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CS131: Computer Vision: Foundations and Applications

Juan Carlos Niebles, Adrien Gaidon, Silvio Savarese

April 13, 2026

Lecture 5�Camera Models

& Geometric

Primitives�

Silvio Savarese & Jeanette Bohg

Lecture 2 -

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13-Apr-26

Intro
Pinhole cameras
Cameras & lenses
The geometry of pinhole
Geometric primitives

Agenda�

Some slides in this lecture are courtesy to Profs. J. Ponce, S. Seitz, F-F Li

Silvio Savarese & Jeanette Bohg

Lecture 2 -

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3D Computer Vision

Slide credit: Andreas Geiger

Silvio Savarese

Lecture 2 -

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How do we see the world?

Let’s design a camera

Idea 1: put a piece of film in front of an object
Do we get a reasonable image?

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Pinhole camera

Idea 2: Add a barrier to block off most of the rays

This reduces blurring
The opening known as the aperture

Aperture

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Some history…

Milestones:

Leonardo da Vinci (1452-1519):

first record of camera obscura (1502)

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Some history…

Milestones:

Leonardo da Vinci (1452-1519): first record of camera obscura
Johann Zahn (1685): first portable camera

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Some history…

Photography (Niépce, “La

Table Servie,” 1822)

Milestones:

Leonardo da Vinci (1452-1519): first record of camera obscura
Johann Zahn (1685): first portable camera
Joseph Nicéphore Niépce (1822): first photo - birth of photography

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Some history…

Photography (Niépce, “La

Table Servie,” 1822)

Milestones:

Leonardo da Vinci (1452-1519): first record of camera obscura
Johann Zahn (1685): first portable camera
Joseph Nicéphore Niépce (1822): first photo - birth of photography

Daguerréotypes (1839)
Photographic Film (Eastman, 1889)
Cinema (Lumière Brothers, 1895)
Color Photography (Lumière Brothers, 1908)

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Let’s also not forget…

Motzu

(468-376 BC)

Aristotle

(384-322 BC)

Also: Plato, Euclid

Al-Kindi (c. 801–873)

Ibn al-Haitham

(965-1040)

Oldest existent book on geometry in China

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Pinhole perspective projection

Pinhole camera

f

f = focal length

o = aperture = pinhole = center of the camera

o

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Pinhole camera

Derived using similar triangles

[Eq. 1]

f

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O

P = [x, z]

P’=[x’, f ]

f

i

k

Pinhole camera

[Eq. 2]

f

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Kate lazuka ©

Pinhole camera

Is the size of the aperture important?

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Shrinking

aperture

size

Adding lenses!

What happens if the aperture is too small?

Less light passes through

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A lens focuses light onto the film

Cameras & Lenses

image

P

P’

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A lens focuses light onto the film

There is a specific distance at which objects are “in focus”
Related to the concept of depth of field

Out of focus

Cameras & Lenses

image

P

P’

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A lens focuses light onto the film

There is a specific distance at which objects are “in focus”
Related to the concept of depth of field

Cameras & Lenses

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A lens focuses light onto the film
All rays parallel to the optical (or principal) axis converge to one point (the focal point) on a plane located at the focal length f from the center of the lens.
Rays passing through the center are not deviated

focal point

f

Cameras & Lenses

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f

Paraxial refraction model

z_o

-z

Z’

From Snell’s law:

[Eq. 3]

P = [x,y,z]

p’ = [x’,y’]

[Eq. 1]

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f

Paraxial refraction model

z_o

-z

Z’

From Snell’s law:

[Eq. 3]

[Eq. 4]

P = [x,y,z]

p’ = [x’,y’]

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Issues with lenses: Radial Distortion

Deviations are most noticeable for rays that pass through the edge of the lens

No distortion

Pin cushion

Barrel (fisheye lens)

Image magnification decreases with distance from the optical axis

Because the paraxial refraction model is only an approximation, this creates a number of aberrations. The most common one is referred to as “radial distortion”. The effect of a radial distortion is that the image magnification decreases or increases as function of the distance separating the optical axis from a projected point of interest p’. If the magnification increases we talk about pin cushion distortion; if the magnification decreases we talk about barrel distortion. The latter typically occurs when one uses fisheye lens. The radial distortion is caused by the fact that the focal length of a lens is not uniform but different portions of the lens have different focal lengths.

