New Image Rectification Schemes for 3D Vision Based on Sequential Virtual Rotation

1. New Image Rectification Schemes for 3D Vision Based on Sequential Virtual Rotation Jin Zhou June 16th, 2009 Dissertation Defense

2. Outline • Introduction • Rectification based on Virtual Sequential Rotation • Image Rectification for Stereoscopic Visualization • Camera Calibration • Stereoscopic View Synthesis from Monocular Endoscopic Sequences • Rapid 3D Modeling from Single Images • Robot Vision • Conclusions and Future Work

3. The Geometry of 3D to 2D Images are 2D projections of the 3D world

4. 3D Vision – The Problem How do we extract 3D information from 2D images? ? 3D of the objects ? ? ? ? 3D of the cameras

5. 3D Vision – Applications • Augmented Reality • Scene Modeling • Virtual Touring • 3D Imaging • Robots

6. 3D Vision – A Human Perspective • Size • Linear Perspective • Object Connections • Stereo • Motion • Shading • Texture

7. 3D Vision – Computational Approaches • Different approaches use different cues • Different problems requires different approaches. • Structure from Motion (SfM) • Rely on point correspondences • Single View Based Modeling (SVBM) • Rely on knowledge of the scene • Camera calibration • All approaches requires the images are calibarated first (either manual or automatic)

8. 3D Vision – Practical Challenges • Camera information is unavailable • Point correspondences is not reliable and time-consuming • Image resolution is limited • Degeneracy

9. 3D Vision – Limitations of Current Approaches • Distortion • Degeneracy • Due to high degree of freedom of geometric models • Lack of geometric meaning • Most approaches are purely based on algebraic derivations or imaginery objects. • Not accurate or not convinient (SVBM).

10. 3D Vision – Our Contributions • Novel image rectification schemes are proposed based on sequential virtual rotation • Novel approaches are proposed for the following problems • Image Rectification for Stereoscopic visualization • Camera Calibration • Stereoscopic View Synthesis from Monocular Endoscopic sequences • Rapid Cones and Cylinders Modeling • Monocular Vision Guided Mobile Robot Navigation

11. 3D Vision – Results of Our Approaches • No affine/projective distortion • Can handle degeneracy • Intuitive geometric meanings • Lead to insights of particular problems • Accurate and fast

12. Publications

13. What is Image Rectification? Image rectification is a process to transform the original images to new images which have desired properties.

14. General Image Transformations H? Image transformation can be defined by a 3x3 matrix H, which is called Homography.

15. Image Rectification based on Virtual Rotation • Homography of camera rotation/zooming • If we normalize the coordinates • Assume R = I Camera orientation is determined at the same time

16. Advantages of the New Rectification Schemes • Intuitive geometric meaning • Robust • Rotation parameters can be computed by various basic image features, such as points, lines and circles. • Can be used for camera calibration. • Can be used for 3D information extraction. • Lead to non-distorted results • Reason: Rotation do not introduce affine/projective distortion

17. Rotations are Decomposed as Euler Angles

18. Rotation Parameters Can Be Estimated by Basic Image Features • Each rotation has only one degree of freedom and thus only needs one constraint. • Example: transforming a point on to y axis Normalize Ambiguity!

19. Image Rectification for Stereoscopic Visualization The Principle of Stereoscopic (3D) Visualization

20. Motivation • Stereo content is scarce • Stereo cameras/camcorders are expensive • Common users seldom use stereo cameras/camcorders • We want to generate stereo content from images/videos taken by common cameras

21. The Problem Given two arbitrary images, rectify them so that the results look like a stereo pair.

22. Our Approach – Rectification based on Virtual Rotation We can “rotate” camera to standard stereo setup.

23. Calibrated Case • Constraints of the stereo camera pair: • The two cameras have the same intrinsic parameters (K) and orientation (R) • The camera’s optical axis is perpendicular to the baseline (C1 – C2) • i.e. the camera’s x axis has the same direction with the baseline UnKnown K1 Known Any vector

24. Uncalibrated Case • For the uncalibrated case, all K, R and C are unknown. We can only start from the fundamental matrix and point correspondences. • Estimate H2 (homography for the second image) ?

25. Determine R based on Sequential Virtual Rotation Constraints: First rotate around z axis so that the point is transformed to x axis (i.e. y = 0) Rotate around y axis so that the point is transformed to infinity.

26. Estimate H1

27. Estimate H1 • Determine a, b and c • Property of standard stereo setup: • For two points with the same depth, their projection on different images should have the same distance (Points with the same depth should have the same disparity). • Approach • Group points by similar disparities • Then compute a, b by minimizing

28. Results Shear distortion Hartley’s Method Our Method Original Pair

29. Results Shrink horizontally Hartley’s Method Our Method Original Pair

30. Camera Auto-Calibration from the Fundamental Matrix Traditional Approaches: Kruppa Equations dual image of the absolute conic Huang-Faugeras constraints Cons: Complex and hard to understand! Derivation for degenerate cases are purely algebraic.

31. Our Approach We transform the original pair to a standard stereo pair through sequential virtual rotation and zooming 7 DOFs 7 Parameters

32. Decomposition Illustration

34. Degeneracy Analysis

35. Degeneracy Analysis

36. Results of Monte Carlo Simulation

37. Stereoscopic View Synthesis From Monocular Endoscopic Videos 3D imaging helps to enable faster and safer surgical operations Two view image rectification can not be applied to the new problem Challenges: 1. Image quality is poor 2. Degeneracy

38. The Framework • We proved: • Affine 3D reconstruction is sufficient. • Linear interpolation in normalized disparity field is equal to linear interpolation in 3D space.

39. Strategy for Solving Degeneracy We assume the initial two frames have same orientation (i.e. they are rectified) The assumption makes the DOF of the fundamental matrix from 8 to 2! No Assumption Degeneracy! Assume the two frame are rectified

40. Interpolation a) Shows the disparities based on the SfM results b) We do Delaunay triangulation and interpolate each triangle c) We pick a set of grid points from b) and do bilinear interpolation d) We fill holes using Laplacian interpolation and do smoothing.

41. Results of Synthetic Data Disparity image after triangulation Final disparity image Ground truth Stereo images

42. Results of Real Data

43. Rapid Cylinders and Cones Modeling from A Single Image

44. Overview of Our Approach • Goal: Rapid + Accurate • Camera Calibration (Orientation Estimation) • Vertical lines • Vanishing line of horizontal plane • A Cone • Modeling from Image • Cones (two points / four points) • Cylinders (two points / four points)

45. The Coordinate Systems Observation: Most objects are standing on the ground (X,Y,Z,O) -- World Coordinate System Y is perpendicular to the ground (x,y,z,O) -- Camera Coordinate System Camera center is at the origin

46. Orientation Estimation from Vertical Lines First rotate around z axis so that the point is transformed to y axis (i.e. x = 0) Rotate around x axis so that the point is transformed to infinity.

47. Orientation Estimation from a Cone RzRxRz R Edges are symmetric to y axis Rx Cross section is a circle Rx (π/2)

48. Illustration

49. Metric Rectification of the Ground Plane Original Rectified Original Rectified

50. Modeling Cones Cone Parameters: 2D Control Points for Cones Modeling: Standard view Cones on Ground General Cones