Theory and Practice of Projec-tive Rectification

2013-05-29Ko Dae-Won

The following = det(A)I where I is the identity matrix.

In particular, if A is an invertible matrix, then .

1. Preliminaries

Theory and Practice of Projective Rectification

Given a vector it is convenient to introduce the skew-symmetric matrix

1. Preliminaries

Theory and Practice of Projective Rectification

The matrix is closely related to the cross-product of vectors in that for any vectors and t,

we have and .

Proposition 1. For any 3 x 3 matrix M and vector t,

1. Preliminaries

Theory and Practice of Projective Rectification

If A is a 3 x 3 non-singular matrix representing a projective transformation of then is the correspond-ing line map.

In other words, if and line on a line L, then and : in symbols ) ).

This formula is derived from Proposition 1.

1. Preliminaries

Theory and Practice of Projective Rectification

2. Property of the fundamental matrix

Theory and Practice of Projective Rectification

According to Proposition2, the matrix F determines the epipoles in both images.

Furthermore, F provides the map between points in one image and epipolar lines in the other image.

2. Property of the fundamental matrix

Theory and Practice of Projective Rectification

Goal: Finding a projective transformation H of an image mapping an epipole to a point at infinity.

In fact, if epipolar lines are to be transformed to lines parrallel with x axis, then the epipole should be mapped to the infinite point

If inappropriate H is chosen, severe projective distortion of the image can take place.

3. Mapping the Epipole to Infinity

Theory and Practice of Projective Rectification

)0,0,1(T

In order that the resampled image should look somewhat like the original image, we may put closer Restrictions on the choice of H.

One condition that leads to good results is to insist thatThe transformation H should act as far as possible as a Rigid transformation in the neighborhood of a given selected point of the image.

3. Mapping the Epipole to Infinity

Theory and Practice of Projective Rectification

u0

By this is meant that to first order neighborhood of may undergo rotation and translation only.

An appropriate choice of point may be the cen-tre of the image.

3. Mapping the Epipole to Infinity

Theory and Practice of Projective Rectification

u0

u0

3. Mapping the Epipole to Infinity

Theory and Practice of Projective Rectification

3. Mapping the Epipole to Infinity

Theory and Practice of Projective Rectification