By Tripathi M. M., Kim J., Kim S.
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Additional resources for A basic inequality for submanifolds in locally conformal almost cosymplectic manifolds
Deﬁnition of a simpliﬁed q function Let us deﬁne a simpliﬁed q function by observing the construction of the transformed function θ and the evolution of this construction in every point x of the domain of deﬁnition of the initial function f. To do so, let us come back to the design of the transformations in the binary case by replacing set X by its indicator function kX and by observing how the indicator function kri (x) of the residues at point x evolves for each transformation step. In the case of an indicator function, this evolution is obvious: all krj (x) are equal to zero except the indicator function of the residue ri containing x.
To illustrate this, let us introduce two new residual operators named respectively ultimate opening and quasi-distance. Ultimate opening Let us consider the residual operator ν where the primitives ψi and ζi are respectively an opening by balls of size i and an opening by balls of size i+1: ψ i = γi ζi = γi+1 ν = sup (γ γi − γi+1 ) i∈I The operator ν does not present any interest in the binary domain. Indeed, in that case, it is easy to show that it is equal to the identity. In the numerical domain, it replaces the initial image by an union of the most signiﬁcant cylinders included in the sub-graph of the initial function.
The second step takes the original image as input and the ﬁltered image as a pilot, with less destructive parameters, and preserves even more the ﬁner details, while removing a lot of the noise. In practice, we have found that performing those two steps only once is enough to reduce the noise dramatically (see ﬁgure 7), although further iterations may be required, depending on the image and the noise. This method is also very useful for color images, since the amoeba-based pilot image provides better color coupling through the use of an appropriate color distance than simply merging the results of a Gaussian ﬁltering of each channel independently.