Lecture 7: Camera Models

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1 Lecture 7: Camera Models Professor Fei- Fei Li Stanford Vision Lab Lecture 7 -! 1

2 What we will learn toda? Pinhole cameras Cameras & lenses The geometr of pinhole cameras Reading: [FP] Chapters 1 3 [HZ] Chapter 6 Lecture 7 -! 2

3 What we will learn toda? Pinhole cameras Cameras & lenses The geometr of pinhole cameras Reading: [FP] Chapters 1 3 [HZ] Chapter 6 Lecture 7 -! 3

4 How do we see the world? Let s design a camera Idea 1: put a piece of film in front of an object Do we get a reasonable image? Lecture 7 -! 4

5 Pinhole camera Add a barrier to block off most of the ras This reduces blurring The opening known as the aperture Lecture 7 -! 5

6 Some histor Milestones: Leonardo da Vinci ( ): first record of camera obscura Johann Zahn (1685): first portable camera Lecture 7 -! 6

7 Some histor Milestones: Leonardo da Vinci ( ): first record of camera obscura Johann Zahn (1685): first portable camera Joseph Nicephore Niepce (1822): first photo - birth of photograph Daguerréotpes (1839) Photographic Film (Eastman, 1889) Cinema (Lumière Brothers, 1895) Color Photograph (Lumière Brothers, 198) Photograph (Niepce, La Table Servie, 1822) Lecture 7 -! 7

8 Some histor Motu ( BC) Oldest eistent book on geometr in China Aristotle ( BC) Also: Plato, Euclid Al- Kindi (c ) Ibn al- Haitham (965-14) Lecture 7 -! 8

9 Lecture 7 -! 9 Pinhole camera = = f f ' ' ' ' ʹ ʹ = ʹ P = P Derived using similar triangles Note: is alwas negamve.

10 Pinhole camera P =[, f ] f O P = [, ] ʹ = f ' Lecture 7 -! 1

11 Pinhole camera f f Common to draw image plane in front of the focal point Moving the image plane merel scales the image. ' = ' = f f Lecture 7 -! 11

12 Pinhole camera Is the sie of the aperture important? Kate lauka Lecture 7 -! 12

13 Cameras & Lenses Shrinking aperture sie - Ras are mied up - Wh the aperture cannot be too small? - Less light passes through - Diffracmon effect Adding lenses! Lecture 7 -! 13

14 Cameras & Lenses A lens focuses light onto the film Lecture 7 -! 14

15 Cameras & Lenses focal point f A lens focuses light onto the film Ras passing through the center are not deviated All parallel ras converge to one point on a plane located at the focal length f Lecture 7 -! 15

16 Cameras & Lenses circle of confusion A lens focuses light onto the film There is a specific distance at which objects are in focus [other points project to a circle of confusion in the image] Lecture 7 -! 16

17 Cameras & Lenses Laws of geometric opmcs Light travels in straight lines in homogeneous medium Reflecmon upon a surface: incoming ra, surface normal, and reflecmon are co- planar Refracmon: when a ra passes from one medium to another Snell s law n 1 sin α 1 = n 2 sin α 2 α 1 = incident angle α 2 = refracmon angle n i = inde of refracmon Lecture 7 -! 17

18 Thin Lenses o ' = f + o f = R 2(n 1) Snell s law: n 1 sin α 1 = n 2 sin α 2 Small angles: n 1 α 1 n 2 α 2 n 1 = n (lens) n 1 = 1 (air) ' = ' = ' ' Lecture 7 -! 18

19 Cameras & Lenses Source wikipedia Lecture 7 -! 19

20 Issues with lenses: Chroma>c Aberra>on Lens has different refracmve indices for different wavelengths: causes color fringing f = R 2(n 1) Lecture 7 -! 2

21 Issues with lenses: Chroma>c Aberra>on Ras farther from the opmcal ais focus closer Lecture 7 -! 21

22 Issues with lenses: Chroma>c Aberra>on Deviamons are most nomceable for ras that pass through the edge of the lens No distortion Pin cushion Barrel (fishee lens) Image magnificamon decreases with distance from the opmcal ais Lecture 7 -! 22

23 What we will learn toda? Pinhole cameras Cameras & lenses The geometr of pinhole cameras Lecture 7 -! 23

24 f Pinhole camera c f = focal length c = center of the camera (,,) R 3 E R 2 (f,f ) Lecture 7 -! 24

25 Pinhole camera Is this a linear transformamon? (,,) (f,f ) No division b is nonlinear! How to make it linear? Lecture 7 -! 25

26 Homogeneous coordinates homogeneous image coordinates homogeneous scene coordinates Convermng from homogeneous coordinates Lecture 7 -! 26

27 Lecture 7 -! 27 Homogeneous coordinates = = 1 1 ' f f f f P f f = ' P i P = M P ' M 3 H 4 R R Perspecmve Projecmon Transformamon: Projec>on matri

28 From re>na plane to images Piels, bosom- let coordinate sstems Lecture 7 -! 28

29 From re>na plane to images c 1. Off set C=[c, c ] c (,,) (f + c, f + c) Lecture 7 -! 29

30 From re>na plane to images c c 1. Off set 2. From metric to piels (,, ) ( f k + c, f l + c ) α β C=[c, c ] Units: k,l : piel/m f : m Non- square piels α, β : piel Lecture 7 -! 3

31 From re>na plane to images c (,,) ( α + c, β + c) C=[c, c ] c Matri form? Lecture 7 -! 31

32 Lecture 7 -! 32 Camera matri c c C=[c, c ] ) c, c (,), ( + + β α = + + = 1 1 ' c c c c P β α β α

33 Lecture 7 -! 33 Camera matri c c C=[c, c ] ) c, c (,), ( + + β α = 1 1 ' c c P β α

34 Lecture 7 -! 34 Camera matri ) c, c (,), ( + + β α c c C=[c, c ] ν Skew parameter = 1 1 ' c c s P β α

35 Camera matri α s c P' = = K[ I ] P β c 1 1 P ' = M P Camera matri K K has 5 degrees of freedom! Lecture 7 -! 35

36 Camera & world reference sstem R,T j w k w O w i w The mapping is defined within the camera reference sstem What if an object is represented in the world reference sstem? Lecture 7 -! 36

37 Camera & world reference sstem R,T j w k w O w i w ' = K[ R T ] P w P = M P w In 4D homogeneous coordinates: P = [ R T ] P w Internal parameters Eternal parameters Lecture 7 -! 37

38 Projec>ve cameras R,T j w k w O w i w P = M P 3 1 w = K R T Pw ' [ ] K = α s β c c 1 How man degrees of freedom? =11! Lecture 7 -! 38

39 Projec>ve cameras R,T j w k w O w P = M P 3 1 w (,, ) w m ( m = K R T Pw ' [ ] P P w w, m m 2 3 P P w w ) 3 4 i w M = m m m M is defined up to scale! Mulmpling M b a scalar won t change the image Lecture 7 -! 39

40 Theorem (Faugeras, 1993) [ T] = [ K R KT ] [ A b] M = K R = A = a a a K α = s c c 1 β α = f β = f k; l Lecture 7 -! 4

41 Proper>es of Projec>on Points project to points Lines project to lines Lecture 7 -! 41

42 Proper>es of Projec>on Angles are not preserved Parallel lines meet Vanishing point Lecture 7 -! 42

43 What we have learned toda? Pinhole cameras Cameras & lenses The geometr of pinhole cameras Reading: [FP] Chapters 1 3 [HZ] Chapter 6 Lecture 7 -! 43

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