CS4670/5670: Computer Vision Kavita Bala. Lec 19: Single- view modeling 2

Transcription

1 CS4670/5670: Computer Vision Kavita Bala Lec 19: Single- view modeling 2

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3 Today Vanishing points in images are useful Recover size Camera calibrakon

4 Vanishing points image plane vanishing point V camera center C line on ground plane line on ground plane ProperKes Any two parallel lines (in 3D) have the same vanishing point v The ray from C through v is parallel to the lines An image may have more than one vanishing point in fact, every image point is a potenkal vanishing point

5 Vanishing lines v 1 v 2 MulKple Vanishing Points Any set of parallel lines on the plane define a vanishing point The union of all of these vanishing points is the horizon line also called vanishing line Note that different planes (can) define different vanishing lines

6 CompuKng vanishing lines v 1 l v 2 C l ground plane ProperKes l is interseckon of horizontal plane through C with image plane Compute l from two sets of parallel lines on ground plane All points at same height as C project to l points higher than C project above l Provides way of comparing height of objects in the scene

7 Comparing heights Vanishing Point

8 Measuring height How high is the camera? Camera height

9 CompuKng vanishing points (from lines) v q 1 q 2 p 2 p 1 Intersect p 1 q 1 with p 2 q 2 Least squares version Be]er to use more than two lines and compute the closest point of interseckon See notes by Bob Collins for one good way of doing this: h]p://www- 2.cs.cmu.edu/~ph/869/www/notes/vanishing.txt

10 Measuring height without a ruler H R

11 Measuring height without a ruler C Z ground plane Compute Z from image measurements Actually get a scaled version of z

12 The cross rako A ProjecKve Invariant Something that does not change under projeckve transformakons (including perspeckve projeckon) P 1 P 2 P 3 P P P P P P P P P The cross- ra0o of 4 collinear points Can permute the point ordering 4! = 24 different orders (but only 6 disknct values) This is the fundamental invariant of projeckve geometry = 1 i i i i Z Y X P P P P P P P P P

13 Measuring height C b r t H T (top of object) R (reference point) R T B R R B T scene cross ra9o t b v Z r r b v Z t image cross ra9o = = H R H R v Z B (bo]om of object) ground plane scene points represented as P = X Y Z 1 image points as p = x y 1

14 Point and line duality A line l is a homogeneous 3- vector l p 1 p 2 l 2 l 1 p What is the line l spanned by rays p 1 and p 2? l is to p 1 and p 2 l = p 1 p 2 l can be interpreted as a plane normal What is the interseckon of two lines l 1 and l 2? p is to l 1 and l 2 p = l 1 l 2 Points and lines are dual in projeckve space

15 Measuring height v z r vanishing line (horizon) v x v t 0 H t R v y b 0 t r b b v v Z Z r t image cross ra9o = H R b

16 Measuring height v z r vanishing line (horizon) t 0 v x v v y m 0 t 1 b 1 b 0 What if the point on the ground plane b 0 is not known? Here the guy is standing on the box, height of box is known Use one side of the box to help find b 0 as shown above b

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18 3D Modeling from a photograph St. Jerome in his Study, H. Steenwick

19 3D Modeling from a photograph

20 3D Modeling from a photograph Flagella0on, Piero della Francesca

21 3D Modeling from a photograph video by Antonio Criminisi

22 3D Modeling from a photograph

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24 Some Related Techniques Image- Based Modeling and Photo EdiKng Mok et al., SIGGRAPH 2001 Single View Modeling of Free- Form Scenes Zhang et al., CVPR 2001 Tour Into The Picture Anjyo et al., SIGGRAPH 1997

25 Camera calibrakon Goal: eskmate the camera parameters Version 1: solve for projeckon matrix wx * * Y X * Z 1 x = wy = * * * * = ΠX w * * * Version 2: solve for camera parameters separately intrinsics (focal length, principle point, pixel size) extrinsics (rotakon angles, translakon) radial distorkon * *

26 CalibraKon using a reference object Place a known object in the scene idenkfy correspondence between image 2D and scene 3D compute mapping from scene to image Issues must know geometry very accurately must know 3D- >2D correspondence

27 EsKmaKng the projeckon matrix Place a known object in the scene idenkfy correspondence between image and scene compute mapping from scene to image

28 Direct linear calibrakon

29 Direct linear calibrakon Can solve for m ij by linear least squares use eigenvector trick that we used for homographies. A x = 0

30 Advantage: Direct linear calibrakon Very simple to formulate and solve Disadvantages: Doesn t tell you the camera parameters Doesn t model radial distorkon Hard to impose constraints (e.g., known f) Doesn t minimize the right error funckon Nonlinear methods are preferred Define error funckon E between projected 3D points and image posikons: nonlinear funckon of intrinsics, extrinsics, radial distorkon Minimize E using nonlinear opkmizakon techniques

31 Summary Known correspondences (ui, vi) and (Xi, Yi, Zi) Compute mij solving system of linear equakons May use this to inikalize non linear error minimizakon problem to recover more accurate mij

32 CalibraKon from vanishing points

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35 RotaKon from vanishing points

36 Vanishing points and projeckon matrix * * * * Π = * * * * = [ π ] 1 π2 π3 π4 * * * * π π1 π2 π3 4 ProjecKon of x axis = v x (X vanishing point) similarly, π π = π = 2 v Y, 3 v Z T [ ] projection of world origin 4 = Π = Π = [ v v v o] X Not So Fast! We only know v s up to a scale factor Π = Y [ a v bv cv o] X Can fully specify by providing 3 reference points Y Z Z

37 AlternaKve: mulk- plane calibrakon Advantage Only requires a plane Images courtesy Jean- Yves Bouguet, Intel Corp. Don t have to know posikons/orientakons Good code available online! (including in OpenCV) Matlab version by Jean- Yves Bouget: h]p:// Zhengyou Zhang s web site: h]p://research.microsou.com/~zhang/calib/

38 Stereo Next Kme