Gestalt psychologists [1,2] observed and emphasized the importance of organizations in vision. They recognized what is computed of the organizations, for example, proximity, continuity, similarity, closure symmetry, etc., but they did not convincingly answer why it is calculated or how. A mathematical model to represent these Gestalt laws still eludes us.
For example, suppose we observe the rectangles in each of the images in Figure 1. We definitely perceive the spatial patterns for each of them. How do we model such spatial patterns? What are sufficient statistics for these patterns? What kind of mathematical model we could use to specify these pattern?
Firstly, we build a descriptive model, which is called Gestalt Ensemble , which is a mixed Markov model  with a dynamic neighborhood.
Secondly, we extend our Gestalt Ensemble model to a general case, Gestalt Fields , which is still a mixed Markov model, but with a wider neighborhood system and inhomogeneous feature definition.
Figure 1. Three examples of spatial patterns. Each rectangle is considered as a basic element here.
This is a descriptive model for the spatial arrangement of the elements, which are marked points and we called textons here. The neighborhood system is defined as shown in Figure 2. The neighbors are decided dynamically during computation with a greedy method. The neighbor could be NULL.
Figure 2. Texton neighborhood system.
For a texton t1 and its neighbor t2, we measure five features (four shown in Figure2 (b)), which capture various Gestalt properties:
Thus a total of 4*5=20
pair-wise features are computed for each texton plus two features of each texton
itself (orientation and joint of scale&stretch). We compute 21 one dimensional marginal histograms and a two-dimensional
Figure 3. The learning process of Gestalt Ensemble model for a cheetah-dots pattern.
Figure 4. The learning process of Gestalt Ensemble model for a lattice pattern.
Figure 5. The learning process of Gestalt Ensemble model for a woods pattern.
Figure 6. The learning process of Gestalt Ensemble model for a dry-land pattern.
Gestalt Fields is a general model from the Gestalt Ensemble. It is a mixed Markov model in which each marked points have ten addressed variables in which five for one end are shown in Figure 7.
Figure 7. Five addressed neighbors for each elements at one end. They are named 1) co-line, 2) co-curve upper, 3) co-curve lower, 4) perpendicular and 5) parallel neighbor respectively.