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CS131: Computer Vision: Foundations and Applications

Juan Carlos Niebles, Adrien Gaidon, Silvio Savarese

April 13, 2026

Lecture 7�Single View Metrology�

Silvio Savarese & Jeanette Bohg

Lecture 4 -

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17-Apr-26

Review calibration and 2D transformations
Vanishing points and lines
Estimating geometry from a single image
Extensions

Lecture 7�Single View Metrology�

Silvio Savarese & Jeanette Bohg

Lecture 4 -

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Calibration Problem

In pixels

World ref. system

Need at least 6 correspondences

11 unknowns

j_C

Calibration rig

p_i

image

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Pinhole perspective projection

Once the camera is calibrated...

C

O_w

Internal parameters K are known
R, T are known – but these can only relate C to the calibration rig

P

p

Can I estimate P from the measurement p from a single image?

No - in general ☹ (P can be anywhere along the line defined by C and p)

Line of sight

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http://www.robots.ox.ac.uk/~vgg/projects/SingleView/models/hut/hutme.wrl

Recovering structure from a single view

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Transformation in 2D

Isometries

Similarities

Affinity

Homographic /Projective

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Transformation in 2D

Isometries:

Preserve distance (areas)
3 DOF
Regulate motion

of rigid object

[Euclideans]

[Eq. 4]

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Transformation in 2D

Similarities:

Preserve

ratio of lengths
angles

4 DOF

[Eq. 5]

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Transformation in 2D

Affinities:

[Eq. 6]

[Eq. 7]

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Transformation in 2D

Affinities:

Preserve:

Parallel lines
Ratio of areas
Ratio of lengths on

collinear lines

- others…

6 DOF

[Eq. 6]

[Eq. 7]

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Transformation in 2D

Affinities:

[Eq. 6]

[Eq. 7]

A = UDV^T = UV^TVDV^T = (UV^T) (V )(D) (V^T)

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Transformation in 2D

Homographies

(Projectivities)

8 DOF
Preserve:

collinearity
and a few others…

[Eq. 8]

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17-Apr-26

Review calibration and 2D transformations
Vanishing points and lines
Estimating geometry from a single image
Extensions

Lecture 4�Single View Metrology�

Silvio Savarese & Jeanette Bohg

Lecture 4 -

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Lines in a 2D plane

-c/b

-a/b

If x = [ x₁, x₂]^T ∈ l

x

y

[Eq. 10]

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Lines in a 2D plane

Intersecting lines

Proof

x

→ x is the intersection point

x

y

x

[Eq. 12]

[Eq. 13]

[Eq. 11]

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2D Points at infinity (ideal points)

Let’s intersect two parallel lines:

In Euclidian coordinates this point is at infinity
Agree with the general idea of two lines intersecting at infinity

[Eq.13]

= ideal point!

Let now us introduce the concept of points at infinity. In general a point in homogenous coordinate is expressed as x = [x1 x2 x3] where x3 different from zero.

<CLICK> A point at infinity x_inf can be expressed in homogenous coordinate where the third coordinate is equal to zero. In fact, if we convert this point to Euclidean coordinate…. EXPLAIN WHAT HAPPENS

<CLICK> Let’s try to further justify this expression. Let us consider two parallel lines l and l, whose parameters are…. READ SLIDE

Q: Do you recall what’s the point of intersection of two parallel lines? At infinity!

<CLICK> Let’s use this property to derive x_inf.

When two lines are parallel, their slope is equal and thus –a/b=-a’/b’.

<CLICK> Let’s now compute the point of intersection of these two lines. Using Eq. 11 we obtain Eq. 13 which is exactly the expression of a point at infinity (in homogenous coordinates). This confirms the intuition that two parallel lines intersect at infinity.

<CLICK> The point of intersection of two parallel lines returns a point at infinity which is also called ideal point.

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2D Points at infinity (ideal points)

Note: the line l = [a b c]^T pass trough the ideal point

So does the line l’ since a b’ = a’ b

[Eq. 15]

= [b –a 0]^T

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Lines infinity

Set of ideal points lies on a line called the line at infinity.

How does it look like?

Indeed:

A line at infinity can be thought of the set of “directions” of lines in the plane

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Projective transformation of a point at infinity

is it a point at infinity?

…no!

An affine transformation of a point

at infinity is still a point at infinity

[Eq. 17]

[Eq. 18]

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Projective transformation of a line (in 2D)

is it a line at infinity?

…no!

