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This paper extends the BV-L1 variational nonlinear image-decomposition approach, useful for image processing, to a genuine color-image decomposition approach. For utilizing inter-channel color cross-correlations, we introduce TV norms of color differences and TV norms of color sums into the BV-L1 energy functionals to be minimized, and then derive denoising-type decomposition-algorithm with an over-complete wavelet transform, by applying the Besov-norm approximation to the variational problem. Our method decomposes a noisy color image without producing undesirable low-frequency colored artifacts in its separated BV-component, and achieves desirable high-quality color-image decomposition, robust against colored random noise. We apply our decomposition method to color-image denoising.