$ \chi^2 $ test for total variation regularization parameter selection

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges. The scaling or regularization parameter in the TV process defines the amount of denoising, with a value of zero giving a result equivalent to the input signal. The discrepancy principle is a classical method for regularization parameter selection whereby data is fit to a specified tolerance. The tolerance is often identified based on the fact that the least squares data fit is known to follow a \begin{document}$ \chi^2 $\end{document} distribution. However, this approach fails when the number of parameters is greater than or equal to the number of data. Typically, heuristics are employed to identify the tolerance in the discrepancy principle and this leads to oversmoothing. In this work we identify a \begin{document}$ \chi^2 $\end{document} test for TV regularization parameter selection assuming the blurring matrix is full rank. In particular, we prove that the degrees of freedom in the TV regularized residual is the number of data and this is used to identify the appropriate tolerance. The importance of this work lies in the fact that the \begin{document}$ \chi^2 $\end{document} test introduced here for TV automates the choice of regularization parameter selection and can straightforwardly be incorporated into any TV algorithm. Results are given for three test images and compared to results using the discrepancy principle and MAP estimates.