Image Formation: Geometry Thurs. Jan. 11, 2018 1 Origins of - - PowerPoint PPT Presentation

β–Ά
image formation geometry
SMART_READER_LITE
LIVE PREVIEW

Image Formation: Geometry Thurs. Jan. 11, 2018 1 Origins of - - PowerPoint PPT Presentation

Apr 03, 2023 155 likes β€’622 views

COMP 546 Lecture 1 Image Formation: Geometry Thurs. Jan. 11, 2018 1 Origins of spatial vision (500 million years ago?) photoreceptor array (eye) brain legs Origins of spatial vision Origins of spatial vision Origins of spatial

slide-1
SLIDE 1

1

COMP 546

Lecture 1

Image Formation: Geometry

  • Thurs. Jan. 11, 2018
slide-2
SLIDE 2

Origins of spatial vision (500 million years ago?)

β€œbrain”

photoreceptor array (eye)

legs

slide-3
SLIDE 3

Origins of spatial vision

slide-4
SLIDE 4

Origins of spatial vision

slide-5
SLIDE 5

Origins of spatial vision

predator

Predator arrives, but no change in light level received by this cell.

slide-6
SLIDE 6

predator

Origins of spatial vision

Some change in light level received by this cell.

slide-7
SLIDE 7

predator

Origins of spatial vision

If right cell measures decrease in light, then move right.

slide-8
SLIDE 8

Evolution of eyes

As pit becomes more concave, angular resolution improves (but amount of light decreases)

eye

slide-9
SLIDE 9

large aperture small aperture

good angular resolution poor angular resolution

slide-10
SLIDE 10

Radians

q

πœ„ π‘ π‘π‘’π‘—π‘π‘œπ‘‘ = π‘π‘ π‘‘π‘šπ‘“π‘œπ‘•π‘’β„Ž π‘π‘œ π‘‘π‘—π‘ π‘‘π‘šπ‘“ 𝑠𝑏𝑒𝑗𝑣𝑑 𝑝𝑔 π‘‘π‘—π‘ π‘‘π‘šπ‘“

slide-11
SLIDE 11

Radians vs. degrees

q

πœ„ π‘ π‘π‘’π‘—π‘π‘œπ‘‘ βˆ—

180 𝑒𝑓𝑕𝑠𝑓𝑓𝑑 𝜌 π‘ π‘π‘’π‘—π‘π‘œπ‘‘

= πœ„ *

180 𝜌

𝑒𝑓𝑕𝑠𝑓𝑓𝑑 1 π‘ π‘π‘’π‘—π‘π‘œ β‰ˆ 57 𝑒𝑓𝑕

slide-12
SLIDE 12

Small angle approximation

12

πœ„ 2

πœ„ β‰ˆ 2 π‘’π‘π‘œ πœ„

2

πœ„ 2

slide-13
SLIDE 13

eye camera

13

Aperture angle from a few slides ago….

slide-14
SLIDE 14

camera

β€œF numberβ€œ (photography)

aperture A β€œfocal length” f

𝐺 π‘œπ‘£π‘›π‘π‘“π‘  ≑ 𝑔 𝐡 β‰ˆ 1

q

q

14

slide-15
SLIDE 15

ASIDE: camera

5 mm aperture A β€œfocal length” f 50 mm

𝐺 π‘œπ‘£π‘›π‘π‘“π‘  ≑ 𝑔 𝐡 = 50 5 = 10

15

slide-16
SLIDE 16

eye (ignore lens)

