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Univariate Polynomial Equations:
 
Univariate Polynomial Equations:
  
[[Image:Algorithm Univariate.jpg|550px]]
+
[[Image:Algorithm Univariate.jpg|900px]]
  
 
Multivariate Polynomial Equations:
 
Multivariate Polynomial Equations:
  
 
[[Image:Algorithm Multivariate.jpg|900px]]
 
[[Image:Algorithm Multivariate.jpg|900px]]
 +
 +
'''Theorems'''
 +
 +
'''Numerical Results'''

Revision as of 09:55, 22 April 2010

A Solution Method For Zero-Dimensional Polynomial Equation System

Motivation

Consider the problem of curve registration, that is, finding the rotation and translation that best maps (i.e., registers) a cloud of points onto a template object, as described on the right.

We first approximate the curve defined by the contour of the template object by an implicit polynomial equation. This yields a bivariate polynomial equation p(x,y) = 0 whose solution set approximates the template contour.

Let (x_i,y_i) , i=1, ..., N be the points of the point cloud. We are looking for the rotation R and the translation T such that p((xi, yi)R + T) = 0 for all i = 1, ..., N. Then we have an overdetermined polynomial equation system with noisy coefficient, which contains N equations and unknown variables R and T. We need to solve this overdetermined polynomial system.

Butterfly model.jpg

Pipeline of the Solution Method Schematic Rep.jpg

Algorithm

Univariate Polynomial Equations:

Algorithm Univariate.jpg

Multivariate Polynomial Equations:

Algorithm Multivariate.jpg

Theorems

Numerical Results

Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva