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