What a single image tells you about the 3D world. Vanishing points, vanishing lines, the image of the absolute conic, and camera calibration from a single photograph.
You might think a single photograph is "flat" — that all 3D information is lost. But that is not true. A single image encodes a surprising amount of 3D structure, provided you know what to look for.
Parallel lines in the world converge to a vanishing point in the image. A set of parallel planes shares a vanishing line. These structures let you measure angles, distance ratios, and even recover the camera's internal calibration — all from one photograph.
Parallel lines in 3D converge to a vanishing point in the image. The vanishing line of the ground plane passes through the vanishing points of all horizontal directions.
The camera matrix P acts not only on points, but on planes, lines, and conics. These actions form the vocabulary for single-view geometry.
| Entity | Action of P | Result |
|---|---|---|
| 3D point X | x = PX | Image point |
| 3D plane π | π = PTl (back-projection) | The plane through camera centre and image line l |
| 3D line (as join of two points) | Image of two endpoints | Image line |
| 3D conic Q on a plane | C = (Pπ)−T Q (Pπ)−1 | Image conic |
The key to single-view geometry is exploiting these mappings. Vanishing points arise from projecting points at infinity (directions), and vanishing lines from projecting the line at infinity on a plane.
A vanishing point is the image of a point at infinity — that is, the image of a direction in 3D. If the direction vector is d, then the vanishing point is v = Md, where M is the left 3×3 submatrix of P.
Conversely, given a vanishing point v, the 3D direction is d = M−1v. This tells us the actual 3D direction of any set of parallel lines that converge to v.
An important property: the camera centre, the vanishing point, and any line in the corresponding 3D direction are coplanar. The ray from the camera centre through the vanishing point is parallel to the 3D lines.
A vanishing line is the image of the line at infinity on a plane. All vanishing points of directions lying in that plane fall on the vanishing line.
The vanishing line of a plane π is computed as l = (KKT)−1 v⊥ when you know the plane's normal direction, or more directly: if you have two vanishing points v1, v2 for two independent directions in the plane, then l = v1 × v2.
The vanishing line encodes the orientation of the plane relative to the camera. From the vanishing line and one vanishing point perpendicular to the plane, you can determine the complete camera orientation.
The absolute conic Ω∞ is a conic on the plane at infinity in 3-space. It is invariant under all Euclidean transformations (rotations and translations). Its image through a camera P = K[R|t] is:
This is the image of the absolute conic (IAC). It depends only on K, not on R or t. So ω tells you the camera's intrinsics regardless of its pose.
ω is a 3×3 symmetric matrix, defined up to scale, so it has 5 DOF — matching the 5 intrinsic parameters of K. To determine ω, we need 5 independent constraints.
The power of ω is measuring angles. Two vanishing points v1, v2 correspond to orthogonal directions in 3D if and only if:
This is a linear constraint on ω. Every pair of orthogonal vanishing points gives one equation. Scenes are full of orthogonal structures (walls meet floors at right angles, window edges are perpendicular), providing rich constraints.
| Source of constraint | Type | # constraints on ω |
|---|---|---|
| Orthogonal vanishing points v1, v2 | Linear | 1 |
| Vanishing point v + vanishing line l at right angles | Linear | 2 |
| Known metric plane (homography H) | Linear | 2 |
| Zero skew constraint | Linear | 1 |
| Square pixels constraint | Linear | 2 |
Given enough constraints on ω, we can solve for it and then extract K via Cholesky factorization of ω−1 = KKT.
All the constraint types from the previous chapter are linear in the entries of ω. So the system Aw = 0 can be solved by SVD. With 5 or more constraints, ω is determined.
If you also know the principal point is at the image centre and pixels are square, then only one unknown remains (the focal length f). A single pair of orthogonal vanishing points suffices to determine f.
Degeneracies arise when the constraints are not independent. For example, three metric planes that are all parallel provide only 2 independent constraints instead of 6.
Single-view geometry enables real-world measurements from photographs. The most powerful technique measures length ratios of parallel line segments.
Given the vanishing line of the ground plane l, the vertical vanishing point v, and two vertical line segments in the image with known base positions:
| Step | Action |
|---|---|
| 1 | Transfer the length of one segment onto the other using a line through the vanishing point of the base-to-base direction |
| 2 | Use the four collinear image points (base, transferred top, actual top, vanishing point) to compute the cross-ratio and hence the length ratio |
No knowledge of the camera calibration K is needed for these ratio measurements. The vanishing points and vanishing line carry all the necessary information.
Given the vanishing line of a plane, we can rectify the image of that plane — warp the image so that the plane appears fronto-parallel (as if photographed straight-on).
The rectifying homography is determined by:
If K is known, the vanishing line determines the plane orientation, and we can compute Rcorrect to rotate the plane to fronto-parallel. Even without K, if we know the aspect ratio of a rectangle on the plane, we can compute the full rectification.
Single-view geometry is the bridge between camera models and multi-view geometry. It shows how much information a camera matrix P encodes about the 3D world.
| Concept | Where it leads |
|---|---|
| Vanishing points | Ch 9: Epipoles are vanishing points of the translation direction |
| IAC ω | Ch 19: Auto-calibration recovers ω from multiple views without a calibration target |
| Plane rectification | Ch 11: Image rectification makes epipolar lines horizontal for stereo matching |
| Back-projection | Ch 12: Triangulation intersects back-projected rays from two views |