Histograms of images valued in the manifold of colours endowed with perceptual metrics

28/10/2015
Publication GSI2015
OAI : oai:www.see.asso.fr:11784:14312
DOI : http://dx.doi.org/10.1007/978-3-319-25040-3_81You do not have permission to access embedded form.

Résumé

We address here the problem of perceptual colour histograms. The Riemannian structure of perceptual distances is measured through standards sets of ellipses, such as Macadam ellipses. We propose an approach based on local Euclidean approximations that enables to take into account the Riemannian structure of perceptual distances, without introducing computational complexity during the construction of the histogram.

Histograms of images valued in the manifold of colours endowed with perceptual metrics

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            <title>Histograms of images valued in the manifold of colours endowed with perceptual metrics</title></titles>
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We address here the problem of perceptual colour histograms. The Riemannian structure of perceptual distances is measured through standards sets of ellipses, such as Macadam ellipses. We propose an approach based on local Euclidean approximations that enables to take into account the Riemannian structure of perceptual distances, without introducing computational complexity during the construction of the histogram.

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Color Histograms using the perceptual metric October 28, 2015 Emmanuel Chevalliera, Ivar Farupb, Jesús Anguloa a CMM-Centre de Morphologie Mathématique, MINES ParisTech; France b Gjovik University College; France emmanuel.chevallier@mines-paristech.fr 1/16 Color Histograms using the perceptual metric Plan of the presentation Formalization of the notion of image histogram Perceptual metric and Macadam ellipses Density estimation in the space of colors 2/16 Color Histograms using the perceptual metric Image histogram : formalization I : Ω → V p → I(p) Ω: support space of pixels: rectangle/parallelepiped. V: the value space (Ω, µΩ), (V , µV ), µΩ and µV are induced by the choosen geometries on Ω and V . Transport of µΩ on V : I∗(µΩ) Image histogram: estimation of f = dI∗(µΩ) dµV 3/16 Color Histograms using the perceptual metric pixels: p ∈ Ω, uniformly distributed with respect to µΩ {I(p), p a pixel }: set of independent draws of the "random variable" I Estimation of f = dI∗(µΩ) dµV from {I(p), p a pixel }: → standard problem of probability density estimation 4/16 Color Histograms using the perceptual metric Perceptual color histograms I : Ω → (M = colors, gperceptual ) p → I(p) Assumption: the perceptual distances between colors is induced by a Riemannian metric The manifold of colors was one of the rst example of Riemannian manifold, suggested by Riemann 5/16 Color Histograms using the perceptual metric Macadam ellipses: just noticeable dierences Chromaticity diagram (constant luminance): Ellipses: elementary unit balls → local L2 metric 6/16 Color Histograms using the perceptual metric Lab space The Euclidean metric of the Lab parametrization is supposed to be more perceptual than other parametrizations Figure: Macadam ellipses in the ab plan However, the ellipses are clearly not balls 7/16 Color Histograms using the perceptual metric Modiction of the density estimator Density → local notion. No need of knowing long geodesics Small distances → local approximation by an Euclidean metric Notations: dR: Perceptual metric ||.||Lab: Canonical Euclidean metric of Lab ||.||c: Euclidean metric on Lab induced by the ellipse at c Small distances around c: ||.||c is "better" than ||.||Lab 8/16 Color Histograms using the perceptual metric Modiction of the density estimator Standard kernel estimator: ˆf (x) = 1 k pi ∈{pixels} 1 r2 K ||x − I(pi )||Lab r Possible modication K ||x − I(pi )||Lab r → K ||x − I(pi )||I(pi ) r where ||.||I(pi ) is an Euclidean distance dened by the interpolated ellipse at I(pi ). 9/16 Color Histograms using the perceptual metric Generally, at c a color: limx→c ||x − c||c dR(x, c) = 1 = limx→c ||x − c||Lab dR(x, c) Thus, ∃A > 0 such that, ∀R > 0, ∃x ∈ BLab(c, R), A < ||x − c|| dR(x, c) − 1 . while ∃Rc = Rc,A such that, ∀x ∈ BLab(c, Rc), ||x − c||c dR(x, c) − 1 < A. hence supBLab(c,Rc ) ||x − c||c dR(x, c) − 1 < A < supBLab(c,Rc ) ||x − c|| dR(x, c) − 1 . 10/16 Color Histograms using the perceptual metric When the scaling factor r is small enough: r ≤ Rc and Bc(c, r) ⊂ BLab(c, Rc) x ∈ B(c, Rc), K ||x−c||c r better than K ||x−c||Lab r . x /∈ B(c, Rc), K ||x−c||c r = K ||x−c||Lab r = 0 11/16 Color Histograms using the perceptual metric Interpolation of a set of local metric: a deep question... What is a good interpolation? Interpolating a function: minimizing variation with respect to a metric. Interpolating a metric? No intrinsic method: depends on a choice of parametrization. Subject of the next study 12/16 Color Histograms using the perceptual metric Barycentric interpolation in the Lab space 13/16 Color Histograms using the perceptual metric Volume change (a) (b) Figure: (a): color photography (b): Zoom of the density change adapted to colours present in the photography 14/16 Color Histograms using the perceptual metric experimental results (a) (b) (c) Figure: The canonical Euclidean metric of the ab projective plane in (a), the canonical metric followed by a division by the local density of the perceptual metric in (b) and the modied kernel formula in (c). 15/16 Color Histograms using the perceptual metric Conclusion A simple observation which improve the consistency of the histogram without requiring additional computational costs Future works will focus on: The interpolation of the ellipses The construction of the geodesics and their applications Thank you for your attention 16/16 Color Histograms using the perceptual metric