Summary: This paper studies the total variation regularization with an \(L^1\) fidelity term (TV-\(L^1\)) model for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry subproblems. Using this result we show that the TV-\(L^1\) model is able to separate image features according to their scales, where the scale is analytically defined by the \(G\)-value. A number of other properties including the geometric and morphological invariance of the TV-\(L^1\) model are also proved and their applications discussed.