Perspective Projection Direct Linear Transformation Traiangulation PERSPECTIVE PROJECTION
- Slides: 25
• Perspective Projection • Direct Linear Transformation • Traiangulation
PERSPECTIVE PROJECTION
Perspective projection Image plane Focal length Camera frame Scene point Optical axis Image coordinates Thus far, in camera’s reference frame only.
Camera parameters • Extrinsic: location and orientation of camera frame with respect to reference frame • Intrinsic: how to map pixel coordinates to image plane coordinates Reference frame Camera 1 frame
Extrinsic camera parameters Camera reference frame World reference frame
Camera parameters • Extrinsic: location and orientation of camera frame with respect to reference frame • Intrinsic: how to map pixel coordinates to image plane coordinates Reference frame Camera 1 frame
Intrinsic camera parameters Ignoring any geometric distortions from • optics, we can describe them by: Coordinates of projected point in camera reference frame Coordinates of image point in pixel units Coordinates of image center in pixel units Effective size of a pixel (mm)
Camera parameters • We know that in terms of camera reference frame: and • Substituting previous eqns describing intrinsic and extrinsic parameters, can relate pixels coordinates to world points: Ri = Row i of rotation matrix
Projection matrix point in camera coordinates • This can be rewritten as a matrix product using homogeneous coordinates: • The motion of the camera is equivalent to the inverse motion of the scene Internal parameters K= wxim wyim w = K[R t] External parameters Xw Yw Zw 1
Calibrating a camera • Compute intrinsic and extrinsic parameters using observed camera data • Main idea • Place “calibration object” with known geometry in the scene • Get correspondences • Solve for mapping from scene to image: estimate M=K[R t]
Projection matrix Pw in homog. • This can be rewritten as a matrix product using homogeneous coordinates: wxim wyim w = K[R t] Xw Yw Zw 1 product M is single projection matrix encoding both extrinsic and intrinsic parameters Let Mi be row i of matrix M
Projection matrix Pw in homog. • This can be rewritten as a matrix product using homogeneous coordinates: wxim wyim w = K[R t] Xw Yw Zw 1 product M is single projection matrix encoding both extrinsic and intrinsic parameters Let Mi be row i of matrix M
Estimating the projection matrix For a given feature point
Estimating the projection matrix Expanding this first equation, we have:
Estimating the projection matrix
Estimating the projection matrix This is true for every feature point, so we can stack up n observed image features and their associated 3 d points Pm = 0 in single equation: P … … … m … … Solve for mij’s (the calibration information) [F&P Section 3. 1]
Summary: camera calibration • Associate image points with scene points on object with known geometry • Use together with perspective projection relationship to estimate projection matrix • (Can also solve for explicit parameters themselves)
DIRECT LINEAR TRANSFORMATION
Direct Linear Transformation from 3 D to 2 D • Given two cameras (a stereo pair) and a 3 D object with known 3 D coordinates we can: 1) For each camera compute the direct linear transformation from 3 D to 2 D • Now we can move the stereo pair to a different place, take a pair of pictures and recover 3 D information: 1) Given two camera matrices and a pair of matching points, we can triangulate to get its depth. 2) We’ll talk about triangulation later in class
When would we calibrate this way? • Makes sense when geometry of system is not going to change over time • …When would it change?
TRIANGULATION
Triangulation • Problem: Given some points in correspondence across two or more images (taken from calibrated cameras), {(uj, vj)}, compute the 3 D location X 24
Triangulation • Method I: intersect viewing rays in 3 D, minimize: X – X is the unknown 3 D point – Cj is the optical center of camera j – Vj is the viewing ray for pixel (uj, vj) – sj is unknown distance along Vj • Advantage: geometrically intuitive 25 Vj Cj
Triangulation • Method II: solve linear equations in X – advantage: very simple • Method III: non-linear minimization – advantage: most accurate (image plane error) 26
- Traiangulation
- Perspective projection
- What is a linear transformation linear algebra
- Linear algebra linear transformation
- Parts of isometric drawing
- System of orthographic projection
- Receding axis
- Scalar vs vector projection
- First and third angle projection
- Yyzzxx
- Vertically stretched or compressed
- Linear transformation
- Transformasi linear
- Isomorphism 線性代數
- Eigenspace
- Range of a linear transformation
- Intensity transformation in digital image processing
- Linear transformation
- Piecewise linear transformation in digital image processing
- Dimensions of a vector space
- Perspective projection
- Weak perspective projection
- Auxiliary vanishing points
- Perspective projection matrix
- Perspective projection
- Digital image formation