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Correcting Projector Distortions on Planar Screens via Homography Daniel Hirt

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

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A Common Setup of a Projector Device Mounted on ceiling

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Another Common Setup Placed on table

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

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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.

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

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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

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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

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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)

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

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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

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Using Homography In homogenous coordinates we get the following pinhole model

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

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Method and Setup

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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

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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.

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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

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

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Results

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