Decomposing an image into structure and texture is an important procedure for image understanding and analysis. Structure retains object hues and sharp edges whilst texture contains oscillating patterns of an observed image. The classical Vese–Osher model has been used for image decomposition, but its resulting structure image tends to show the undesirable staircase effect. Second order variational models that use a bounded Hessian regulariser have been proposed to remedy this side effect, but they tend to blur edges of objects in structure components. In this paper, we propose an edge-weighted second order variational model for image decomposition, which is able to eliminate staircase effects and preserve object edges. To avoid directly calculating the high order nonlinear partial differential equations of the proposed model, a fast split Bregman algorithm is developed, which uses the fast Fourier transform and analytical generalised soft thresholding equations. Extensive experiments demonstrate that the proposed variational image decomposition model outperforms state-of-the-art first and second order image decomposition models. By removing the texture component from the original noisy image, the effectiveness of the proposed model for image denoising has also been validated.