09 image formation – Welcome to alumni.media.mit.edu

 

 

Lecture 9
Lecture 9
Lecture 9

Image Formation
Image Formation

How do we see the world?

object

film

(cid:132) Let’s design a camera

(cid:137) Idea 1: put a piece of film in front of an object
(cid:137) Do we get a reasonable image?

Slide by Steve Seitz

It receives light from all directions

Pinhole camera

object

barrier

film

From Photography, London et al.

(cid:132) Add a barrier to block off most of the rays

(cid:137) This reduces blurring
(cid:137) The opening known as the aperture
(cid:137) How does this transform the image?

Slide by Steve Seitz

Pinhole

Pinhole camera model

From Photography, London et al.

(cid:132) Pinhole model:

(cid:137) Captures pencil of rays – all rays through a single point
(cid:137) The point is called Center of Projection (COP)
(cid:137) The image is formed on the Image Plane
(cid:137) Effective focal length f is distance from COP to Image Plane

Slide by Steve Seitz

1

Pinhole size?

Lenses

(cid:132) gather more

light!

(cid:132) But need to be

focused

From Photography, London et al.

From Photography, London et al.

Dimensionality Reduction Machine
(3D to 2D)

3D world

2D image

Funny things happen…

Point of observation

(cid:132) What have we lost?

(cid:137) Angles
(cid:137) Distances (lengths)

Figures © Stephen E. Palmer, 2002

Parallel lines aren’t…

Distant objects are smaller

…but humans adopt!

length of B = 2*length of C
length of B’ = length of C’

Müller-Lyer Illusion

Forsyth&Ponce

We don’t make measurements in the image plane

http://www.michaelbach.de/ot/sze_muelue/index.html

2

Perspective projection

The equation of projection

(cid:132) Abstract camera model – box with a small hole in it
(cid:132) In an ideal pinhole camera everything is in focus

Forsyth&Ponce

2D image

3D world

The equation of projection

(cid:132) Cartesian coordinates:

(cid:137)We have, by similar triangles, that

(

zyx
,
,

)

→

(

f

f

,

−

,

f

)

x
z

y
z

(cid:137)Ignore the third coordinate, and get

(

zyx
,
,

)

→

(

f

f

,

x
z

y
z

)

The camera matrix

(cid:132) Turn previous expression into HC’s

(cid:137)HC’s for 3D point are (x,y,z,1)
(cid:137)HC’s for point in image are (u,v,w)

u

01
















(cid:132) Position of the point in the image from HC























00

10

w

=

0

0

1

0

0

0

v

f

x

y

z

1










normalize by w







u

v

w







=







x

y

fz







1
w







u

v

w







=

f
z

x

y

fz













=







fx

fy

z

z

1







Weak perspective

(cid:132) Issues:

(cid:137) perspective effects, but not over the scale of individual objects
(cid:137) collect points into a group at about the same depth, then divide

each point by the depth of its group

(cid:137) Adv: easy
(cid:137) Disadv: wrong

Orthographic projection

Telescope projection can be modeled by orthographic projection

3

Orthographic Projection

(cid:132) Special case of perspective projection
(cid:137) Distance from the COP to the PP is infinite

Image

World

Building a real camera

(cid:137) Also called “parallel projection”
(cid:137) What’s the projection matrix?

Slide by Steve Seitz

Camera Obscura

Camera Obscura, Gemma Frisius, 1558

Home-made pinhole camera

(cid:132) The first camera

(cid:137) Known to Aristotle
(cid:137) Depth of the room is the effective focal length

http://www.debevec.org/Pinhole/

Why so
blurry?

Shrinking the aperture

Shrinking the aperture

(cid:132) Why not make the aperture as small as possible?

(cid:137) Less light gets through
(cid:137) Diffraction effects…

Slide by Steve Seitz

4

The reason for lenses

Focus

Focus and Defocus

Thin lenses

object

lens

film

(cid:132) A lens focuses light onto the film

(cid:137) There is a specific distance at which objects are “in focus”
(cid:132) other points project to a “circle of confusion” in the image
(cid:137) Changing the shape of the lens changes this distance

(cid:132) Thin lens equation:

(cid:137) Any object point satisfying this equation is in focus
(cid:137) What is the shape of the focus region?
(cid:137) How can we change the focus region?
(cid:137) Thin lens applet: http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html (by Fu-Kwun Hwang )

Slide by Steve Seitz

Slide by Steve Seitz

Varying Focus

Depth Of Field

Slide by Steve Seitz

“circle of
confusion”

Ren Ng

5

Depth of Field

Aperture controls Depth of Field

http://www.cambridgeincolour.com/tutorials/depth-of-field.htm

(cid:132) Changing the aperture size affects depth of field

(cid:137) A smaller aperture increases the range in which the object is

(cid:137) But small aperture reduces amount of light – need to increase

approximately in focus

exposure

Varying the aperture

Nice Depth of Field effect

Large apeture = small DOF

Small apeture = large DOF

Field of View (Zoom)

Field of View (Zoom)

6

Field of View (Zoom)

FOV depends of Focal Length

f
f

Smaller FOV = larger Focal Length

Field of View / Focal Length

Large FOV, small f
Camera close to car

Small FOV, large f
Camera far from the car

Lens Flaws: Chromatic Aberration

(cid:132) Dispersion: wavelength-dependent refractive index
(cid:137) (enables prism to spread white light beam into rainbow)

(cid:132) Modifies ray-bending and lens focal length: f(λ)

(cid:132) color fringes near edges of image
(cid:132) Corrections: add ‘doublet’ lens of flint glass, etc.

7

From Zisserman & Hartley

Lens Flaws

Chromatic Aberration

Radial Distortion (e.g. ‘Barrel’ and
‘pin-cushion’)

straight lines curve around the image center

Near Lens Center
Near Lens Center

Near Lens Outer Edge
Near Lens Outer Edge

Radial Distortion

Radial Distortion

No distortion

Pin cushion

Barrel

(cid:132) Radial distortion of the image
(cid:137) Caused by imperfect lenses
(cid:137) Deviations are most noticeable for rays that pass

through the edge of the lens

8