Chapter 2
Low level processing is performed on regular lattices of images. The set
of sites 
index image pixels in the image plane. The
set 
contains continuous or discrete labels for the pixels.
Therefore the problems fall into categories LP1 and LP2. Most existing
MRF vision models are for low level processing. Image restoration and
segmentation is one of them which has been studied most
comprehensively in MRF modeling.
Surface reconstruction can be viewed as a more general case than restoration in that the data can be sparse, i.e. available at certain locations of the image lattice.
MRFs can play a fuller role in texture analysis because textured images present anisotropic properties.
The treatment of optical flow as MRF is similar to that of restoration and reconstruction. Edge detection is often addressed along with restoration, reconstruction and analysis of other image properties such as texture, flow and motion. We can also find low level applications of MRFs such as active contours, deformable templates, data fusion and visual integration.
In this chapter, we formulate various MAP-MRF models for low level vision, following the procedure summarized in Section 1.5.4. We begin with the prototypical MAP-MRF models for image restoration, The presentation therein introduces the most important concepts in MRF modeling. After that, the formulations for the image restoration are extended to a closely related problem, surface reconstruction, in which the observation may be sparser. The MRF models for boundary detection, texture and optical flow are described subsequently. How to impose the smoothness constraint while allowing discontinuities is an important issue in computer vision [Terzopoulos 1983b ; Geman and Geman 1984 ; Blake and Zisserman 1987] which deserves a thorough investigation; it is the topic of Chapter 3. Another important issue, MRF parameter estimation in low level vision, will be discussed in Chapter 6.