CS代写 CMT107 Visual Computing

CMT107 Visual Computing

Camera Calibration

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School of Computer Science and Informatics

Cardiff University

• Pinhole cameras
• Vanishing points

• Real camera
• Aperture adjustment

• Thin lens formula

• Lens flaws

• Pinhole camera model
• Camera parameters: intrinsic parameters, extrinsic parameters

• Camera calibration
• Linear method

Acknowledgement
The majority of the slides in this section are from at University of Illinois at Urbana-Champaign

Let’s Design a Camera

• Idea 1: put a piece of film in front of an object?

• Do we get a reasonable image?

Let’s Design a Camera

• Add a barrier to block off most of the rays
• This reduces blurring

• The opening is know as the aperture

Pinhole Camera Model

• Pinhole model:
• Captures pencil of rays – all rays through a single point (pinhole)

• The point is called centre of projection (focal point)

• The image is formed on the image plane

• A virtual image plane is used as mathematical description of the real image plane

Image plane

Vanishing Points

• Parallel lines are no longer parallel after projection. They converge at a single
point on the image plane – vanishing point

• Each direction in space has its own vanishing point

• Exception: directions parallel to the image plane

vanishing point

Building a Real Camera

Home-made Pinhole Camera

• Why so blurry?
“a larger pinhole to compensate for
the smaller amount of light. The result
is an image with more blur.”

http://www.debevec.org/Pinhole/

http://www.debevec.org/Pinhole/35mm-pinhole-camera.jpg
http://www.debevec.org/Pinhole/

Shrinking the Aperture

• Why not make the aperture as small as possible?
• Less light get through

• Diffraction effects …

Shrinking the Aperture

Adding a Lens

• A lens focuses light onto the film

• Thin lens model:
• Rays passing through the centre are not deviated (pinhole projection model still holds)

• A lens focuses light onto the film

• Thin lens model:
• Rays passing through the centre are not deviated (pinhole projection model still holds)

• All parallel rays converge to one point on a plane located at the focal length 𝑓

Adding a Lens

focal point

Adding a Lens

• A lens focuses light onto the film
• There is a specific distance at which an object is “in focus”, other points project to a

“circle of confusion” in the image

“circle of
confusion”

Thin Lens Formula

• What is the relation between:

The focal length 𝑓

The distance of the object
from the optical centre 𝐷

The distance at which the
object will be in focus 𝐷′

objectimage

Thin Lens Formula

• Similar triangles everywhere!

objectimage

Thin Lens Formula

• Similar triangles everywhere!

objectimage

Thin Lens Formula

• Similar triangles everywhere!

objectimage

Thin Lens Formula

• Any point satisfying the thin
lens equation is in focus

• As 𝑓 is fixed, the farther the
object, the closer the plane of

objectimage

Real Lenses

Lens Flaws: Chromatic Aberration

• Lens has different refractive indices for different
wavelengths, causes colour fringing

Near Lens centre Near Lens Outer Edge

Lens Flaws: Spherical Aberration

• Spherical lenses do not focus light perfectly

• Rays farther from the optical axis focus closer

Lens Flaws: Vignetting

Lens Flaws: Radial Distortion

• Caused by imperfect lenses

• Deviations are most noticeable near the edge of the lens

No distortion Pin cushion Barrel

Pinhole Camera Model Revisit

• Principal axis: line from the camera centre perpendicular to the image plane

• Camera coordinate system: camera centre is at the origin and the principal
axis is the z-axis

Pinhole Camera Model Revisit

)/,/(),,( ZYfZXfZYX 

Principal Point

• Principal point (p): point where principal axis intersects the image plane

• Normalised coordinate system: origin is at the principal point

• Image coordinate system: origin is in the corner

• How to go from normalized coordinate system to image coordinate system?

Principal Point Offset

Principal point:

pZYfpZXfZYX ++

Principal Point Offset

K  0|IKP =Calibration Matrix

Pixel Coordinates

• 𝑚𝑥 pixels per meter in horizontal direction;

𝑚𝑦 pixels per meter in vertical direction

Pixel size:

Camera Rotation and Translation

• In general, the camera coordinate
frame will be related to the world
coordinate frame by a rotation and
a translation

( )C~-X~RX~

coords. of point
in camera frame

coords. of camera centre
in world frame

coords. of a point
in world frame (nonhomogeneous)

Camera Rotation and Translation

• In non-homogenous coordinates

• In homogenous coordinates

( )C~-X~RX~

   XC~R|RKX0|IKx

−==  ,t|RKP = C

Camera Parameters

• Intrinsic parameters
• Principal point coordinates

• Focal length

• Pixel magnification factors

• Skew (non-rectangular pixels)

• Radio distortion

Camera Parameters

• Intrinsic parameters
• Principal point coordinates

• Focal length

• Pixel magnification factors

• Skew (non-rectangular pixels)

• Radio distortion

• Extrinsic parameters
• Rotation and translation relative to world coordinate system

Camera Calibration

𝐱 = 𝐏𝐗 = 𝐊 𝐑 𝐓 𝐗

Camera Calibration

• Given 𝑛 points with known 3D coordinates 𝑋𝑖 and known image projections
𝑥𝑖, estimate the camera parameters

Camera Calibration: Linear Method

T = 𝑃11 𝑃12 𝑃13 𝑃14

T = 𝑃21 𝑃22 𝑃23 𝑃24
T = 𝑃31 𝑃32 𝑃33 𝑃34

Two linearly independent equations

xi × PXi = 0,

Camera Calibration: Linear Method

= 0 AP = 0

• P has 11 degrees of freedom (12 parameters, but scale is arbitrary

• One 2D/3D correspondence gives two linearly independent equations

• At least 6 correspondences are needed for a solution

• Homogeneous least squares
• The eigenvector corresponding to the smallest eigenvalue of ATA

Camera Calibration: Linear Method

= 0 AP = 0

• Note: for coplanar points that satisfy Π𝑇X = 0, we will get the degenerate
solutions: Π, 0, 0 , 0, Π, 0 , or 0, 0, Π .

Camera Calibration: Linear Method

• Advantages
• Easy to formulate and solve

• Disadvantages
• Doesn’t directly tell you camera parameters

• Can’t impose constraints, such as known focal length and orthogonality

• Doesn’t model radial distortion

• Only an approximate solution

• Non-linear methods are preferred
• Define error as difference between projected points and measured points

• Minimise error using Newton’s method or other non-linear optimisation

• Describe pinhole model.

• What is vanishing point?

• What are intrinsic/extrinsic camera parameters?

• Describe the linear camera calibration method.

• What are the advantages and disadvantages of linear method for camera
calibration?

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