3-D Euclidean Space - Vectors
Rigid Body Motion and A “free” vector is defined by a pair Image Formation of points :
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Coordinates of the vector :
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Rotation Matrices in 3D 3D Rotation of Points – Euler angles
Rotation around the coordinate axes, counter-clockwise: 1 0 0 •3 by 3 matrices R (α) = 0 cosα − sinα • 9 parameters – only three degrees of freedom x • Representations – either three Euler angles 0 sinα cosα • or axis and angle represntation P’ cos β 0 sin β Y’ γ R (β ) = 0 1 0 y − β β y P sin 0 cos x cosγ − sinγ 0 X’ • Properties of rotation matrices (constraints between the R (γ ) = sinγ cosγ 0 z elements) z 0 0 1
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1 Rotation Matrices in 3D Canonical Coordinates for Rotation
•3 by 3 matrices Property of R • 9 parameters – only three degrees of freedom • Representations – either three Euler angles Taking derivative • or axis and angle representation
Skew symmetric matrix property
By algebra • Properties of rotation matrices (constraints between the elements) By solution to ODE Columns are orthonormal
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3D Rotation (axis & angle) Rotation Matrices
Solution to the ODE Given
How to compute angle and axis
with
or
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2 Rigid Body Motion – Homogeneous 3D Translation of Points Coordinates
Translate by a vector 3-D coordinates are related by: Homogeneous coordinates: P’ t Y’ x’ x P Homogeneous coordinates are related by: z’ y z
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Rigid Body Motion – Homogeneous Properties of Rigid Body Motions Coordinates
Rigid body motion composition 3-D coordinates are related by: Homogeneous coordinates:
Rigid body motion inverse
Homogeneous coordinates are related by: Rigid body motion acting on vectors
CS682, Jana Kosecka Vectors are only affected by rotationCS682, Jana – 4Koseckath homogeneous coordinate is zero
3 Rigid Body Transformation Rigid Body Motion
• Camera is moving
• Notion of a twist
• Relationship between velocities Coordinates are related by: Camera pose is specified by: CS682, Jana Kosecka CS682, Jana Kosecka
Image Formation – Perspective Projection Pinhole Camera Model
“The Scholar of Athens,” Raphael, 1518 Pinhole
Frontal pinhole
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4 More on homogeneous coordinates Pinhole Camera Model
In homogenous coordinates – these represent the 2-D coordinates Same point in 3D Homogeneous coordinates
The first coordinates can be obtained from the second by division by W
What if W is zero ? Special point – point at infinity – more later In homogeneous coordinates – there is a difference between point and vector
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Image Coordinates Calibration Matrix and Camera Model
Pinhole camera Pixel coordinates metric coordinates
Linear transformation
Calibration matrix pixel (intrinsic parameters) coordinates Projection matrix
Camera model CS682, Jana Kosecka CS682, Jana Kosecka
5 Calibration Matrix and Camera Model Radial Distortion
Pinhole camera Pixel coordinates Nonlinear transformation along the radial direction
More compactly
Transformation between camera coordinate Systems and world coordinate system Distortion correction: make lines straight
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Image of a line – homogeneous Image of a point representation
Homogeneous coordinates of a 3-D point Homogeneous representation of a 3-D line
Homogeneous coordinates of its 2-D image
Projection of a 3-D point to an image plane Homogeneous representation of its 2-D image
Projection of a 3-D line to an image plane
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6 Image of a line – 2D representations Visual Illusions
Representation of a 3-D line
Projection of a line - line in the image plane
Special cases – parallel to the image plane, perpendicular When λ -> ∞ - vanishing points In art – 1-point perspective, 2-point perspective, 3-point perspective
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Vanishing points Ames Room Illusions
Different sets of parallel lines in a plane intersect at vanishing points, vanishing points form a horizon line
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7 More Illusions
Which of the two monsters is bigger ?
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