More single view geometry Describes the images of planes, lines,conics and quadrics under perspective projection and their forward and backward properties

Camera properties Images acquired by the cameras with the same centre are related by a plane projective transformation Image entities on the plane at infinity,  inf, do not depend on camera position, only on camera rotation and internal parameters, K

Camera properties 2 The image of a point or a line on  inf, depend on both K and camera rotation. The image of the absolute conic, , depends only on K; it is unaffected by camera rotation and position.  = ( KK T ) -1

Camera properties 2  defines the angle between the rays back- projected from image points Thus camera rotation can be computed from vanishing points independent from camera position. In turn, K may be computed from the known angle between rays; in particular, K may be computed from vanishing points corresponding to orthogonal scene directions.

Perspective image of points on a plane

Action of a projective camera on planes

Action of a projective camera on lines

Line projection

Action of a projective camera on conics

Action of a projective camera on conics 2

On conics

Images of smooth surfaces

Images of smooth surfaces 2

Contour generator and apparent contour: for parallel projection

Contour generator and apparent contour: for central projection

Action of a projective camera on quadrics Since intersection and tangency are preserved, the contour generator is a (plane) conic. Thus the apparent contour of a general quadric is a conic, so is the contour generator.

Result 7.8

On quadrics

Result 7.9 The cone with vertex V and tangent to the quadric is the degenerate quadric Q CO = (V T QV) Q – (QV)(QV) T Note that Q CO V = 0, so that V is the vertex of the cone as assumed.

The cone rays of a quadric

The cone rays with vertex the camera centre

Example 7.10

The importance of the camera centre

The camera centre

Moving image plane

Moving image plane 2

Moving image plane 3

Camera rotation

Example

(a), (b) camera rotates about camera centre. (c) camera rotates about camera centre and translate

Synthetic views

Synthetic views. (a) Source image (b) Frontal parallel view of corridor floor

Synthetic views. (a) Source image (c) Frontal parallel view of corridor wall

Planar panoramic mosaicing

Three images acquired by a rotating camera may be registered to the frame of the middle one

Planar panoramic mosaicing 1

Planar panoramic mosaicing 2

Planar panoramic mosaicing 3

Projective (reduced) notation

Moving camera centre

Parallax Consider two 3-space points which has coincident images in the first view( points are on the same ray). If the camera centre is moved (not along that ray), the iamge coincident is lost. This relative displacement of image points is termed Parallax. An important special case is when all scene points are coplanar. In this case, corresponding image points are related by planar homography even if the camera centre is moved. Vanishing points, which are points on  inf are related by planar homography for any camera motion.

Motion parallax

Camera calibration and image of the absolute conic

The angles between two rays

The angle  between two rays

Relation between an image line and a scene plane

The image of the absolute conic

The image of the absolute conic 2

The image of the absolute conic 3

The image of the absolute conic 4

The image of the absolute conic 5

Example: A simple calibration device

Calibration from metric planes

Outline of the calibration algorithm

Orthogonality in the image

Orthogonality in the image 2

Orthogonality represented by pole- polar relationship

Reading the internal parameters K from the calibrated conic

To construct the line perpendicular to the ray through image point x

Vanishing point formation (a) Plane to line camera

Vanishing point formation: 3-space to plane camera

Vanishing line formation(a)

Vanishing line formation (b)

Vanishing points and lines

Image plane and principal point

The principal point is the orthocentre of an orthongonal triad of vanishing points in image (a)

The principal point is the orthocentre of the triangle with the vanishing point as the vertices

The calibrating conic computed from the three orthogonal vanishing point

The calibrating conic for the image (a)