Dynamic Perspective CS 543 / ECE 549 Saurabh Gupta Spring 2020, - - PowerPoint PPT Presentation

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Dynamic Perspective CS 543 / ECE 549 Saurabh Gupta Spring 2020, - - PowerPoint PPT Presentation

Dynamic Perspective CS 543 / ECE 549 Saurabh Gupta Spring 2020, UIUC http://saurabhg.web.illinois.edu/teaching/ece549/sp2020/ Many slides adapted from J. Malik. Perspective Projection P Y &' ( , &) , , ( Z f


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SLIDE 1

Dynamic Perspective

CS 543 / ECE 549 – Saurabh Gupta Spring 2020, UIUC

http://saurabhg.web.illinois.edu/teaching/ece549/sp2020/

Many slides adapted from J. Malik.

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SLIDE 2

Perspective Projection

π‘Œ, 𝑍, π‘Ž β†’

&' ( , &) (

f Z P y O p Y

Suppose the camera moves with respect to the world…

  • Point P (π‘Œ, 𝑍, π‘Ž) in the world moves relative to the

camera, its projection in the image (𝑦, 𝑧) moves as well.

  • This movement in the image plane is called optical flow.
  • Suppose the image of the point (𝑦, 𝑧) moves to

(𝑦 + βˆ†π‘¦, 𝑧 + βˆ†π‘§) in time βˆ†π‘’, then 12

13 , 14 13

are the two components of the optical flow.

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SLIDE 3

Outline

  • Relate optical flow to camera motion
  • Special cases
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SLIDE 4

How does a point X in the scene move?

  • Assume that the camera moves with a

translational velocity 𝑒 = (𝑒2, 𝑒4, 𝑒6) and angular velocity πœ• = πœ•2, πœ•4, πœ•6 .

  • Linear velocity of point 𝑄 = π‘Œ, 𝑍, π‘Ž is given

by Μ‡ 𝑄 = βˆ’π‘’ βˆ’ πœ•Γ—π‘„. Μ‡ π‘Œ Μ‡ 𝑍 Μ‡ π‘Ž = βˆ’ 𝑒2 𝑒4 𝑒6 βˆ’ πœ•4π‘Ž βˆ’ πœ•6𝑍 πœ•6π‘Œ βˆ’ πœ•2π‘Ž πœ•2𝑍 βˆ’ πœ•4π‘Œ

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SLIDE 5

Now, lets consider the effect of projection

Μ‡ π‘Œ Μ‡ 𝑍 Μ‡ π‘Ž = βˆ’ 𝑒2 𝑒4 𝑒6 βˆ’ πœ•4π‘Ž βˆ’ πœ•6𝑍 πœ•6π‘Œ βˆ’ πœ•2π‘Ž πœ•2𝑍 βˆ’ πœ•4π‘Œ

  • Assume, 𝑔 = 1, 𝑦 =

' ( , 𝑧 = ) ( .

  • Μ‡

𝑦 =

Μ‡ '(@ Μ‡ (' (A

, Μ‡ 𝑧 =

Μ‡ )(@ Μ‡ () (A

  • Substitute Μ‡

π‘Œ, Μ‡ 𝑍, Μ‡ π‘Ž , from equation above:

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6

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SLIDE 6

Dynamic Perspective Equations

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6 Translation Component Rotation Component

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SLIDE 7

Optical flow for pure rotation

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6

  • 𝑣

𝑀 = 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6

  • We can determine πœ• from the flow field.
  • Flow field is independent of π‘Ž(𝑦, 𝑧).
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SLIDE 8

Optical flow for pure translation along Z-axis

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6

  • 𝑣

𝑀 =

3F ((2,4)

𝑦 𝑧

  • Optical flow vector is a scalar

multiple of position vector.

  • Scale factor ambiguity, if 𝑒6 β†’

𝑙𝑒6, and π‘Ž β†’ π‘™π‘Ž, optical flow remains unchanged.

  • But, you can get time to

collision, π‘Ž/𝑒6.

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SLIDE 9
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SLIDE 10
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SLIDE 11
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SLIDE 12
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SLIDE 13

Optical flow for general translation

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6

  • 𝑣 = @3IJ23F

((2,4) , v = @3LJ43F ((2,4)

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SLIDE 14

Optical flow for points on a road

Slide by J. Malik

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SLIDE 15

Translating along X-axis in front of a wall

Slide by J. Malik

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SLIDE 16

Estimating Optical Flow from Images

http://en.wikipedia.org/wiki/Barberpole_illusion

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SLIDE 17

Estimating Optical Flow from Images

http://en.wikipedia.org/wiki/Barberpole_illusion

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SLIDE 18

Estimating Optical Flow from Images

Aperture Problem

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SLIDE 19

Recap

  • Relate optical flow to camera motion
  • Special cases
  • Pure rotation / pure translation / time to collision

𝑣 𝑀 = Μ‡ 𝑦 Μ‡ 𝑧 = 1 π‘Ž βˆ’1 𝑦 βˆ’1 𝑧 𝑒2 𝑒4 𝑒6 + 𝑦𝑧 βˆ’ 1 + 𝑦E 𝑧 (1 + 𝑧E) βˆ’π‘¦π‘§ βˆ’π‘¦ πœ•2 πœ•4 πœ•6