Curvature Measures

Federer, Herbert

doi:10.2307/1993504

Cited by 506 publications

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“…A classical result of Federer [8] or, alternatively, a recent result of Schneider [26] then imply that ∂K and g∂L intersect almost transversally for µ -almost all g ∈ G(n). In fact, Schneider's result even yields that the boundaries intersect transversally for µ -almost all g ∈ G(n).…”

Section: Introductionmentioning

confidence: 94%

“…For a convex body C ⊂ IR n and r ≥ 0, we write C r := C + rB and call C r a parallel body of C at distance r. It is easy to see that for convex bodies D, E ⊂ IR n , r > 0, K := D r and L := E r the boundaries ∂K and ∂L are sets of positive reach. It can be deduced from a special case of Federer's kinematic formula for sets with positive reach [8], Theorem 6.11, that the kinematic formulas (1.6) -(1.8) hold for such a special choice of K and L. Then one passes to the limit r 0 in these formulas. The right -hand sides behave continuously with respect to this limit.…”

Section: Introductionmentioning

confidence: 99%

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Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Hug

Schätzle

2001

Math. Nachr.

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Let K, L ⊂ IR n be two convex bodies with non-empty interiors and with boundaries ∂K, ∂L, and let χ denote the Euler characteristic as defined in singular homology theory. We prove two translative integral formulas involving boundaries of convex bodies. It is shown that the integrals of the functions t → χ(∂K ∩ (∂L + t)) and t → χ(∂K ∩ (L + t)), t ∈ IR n , with respect to an ndimensional Haar measure of IR n can be expressed in terms of certain mixed volumes of K and L. In the particular case where K and L are outer parallel bodies of convex bodies at distance r > 0, the result will be deduced from a recent (local) translative integral formula for sets with positive reach. The general case follows from this and from the following (global) topological result. Let Kr, Lr denote the outer parallel bodies of K, L at distance r ≥ 0. Establishing a conjecture of Firey (1978), we show that the homotopy type of ∂Kr ∩ ∂Lr and ∂Kr ∩ Lr, respectively, is independent of r ≥ 0 if K • ∩ L • = ∅ and if ∂K and ∂L intersect almost transversally. As an immediate consequence of our translative integral formulas, we obtain a proof for two kinematic formulas which have also been conjectured by Firey.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Hug

Schätzle

2001

Math. Nachr.

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“…(1) below) for certain geometric quantities associated to subspaces A, B ⊂ M ; the case G = SO(n) R n , M = R n being the classical kinematic formulas of Blaschke-Santaló-Federer-Chern [26], [30]. The proof was a distillation of the geometric method used in [14] and [16] to establish the classical case.…”

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confidence: 99%

Hermitian integral geometry

2011

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We give in explicit form the principal kinematic formula for the action of the affine unitary group on C n , together with a straightforward algebraic method for computing the full array of unitary kinematic formulas, expressed in terms of certain convex valuations introduced, essentially, by H. Tasaki. We introduce also several other canonical bases for the algebra of unitary-invariant valuations, explore their interrelations, and characterize in these terms the cones of positive and monotone elements.

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“…where the collection of curves are the contours of a function ψ : Ω → R. Provided regularity conditions on u and ψ, the coarea formula [1] allows the integral to be rewritten as…”

Section: Discretizationmentioning

confidence: 99%

“…This ensures that a perturbation v of v with v − v ∞ < T does not change the best-fitting stencil. The threshold is set at T = 10 −4 relative to pixel intensities in the range [0,1].…”

Section: Lines Parabolas Corners Circlementioning

confidence: 99%

Image Interpolation with Geometric Contour Stencils

Getreuer¹

2011

Image Processing on Line

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We consider the image interpolation problem where, given an image v with uniformly-sampled pixels v m,n and point spread function h, the goal is to find a function u(x, y) satisfying v m,n = (h * u)(m, n) for all m, n ∈ Z (1) so that u approximates the underlying function from which v was sampled. This article improves upon the IPOL article "Image Interpolation with Contour Stencils" [7]. In the previous work, contour stencils are used to estimate the image contours locally as short line segments. This article begins with a continuous formulation of total variation integrated over a collection of curves and defines contour stencils as a consistent discretization. This discretization is more reliable than the previous approach and can effectively distinguish contours that are locally shaped like lines, curves, corners, and circles. These improved contour stencils sense more of the geometry in the image. Interpolation is performed using an extension of the method described in the previous article. Using the improved contour stencils, there is an increase in image quality while maintaining similar computational efficiency. Source CodeANSI C source code to produce the same results as the demo is accessible on the article web page http://dx.doi.org/10.5201/ipol.2011.g_igcs. Future software releases and updates will be posted at http://dev.ipol.im/~getreuer/code.

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Curvature Measures

Cited by 506 publications

References 0 publications

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Intersections and Translative Integral Formulas for Boundaries of Convex Bodies

Hermitian integral geometry

Image Interpolation with Geometric Contour Stencils

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