[Eq. 19]

[Eq. 20]

[Eq. 21]

This time, let’s apply a projective transformation H to a line. The projective transformation of a line l is l’ = H^-Tl (Eq 19).

<CLICK> Let’s now apply the projective transformation H to a line at infinity l_inf. Is the projected line still at infinity?

<CLICK> No.

<CLICK> Let’s now apply the affine transformation H_A to a line at infinity l_inf. Is the projected line still at infinity?

<CLICK> Yes, as the derivation next to Eq 21 shows.

Let’s derive this equation. All points that pass through a line l must satisfy the line equation: x^Tl = 0; —> x^TH^TH^-Tl = 0

Since x’ = H x (and x’^T = x^TH^T), then x’^TH^-Tl = 0; Because, after the transformation, a projected point x’ must still belong to the projected line l’ (x’^Tl’ = 0), it implies that l’ = H^-Tl.

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Transformation in 2D

Affinities:

Preserve:

Parallel lines
Points at infinity
Lines at infinity

[Eq. 6]

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Points and planes in 3D

x

y

z

[Eq. 23]

[Eq. 22]

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Lines have 4 degrees of freedom - hard to represent in 3D-space

Can be defined as intersection of 2 planes

Lines in 3D

d = direction of the line

= [a, b, c]^T

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Points at infinity in 3D

Points where parallel lines intersect in 3D

world

Parallel lines

point at infinity

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Vanishing points

The projective projection of a point at infinity into the image plane defines a vanishing point.

M

world

Parallel lines

point at infinity

= direction of a pair of parallel lines in 3D

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d = direction of the line

= [a, b, c]^T

d

C

v

M

[Eq. 24]

[Eq. 25]

Proof:

Vanishing points and directions

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Vanishing (horizon) line

horizon

Projective transformation M

Image

π

[Eq. 26]

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Example of horizon line

The orange line is the horizon!

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Are these two lines parallel or not?

- Recognize the horizon line

Measure if the 2 lines meet at the horizon
if yes, these 2 lines are // in 3D

Recognition helps reconstruction!
Humans have learnt this

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Vanishing points and planes

C

n

…

π

[Eq. 27]

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Angle between 2 vanishing points

C

d₁

v₂

v₁

d₂

If

[Eq. 28]

x_1∞

x_2∞

[Eq. 29]

Scalar equation

[Eq. 30]

The last property we introduce allows to relate the direction of lines in 3D with the corresponding vanishing points in the image.

<CLICK> Suppose that two pairs of parallel lines have directions d₁ and d₂, and are associated to the points at infinity x₁_∞ and x₂_∞_,respectively. Let v₁ and v₂ be the corresponding vanishing points.

<CLICK> It is possible to prove (but we won’t this in this lecture) that the angle between d₁ and d₂ and the vanishing points v₁ and v₂ are related by the Eq. 28., where ω is defined as the matrix (K K^T)^-1.

<CLICK> One special case which is useful in practice is when the two set of parallel lines are orthogonal to each other. In this case, d₁ and d₂are orthogonal, which gives us [Eq. 29]. Notice that this is a scalar equation.

NOTE: The matrix ω has a special geometrical meaning in that it is the projective transformation in the image plane of an absolute conic Ω_∞ in 3D.

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Properties of

symmetric and known up scale

zero-skew

square pixel

1.

2.

3.

[Eq. 30]

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Summary

[Eq. 24]

[Eq. 27]

[Eq. 28]

[Eq. 29]

To calibrate the camera
To estimate the geometry of the 3D world

Useful to:

[Eq. 30]

So what we have derived so far?

<CLICK> A relationship between parallel lines in 3D and the corresponding vanishing point in the image
<CLICK> A relationship that connects the normal of a plane in 3D with the corresponding horizon line in the image.
<CLICK> A relationship between the direction of lines in 3D and the corresponding vanishing points in the image.

Notice that in each of these relationships we have K that captures the intrinsic camera parameters.

<CLICK>

These properties are useful for two reasons:

First. To calibrate the camera: By using Eq 28 or Eq 29, we can set up a system of equations that allows us to calibrate our camera – that is, to estimate internal parameters of the camera
Second. To estimate the geometry of the 3D world: Once K is estimated or K is known, we can use equation 27 to estimate the orientation of planes in 3D w.r.t. to the camera reference system

Let’s see some examples next.

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17-Apr-26

Review calibration
Vanishing points and line
Estimating geometry from a single image
Extensions

Lecture 4�Single View Metrology�

Silvio Savarese & Jeanette Bohg

Lecture 4 -

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v₁

Do we have enough constraints to estimate K?

