Correcting Projector Distortions on Planar Screens via Homography

1. Correcting Projector Distortions on Planar Screens via Homography Daniel Hirt

2. Projector Devices Today • More affordable • Smaller • Some are even low-cost and compact

3. A Common Setup of a Projector Device Mounted on ceiling

4. Another Common Setup Placed on table

5. Deviations from Recommeded Setup • Cause distortions • Mild deviations may cause mild distortions, oftenly referred to as “keystone effect”

6. Basic Distortion Correction • Most projectors offer a limited range of methods to correct a distorted image. • Usually only “keystone correction” is available, and require manual operation.

7. Problem Solved? • “Keystone Correction” features in projectors does not overcome all distortions. • Some distortions might be caused by extreme conditions of projector placement.

8. Projection Correction On Planar Screens • Recall the perspective projection formula, give a 3D point (x,y,z). • We can use this to correct our image, but... • We do not have any 3D information

9. Approaches to Get 3D Information • Rectified Calibrated Stereo (two cameras) • Determine calibration values for: • Projector • Camera • Each of the above can give enough information for us to correct the distorted image

10. However, We Need to Also Know • Intrinsic Parameters • Focal length • Principal point • Lens distortion • Extrinsic Parameters • Translation • Rotation Note: not all are actually required to be able to get a correction, but we need to have most for each of the participants (camera, projector)

11. Chosen Approach - Homography • Popular in image and video analysis • Offers a simpler approach for planar-to-planar projection problems

12. Using Homography • A point (x1,y1) is projected from one plane to another point (x2,y2) x1,y1 x2,y2 • We represent these points in homogenous coordinates

13. Using Homography • In homogenous coordinates we get the following pinhole model

14. Using Homography • Applying properties of homogenous representation where z=0 in points on planars, we get: 1 Solving an 8-DOF system

15. Method and Setup

16. Steps • Get at least 4 correspondence points (usually the four corners) to solve 8-DOF system. • Solve the homography matrix from corresponding points in captured projected image (webcam) to reference straight image. • Apply persepective warp: H*(reference image) • “pre-warping” • Re-project the pre-warped image

17. Raised Issues The model is a good approximation • Some factors are added but are not considered in the model: • Projector and webcam’s native distortions • In practice, we need to improve the process, for more flexibility.

18. Improvement • Project a chessboard pattern: 6x8 squares (5x7 inner corners) • Detects 35 corresponding points • Scale-down the reference image to approximate to the size of the captured image (factor of resize: diagonals on inner corners) • Solves the Homography using RANSAC with the 35 sample points

19. Setup and Results -Laptop Computer -Webcam -Pico Projector -Program using OpenCV 2.4.3 library (Linux OS)

20. Results