humanoid robotics projective geometry, homogeneous …
TRANSCRIPT
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Humanoid Robotics
Projective Geometry, Homogeneous Coordinates
(brief introduction)
Maren Bennewitz
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Motivation § Cameras generate a projected image of the
3D world § In Euclidian geometry, the math for
describing the transform can get difficult § Projective geometry: alternative algebraic
representation of geometric transformations § So-called homogeneous coordinates are
often used in robotics § All affine transformations and even
projective transformations can be expressed with one matrix multiplication
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Pinhole Camera Model § Describes the projection of a 3D world point
into the camera image § A box with an infinitesimal small hole § The camera center is the intersection point
of the rays (pinhole) § The back wall is the image plane § The distance between the camera center
and image plane is the camera constant
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Pinhole Camera Model
image plane
object
camera center (pinhole)
camera constant (focal length)
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Geometry and Images
Image courtesy: Förstner
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§ Straight lines stay straight § Parallel lines may intersect
Image courtesy: Förstner
Geometry and Images
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Pinhole Camera Model: Properties § Line-preserving: straight lines are
mapped to straight lines § Not angle-preserving: angles
between lines change § Not length-preserving: size of
objects is inverse proportional to the distance
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Information Loss Caused by the Projection § A camera projects from 3D to 2D § 3D information can only be recovered
if additional information is available
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Information Loss Caused by the Projection § A camera projects from 3D to 2D § 3D information can only be recovered
if additional information is available § Multiple images
§ Details about the camera § Background knowledge § Known size of objects
§ …
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Homogeneous Coordinates (H.C.)
§ H.C. are a system of coordinates used in projective geometry
§ Formulas using H.C. are often simpler than using Euclidian coordinates
§ A single matrix can represent affine and projective transformations
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Notation
Point (or or ) § in homogeneous coordinates § in Euclidian coordinates 2D vs. 3D space § lowercase = 2D § capitalized = 3D
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Homogeneous Coordinates
Definition The representation of a geometric object is homogeneous if and represent the same object for
Example
homogeneous Euclidian
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Homogeneous Coordinates
§ H.C. use a n+1 dimensional vector to represent the same (n-dim.) point
§ Example:
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Definition
§ Homogeneous coordinates of a point in is a 3-dim. vector
§ Corresponding Euclidian coordinates
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From Homogeneous to Euclidian Coordinates
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From Homogeneous to Euclidian Coordinates
Image courtesy: Förstner
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3D Points Analogous for 3D points in Euclidian space:
homogeneous Euclidian
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Origin of the Euclidian Coordinate System in H.C.
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Projective Transformation
§ Invertible linear mapping
§ Also called “homography”
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Example: Homography
source: openMVG
in the following: 3D transformations relevant for this lecture
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§ Projective mapping
§ Translation: 3 parameters (3 translation)
Important 3D Transformations
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Important 3D Transformations
§ Rotation: 3 parameters (3 rotation)
rotation matrix
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Reminder: Rotation Matrices § 2D:
§ 3D:
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Important 3D Transformations
§ Rotation: 3 parameters (3 rotation)
§ Rigid body transformation: 6 params (3 translation + 3 rotation)
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Important 3D Transformations
§ Similarity transformation: 7 params (3 trans + 3 rot + 1 scale)
§ Affine transformation: 12 parameters (3 trans + 3 rot + 3 scale + 3 sheer)
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Important 3D Transformations
§ Projective transformation: 15 params (affine transformation + 3 parameters)
§ These 3 parameters are the projective
that lead to the fact that parallel lines may not stay parallel
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Transformations Hierarchy (2D/3D)
projective transformation (8/15 parameters)
affine transformation (6/12 parameters)
similarity transformation (4/7 parameters)
motion transformation (3/6 parameters)
translation (2/3 parameters)
rotation (1/3 parameters)
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Inverting and Chaining
§ Inverting a transformation
§ Chaining transformations via matrix products (not commutative)
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Conclusion
§ Homogeneous coordinates are an alternative representation for transforms
§ Can simplify mathematical expressions § Allow for easy chaining and inversion
of transformations § Modeled through an extra dimension § Equivalence up to scale
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Literature § Multiple View Geometry in Computer Vision,
R. Hartley and A. Zisserman, Ch. 2, 3
§ Slides based on Chapter 15 “Projective Geometry (Homogeneous Coordinates)”, Photogrammetry I by C. Stachniss