The total variation (TV) model is attractive in that it is able to preserve sharp attributes in images.
However, the restored images from TV-based methods do not usually stay in a given dynamic range,
and hence projection is required to bring them back into the dynamic range for visual presentation
or for storage in digital media. This will affect the accuracy of the restoration as the projected image
will no longer be the minimizer of the given TV model. In this paper, we show that one can get much
more accurate solutions by imposing box constraints on the TV models and solving the resulting
constrained models. Our numerical results show that for some images where there are many pixels
with values lying on the boundary of the dynamic range, the gain can be as great as 10.28 decibel
in the peak signal-to-noise ratio. One traditional hindrance using the constrained model is that it
is difficult to solve. However, in this paper, we propose using the alternating direction method of
multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm
that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM
algorithm is better than some state-of-the-art algorithms for unconstrained models in terms of both
accuracy and robustness with respect to the regularization parameter.