Geometrical Primitives, Transformations and Image Formation
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1 Geometrical Primitives, Transformations and Image Formation EECS Fall 2014! Foundations of Computer Vision!! Instructor: Jason Corso! Materials on these slides have come from many sources in addition to myself; I am infinitely grateful to these, especially Greg Hager and Silvio Savarese.!
2 Plan 2 Geometric Primitives! Points, Lines in 2D and 3D! Transformations in 2D and 3D! Basic Image Formation! Camera Parameters! Lens Distortion!
3 Geometric Primitives 3 2D points: pixel coordinates! Using homogeneous coordinates! Vectors differing by scale are equivalent.! 2D Projective Space! augmented vector! When the last element, call it an ideal point.!
4 Geometric Primitives 4 2D lines with homogeneous coordinates! Normalized coordinates! normal vector! Polar coordinates! The image on this slides is sourced from the Szeliski book.!
5 Geometric Primitives 5 Intersection of two lines! Line connecting two points!
6 Geometric Primitives 6 3D points!
7 Geometric Primitives 7 3D planes! Spherical coordinates! can be written as a function of two angles.! Images on this slides are sourced from the Szeliski book and Wolfram website.!
8 Geometric Primitives 8 3D lines! Consider two points on the line.! For the case of homogeneous coordinates:! When the second point is at infinity,! The image on this slides is sourced from the Szeliski book.!
9 Geometric Transformations 9 2D translation! Identity matrix!! 2D rotation and translation! 2D rigid body or Euclidean transformation! Rotation! matrix!
10 Geometric Transformations 10 2D scaled rotation or similarity transform! Constraint is not enforced.! 2D affine transformation!
11 Geometric Transformations 11 2D projective, also called the homography! Projective matrix is defined up to scale.! Inhomogeneous results are computed after homogeneous operation.!
12 Hierarchy of 2D Planar Transformations 12 The images on this slides are sourced from the Szeliski book.!
13 Hierarchy of 3D Coordinate Transformations 13
14 Projective Geometry 14 These geometry basics are but the surface of an area important to computer vision called projective geometry.! Further reading: An Introduction to Projective Geometry by Stan Birchfield.!
15 15
16 Light 16 Getting light to the sensor.! What does this image look like?! Source: S. Savarese, GD Hager and S Seitz slides.!
17 Light through a pinhole 17 Place a barrier in front of the film.! Let a small pinhole of light through.! aperture! Leonardo da Vinci ( ): Camera Obscura! Source: D. Forsyth, S. Savarese, GD Hager and S Seitz slides.!
18 Light through a pinhole 18 Pinhole: box with a small hole in it.! Abstract model that does indeed work in practice.! aperture! focal length! Source: D. Forsyth, S. Savarese slides.!
19 Pinhole, or Central, Perspective 19 Points are collinear.! Therefore, we have and.! Source: D. Forsyth, S. Savarese slides.!
20 Properties of Pinhole Perspective Projection 20 Distant objects appear smaller! Source: D. Forsyth, S. Savarese slides.!
21 Properties of Pinhole Perspective Projection 21 Points project to points! Lines project to lines! Vanishing Point! Angles are not preserved.! Parallel lines meet!! Source: S. Savarese slides.!
22 Fun with vanishing points 22 Source: S. Seitz slides.!
23 Perspective cues 23 Source: S. Seitz slides.!
24 Perspective cues 24 Source: S. Seitz slides.!
25 Perspective cues 25 Source: S. Seitz slides.!
26 Pinhole, or Central, Perspective 26 It is common to draw the image plane in front of the focal point.! Source: D. Forsyth, S. Savarese slides.!
27 Weak Perspective 27 A coarser approximation to image formation is called weak perspective, or scaled orthography.! Consider a fronto-parallel plane defined by.! Rewrite projection equations for any point in! Source: D. Forsyth, S. Savarese slides.!
28 Orthographic Projections 28 Further, when the camera will be at a fixed distance from the scene, we can further normalize the coordinates.! Make! Then and! Almost never an acceptable model for image formation.!! Source: D. Forsyth, S. Savarese slides.!
29 Projection Matrices 29 Can formulate the perspective projections as matrix operations with homogeneous coordinates.!! Why are homogeneous coordinates necessary here?! Can also formulate as a 4x4 projection.!
30 Projection Matrices 30 How does scaling affect the projection?!!
31 Role of aperture size 31 When aperture is big, what happens?!!!! Why not make the aperture as small as possible?! Not enough light gets through.! Diffraction.! Source: S. Seitz, S. Savarese slides.!
32 Adding a lens 32 A lens focuses the light onto the film/ccd.! Rays passing through the center are not deviated.! All parallel rays converge to one point on a plane located at the focal length f.! focal point f optical center (Center Of Projection) focal point Source: GD Hager and S Seitz slides.!
33 Pinhole vs. lens 33 Source: GD Hager and S Seitz slides.!
34 Thin lens equation 34 How to relate distance of object from optical center ( ) to the distance at which it will be in focus ( ), given focal length?! Source: GD Hager and S Seitz slides.!
35 Thin lens equation 35 How to relate distance of object from optical center ( ) to the distance at which it will be in focus ( ), given focal length?! Source: GD Hager and S Seitz slides.!
36 Thin lens equation 36 How to relate distance of object from optical center ( ) to the distance at which it will be in focus ( ), given focal length?! Source: GD Hager and S Seitz slides.!
37 Thin lens equation 37 Any object point satisfying this equation is in focus! Source: GD Hager and S Seitz slides.!
38 Depth of field f / 5.6 Source: S Seitz slides.! f / 32 Changing the aperture size affects depth of field A smaller aperture increases the range in which the object is approximately in focus Flower images from Wikipedia
39 Projection equation The projection matrix models the cumulative effect of all parameters Useful to decompose into a series of operations x = ΠX = = 1 * * * * * * * * * * * * Z Y X s sy sx = ' 0 ' x x x x x x c y c x y fs x fs T I R Π projection intrinsics rotation translation identity matrix Camera parameters A camera is described by several parameters Translation T of the optical center from the origin of world coords Rotation R of the image plane focal length f, principle point (x c, y c ), pixel size (s x, s y ) blue parameters are called extrinsics, red are intrinsics The definitions of these parameters are not completely standardized especially intrinsics varies from one book to another Source: S Seitz slides.!
40 40 Radial Distortion Pin Cushion! Barrel / Fisheye! Source: S Savarese slides.!
41 Radial Distortion No distortion Pin cushion Barrel Radial distortion of the image Caused by imperfect lenses Deviations are most noticeable for rays that pass through the edge of the lens Source: S Seitz slides.!
42 Correcting radial distortion Source: S Seitz slides.! from Helmut Dersch
43 Distortion Source: S Seitz slides.!
44 Modeling distortion Project to normalized image coordinates Apply radial distortion Apply focal length translate image center To model lens distortion Use above projection operation instead of standard projection matrix multiplication Source: S Seitz slides.!
45 45
46 Other types of projection Lots of intriguing variants (I ll just mention a few fun ones) Source: S Seitz slides.!
47 360 degree field of view Basic approach Take a photo of a parabolic mirror with an orthographic lens (Nayar) Or buy one a lens from a variety of omnicam manufacturers See Source: S Seitz slides.!
48 Tilt-shift Source: S Seitz slides.! Titlt-shift images from Olivo Barbieri and Photoshop imitations
49 Rotating sensor (or object) Rollout Photographs Justin Kerr Also known as cyclographs, peripheral images Source: S Seitz slides.!
50 Photofinish Source: S Seitz slides.!
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