5 mm aperture A length f 25 mm

𝐺 π‘œπ‘£π‘›π‘π‘“π‘  ≑ 𝑔 𝐡 = 25 5 = 5

16

slide-17
SLIDE 17

Visual Angle

𝛽 𝛽 β‰ˆ π‘π‘π‘˜π‘“π‘‘π‘’ β„Žπ‘“π‘—π‘•β„Žπ‘’ π‘’π‘—π‘‘π‘’π‘π‘œπ‘‘π‘“

17

slide-18
SLIDE 18

Visual Angle

𝛽 𝛽 β‰ˆ 𝑗𝑛𝑏𝑕𝑓 𝑑𝑗𝑨𝑓 𝑝𝑔 π‘π‘π‘˜π‘“π‘‘π‘’ 𝑒𝑗𝑏𝑛𝑓𝑒𝑓𝑠 𝑝𝑔 π‘“π‘§π‘“π‘π‘π‘šπ‘š

18

slide-19
SLIDE 19

Two different concepts

Aperture angle Visual angle

19

slide-20
SLIDE 20

Visual Angle Example 1

𝛽 𝛽 β‰ˆ

π‘π‘π‘˜π‘“π‘‘π‘’ β„Žπ‘“π‘—π‘•β„Žπ‘’ π‘’π‘—π‘‘π‘’π‘π‘œπ‘‘π‘“

=

1 𝑑𝑛

180 𝜌

𝑑𝑛

= 1 degree

1 𝑑𝑛 𝑛

57 𝑑𝑛 (armβ€²s length)

Finger nail

20

slide-21
SLIDE 21

Visual Angle Example 2

𝛽

𝛽 β‰ˆ

π‘π‘π‘˜π‘“π‘‘π‘’ β„Žπ‘“π‘—π‘•β„Žπ‘’ π‘’π‘—π‘‘π‘’π‘π‘œπ‘‘π‘“

=

𝜌 10 𝑛

18 𝑛 =

𝜌 180 π‘ π‘π‘’π‘—π‘π‘œπ‘‘ = 1 degree

31.4 𝑑𝑛

18 𝑛

21

slide-22
SLIDE 22

Example 3: moon

22

Visual angle of moon is about

1 2 𝑒𝑓𝑕.

slide-23
SLIDE 23

Units of visual angle

23

1 radian =

180 𝜌

deg 1 deg = 60 minutes (or β€œarcmin”) 1 minute = 60 seconds (or β€œarcsec”)

slide-24
SLIDE 24

pinhole camera mode

Image position

(X, Y, Z ) (x, y)

24

slide-25
SLIDE 25

(X, Y, Z ) (x, y) (0, 0) image plane behind pinhole

π‘Ž 𝑍 π‘Œ

Pinhole camera

25

slide-26
SLIDE 26

(Y, Z ) image plane Z = - f y

View from side (YZ)

pinhole position (0, 0, 0 )

π‘Ž 𝑍

𝑧 𝑔 = 𝑍 π‘Ž

26

𝑧

slide-27
SLIDE 27

(X, Z ) image plane Z = - f x

View from above (XZ)

pinhole position (0, 0, 0 )

𝑦 𝑔 = π‘Œ π‘Ž

π‘Ž π‘Œ

27

𝑦

slide-28
SLIDE 28

(X, Y, Z ) 𝑦, 𝑧 (0, 0) image plane Z=-f behind pinhole

π‘Ž 𝑍 π‘Œ

𝑦 𝑔 , 𝑧 𝑔 = π‘Œ π‘Ž , 𝑍 π‘Ž

Image position in radians*

28

*assuming small angle approximation

slide-29
SLIDE 29

(X, Y, Z )

π‘Ž 𝑍 π‘Œ

𝑦, 𝑧 (0, 0) image plane in front of pinhole (0, 0) 𝑦, 𝑧

𝑦 𝑔 , 𝑧 𝑔 = π‘Œ π‘Ž , 𝑍 π‘Ž

Visual direction in radians*

slide-30
SLIDE 30

Example (ground and horizon)

30

slide-31
SLIDE 31

Image projection

(upside down and backwards)

31

slide-32
SLIDE 32

32

𝑦 𝑧 𝑦 𝑧

Visual direction (image plane in front of pinhole) Image projection (image plane behind pinhole)

slide-33
SLIDE 33

(X, Y, Z )

Depth Map

33

The mapping π‘Ž 𝑦, 𝑧 from image positions 𝑦, 𝑧 to depth π‘Ž values on a 3D surface is called a β€œdepth map”.

