Projective Geometry and Camera Models

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3 Projective Geometry and Camera Models Computer Vision CS 43 Brown James Hays Slides from Derek Hoiem, Alexei Efros, Steve Seitz, and David Forsyth

4 Administrative Stuff My Office hours, CIT 375 Monday and Friday 2-3 TA Office hours, CIT 29 Sunday 4-6 Monday 6-8 Monday 8- Tuesday 6-8 Thursday 6-8 Project is out

5 Previous class: Introduction Overview of vision, examples of state of art, preview of projects Robotics Machine Learning Computer Vision Human Computer Interaction Graphics Computational Photography Image Processing Feature Matching Recognition Neuroscience Medical Imaging Optics

6 What do you need to make a camera from scratch?

7 Today s class Mapping between image and world coordinates Pinhole camera model Projective geometry Vanishing points and lines Projection matrix

8 Today s class: Camera and World Geometry How tall is this woman? How high is the camera? What is the camera rotation? What is the focal length of the camera? Which ball is closer?

9 Image formation Slide source: Seitz Let s design a camera Idea : put a piece of film in front of an object Do we get a reasonable image?

10 Pinhole camera Idea 2: add a barrier to block off most of the rays This reduces blurring The opening known as the aperture Slide source: Seitz

11 Pinhole camera f c f = focal length c = center of the camera Figure from Forsyth

12 Camera obscura: the pre-camera Known during classical period in China and Greece (e.g. Mo-Ti, China, 47BC to 39BC) Illustration of Camera Obscura Freestanding camera obscura at UNC Chapel Hill Photo by Seth Ilys

13 Camera Obscura used for Tracing Lens Based Camera Obscura, 568

14 First Photograph Oldest surviving photograph Took 8 hours on pewter plate Photograph of the first photograph Joseph Niepce, 826 Stored at UT Austin Niepce later teamed up with Daguerre, who eventually created Daguerrotypes

15 Dimensionality Reduction Machine (3D to 2D) 3D world 2D image Point of observation Figures Stephen E. Palmer, 22

16 Projection can be tricky Slide source: Seitz

17 Projection can be tricky Slide source: Seitz

18 Projective Geometry What is lost? Length Who is taller? Which is closer?

19 Length is not preserved A C B Figure by David Forsyth

20 Projective Geometry What is lost? Length Angles Parallel? Perpendicular?

21 Projective Geometry What is preserved? Straight lines are still straight

22 Vanishing points and lines Parallel lines in the world intersect in the image at a vanishing point

23 Vanishing points and lines Vanishing Line Vanishing Point o Vanishing Point o

24 Slide from Efros, Photo from Criminisi Vanishing points and lines Vertical vanishing point (at infinity) Vanishing line Vanishing point Vanishing point

25 Projection: world coordinates image coordinates Optical Center (u., v ) f Z Y.. P = X Y Z. u v u p = v Camera Center (t x, t y, t z )

26 Homogeneous coordinates Conversion Converting to homogeneous coordinates homogeneous image coordinates homogeneous scene coordinates Converting from homogeneous coordinates

27 Homogeneous coordinates Invariant to scaling x k y w = kx ky kw Homogeneous Coordinates kx kw ky kw = x w y w Cartesian Coordinates Point in Cartesian is ray in Homogeneous

28 Projection matrix Slide Credit: Saverese R,T j w k w O w i w x = K[ R t] X x: Image Coordinates: (u,v,) K: Intrinsic Matrix (3x3) R: Rotation (3x3) t: Translation (3x) X: World Coordinates: (X,Y,Z,)

29 Interlude: why does this matter?

30 Relating multiple views

31 Object Recognition (CVPR 26)

32 Inserting photographed objects into images (SIGGRAPH 27) Original Created

33 [ ] X x = K I = z y x f f v u w K Slide Credit: Saverese Projection matrix Intrinsic Assumptions Unit aspect ratio Optical center at (,) No skew Extrinsic Assumptions No rotation Camera at (,,)

34 Remove assumption: known optical center [ ] X x = K I = z y x v f u f v u w Intrinsic Assumptions Unit aspect ratio No skew Extrinsic Assumptions No rotation Camera at (,,)

35 Remove assumption: square pixels [ ] X x = K I = z y x v u v u w β α Intrinsic Assumptions No skew Extrinsic Assumptions No rotation Camera at (,,)

36 Remove assumption: non-skewed pixels [ ] X x = K I = z y x v u s v u w β α Intrinsic Assumptions Extrinsic Assumptions No rotation Camera at (,,) Note: different books use different notation for parameters

37 Oriented and Translated Camera R j w t k w O w i w

38 Allow camera translation [ ] X t x = K I = z y x t t t v u v u w z y x β α Intrinsic Assumptions Extrinsic Assumptions No rotation

39 3D Rotation of Points Rotation around the coordinate axes, counter-clockwise: = = = cos sin sin cos ) ( cos sin sin cos ) ( cos sin sin cos ) ( γ γ γ γ γ β β β β β α α α α α z y x R R R p p γ y z Slide Credit: Saverese

40 Allow camera rotation [ ] X t x = K R = z y x t r r r t r r r t r r r v u s v u w z y x β α

41 Degrees of freedom [ ] X t x = K R = z y x t r r r t r r r t r r r v u s v u w z y x β α 5 6

42 Orthographic Projection Special case of perspective projection Distance from the COP to the image plane is infinite Also called parallel projection What s the projection matrix? Image World Slide by Steve Seitz = z y x v u w

43 Field of View (Zoom, focal length)

44 Beyond Pinholes: Radial Distortion Corrected Barrel Distortion Image from Martin Habbecke

45 Things to remember Vanishing points and vanishing lines Vanishing line Vanishing point Vertical vanishing point (at infinity) Vanishing point Pinhole camera model and camera projection matrix Homogeneous coordinates x = K[ R t] X

46 Next class Light, color, and sensors

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