3-D Euclidean Space - Vectors

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Image Formation

Jana Kosecka

A “free” vector is defined by a pair of points :

Coordinates of the vector :

1 3D Rotation of Points – Euler angles Rotation around the coordinate axes, counter-clockwise:

P ’ ’ Y γ y P X’ x z

Rotation Matrices in 3D

• 3 by 3 matrices • 9 parameters – only three degrees of freedom • Representations – either three Euler angles • or axis and angle representation

• Properties of rotation matrices (constraints between the elements)

2 Rotation Matrices in 3D

• 3 by 3 matrices • 9 parameters – only three degrees of freedom • Representations – either three Euler angles • or axis and angle representation

• Properties of rotation matrices (constraints between the elements)

Columns are orthonormal

Canonical Coordinates for Rotation

Property of R

Taking derivative

Skew symmetric matrix property

By algebra

By solution to ODE

3 3D Rotation (axis & angle)

Solution to the ODE

with

Rotation Matrices

Given

How to compute angle and axis

4 3D Translation of Points

Translate by a vector

P ’ t ’ Y x’ x P z’ y z

Rigid Body Motion – Homogeneous Coordinates

3-D coordinates are related by: Homogeneous coordinates:

Homogeneous coordinates are related by:

5 Rigid Body Motion – Homogeneous Coordinates

3-D coordinates are related by: Homogeneous coordinates:

Homogeneous coordinates are related by:

Properties of Rigid Body Motions

Rigid body motion composition

Rigid body motion inverse

Rigid body motion acting on vectors

Vectors are only affected by rotation – 4th homogeneous coordinate is zero

6 Rigid Body Transformation

Coordinates are related by: Camera pose is specified by:

Image Formation

• If the object is our lens the refracted light causes the images

• How to integrate the information from all the rays being reflected from the single point on the surface ?

• Depending in their angle of incidence, some are more refracted then others – refracted rays all meet at the point – basic principles of lenses

• Also light from different surface points may hit the same lens point but they are refracted differently - Kepler’s retinal theory

7 Let’s design a camera

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

Slide by Steve Seitz

Pinhole camera

• Add a barrier to block off most of the rays – This reduces blurring – The opening is known as the aperture

Slide by Steve Seitz

8 Pinhole camera model

• Pinhole model: – Captures pencil of rays – all rays through a single point – The point is called Center of Projection (focal point) – The image is formed on the Image Plane

Slide by Steve Seitz

Camera Obscura

• Basic principle known to Mozi (470-390 BCE), Aristotle (384-322 BCE) • Drawing aid for artists: described by Leonardo da Vinci (1452-1519)

Gemma Frisius, 1558

Source: A. Efros

9 Home-made pinhole camera

Why so blurry?

Slide by A. Efros http://www.debevec.org/Pinhole/

Shrinking the aperture

• Why not make the aperture as small as possible? – Less light gets through – Diffraction effects…

Slide by Steve Seitz

10 Shrinking the aperture

Adding a lens

• A lens focuses light onto the film – Thin lens model: • Rays passing through the center are not deviated (pinhole projection model still holds)

Slide by Steve Seitz

11 Adding a lens

• A lens focuses light onto the film – Thin lens model: • Rays passing through the center are not deviated (pinhole projection model still holds) • All parallel rays converge to one point on a plane located at the focal length f

Slide by Steve Seitz

Adding a lens

• A lens focuses light onto the film – There is a specific distance at which objects are “in focus” • other points project to a “circle of confusion” in the image

Slide by Steve Seitz

12 Thin lens formula

Similar triangles everywhere! y’/y = D’/D

D’ D f y y’

image lens object plane Slide by Frédo Durand

Thin lens formula

Similar triangles everywhere! y’/y = D’/D y’/y = (D’-f)/f D’ D f y y’

image lens object plane Slide by Frédo Durand

13 Thin lens formula Any point satisfying the thin lens 1 + 1 = 1 D’ D f equation is in focus.

D’ D f

image lens object plane Slide by Frédo Durand

Depth of Field

• in real camera lenses, there is a range of points which are brought into focus at the same distance • depth of field of the lens , as Z gets large – z’ approaches f • human eye – power of accommodation – changing f

14 How can we control the depth of field?

• Changing the aperture size affects depth of field – A smaller aperture increases the range in which the object is approximately in focus – But small aperture reduces amount of light –

need to increase exposure Slide by A. Efros

Varying the aperture

Large aperture = small DOF Small aperture = large DOF Slide by A. Efros

15 Field of View

Slide by A. Efros

Field of View

What does FOV depend on? Slide by A. Efros

16 Field of View

f f

FOV depends on focal length and size of the camera retina

Smaller FOV = larger Focal Length Slide by A. Efros

Field of View / Focal Length

Large FOV, small f Camera close to car

Small FOV, large f Camera far from the car Sources: A. Efros, F. Durand

17 Same effect for faces

wide-angle standard telephoto

Source: F. Durand

Image Formation – Perspective Projection

“The School of Athens,” Raphael, 1518

18 Pinhole Camera Model

Pinhole

Frontal pinhole