Costum Prior Distribution For TV Denoising

Hello I am working on an inverse problem, using the Total Variation (TV) regularization. However, In the literature, the problems are mainly solved using optimization schemes. Nevertheless, the problem can be thought of in the Bayesian case as problem where the likelihood is normal and the prior is proportional to the exponential of the negative TV. In summary I am trying to create a custom continuous prior distribution that is : p(\theta|\alpha, \beta)\propto (\alpha \beta)^p exp\{-\sum_{i=1}^{i=p} [\sqrt{(\alpha\nabla_i^h\theta)^2+(\beta\nabla_i^v\theta)^2}]\}. where \nabla^v\theta and \nabla^h\theta are respectively the vertical and the horizontal gradients of the image \theta (I have already coded these two gradients). My main question is just the part of coding the distribution p(\theta|\alpha, \beta). Any help will be much appreciated.