Transcript Camera Parameters CS485/685 Computer Vision Prof. George Bebis

Camera Parameters
CS485/685 Computer Vision
Prof. George Bebis
CCD array and frame buffer
• The physical image plane is the CCD array of n x m
rectangular grid of photo-sensors.
• The pixel image plane (frame buffer) is an array of
N x M integer values (pixels).
CCD array and frame buffer (cont’d)
• The position of the same point on the image plane will
be different if measured in CCD elements (x, y) or
image pixels (xim, yim).
(assuming that the origin in both cases is the upper-left corner)
(xim, yim) measured in pixels
(x, y) measured in millimeters.
Reference Frames
• Five reference frames are needed in general for 3D
scene analysis.
–
–
–
–
–
Object
World
Camera
Image
Pixel
(1) Object Coordinate Frame
• 3D coordinate system: (xb, yb, zb)
• Useful for modeling objects (i.e., check if a particular
hole is in proper position relative to other holes)
• Object coordinates do not change regardless how the
object is placed in the scene.
Our notation: (Xo, Yo, Zo)T
(2) World Coordinate Frame
• 3D coordinate system: (xw, yw, zw)
• Useful for interrelating objects in 3D
Our notation: (Xw, Yw, Zw)T
(3) Camera Coordinate Frame
• 3D coordinate system: (xc, yc, zc)
• Useful for representing objects with respect to the location
of the camera.
Our notation: (Xc, Yc, Zc)T
(4) Image Plane Coordinate Frame (i.e., CCD plane)
• 2D coordinate system: (x f , y f )
• Describes the coordinates of 3D points projected on the
image plane.
Our notation: (x, y)T
(5) Pixel Coordinate Frame
• 2D coordinate system: (c, r)
• Each pixel in this frame has integer pixel coordinates.
y
Our notation: (xim, yim)T
x
Transformations between frames
World and Camera coordinate systems
• In general, the world and camera coordinate systems
are not aligned.
center of projection
optical axis
World and Camera coordinate systems (cont’d)
• To simplify mathematics, let’s assume:
(1) The center of projection coincides with the origin of the
world coordinate system.
(2) The optical axis is aligned with the world’s z-axis and
x,y are parallel with X, Y
Y
X
Z
x
y
World and Camera coordinate systems (cont’d)
(3) Avoid image inversion by assuming that the image plane
is in front of the center of projection.
(4) The origin of the image plane is the principal point.
center of
projection
Terminology - Summary
• The model consists of a plane (image plane) and a 3D
point O (center of projection).
• The distance f between the image plane and the center
of projection O is the focal length (e.g., the distance
between the lens and the CCD array).
center of
projection
Terminology - Summary (cont’d)
• The line through O and perpendicular to the image
plane is the optical axis.
• The intersection of the optical axis with the image
plane is called principal point.
center of
projection
Note: the principal point is not necessarily the image center.
The equations of perspective projection
Y
X
Z
The equations of perspective projection (cont’d)
• Using matrix notation:
1
or
1
• Verify the correctness of the above matrix
– homogenize using w = Z
1
1/f
Properties of perspective projection
• Many-to-one mapping
– The projection of a point is not unique
– Any point on the line OP has the same projection
Properties of perspective projection (cont’d)
• Scaling/Foreshortening
– Object’s image size is inversely proportional to the distance
of the object from the camera.
Properties of perspective projection (cont’d)
• When a line (or surface) is parallel to the image plane, the
effect of perspective projection is scaling.
• When an line (or surface) is not parallel to the image plane,
the effect is foreshortening (i.e., perspective distortion).
Properties of perspective projection (cont’d)
• Effect of focal length
– As f gets smaller, more points project onto the image plane
(wide-angle camera).
– As f gets larger, the field of view becomes smaller (more
telescopic).
Properties of perspective projection (cont’d)
• What happens to lines, distances, angles and parallelism?
– Lines in 3D project to lines in 2D (with an exception …)
– Distances and angles are not preserved.
– Parallel lines do not in general project to parallel lines due
to foreshortening (unless they are parallel to the image plane).
Properties of perspective projection (cont’d)
• 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 infinity).
– The vanishing point of a line depends on the orientation of
the line and not on the position of the line.
Note: vanishing
points might lie
outside of the
image plane!
Properties of perspective projection (cont’d)
• Alternative definition for vanishing point:
– The vanishing point of any given line in space is located at
the point in the image where a parallel line through the
center of projection intersects the image plane.
