# Question about prior in bayesian image processing

I am learning Bayesian image processing. Bayesian approach will take prior knowledge about image into account. From one material, it says knowledge is expressed via probability functions. I understand noise can be expresses as probability distribution. How can one image is expressed via probability?

From another material, it mentions Markov random field (MRF). Is it used for modeling $f$? What is the relation between Markov random field and Gaussian mixture model?

Your question is about how to express $p(f)=p((f_i))$ ($i$ indexing pixel) the prior probability on the unknown e.g. segmentation mask (the unknown parameter of interest).
In fact in image processing, prior information $p(f)$ is crucial and consists mainly in introducing dependencies between the values of the (unknown) parameter of neighbour pixels. As an example, in segmentation task, neighbour pixels can be more likely to have the same label : segmentation mask with large regions of a given label are more likely than patchy mask with many isolated different labels. This is typically done by introducing a Markov random field structure to the parameter of interest. Practically, it consists in introducing the pixel-wise prior probabilities of adjacent pixels: $$P(f_i|(f_j)_{j \in V(i)})$$ (where $V(i)$ are the neighboor pixels of pixel $i$) that will be used to define completelty $p(f)=p((f_i))$. I suggest you to look for a dedicated lecture e.g. http://www.inf.u-szeged.hu/ssip/2008/presentations2/Kato_ssip2008.pdf for more details.
• Joint probability distribution has nothing to do with modeling: it is the name used to say that we consider the density of the e.g. parameters together e.g. $p(\theta_1,\theta_2|x)$ is the joint posterior. Mixture model refers to a typical hierarchical model (en.wikipedia.org/wiki/Mixture_model) – peuhp Feb 25 '16 at 10:22