Epipolar geometry

From Wikipedia, the free encyclopedia

Epipolar geometry refers to the geometry of stereo vision. When two cameras view a 3D scene, there are a number of geometric facts that dictate relationships between these views. The precise type of relationship depends on the particular camera model used. Common camera models are orthographic cameras and perspective or pinhole cameras. Roughly speaking an orthographic camera model describes the basic properties of a camera with a telephoto lens while a perspective camera model describes the basic properties of a camera with a wide angle lens.

The ﬁgure below represents two cameras looking at point P. In real cameras, the focal plane is behind the focal point, and has an inverted image. However, we can also think of a front image plane that is not inverted and that is easier to understand. In this case the two upright planes represent the front image planes of two cameras. O_L and O_R represent the focal points of the two cameras given a pinhole camera representation of the cameras. P represents the point of interest in both cameras. p_L and p_R represent where point P is projected onto the image plane.

Each camera takes a 2D image of the 3D world. This conversion from 3D to 2D is referred to as a perspective projection. It is common to model this projection operation by rays that emanate from the camera. Note that each emanating ray in 3D corresponds to a single point in the image.

So, line O_L-P is seen by the left camera as a point because it is directly in line with the camera's focal point. However, the right camera sees it as a line on the focal plane. That line (E_R-p_R) in the right camera is called an epipolar line. Symmetrically, the line O_R-P seen by the right camera as a point is seen as epipolar line E_L-p_Lby the left camera (Shapiro & Stockman 2001:403).

As an alternative visualization, consider the points P, O_L & O_R that form a plane called the epipolar plane. The epipolar plane intersects each camera's focal plane and forms a line - the epipolar lines.

Because the camera focal planes (in this example) are not the same plane, cameras can "see" each other along the line O_L-O_R. The point of intersection on each focal plane is called the epipole (denoted by E_L and E_R) (Shapiro & Stockman 2001:403).

If the position of the cameras is precisely known (position and angle in relation to each other), this geometry may be used to calculate the position and distance of points viewed by the cameras.

The epipolar geometry is simplified if the camera focal planes are on the same plane. In that case, the epipolar lines are the same line (E_L-p_L = E_R-p_R). The epipolar line also is parallel to the line Ol-Or between the focal points. Therefore also, all common points in the image can be found by looking along a line (Shapiro & Stockman 2001:401). (In the previous example, a point from one camera's image would have to be found in the other camera's image by looking in two dimensions - a much more complex task.) If the cameras cannot be positioned in this way, the images from the cameras may be transformed to emulate having a common image plane. This process is called image rectification.