Application of Nonlinear Dynamics to Image Recognition
A stable non-linear dynamic system has three kinds of attractors:
Among these three attractors, the strange attractor displays some interesting behaviors. Strange attractors are results of non-linear dynamics behaving chaotically. Some interesting characteristics of chaos are presented below:
- Point attractor
- Periodic attractor (limit cycle)
- Strange attractor
Characteristic No. 7, 8, and 9 suggest that chaotic dynamics are capable of producing complex structures. Since the state trajectories never repeat, yet they are bounded to the attractor in the phase space, chaotic dynamics can create infinite variety within a bound.
- Chaos results from a deterministic process.
- Chaos happens only in non-linear systems.
- Chaos happens in feedback systems.
- Chaos is entirely self-generated.
- Chaos can usually pass all statistical test for randomness.
- The Fourier spectrum is "broad" but with some periodicities.
- Chaos includes one or more types order.
- Chaos can have large number of symmetry, including scale symmetry.
- The phase space trajectories may have some fractal properties.
- Fractals can be scale invariant.
- Details of the chaotic behavior are hypersensitive to changes in initial conditions. Chaotic trajectories never repeat, and they never cross each other.
- The ranges of the variables have finite bounds. The state trajectories are bounded in phase space. The region where the trajectories are bounded is called the attractor for the system.
This situation is analogous to a biological system such as an apple tree or a human face. There are no two apple trees that are exactly alike, yet there is some thing easily recognizable about an apple tree. Even when we come a cross an apple tree which we have never seen before, we can still recognize that as an apple tree. The same is true for an orange tree. However, apple trees are distinct from orange trees. It seems that there are some invariant features are common to all apple trees, and there are some invariant features common to all orange trees. Moreover, the common features of apple trees are different from the common features of orange trees. Apple trees create infinite variety within a bound. This boundary defines apple trees' attractor. Some underlying dynamics give rise to this attractor. It can be assumed that orange trees' attractor is different from apple trees' attractor.
So, it seems that if we can identify the attractor associated with the apple trees we will be able to identify an apple tree. Therefore, it seems feasible that chaotic dynamics can be used to solve recognition problem. We are investigating this possibility.
We describe a nonlinear system capable of use as a recognition system. This system is composed of a set of coupled oscillators, connected by linear springs. Images are overlaid on the system by altering the masses and spring constants of the oscillators, thereby modifying the detailed behavior of the system. Signatures extracted from the system using FFT of the individual oscillator positions show coherance, relative continuity, and translational and rotational invariance. These properties are discussed in the context of the eventual use of this system as a general identification system.
S. Hoque, S. Kazadi, A. Li, W. Chen, and E. Sadun. Identification of Shapes Using a Nonlinear Dynamic System . Lecture Notes in Computer Science , 2095, pp.236-245, 2001. (Postscript) (PDF)