ONLY if there is question: for instance, α can be in general large and sinα cannot approximated by α)

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Intro
Pinhole cameras
Cameras & lenses
The geometry of pinhole cameras

Intrinsic
Extrinsic

Geometric primitives

Agenda�

Some slides in this lecture are courtesy to Profs. J. Ponce, S. Seitz, F-F Li

Silvio Savarese & Jeanette Bohg

Lecture 2 -

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Pinhole perspective projection

Pinhole camera

f = focal length

o = center of the camera

[Eq. 1]

f

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From retina plane to images

Pixels, bottom-left coordinate systems

f

Retina plane

Digital image

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Coordinate systems

f

x

y

x^c

y^c

C’’=[c_x, c_y]

Off set

[Eq. 5]

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Converting to pixels

Off set
From metric to pixels

x

y

x^c

y^c

C=[c_x, c_y]

: pixel

Non-square pixels

f

[Eq. 6]

Units:

k,l : pixel/m

f : m

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Is this a linear transformation?

x

y

x^c

y^c

C=[c_x, c_y]

Is this projective transformation linear?

f

[Eq. 7]

No — division by z is nonlinear

Can we express it in a matrix form?

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Homogeneous coordinates

homogeneous image

coordinates

homogeneous scene

coordinates

Converting back from homogeneous coordinates

E🡪H

H🡪E

One way to get around this problem is to change coordinate system and go from the Euclidean reference system (our original coordinate system) to the so called Homogenous coordinate system. The process of going back and forth from Euclidean and Homogenous coordinate system is non-linear, but the good news is that once we are in the homogenous coordinate system a projective transformation like the one expressed by Eq. 7 becomes linear. The change of reference system from Euclidean to Homogenous goes as follows:

A point X in Rⁿ (in the Euclidean coordinate system) is represented as Rⁿ⁺¹ vector (in the Homogenous coordinate system) by adding a final coordinate of 1. See slide for examples.

<CLICK> How do we go back from H to E? A generic point X in Rⁿ⁺¹(in the Homogenous coordinate system) is represented as a Rⁿvector (in the Euclidean coordinate system) by suppressing the last coordinate and dividing the first n coordinates of X by the n+1^th coordinate. See slide for examples.

The benefit of working with homogenous coordinates will become more evident throughout the upcoming lectures. For now let’s go back to Eq. 7 and propose an alternative formulation in the homogenous coordinate systems.

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Projective transformation in the homogenous coordinate system

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Projective transformation in the homogenous coordinate system

[Eq.8]

Homogenous

Euclidian

Let’s multiple the matrix M (which contains an arrangement of the camera parameters) by P_h: the result is the vector P’_h as expressed by Eq. 8 (again, we are in homogeneous coordinate systems). Now let’s convert P’_h back to the Euclidean reference system. How do we do this?

<CLICK> The result is exactly the point P’ in Eq. 7. As this process illustrates:

The projective transformation can be expressed as a linear transformation in the Homogenous reference system
This transformation is expressed as a multiplication of the matrix M and the vector P_h.
The conversion from Homogenous to Euclidean returns the actual point P’ (that’s where the non-linearity takes place).

The key advantage of this representation is that we have obtained a much more compact expression for capturing the projective transformation. As we’ll see next, as we consider additional linear transformations to characterize the camera model, these can be nicely and compactly added as matrix multiplications to the expression in Eq. 8.

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The Camera Matrix

Camera

matrix K

f

[Eq.9]

From this point on, let’s assume that we work on homogenous coordinates and that a generic point P in 3D is expressed in such coordinate system. To keep the notation light, we drop the index h (we’ll add the index back when the discussion becomes ambiguous).

By working In homogenous coordinates, we can write a very compact expression that regulates the projective transformation (see Eq. 9) <CLICK>, where M is defined in the previous slide; M itself can be expressed as here, where I is a 3x3 identity matrix and and K is defined as the camera matrix.

The camera matrix contains some of the critical parameters that are useful to characterize a camera model. Two parameters are missing: skewness and distortion. We will talk about skewness in the next slide and distortion in the next lecture.

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Camera Skewness

x

y

x^c

y^c

C=[c_x, c_y]

f

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Degrees of freedom of K

x

y

x^c

y^c

C=[c_x, c_y]

How many degrees of freedom does K have?

f

5 degrees of freedom!

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Canonical Projective Transformation

M

[Eq.10]

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13-Apr-26

Pinhole cameras
Cameras & lenses
The geometry of pinhole cameras

Intrinsic
Extrinsic

Geometric primitives

Lecture 2�Camera Models�

Silvio Savarese & Jeanette Bohg

Lecture 2 -

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World reference system

O_w

i_w

k_w

j_w

R,T

The mapping so far is defined within the camera reference system

What if an object is represented in the world reference system?

