Summation Invariants Of Objects Under Projective Transformation Group With Application

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Date
2016-02
Authors
Azhir, Nasereh
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Universiti Sains Malaysia
Abstract
Geometric invariants are features which unchanged under a variety of transformations and they can be used as the shape descriptors to overcome many of problems of object recognition problems in computer vision. The role of invariants in computer vision has been advocated for various applications such as shape representation, shape matching, object recognition and robotic. This thesis solving problems associated with deriving invariants of two dimensional objects under projective transformation groups in Euclidean space. In this thesis, a method is given to determine projective invariants for planar objects under projective transformation groups and an algorithm is given to apply the derived invariants in order to solve some issues of object recognition under transformation groups. The Cartan’s method of moving frame is applied to derive these invariants. Novel integral potentials for 2D curves are proposed to derive integral invariants under the action of a subgroup of projective transformation with 6 degrees of freedom. In case of discrete data, new summation invariants are proposed under subgroup of projective transformation of planar objects with 6 degrees of freedom. Besides, comparison analysis has been facilitated for the derived invariants and previous invariants for objects under Euclidean, affine and projective group actions. Moreover, the performance of the method has been discussed for objects under white Gaussian-distributed noise levels. This can solve the problem of deriving invariants for objects under local projective transformation groups happening because of the existing x and y in the dominator of this action in Euclidean space. Application of these invariants to discrete data, obtained from a sample of boundary of car contour, generates a pattern of similar classification under projective transformation. In addition, a deductive method is proposed to derive invariants for projective transformation by splitting this transformation group with 8 degrees of freedom to subgroups of the projective transformation with lower 6 degrees of freedom whose invariants can be derived based on the potential variables for the standard actions of projective groups on R2. As image of an object under projective transformation in some perspective view may largely deformed, solving some problems associated with finding missing data in a planar object which undergoing two different projective transformation groups is of interest. Thereby a new algorithm is presented to find the data points in a planar object which are not available to any reason while two different perspective images of the object are available under two different types of projective transformation group. The method is applied to generate missing data from the boundary of a computed tomography image of a skull and a magnetic resonance imaging representation of the brain. The robustness of the algorithm is examined in the condition of white Gaussian noises to the sample data which shows the good performance of the method.
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Invariants
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