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? enter image description here

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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?


A model relates your observations and unknown parameters we are interested in. In the case of image processing the observation is most of the time the acquired image and the unknown parameters can be e.g. segmentation labels image for a segmentation task, true intensity image for a denoising task.

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.

Another example of prior could consists in having some shape prior e.g. a region with label 1 looks like a circle, a region with label 2 looks like a dog, ... (e.g. http://www.ece.ucsb.edu/~manj/ManjBio2008/07_Vu_CVPR2008.pdf)

  • $\begingroup$ Thanks. Currently I am working on image denoising. Does Markov random field model the dependencies between current pixel and its neighborhood? Is the size of the neighborhood is 4 or 8 in practical applications? Can you provide some links to documents or implementationson image denoising? $\endgroup$ – Jogging Song Feb 24 '16 at 9:11
  • $\begingroup$ Happy to help. MRF can be used for modeling image denoising. Nevertheless, modern approache to Bayesian denoising, model the prior in the wavelet domain (course at 6.869.csail.mit.edu/fa13/lectures/lecture6.pdf). The size of the neighboorhood depends on the application. $\endgroup$ – peuhp Feb 24 '16 at 9:57
  • $\begingroup$ Thanks for the valuable material. Mixture model is mentioned. It is difficult to understand the term. What is the difference between mixture model and joint probability distribution? $\endgroup$ – Jogging Song Feb 25 '16 at 1:25
  • $\begingroup$ 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) $\endgroup$ – peuhp Feb 25 '16 at 10:22

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