Three dimensional measurement as optimization of parameters of a curve on epipolar plane by stereo camera - A basic theory based on epipolar geometry

This paper describes a method to solve the binocular stereo correspondance problem on an epipolar plane. In porposed method, a parametric curve is on the epipolar plane which is calcurated using world coordinates of lens centers and a image point. And a coordinate frame is set up on the epipoar plane(shn), and a transformation matrix between the world coordinate system and the coordinate frame (shn) is calcurated. Coordinates of the parametric curve is transformed from the coordinate frame (shn) to the world coordinate system. The parametric curve is projected to two cameras, and these projected lines have correspondance area on the images. Pixel values in area around these correspondance line on each images are compared to calcurate errors. And an error-function is defined the errors. Parameters of the curve are optimzed to let the error function be smaller. The laid parameteric curve is a polynominal and the polytope method is used as optimizing method. Pixel values of epipolar line on each two images are fitted each other on the images. A board shape object and a column shape object are used as experimentation. And the shape of the parametric curve is shown as a result.