      2.4
Essentiality (of an edge set)      (14.17)
Excision lemmas      (14.12) (14.15) (14.17)
Excision method      (14.1)
Existence of a curve containing the edge set      16.3 16.4
Existence of Mumford — Shah minima, general case      (15.44)
Existence of Mumford — Shah minima, piecewise constant case      (5.1)
Fubini lemma      (12.24) (14.19) (14.43)
Gauss function      2.1
Gestalt psychology      4.1
Golab's theorem      (10.19)
Graduated Non Convexity algorithm      4.4
Green formulas      13.2 (13.9) (13.10) (13.13) (13.15)
Haralick      3.3
Haralick — Shapiro      1.1
Harmonic function      13.3
Hausdorff distance      10.1
Hausdorff outer measure      6.1 (6.1)
Heat equation      2.1
Hildreth — Marr      2.1
Hybrid (edge and region growing) method      3.3
I(u), I(K), $I_{B}(u)$, terms of the Mumford — Shah energy      13.1
Illusory contours      1.1
Image      5.1
Image, Multichannel image      5.1 5.4
Irregular set      6.4 (8.23)
Irregular set, example of      (8.30)
Isodiametric inequality      7.3
Isoperimetric inequality      5.2
Jones      3.2
Jordan curve      5.2
Jordan curve lemma      (5.2)
Kass — Witkin — Terzopoulos      3.2
Koepfler — Lopez      5.4
Koepfler — Morel — Solimini      5.4
Lebesgue measure      7.3
Leclerc      4.3
Length (of a rectifiable curve, of a segmentation)      5.2
Length, relation between length and Hausdorff measure      11.3
Leonardis — Gupta — Bajcsy      3.1
Lowe      3.2
Lower semicontinuity of Hausdorff measure...      10.4 (10.14)
Lower semicontinuity of the general Mumford — Shah energy      (13.6)
Lower semicontinuity of the Mumford — Shah energy for 2-normal segmentations      (5.6)
Mallat — Zhong      2.1
Maximum principle      (13.5)
Mean curvature motion      2.4
Measurability of sets defined from density properties      8.29
Measurability, (in the sense of Caratheodory)      (6.6)
Merging      5.2 5.3
Minimality assumption (M)      (14.4)
Minimum description length      4.3
Montaneri — Martelli      3.2
Multiscale edge linking      3.2
Multiscale region growing      3.1
Mumford — Shah      3.1
Negligible set      6.1
Neumann boundary condition      (13.1)
Nitzberg — Mumford      4.1
Nitzberg — Shiota      2.2
Nonuniqueness of Mumford — Shah minima      (15.49)
Nordstroem      4.4
Osher — Rudin      2.1 2.4
Osher — Sethian      2.3
Outer measure      (6.2) (6.3)
Pavlidis      3.3
Pavlidis — Horowitz      3.1
Pavlidis — Liow      3.3
Perkins      1.1
Perona — Malik      2.2
Precompact metric space      10.1
Projection density arguments      (9.36) (9.47) (9.53)
Projection property (first and second)      15.4
Projection property of unrectifiable sets      (6.36) 12.3
Pyramidal data structure      3.1
Pyramidal segmentation algorithm      5.4
Rectifiability properties, atomized curve property      16.1
Rectifiability properties, concentrated      16.1
Rectifiability properties, quantified nonconnectedness property      16.1
Rectifiable curve      5.2 11.3
Rectifiable, $\alpha$-rectifiable surface, set      11.1
Rectifiable, $\alpha$-simply rectifiable curves and sets      11.3
Reflected of a set      9.1
Reflection arguments (for regular sets)      (9.4) (9.6)
Reflection lemmas      (9.9) (9.14)
Region growing      3.1
Regions      1.1
Regions of a segmentation      5.2
Regions, overlapping regions      4.1
Regular set      (8.23)
Regularity of rectifiable sets      11.2 (11.19)
Regularity, equivalence with rectifiability      (12.10)
Regularity, regularity of Mumford — Shah minimal edge set      15.4
Relative boundary      5.2
Reliable set      (13.12)
Richardson      4.4
Saliency map      4.2
Sard lemma      (11.11)
Scale in multiscale analysis      1.2
Scale in piecewise constant segmentation      5.1
Scale space      2.1
Segmentation      1.1
Segmentation as a partition      5.2
Segmentation, affine invariant      4.5
Selective smoothing      2.2
Sha'ashua — Ullman      4.2
Small oscillation covering      14.2
Small oscillation covering lemma      16.1
Snakes method      3.3
Sobolev space      13.1
Split and merge      3.1
T-junctions      4.1
Tangent space to a regular set      9.3 (9.66) (9.68)
Tangent space to the graph of a differentiable map      (11.4)
Tangent space, nonexistence for an unrectifiable set      (12.9)
Terzopoulos      3.1
Texture density      3.1
Uniformly concentrated sets      10.4
Unrectifiable      6.4
Unrectifiable, purely unrectifiable set      11.1
Variational formulation      2.1
Variational formulation of edge linking methods      3.2
Variational formulation of region growing methods      3.1
Viscosity solution      2.3
Vitali Covering Lemma      7.14
Vitali covering, approximate covering      7.2
Weak membrane model      4.4
Weiss — Boldt      3.2
Witkin — Koenderink      2.1
Zero crossing of laplacian      3.2
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