CV: 3D to 2D mathematics. Perspective transformation; camera calibration; stereo computation; and more

Transcription

1 CV: 3D to 2D mathematics Perspective transformation; camera calibration; stereo computation; and more

2 Roadmap of topics n Review perspective transformation n Camera calibration n Stereo methods n Structured light methods n Depth-from-focus n Shape-from-shading

3 Review coordinate systems Camera or sensor C Camera or sensor D World or global W Object or model M

4 Convenient notation for points and transformations This point P has 2 real coordinates in the image This transformation maps each point in the real world W to a point in the image I This point P has 3 real world coordinates in coordinate system W

5 Current goal Develop the theory in terms of modules (components) so that concepts are understood and can be put into practical application

6 Perspective transformation Camera origin is center of projection, not lens X and Y are scaled by the ratio of focal length to depth Z

7 In next homework & project n n n n n fit camera model to image with jig jig has known precise 3D coordinates examine accuracy of camera model use camera model to do graphics use two camera models to compute depth from stereo

8 Notes on perspective trans. n n n n 3D world scaled according to ratio of depth to focal length scaling formulas are in terms of real numbers with the same units e.g. mm in the 3D world and mm in the image plane real image coordinates must be further scaled to pixel row and column entire 3D ray images to the same 2D point

9 Goal: General perspective trans to be developed (accept for now) Camera matrix C transforms 3D real world point into image row and column using 11 parameters

10 The 11 parameters Cij model n n internal camera parameters: focal length f ratio of pixel height and width any shear due to sensor chip alignment external orientation parameters: rotation of camera frame relative to world frame translation of camera frame relative to world The 11 parameters of this model are NOT independent. Radial distortion is not linear and is not modeled.

11 Camera matrix via least squares Minimize the residuals in the image plane. Get 2 equations for each pair ((r, c), (x, y, z))

12 2 equations for each pair Known 3D points Here, (u, v) is the point in the image where 3D point (x,y,z) is projected. The 11 unknowns d jk form the camera matrix. Camera parameters Known image points

13 2n linear equations from n pairs ((u,v) (x,y,z)) Standard linear algebra problem; easily solved in Matlab or by using a linear algebra package. Often, package replaces b s with the residuals.

14 Use a jig for calibration Get pairings ((r, c) (x, y, z)) n Jig has known set of points n Measure points in world system W or use the jig to define W n Take image with camera and determine 2D points

15 Example calibration data # # IMAGE: g1view1.ras # # INPUT DATA OUTPUT DATA # Point Image 2-D (U,V) 3-D Coordinates (X,Y,Z) 2-D Fit Data Residuals X Y A B C D N O P # CALIBRATION MATRIX

16 3D points on jig Dimensions in inches

17 Jig set in workspace Mapping is established between 3D points (x, y, z) and image points (u, v)

18 Other jigs used at MSU n n frame with wires and beads placed in car instead of the driver seat (to do stereo measurements of car driver) frame with wires and beads as big as a harp to calibrate space for people walking (up to 6 cameras, persons wear tight body suit with reflecting disks, cameras compute 3D motion trajectory)

19 Least squares set up A X = B 2n x x 1 = 2n x 1

20 Least squares abstraction

21 Justify the form of camera matrix n Another sequence of slides n Rotation, scaling, shear in 3D real world as a 3x3 (or 4x4) matrix n Projection to real 2D image as 4x4 matrix n Scaling real image coordinates to [r, c] coordinates as 4x4 matrix n Combine them all into one 4x4 matrix

22 Other mathematical models Two camera stereo

23 Baseline stereo: carefully aligned cameras

24 Computing (x, y, z) in 3D from corresponding 2D image points

25 2 calibrated cameras view the same 3D point at (r1,c1)(r2,c2)

26 Compute closest approach of the two rays: use center point V Shortest line segment between rays

27 Connector is perpendicular to both imaging rays

28 Solve for the endpoints of the connector Scaler mult. Fix book

29 Geometric interpretation n Create a plane perpendicular to ray P; it can slide along the ray. All lines normal to ray P are in this plane. n Create a plane perpendicular to ray Q; it can slide along the ray. All normals to ray Q are in this plane. n Slide the planes and find that shortest line segment simultaneously in both planes.

30 Correspondence problem: more difficult aspect

31 Correspondence problem is difficult n Can use interest points and cross correlation n n n Can limit search to epipolar line Can use symbolic matching (Ch 11) to determine corresponding points (called structural stereopsis) apparently humans don t need it

32 Epipolar constraint With aligned cameras, search for corresponding point is 1D along corresponding row of other camera.

33 Epipolar constraint for non baseline stereo computation Need to know relative orientation of cameras C1 and C2 If cameras are not aligned, a 1D search can still be determined for the corresponding point. P1, C1, C2 determine a plane that cuts image I2 in a line: P2 will be on that line.

34 Measuring driver body position 4 cameras were used to measure driver position and posture while driving: 2mm accuracy achieved