Properties of perspective projection (cont’d)
• Vanishing line:
– The vanishing points of all the lines that lie on the same
plane form the vanishing line.
– Also defined by the intersection of a parallel plane through
the center of projection with the image plane.
vanishing
line
Orthographic Projection
• The projection of a 3D object onto a plane by a set of
parallel rays orthogonal to the image plane.
• It is the limit of perspective projection as
Orthographic Projection (cont’d)
• Using matrix notation:
• Verify the correctness of the above matrix (homogenize
using w=1):
Properties of orthographic projection
• Parallel lines project to parallel lines.
• Size does not change with distance from the camera.
Weak-perspective projection
• Approximate perspective projection by scaled
orthographic projection (i.e., linear transformation).
• Good approximation if:
(1) the object lies close to the optical axis.
(2) the object’s dimensions are small compared to its average
distance
from the camera
Weak perspective projection (cont’d)
• The term
is a scale factor now (e.g., every point is scaled
by the same factor).
• Using matrix notation:
• Verify - homogenize using
What assumptions have we made so far?
• Camera and world coordinate systems have been aligned
(i.e., all distances are measured in the camera’s reference
frame).
• The origin of the image plane is the principal point.
World – Pixel Coordinates
• In general, world and pixel coordinates are related by
additional parameters such as:
–
–
–
–
the position and orientation of the camera
the focal length of the lens
the position of the principal point
the size of the pixels
Types of parameters
• Extrinsic: the parameters that define the location and
orientation of the camera reference frame with respect
to a known world reference frame.
• Intrinsic: the parameters necessary to link the pixel
coordinates of an image point with the corresponding
coordinates in the camera reference frame.
Types of parameters (cont’d)
Extrinsic camera parameters
• Describe the transformation between the unknown camera
reference frame and the known world reference frame.
• Typically, determining these parameters means:
(1) find the translation
vector that maps the camera’s
origin to the world’s origin.
RT, T
(2) find the rotation matrix
that aligns the camera’s axes
with the world’s axes.
R, -T
Extrinsic camera parameters (cont’d)
• Using the extrinsic camera parameters, we can find the
relation between the coordinates of a point P in world
(Pw) and camera (Pc) coordinates:
Extrinsic camera parameters (cont’d)
or
where RiT corresponds to the i-th row of the rotation matrix
Intrinsic camera parameters
• Characterize the geometric, digital, and optical
characteristics of the camera:
(1) the perspective projection (focal length f ).
(2) the transformation between image plane coordinates
and pixel coordinates.
(3) the geometric distortion introduced by the optics.
Intrinsic camera parameters
(1) From Camera Coordinates to Image Plane Coordinates:
perspective projection:
Intrinsic camera parameters (cont’d)
(2) From Image Plane Coordinates to Pixel coordinates
(ox , oy) are the coordinates of the principal point
e.g., ox = N/2, oy = M/2 if the principal point is
the center of the image
sx , sy correspond to the effective size of the
pixels in the horizontal and vertical directions
(in millimeters)
Intrinsic camera parameters (cont’d)
• Using matrix notation:
Intrinsic camera parameters (cont’d)
(3) Relating pixel coordinates to world coordinates
or
f/sx
f/sy
Intrinsic camera parameters (cont’d)
Image distortions due to optics
(1) Radial distortion:
r x y
2
2
2
k1, k2, and k3 are intrinsic parameters
Correcting radial distortion
Intrinsic camera parameters (cont’d)
Image distortions due to optics
(2) Tangential distortion:
p1 and p2 are intrinsic parameters
Combine extrinsic with intrinsic camera parameters
• The matrix containing the intrinsic camera parameters
(not including distortion parameters for simplicity):
• The matrix containing the extrinsic camera parameters:
Combine extrinsic with
intrinsic camera parameters (cont’d)
• Using homogeneous coordinates:
• M is called the projection matrix (i.e., 3 x 4 matrix).
Combine extrinsic with
intrinsic camera parameters (cont’d)
• Warning: homogenization is required to obtain the pixel
coordinates:
Perspective projection - revisited
• Assuming ox = oy = 0 and sx = sy = 1
M  Mp
• Verify:
Homogenize:
√
Weak-perspective projection - revisited
M  Mwp
where
is the centroid of the object
(i.e., average distance from the camera)
• Verify:
Homogenize:
√