1 Introduction

1.1 Epipolar Geometry

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Two cameras take a picture of the same scene from different points of view. The epipolar geometry then describes the relation between the two resulting views.

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Epipole (epipolar point)

1.2 Essential Matrix

a 3x3 matrix that encodes the epipolar geometry of two views where the cameras are already calibrated.

Since CC’, Cq, and C’q’ vectors are three vectors that lie on the same epipolar plane we have the following equations:

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  1. q=Cq in the frame of camera C (also use C for the 3d point), denoted by Xc;

  2. CC’ in frame C is t;

  3. Rq’=C’q’ since

    1. C’Q’=q’ in frame Xc’;
    2. its coordinate in frame C is: R(q’+t);
    3. txt=0;

  4. matrix representation of the cross product

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1.2.1 Longuet-Higgins equation, 1981

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  1. Now, q and q’ are in two 3d camera frames, respectively. next step is ⇒ homogeneous representations of the 2D image frames, respectively
    1. but still not in the image coordinates. but called normalized image coordinates.
  2. fl focus of left camera, Zl: Pl=[Xl, Yl, Zl]

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