Image Segmentation Based on the Poincaré Map Method

ABSTRACT:
Active contour models (ACMs) integrated with various kinds of external force fields to pull the contours to the exact boundaries have shown their powerful abilities in object segmentation. However, local minimum problems still exist within these models, particularly the vector field’s “equilibrium issues.” Different from traditional ACMs, within this paper, the task of object segmentation is achieved in a novel manner by the Poincaré map method in a defined vector field in view of dynamical systems. An interpolated swirling and attracting flow (ISAF) vector field is first generated for the observed image. Then, the states on the limit cycles of the ISAF are located by the convergence of Newton–Raphson sequences on the given Poincaré sections. Meanwhile, the periods of limit cycles are determined. Consequently, the objects’ boundaries are represented by integral equations with the corresponding converged states and periods. Experiments and comparisons with some traditional external force field methods are done to exhibit the superiority of the proposed method in cases of complex concave boundary segmentation, multiple-object segmentation, and initialization flexibility. In addition, it is more computationally efficient than traditional ACMs by solving the problem in some lower dimensional subspace without using level-set methods.




EXISTING SYSTEM:
The former type usually establishes energy functional composed of internal and external energy terms. It is not intrinsic and has difficulty in determining and handling the topological changes of evolving contours. The latter one constructs an intrinsic Riemannian metric-based functional. Then, the active contours are treated as the zero crossings of level-set functions. It can tackle topological changes elegantly at the cost of more computational costs.

PROPOSED SYSTEM:
In this paper, an interpolated swirling and attracting flow (ISAF) field is generated by extending a so-called edge tangent flow (ETF) only with a nonzero value at the boundaries to the whole image domain. It is a static vector field. Different from traditional vector fields, the components in this vector field near the boundary are not perpendicular but tangent to the boundary. Thus, in the proposed vector field, it is possible for evolution to be carried out along the boundaries. Then, the proposed time-invariant vector field is considered as the right-hand-side vector-valued function of an autonomous dynamical system. As a result, the segmentation problem is translated to the problem of the limit cycle location by applying the related theory in dynamical systems.




MODULES:
ü ISAF Field
ü Within the Framework of Dynamical Systems
ü Poincaré Map
ü Multiple-Object Segmentation

MODULES DESCRIPTION:

ISAF Field

Inspired by the idea, an ISAF field for the observed image is generated in this part. It is composed of two components, namely, diffused ETF (DETF; swirling component) and diffused edge perpendicular flow (DEPF; attracting component). They are directly described here by the evolution equation form for simplicity.

Within the Framework of Dynamical Systems

Within the above part, an ISAF field for the observed image is first constructed, where its eddies correspond to the objects’ boundaries. Naturally, the properties of the vector field remind us of the limit cycle theory in dynamical systems. As far as we know, a dynamical system is usually defined by an array of ordinary differential equations. In nonlinear dynamical systems, limit cycles are isolated closed periodical trajectories of the vector field in the state space, where the neighboring trajectory is not closed and spiral toward or away from these cycles as time goes to infinity. Thus, they share common features with the above eddies. In fact, the above eddies can be considered as limit cycles of a vector field that belongs to some dynamical systems. Consequently, with the knowledge of dynamical systems, the limit cycles are located, including their periods and candidate points on the limit cycles. In nonlinear dynamical systems, limit cycles are isolated closed periodical trajectories of the vector field in the state space, where the neighboring trajectory is not closed and spiral toward or away from these cycles as time goes to infinity. Thus, they share common features with the above eddies. In fact, the above eddies can be considered as limit cycles of a vector field that belongs to some dynamical systems. Consequently, with the knowledge of dynamical systems, the limit cycles are located, including their periods and candidate points on the limit cycles. In nonlinear dynamical systems, limit cycles are isolated closed periodical trajectories of the vector field in the state space, where the neighboring trajectory is not closed and spiral toward or away from these cycles as time goes to infinity. Thus, they share common features with the above eddies. In fact, the above eddies can be considered as limit cycles of a vector field that belongs to some dynamical systems. Consequently, with the knowledge of dynamical systems, the limit cycles are located, including their periods and candidate points on the limit cycles.

Poincaré Map
In this part, the elementary knowledge of the Poincaré map is introduced. In the state space of the dynamical system, every limit cycle (if there exists) has its basin of attraction, where all the streamlines in the region flow to the limit cycle. In addition, within the state space, its Poincaré section.  can be defined as a lower dimensional hyperplane transversal to the flow of the system. In particular, it should intersect with its corresponding limit cycle. The “corresponding limit cycle” means the limit cycle, the basin of attraction for which the Poincaré section lies in.


Multiple-Object Segmentation

In this part, without using the level-set method, the proposed method is applied to multiple-object segmentation. Concretely, the corresponding ISAF field for an observed image is constructed first. However, usually, is chosen more than the number of desired objects. Then, these states will move along the flow to generate trajectories within the flow field, and finally, they will move to the orbits along the objects’ boundaries round and round to finish the object location.

HARDWARE REQUIREMENTS

•                     SYSTEM             : Pentium IV 2.4 GHz
•                     HARD DISK        : 40 GB
•                     FLOPPY DRIVE  : 1.44 MB
•                     MONITOR           : 15 VGA colour
•                     MOUSE               : Logitech.
•                     RAM                    : 256 MB
•                     KEYBOARD       : 110 keys enhanced.

SOFTWARE REQUIREMENTS

•                     Operating system           :-  Windows XP Professional
•                     Front End             :-  Microsoft Visual Studio .Net 2008
•                     Coding Language : - C# .NET.

REFERENCE:
Delu Zeng, Zhiheng Zhou, and Shengli Xie, Senior Member, IEEE, “Image Segmentation Based on the Poincaré Map Method”, IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 3, MARCH 2012.