π‘Ž 𝑍 π‘Œ

𝑦, 𝑧 (0, 0)

slide-34
SLIDE 34

(- β„Ž , Z ) y

What is the depth map of a ground plane ?

π‘Ž 𝑍

𝑍 = βˆ’ β„Ž Ground plane

34

slide-35
SLIDE 35

What is the depth map of a ground plane ?

𝑍 = βˆ’ β„Ž

(- β„Ž , Z ) y

π‘Ž 𝑍

Ground plane

35

𝑧 𝑔 = 𝑍 π‘Ž Thus, π‘Ž = βˆ’β„Žπ‘” 𝑧

slide-36
SLIDE 36

36

𝑦 𝑧

Visual direction (image plane in front of pinhole)

π‘Ž = βˆ’β„Ž 𝑔 𝑧

slide-37
SLIDE 37

right eye

Binocular Vision

37

(0, 0, f ) left eye (π‘ˆ

𝑦, 0, f )

π‘Ž π‘Ž π‘Œπ‘š π‘Œπ‘  𝑍 𝑍

Assume eyes are separated by π‘ˆ

π‘Œ in the X direction.

π‘ˆ

π‘Œ is the interocular distance.

π‘ˆ

𝑦

slide-38
SLIDE 38

38

What is the difference in or visual direction (or image position) of each 3D object in the left and right images? How does this difference depend on depth ?

slide-39
SLIDE 39

(π‘Œ0, π‘Ž0)

π‘Ž π‘Œπ‘  π‘Ž

(π‘¦π‘š , 𝑔 )

π‘Œπ‘š

View from above (XZ)

39

π‘ˆ

𝑦

(𝑦𝑠 , 𝑔)

slide-40
SLIDE 40

40

Binocular disparity ≑

π‘¦π‘š 𝑔 βˆ’ 𝑦𝑠 𝑔

is the difference in visual direction of a 3D point as seen by two eye.

slide-41
SLIDE 41

41

π‘¦π‘š 𝑔 = π‘Œ0 π‘Ž0 𝑦𝑠 𝑔 = π‘Œ0 βˆ’ π‘ˆ

𝑦

π‘Ž0

Thus, binocular disparity =

π‘ˆ

𝑦

π‘Ž0

Binocular disparity ≑

π‘¦π‘š 𝑔 βˆ’ 𝑦𝑠 𝑔

slide-42
SLIDE 42

Superimposing left and right eye images

42

𝑦 𝑧

cloud

Zero disparity

binocular disparity =

π‘ˆ

𝑦

π‘Ž0 = π‘ˆ

𝑦

βˆ’β„Ž 𝑧 𝑔

slide-43
SLIDE 43

Vergence (rotating the eyes)

π‘Ž π‘Œπ‘  π‘Ž π‘Œπ‘š

43

Here we assume horizontal rotation

  • nly (β€œpan”).
slide-44
SLIDE 44

Vergence

44

𝑦 𝑧

cloud cloud Positive disparity Negative disparity Zero disparity Positive disparity Example: verge on far person

𝑧 𝑦

slide-45
SLIDE 45

45

Binocular disparity ≑ (

π‘¦π‘š 𝑔 βˆ’ πœ„π‘š) βˆ’ ( 𝑦𝑠 𝑔 βˆ’πœ„π‘ )

Let πœ„π‘š and πœ„π‘  be the rotations of the left and right eyes due to vergence. The rotations can be approximated by a shift in image position.

=

π‘¦π‘š 𝑔 βˆ’ 𝑦𝑠 𝑔

βˆ’ (πœ„π‘š βˆ’ πœ„π‘ )