Error estimation in reconstruction of quadratic curves in 3D space

Abstract

Many natural and man-made objects have planar and curvilinear surfaces. The images of such curves do not usually have sufficient distinctive features to apply conventional feature-based reconstruction algorithms. In this paper, we describe a method for the reconstruction of various kinds of quadratic curves in 3D space as an intersection of two cones containing the respective projected digitized curve images in the presence of Gaussian noise. The advantage of this method is that it overcomes the correspondence problem that occurs in pairs of projections of the curve. Using nonlinear least-squares curve fitting, the parameters of a curve in 2D digitized image planes are determined. From this we reconstruct the 3D quadratic curve. Relevant mathematical formulations and analytical solutions for obtaining the equation of the reconstructed curve are given. Simulation studies have been conducted to observe the effect of noise on errors in the process of reconstruction. Results for various types of quadratic curves are presented using simulation studies. These are the main contributions of this work. The angle between the reconstructed and the original quadratic curves in 3D space has been used as the criterion for the measurement of the error. The results of this study are useful for the design of a stereo-based imaging system (such as the LBW decision in cricket, the path of a missile, robotic vision, path planning, etc.) and for the best reconstruction with minimum error.

Keywords:

• Curve reconstruction
• Perspective projection
• Gaussian noise
• Collinearity equation
• Curve fitting

Acknowledgements

SK gratefully acknowledges the financial support of the Council of Scientific and Industrial Research, New Delhi, India through a Junior Research Fellowship (JRF) scheme (CSIR Award No. 9/143(503)/2004-EMR). RB also gratefully acknowledges the financial support of the Sponsored Research and Industrial Consultancy (SRIC), the Indian Institute of Technology Roorkee, India under grant No. FIG-100421-MAT.