  The presence of noise is an intrinsic problem in acquisition processes for
digital images. One way to enhance images is to combine the forward and
backward diffusion equations. However, the latter problem is well known to be
exponentially unstable with respect to any small perturbations on the final
data. In this scenario, the final data can be regarded as a blurred image
obtained from the forward process, and that image can be pixelated as a
network. Therefore, we study in this work a regularization framework for the
backward diffusion equation on graphs. Our aim is to construct a spectral
graph-based solution based upon a cut-off projection. Stability and convergence
results are provided together with some numerical experiments.
