[Introduction] [Gestalt Ensemble] [Gestalt Fields]

Introduction:

        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 [3], which is a mixed Markov model [5] with a dynamic neighborhood.

        Secondly, we extend  our Gestalt Ensemble model to a general case, Gestalt Fields [4], which is still a mixed Markov model, but with a wider neighborhood system and inhomogeneous feature definition.

a). Cheetah-dots b). Woods c). Dry-land

        Figure 1. Three examples of spatial patterns. Each rectangle is considered as a basic element here.

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Gestalt Ensemble: 

        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.

a) A texton has four neighbors

b) Four measurements

        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:

  1. dc: Distance between two centers, which measures proximity.

  2. dm: Gap between two textons, which measures connectedness and continuation.

  3. \alpha: Angle of a neighbor relative to the main axis of the reference texton. This is mostly useful in quadrants I and III. \alpha/dc measures the curvature of possible curves formed by the textons, or co-linearity and co-circularity in the Gestalt language.

  4. \gamma: Relative orientations between the two textons. This is mostly useful for neighbors in quadrants II and IV and measures parallelism.

  5. r: Size ratio which denotes the similarity of texton sizes. r is the width of t2 divided by the width of t1 for
    neighbors in quadrants I and III and r is length of t2 divided by the length of t_1 for neighbors in quadrants II and
    IV.

        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
histogram for scale&stretch, averaged over all textons. The Gestalt Ensemble model is a descriptive model built on these histograms and with one more random variable which is the total number of the textons. The parameters of the model is obtained through learning by sampling. Below are the results.

a). observed b). t = 1 c). t = 9
d). t = 69 e). t = 141 f). learning process

         Figure 3. The learning process of Gestalt Ensemble model for a cheetah-dots pattern.

a). observed b). t = 1 c). t = 18
d). t = 51 e). t = 147 f). learning process

         Figure 4. The learning process of Gestalt Ensemble model for a lattice pattern.

a). t = 1 b). t = 30 c). t = 332

        Figure 5. The learning process of Gestalt Ensemble model for a woods pattern.

a). t = 1 b). t = 72 c). t = 202

        Figure 6. The learning process of Gestalt Ensemble model for a dry-land pattern.

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Gestalt Fields

        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.

      
a). Five neighbors at one end b). Neighborhoods modeled as addressed variables

        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.

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References:

  1. K. Koffka, "Principles of Gestalt Psychology", New York: Harcourt, 1935.

  2. W. Kohler, "Gestalt Psychology", New York: Liveright, 1929.

  3. Cheng-en Guo, Song-Chun Zhu, and Ying Nian Wu
    "Modeling Visual Patterns by Integrating Descriptive and Generative Methods" (.pdf 3.2M)
    International Journal of Computer Vision, Vol. 53, No. 1, pp.5-29, June 2003.

  4. Cheng-en Guo, Song-chun Zhu and Yingnian Wu
    "A Mathematical Theory of Primal Sketch and Sketchability" (.pdf 553K)
    Proc. of International Conference on Computer Vision, Nice France, 2003

  5. A. Fridman, ``Mixed Markov Models'', Doctoral dissertation, Division of Applied Math, Brown University. 2000.

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[Introduction] [Gestalt Ensemble] [Gestalt Fields]