  This paper considers the constrained total variation (TV) denoising problem
for complex-valued images. We extend the definition of TV seminorms for
real-valued images to dealing with complex-valued ones. In particular, we
introduce two types of complex TV in both isotropic and anisotropic forms. To
solve the constrained denoising problem, we adopt a dual approach and derive an
accelerated gradient projection algorithm. We further generalize the proposed
denoising algorithm as a key building block of the proximal gradient scheme to
solve a vast class of complex constrained optimization problems with TV
regularizers. As an example, we apply the proposed algorithmic framework to
phase retrieval. We combine the complex TV regularizer with the conventional
projection-based method within the constraint complex TV model. Initial results
from both simulated and optical experiments demonstrate the validity of the
constrained TV model in extracting sparsity priors within complex-valued
images, while also utilizing physically tractable constraints that help speed
up convergence.
