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A camera flattens a 3D scene onto a 2D image. This chapter works that projection backward. You start with the projective geometry of points, lines and homographies. You then measure a scene from one image using vanishing points, and recover depth from two images with stereo and epipolar geometry. The chapter ends with how motion in 3D shows up as motion in the image. Structure from motion and bundle adjustment, which recover camera poses from many images, are still to come. These tools produce the posed images that 3D Gaussian splatting needs in the Sim-to-Real Transfer chapter.

Representing Images and Geometry

Homogeneous coordinates, 2D transformations as matrices, point and line duality, and forward and backward warping.

Homographies and Robust Estimation

Estimate a plane-to-plane mapping with normalized DLT, reject outliers with RANSAC, and rectify a plane.

Measuring a Scene from One Image

Vanishing points, the cross-ratio, and camera calibration from one image of a room.

Depth from Two Views

Disparity and depth, cost volumes and block matching, failure cases, and epipolar geometry.

Estimating Motion by Patch Matching

Recover image motion between two frames by local patch matching, and measure its error.

3D Motion and Its 2D Projection

How the 3D motion of points and cameras projects to motion in the image.
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