Notations
• Let u : Ω → R^n be a discrete image, with n = 1 for a gray level image, n = 3 for a color image, and where Ω ⊂ Z^2 is the bounded image domain.
• Let T(u) be the image after color or contrast modification (Histogram equalization, color transfer, gamma transform, etc).
• We define M(u) := T(u) − u as the transportation map of the image u.
Principle
The image T(u) often exhibits some unpleasant artefacts, such as JPEG blocs, color inconsistancies, noise enhancement or loss of details (texture washing).
We propose to regularize the transportation map M thanks to the operator Y_u, a weighted average with weights depending on the similarity of pixels in the original image u. The effect of this operator on an image v : Ω → R^n with n ≥ 1 is defined as
Y_u (v) : x ∈ Ω → R^n = 1 / c(x) * Σ_{y ∈ N (x)} v(y) · w_u (x, y)
with weights w_u (x, y) = exp { || u(x)−u(y) ||^2 / σ^2 },
where ||.|| stands for the Euclidean distance in R^n ,
where N (x) = x+N (0) ⊂ Ω with N (0) a spatial neighborhood of 0,
where σ is a tuning parameter of the method and
C(x) is the normalization constant c(x) = Σ_{y ∈ N (x)} w_u (x, y) .
Observe that if we apply Y_u to the image u, we obtain the Yaroslavsky filter.
The regularization of the image T(u), referred to as Transportation Map Regularization (TMR), is then defined as
TMR_u ( T(u) ) := u + Y_u ( M(u) ) = Y_u ( T(u) ) + u − Y_u (u) .
The TMR filter is iterated to remove the aforementioned artefacts. In order to control the iterations of the TMR filter, we compute at each iteration a convergence map, written C and defined at each pixel as follows:
C(x) = || Y_u^k [ M(u(x)) ] − Y_u^(k-1) [ M(u(x)) ] || .
We then consider that there is numerical convergence in pixel x when C(x) < τ, and the TMR filter is only
applied to pixels for which the convergence map is greater than the threshold τ.
Experimental settings
For all the following experiments, the parameters have been set to:
for 8n-bits images (i.e. when n=3, the RGB color space is [0, 255]^3).
For more details, see [1] and [2].