K has 5 degrees of freedom and Eq.29 is a scalar equation ☹

Single view calibration - example

[Eq. 28]

[Eq. 29]

v₂

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v₁

Single view calibration - example

[Eq. 28]

v₃

[Eqs. 31]

v₂

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Single view calibration - example

[Eqs. 31]

v₁

v₂

v₃

Square pixels
No skew

🡪

known up to scale

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Single view calibration - example

🡪 Compute !

[Eqs. 31]

v₁

v₂

v₃

Square pixels
No skew

🡪

known up to scale

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Single view calibration - example

(Cholesky factorization)

Once ω is calculated, we get K:

[Eqs. 31]

v₁

v₂

v₃

Square pixels
No skew

🡪

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Single view reconstruction - example

l_h

known

= Scene plane orientation in

the camera reference system

Select orientation discontinuities

[Eq. 27]

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Single view reconstruction - example

Recover the structure within the camera reference system

Notice: the actual scale of the scene is NOT recovered

C

Recognition helps reconstruction!
Humans have learnt this

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17-Apr-26

Review calibration
Vanishing points and lines
Estimating geometry from a single image
Extensions

Lecture 4�Single View Metrology�

Silvio Savarese & Jeanette Bohg

Lecture 4 -

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Criminisi & Zisserman, 99

http://www.robots.ox.ac.uk/~vgg/projects/SingleView/models/merton/merton.wrl

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Criminisi & Zisserman, 99

http://www.robots.ox.ac.uk/~vgg/projects/SingleView/models/merton/merton.wrl

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La Trinita' (1426) Firenze, Santa Maria Novella; by Masaccio (1401-1428)

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La Trinita' (1426) Firenze, Santa Maria Novella; by Masaccio (1401-1428)

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http://www.robots.ox.ac.uk/~vgg/projects/SingleView/models/hut/hutme.wrl

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Manually select:

Vanishing points and lines;
Planar surfaces;
Occluding boundaries;
Etc..

Single view reconstruction - drawbacks

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Automatic Photo Pop-up

Hoiem et al, 05

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Automatic Photo Pop-up

Hoiem et al, 05…

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Automatic Photo Pop-up

Hoiem et al, 05…

http://www.cs.uiuc.edu/homes/dhoiem/projects/software.html

Software:

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Training

Image

Depth

Prediction

Planar Surface

Segmentation

Plane Parameter MRF

Connectivity

Co-Planarity

Make3D

youtube

Saxena, Sun, Ng, 05…

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A software: Make3D

“Convert your image into 3d model”

Make3D

Saxena, Sun, Ng, 05…

http://make3d.stanford.edu/

http://make3d.cs.cornell.edu/

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Depth map reconstruction using deep learning

Depth Map Prediction from a Single Image using a Multi-Scale Deep Network,

Eigen, D., Puhrsch, C. and Fergus, R. Proc. Neural Information Processing Systems 2014,

Eigen et al., 2014

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3D Layout estimation

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Dasgupta, et al. CVPR 2016

Hoiem et al, 2006

Oliva & Torralba, 2007

Rabinovich et al, 2007

Li & Fei-Fei, 2007

Vogel & Schiele, 2007

Herdau et al., 2009

Gupta et al, 2010

Sadeghi & Farhardi, 2011

Li et al, 2012

Fouhey et al, 2012

Desai et al, 2009

Gould et al., 2009

How to estimate such surfaces in general conditions? Many options are possible. Very recently we have shown it is possible to learn to infer the geometry of indoors scenes – floors, walls, ceiling – in a mere end-to-end fashion using a deep learning architecture <click>.

This architecture maps RGB-pixels to one of these geometric classes;

<click> an ensuing optimization process guarantees that this labeling process is globally and geometrically consistent

~~~

when a simple box model representation is used. In this case the goal is to estimate different geometric classes such as floors, walls, ceiling of a room. We have shown this by training an end-to-end deep architecture that maps RGB-pixels to one of these geometric classes. A ensuing optimization process guarantees that this labeling process is globally and geometrically consistent

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3D Layout estimation

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Coherent object detection and scene layout estimation from a single image

Bao, et al., CVPR 2010, BMVC 2010

Y. Bao

M. Sun

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Next lecture:

Multi-view geometry (epipolar geometry)

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Appendix

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Vanishing points - example

C₂

v₂

C₁

v₁

R,T

star

v1, v2: measurements

K = known and constant

Can I compute R?

No rotation around z

In 2D

d

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v₁

v₂