Camera Parameters

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 (x

, y

).

(assuming that the origin in both cases is the upper-left corner)

(x

, y

) 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: (x

, y

, z

)

• 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: (X

, Y

, Z

)

(2) World Coordinate Frame

• 3D coordinate system: (x

, y

, z

)

• Useful for interrelating objects in 3D

Our notation: (X

, Y

, Z

)

(3) Camera Coordinate Frame

• 3D coordinate system: (x

, y

, z

)

• Useful for representing objects with respect to the location of the camera.

Our notation: (X

, Y

, Z

)

(4) Image Plane Coordinate Frame (i.e., CCD plane)

• 2D coordinate system: (x

, y

)

• Describes the coordinates of 3D points projected on the image plane.

Our notation: (x, y)

(5) Pixel Coordinate Frame

• 2D coordinate system: (c, r)

• Each pixel in this frame has integer pixel coordinates.

Our notation: (x

, y

)

y

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

Z Y

X

y x

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.

Note: the principal point is not necessarily the image center.

center of projection

The equations of perspective projection

Y

X

Z

The equations of perspective projection (cont’d)

• Using matrix notation:

• Verify the correctness of the above matrix

– homogenize using w = Z

or

1

1

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.

(2) find the rotation matrix that aligns the camera’s axes with the world’s axes.

R, -T R^T, 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 (P

) and camera (P

) coordinates:

Extrinsic camera parameters (cont’d)

where R

corresponds to the i-th row of the rotation matrix

or

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

(o_x , o_y) are the coordinates of the principal point e.g., o_x = N/2, o_y = M/2 if the principal point is the center of the image

s_x , s_y 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

f/s_x f/s_y

or

Intrinsic camera parameters (cont’d)

Image distortions due to optics

(1) Radial distortion:

2 2 2

r  x  y

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 o

= o

= 0 and s

= s

= 1

• Verify:

Homogenize:

M  M

√

Weak-perspective projection - revisited

where is the centroid of the object (i.e., average distance from the camera)

• Verify:

Homogenize:

M  M

√