Correcting Projector Distortions on Planar Screens via Homography

Daniel Hirt

Projector Devices Today

• More affordable• Smaller• Some are even low-cost and compact

A Common Setup of a Projector Device

Mounted on ceiling

Another Common Setup

Placed on table

• Cause distortions• Mild deviations may cause mild distortions,

oftenly referred to as “keystone effect”

Deviations from Recommeded Setup

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.

Problem Solved?

• “Keystone Correction” features in projectors does not overcome all distortions.

• Some distortions might be caused by extreme conditions of projector placement.

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

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

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)

Chosen Approach - Homography

• Popular in image and video analysis• Offers a simpler approach for planar-to-planar

projection problems

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

Using Homography

• In homogenous coordinates we get the following pinhole model

Using Homography

• Applying properties of homogenous representation where z=0 in points on planars, we get:

1

Solving an 8-DOF system

Method and Setup

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

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.

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

Setup and Results

-Laptop Computer-Webcam-Pico Projector-Program using OpenCV 2.4.3 library (Linux OS)

Results