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The aim is to obtain efficient algorithms for image regularisation optimised for removing different types of noise. One can accomplish this by combining total variation regularisation with a noise-specific way to measure the fidelity between the noisy and the denoised images. To obtain a minimum of the resulting functional, a quasi-Newton method is proposed, which converges faster than the commonly used method of gradient descent. A unified algorithmic and theoretical framework for a general class of data-fidelity terms is presented. As examples, we consider Poisson noise and impulse noise.