Need to introduce an additional mapping from world ref system to camera ref system

f

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2D Translation

P

P'

t

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2D Translation Equation

P

x

y

tx

ty

P’

t

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2D Translation using Homogeneous Coordinates

P

x

y

tx

ty

P’

t

?

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Scaling

P

P'

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Scaling Equation

P

x

y

s_x x

P’

s_y y

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Rotation

P

P'

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Rotation Equations

P

x

y’

P’

θ

x’

y

How many degrees of freedom?

1

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Scale + Rotation + Translation

If s_x= s_y, this is a

similarity

transformation

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3D Translation of Points

A translation vector in 3D has 3 degrees of freedom

P

x

y

Tx

Ty

P’

T

z

I

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3D Rotation of Points

Rotation around the coordinate axes, counter-clockwise:

P

x

y’

P’

x’

y

z

A rotation matrix in 3D has 3 degrees of freedom

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3D Translation and Rotation

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World reference system

O_w

i_w

k_w

j_w

R,T

In 4D homogeneous coordinates:

Internal parameters

External parameters

P

P’

f

[Eq.11]

[Eq.9]

The transformation that allows to relate points from/to the world reference system to/from the camera reference system is a rigid transformation and is regulated by a translation R and a rotation T in 3D. The benefit of working in a homogeneous coordinates is that these two transformations can be compactly described by the matrix [R T; O 1] as the slide illustrates. This is a 4x4 matrix and allows to map any point P_w = [x_w y_w z_w 1]^T in the world reference system to a point P = [x y z 1]^T in the camera reference system (in homogenous coordinates). The projective mapping from P_w to P’ = [x’ y’ z’] in the image can be expressed as follows. <CLICK> Let’s recall eq. 9 that maps P (in the camera system) to P’ in the image plane. Remember K captures internal or physical properties of the camera (e.g. focal length, etc…);

<CLICK> Let’s replace P by [R T; 0 1] Pw;

<CLICK> this gives us equation 11.

<CLICK> [R T] are called the external or extrinsic parameters and are related to pose and location of the camera; Notice that intrinsic and extrinsic parameters are nicely separated in this formulation.

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The projective transformation

O_w

i_w

k_w

j_w

R,T

How many degrees of freedom does M have?

5 + 3 + 3 =11!

P

P’

f

[Eq.11]

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The projective transformation

O_w

i_w

k_w

j_w

R,T

E

P

P’

f

[Eq.12]

The mapping between P_w and P’ is in homogenous coordinates. If we want to obtain P’ in Euclidean coordinates (which corresponds to our actual measurements in the image), we need to change coordinates from homogenous to Euclidean by dropping the last coordinate of P’ and dividing the first two coordinates by the last coordinate. This operation is illustrated as follows.

<CLICK> Let’s express M = [m₁, m₂ m₃]T; Here m₁, m₂ and m₃ are the first, second and third row vectors of M, respectively. Thus, each of the m_i is 1x4 vector

<CLICK> Thus, SEE SLIDE

<CLICK> Let’s now go from homogenous to Euclidean and obtain Eq 12 SEE SLIDE

As Eq. 12 shows, the final outcome of the most generic formulation of a projective transformation in Euclidean coordinates is a rather complex (and non linear) intermix of the intrinsic and extrinsic parameters. However, by deriving this mapping in the homogenous coordinates, we managed to have a good understanding of the geometrical properties of the problem where intrinsic and extrinsic parameters were separated.

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Theorem (Faugeras, 1993)

[Eq.13]

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Properties of projective transformations

Points project to points
Lines project to lines
Distant objects look smaller

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Properties of Projection

Angles are not preserved
Parallel lines meet!

Parallel lines in the world intersect in the image at a “vanishing point”

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Horizon line (vanishing line)

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One-point perspective

Masaccio, Trinity, Santa Maria Novella, Florence, 1425-28

Credit slide S. Lazebnik

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Next lecture

How to calibrate a camera?�

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Supplemental material

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Thin Lenses

Snell’s law:

n₁ sinα₁= n₂ sin α₂

Small angles:

n₁ α₁ ≈ n₂α₂

n₁ = n(lens)

n₁ = 1(air)

z_o

Focal length

[FP] sec 1.1, page 8.

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Horizon line (vanishing line)

horizon