The Geometry of Perspective Projection

Transcription

1 The Geometry o Perspective Projection Pinhole camera and perspective projection - This is the simplest imaging device which, however, captures accurately the geometry o perspective projection. -Rays o light enters the camera through an ininitesimally small aperture. -The intersection o the light rays with the image plane orm the image o the object. -Such a mapping rom three dimensions onto two dimensions is called perspective projection.

2 -2- Asimpliied geometric arrangement -Ingeneral, the world and camera coordinate systems are not aligned. - To simpliy the derivation o the perspective projection equations, we will make the ollowing assumptions: () the center o projection coincides with the origin o the world. (2) the camera axis (optical axis) is aligned with the world s z-axis.

3 -3- (3) avoid image inversion by assuming that the image plane is in ront o the center o projection. Some terminology -The model consists o a plane (image plane) and a 3D point O (center o projection). -The distance between the image plane and the center o projection O is the ocal length (e.g., the distance between the lens and the CCD array). -The line through O and perpendicular to the image plane is the optical axis. -The intersection o the optical axis with the image place is called principal point or image center. (note: the principal point is not always the "actual" center o the image)

4 -4- The equations o perspective projection (notation: (x, y, z) (X, Y, ), r R, (x, y, z ) (x, y, z), r r) -Using the ollowing similar triangles: () rom OA B and OAB: = r R (2) rom A B C and ABC: x X = y Y = r R perspective proj. eqs: x = X y = Y z = -Using matrix notation: x h y h z h w = X Y -Veriy the correctness o the above matrix (homogenize using w = ): x = x h w = X y = y h w = Y z = z h w =

5 -5- Properties o perspective projection Many-to-one mapping -The projection o a point is not unique (any point on the line OP has the same projection). Scaling/Foreshor tening -The distance to an object is inversely proportional to its image size.

6 -6- -When a line (or surace) is parallel to the image plane, the eect o perspective projection is scaling. -When an line (or surace) is not parallel to the image plane, we use the term oreshortening to describe the projective distortion (i.e., the dimension parallel to the optical axis is compressed relative to the rontal dimension). Eect o ocal length -As gets smaller, more points project onto the image plane (wide-angle camera). -As gets larger, the ield o view becomes smaller (more telescopic). Lines, distances, angles -Lines in 3D project to lines in 2D. -Distances and angles are not preserved. -Parallel lines do not in general project to parallel lines (unless they are parallel to the image plane).

7 -7- Vanishing point *parallel lines in space project perspectively onto lines that on extension intersect at a single point in the image plane called vanishing point or point at ininity. *(alternative deinition) the vanishing point o a line depends on the orientation o the line and not on the position o the line. *the vanishing point o any given line in space is located at the point in the image where a parallel line through the center o projection intersects the image plane. Vanishing line *the vanishing points o all the lines that lie on the same plane orm the vanishing line. *also deined by the intersection o a parallel plane through the center o projection with the image plane.

8 -8- Orthographic Projection -Itisthe projection o a 3D object onto a plane by a set o parallel rays orthogonal to the image plane. - It is the limit o perspective projection as > (i.e., / > ) orthographic proj. eqs: x = X, y = Y (drop ) -Using matrix notation: x h y h z h w = X Y -Veriy the correctness o the above matrix (homogenize using w=): x = x h w = X Properties o orthographic projection -Parallel lines project to parallel lines. y = y h w = Y -Size does not change with distance rom the camera.

9 -9- Weak Perspective Projection -Perspective projection is a non-linear transormation. - We can approximate perspective by scaled orthographic projection (i.e., linear transormation) i: () the object lies close to the optical axis. (2) the object s dimensions are small compared to its average distance rom the camera (i.e., z < /2) weak perspective proj. eqs: x = X X y = Y Y (drop ) -The term is a scale actor now (e.g., every point is scaled by the same actor). -Using matrix notation: x h y h z h w = X Y -Veriy the correctness o the above matrix (homogenize using w = ): x = x h w = X y = y h w = Y