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Two-views geometry Outline . Background: Camera, Projection Necessary tools: A taste of projective geometry Two view geometry: Homography Epipolar geometry, the essential matrix Camera calibration, the fundamental matrix 3D reconstruction (Stereo algorithms) next week.
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Two-views geometryOutline • Background: Camera, Projection • Necessary tools: A taste of projective geometry • Two view geometry: • Homography • Epipolar geometry, the essential matrix • Camera calibration, the fundamental matrix • 3D reconstruction (Stereo algorithms) next week. Many of the slides are courtesy of Prof. Ronen Basri
Objective 3-D Scene u’ u What can 2 images tell us about …. Faugeras et. al. ECCV 92
Objective 3-D Scene u’ u Study the mathematical relations between corresponding image points. “Corresponding”means originated from the same 3D point.
World Cup 66: England-Germany Conclusion: no goal (missing 3 inches) (Reid and Zisserman, “Goal-directed video metrology”)
Camera Obscura "Reinerus Gemma-Frisius, observed an eclipse of the sun at Louvain on January 24, 1544, and later he used this illustration of the event in his book De Radio Astronomica et Geometrica, 1545. It is thought to be the first published illustration of a camera obscura..." Hammond, John H., The Camera Obscura, A Chronicle
A few words about Cameras • Camera obscura dates from 15th century • First photograph on record shown in the book - 1822 • Basic abstraction is the pinhole camera • Current cameras contain a lens and a recording device (film, CCD, CMOS) • The human eye functions very much like a camera
Ideal Lenses Lens acts as a pinhole (for 3D points at the focal depth).
Regular Lenses E.g., the cameras in our lab. To learn more on lens-distortion see Hartley & Zisserman Sec. 7.4 p.189. Not part of this class.
Notation • O – Focal center • π – Image plane • Z – Optical axis • f – Focal length
Projection y x f Y X Z
Perspective Projection • Origin (0,0,0)is the Focal center • X,Y (x,y) axis are along the image axis (height / width). • Z is depth = distance along the Optical axis • f – Focal length
Orthographic Projection • Projection rays are parallel • Image plane is fronto-parallel • (orthogonal to rays) • Focal center at infinity
Scaled Orthographic Projection Also called “weak perspective”
Pros and Cons of Projection Models • Weak perspective has simpler math. • Accurate when object is small and distant. • Useful for object recognition. • Pinhole perspective much more accurate. • Used in structure from motion. • When accuracy really matters (SFM), we must model the real camera (exact imaging processes): • Perspective projection, calibration parameters (later), and all other issues (radial distortion).
Two-views geometryOutline • Background: Camera, Projection • Necessary tools: A taste of projective geometry • Two view geometry: • Homography • Epipolar geometry, the essential matrix • Camera calibration, the fundamental matrix • 3D reconstruction from two views (Stereo algorithms) Hartley & Zisserman: Sec. 2 Proj. Geom. of 2D. Sec. 3 Proj. Geom. of 3D.
Reading • Hartley & Zisserman: • Sec. 2 Proj. Geo. of 2D: • 2.1- 2.2.3 point lines in 2D • 2.3 -2.4 transformations • 2.7 line at infinity • Sec. 3 Proj. Geo. of 3D. • 3.1 – 3.2 point planes & lines. • 3.4 transformations
Why projective Geometry (Motivation) • Euclidean Geometry is good for questions like: what objects have the same shape (= congruent) Same shapes are related by rotation and translation
Why Projective Geometry (Motivation) • Answers the question what appearances (projections) represent the same shape Same shapes are related by a projective transformation
Why Projective Geometry (Motivation) Parallel lines meet at the horizon (“vanishing line”) Where do parallel lines meet?
Coordinates in Euclidean Space Not in space 0 1 2 3 ∞
Coordinates in Projective Line P1 Points on a line P1 are represented as rays from origin in 2D, Origin is excluded from space “Ideal point” k(1,1) k(0,1) k(-1,1) k(2,1) -1 0 1 2 ∞ k(1,0)
Coordinates in Projective Plane P2 Take R3 –{0,0,0} and look at scale equivalence class (rays/lines trough the origin). k(0,1,1) k(1,1,1) “Ideal point” k(0,0,1) k(1,0,1) k(x,y,0)
Projective Line vs. the Real Line “Ideal point” k(1,1) k(0,1) k(-1,1) k(2,1) -1 0 1 2 ∞ k(1,0)
k(0,1,1) k(1,1,1) “Ideal line” k(0,0,1) k(1,0,1) k(x,y,0) Projective Plane vs Euclidian plane
2D Projective Geometry: Basics • A point: • A line: we denote a line with a 3-vector • Line coordinates are homogenous • Points and lines are dual: p is on l if • Intersection of two lines/points
Area of parallelogram bounded by u and v Cross Product Every entry is a determinant of the two other entries Hartley & Zisserman p. 581
Cross Product in matrix notation [ ]x Hartley & Zisserman p. 581
Example: Intersection of parallel lines Q: How many ideal points are there in P2? A: 1 degree of freedom family – the line at infinity
u’ u Projective Transformations
Transformations of the projective line Given a 2D linear transformation G:R2 R2 Study the induced transformation on the Equivalents classes. On the realization y=1 we get
Properties: • Invertible (T-1 exists) • Composable (To G is a projective transformation) • Closed under composition • Has 4 parameters • 3 degrees of freedom • Defined by 3 points Every point defines 1 constraint
Perspective mapping Pencil of rays Transformations of the projective line A perspective mapping is a projective transformation T:P1 P1 Perceptivity is a special projective mapping. Hartley & Zisserman p. 632 Lines connecting corresponding points are “concurrent”
Ideal points and projective transformations Projective transformation can map ∞ to a real point
2D Projective Transformation 4 names 3 definitions • Projectivity: An invertible mapping h:P2 P2 S.T: • Homography. A 3x3 (non singular) invertible matrix acting on homogenous 3-vectores. • Collineation A transformations that map lines to lines Hartley & Zisserman p. 32
2D Projective Transformation • H is defined up to scale • 9 parameters • 8 degrees of freedom • Determined by 4 corresponding points how does H operate on lines? Hartley & Zisserman p. 32
Plane Perspective This mapping clearly maps lines to lines
Plane Perspective acting on conics Hartley & Zisserman p. 30 & 36 Not part of this class
Hierarchy of Transformations Projective Affine Similarity Rigid (Isometry) Translation: Rotation: Scale Hartley & Zisserman p. Sec. 2.4
Euclidean Transformations (Isometries) Rotation: Translation:
Hierarchy of Transformations • Isometry (Euclidean), • Similarity, • Affine, general linear • Projective,
Two-views geometryOutline • Background: Camera, Projection • Necessary tools: A taste of projective geometry • Two view geometry: • Homography • Epipolar geometry, the essential matrix • Camera calibration, the fundamental matrix • 3D reconstruction from two views (Stereo algorithms)
Two View Geometry • When a camera changes position and orientation, the scene moves rigidly relative to the camera 3-D Scene u’ u Rotation + translation
Objective: find formulas that links corresponding points 3-D Scene u’ u Rotation + translation
Two View Geometry (simple cases) • In two cases this results in homography: • Camera rotates around its focal point • The scene is planar Then: • Point correspondence forms 1:1mapping • depth cannot be recovered