High-order variational models are powerful methods for image processing and analysis, but they can lead to complicated high-order nonlinear partial differential equations that are difficult to discretise to solve computationally. In this paper, we present some representative high-order variational models and provide detailed descretisation of these models and numerical implementation of the split Bregman algorithm for solving these models using the fast Fourier transform. We demonstrate the advantages and disadvantages of these high-order models in the context of image denoising through extensive experiments. The methods and techniques can also be used for other applications, such as image decomposition, inpainting and segmentation.