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Projective geometry

Projective geometry—what’s it good for?

Applications of projective geometry

Measurements on planes

Image rectification

Solving for homographies

Solving for homographies

The projective plane

Projective lines

Point and line duality

Ideal points and lines

Homographies of points and lines

3D projective geometry

3D to 2D:  “perspective” projection

Vanishing points (2D)

Vanishing points

Vanishing lines

Vanishing lines

Computing vanishing points

Computing vanishing lines

Slide 22

Fun with vanishing points

Perspective cues

Perspective cues

Perspective cues

Comparing heights

Measuring height

Computing vanishing points (from lines)

Measuring height without a ruler

The cross ratio

Measuring height

Measuring height

Measuring height

Computing (X,Y,Z) coordinates

3D Modeling from a photograph

Camera calibration

Vanishing points and projection matrix

Calibration using a reference object

Chromaglyphs

Estimating the projection matrix

Direct linear calibration

Direct linear calibration

Direct linear calibration

Alternative:  multi-plane calibration

Some Related Techniques