where ∥ ⋅ ∥ τ is the total variation norm.
where is the total variation of on and (1.4).
A well-known example of such methods is the Total Variation minimization, which promotes images with sparse gradient.
A measure of the oscillation is the total variation which is given by: {{TV}}left( {Q^{n} } right) = mathop sum limits_{i = - infty }^{ + infty } left| {Q_{i}^{n} - Q_{i - 1}^{n} } right| (10 where TV is total variation.
where the first term is the total variation of u, and the second term is data fitting term, respectively, and λ is the parameter.
The first method that could be considered to fall under the image decomposition framework is the total variation model of Rudin, Osher and Fatemi [7].
There is also the polar decomposition D s u = D s u → | D s u |, where | D s u | is the total variation measure of D s u.
In Equation 22a, ∫ Ω |∇ϕ|δ ϵ dx is not the total variation of ϕ, but its equivalent formula ∫ Ω |∇H ϵ |dx is the total variation of H ϵ. Based on this observation, we can introduce a dual variable to replace ∫ Ω |∇H ϵ |dx with its dual formula Sup p → : p → ≤ 1 ∫ Ω H ϵ ϕ ∇ ⋅ p → dx.
where p, q are continuous periodic functions with period T > 0, p ∈ C n ( R, R ) with | p ( t ) | ≠ 1, f, g ∈ C ( R, R ), r > 0, n is a positive integer, σ ∈ R, α : [ − r, 0 ] → R + is a bounded variation function, ⋁ − r 0 = 1 and α ( 0 ) ≠ α ( − r ), where ⋁ − r 0 is the total variation of α ( s ) over [ − r, 0 ].
