Lecture 21:
Cameras & Lenses II
Computer Graphics and Imaging
UC Berkeley CS184/284A, Spring 2017
Cameras & Lenses Lectures
Last time:
•
•
Field of view, composition & perspective
Shutter / exposure durations
Today
•
•
Thin lens approximation, Gauss ray diagrams
Lens focus effects: defocus, depth of field, bokeh
Next time:
•
•
Real lens designs and effects
Lens exposure considerations
CS184/284A
Ren Ng
Real Lens Designs Are Highly Complex
[Apple]
CS184/284A
Topic of next lecture
Ren Ng
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Real plano-convex lens (spherical surface shape).
Lens does not converge rays to a point anywhere.
CS184/284A
More discussion next lecture
Ren Ng
aberration.
Figure �.�: Spherical
ried to remarkable extremes. Zoom l
Real Lens Elements Are Not Ideal – Aberrations
Today: Thin Lens Approximation
Ideal Thin Lens – Focal Point
Focal
Point
Credit: Karen Watson
Focal
Length
Assume all parallel rays entering a lens pass through its focal point.
CS184/284A
Ren Ng
Lens Focusing – Conjugate Points
•
Rays from a point in object space
intersect at a point in image space
•
Object space
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Image space
These are called conjugate points
•
We create images focused at
different depths by placing a
sensor at the conjugate distances
•
Question: what is the relationship
between the position of a lens’
conjugate points?
Ren Ng
Gauss’ Ray Diagrams
Gauss’ Ray Tracing Construction
Parallel Ray
Chief Ray
Focal Ray
Object
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Image
Ren Ng
Gauss’ Ray Tracing Construction
f
zo
f
zi
What is the relationship between conjugate depths zo , zi ?
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Ren Ng
Gauss’ Ray Tracing Construction
ho
ho
zo
hi
=
f
f
ho
zo
CS184/284A
f
ho
hi
hi
=
f
f
ho
hi
=
f
zi f
f
hi
ho
hi
=
f
zi f
Ren Ng
Gauss’ Ray Tracing Construction
ho
hi
=
f
zi f
ho
f
=
hi
zi f
ho
zo
hi
=
f
f
ho
zo f
=
hi
f
zo
f
f
(zo
z o zi
f )(zi
f
=
zi
2
f) = f
2
(zo + zi )f + f = f
f
Newtonian Thin Lens Equation
2
zo zi = (zo + zi )f
1
1
1
=
+
f
zi
zo
CS184/284A
Object / image heights
factor out - applies to all rays
Gaussian Thin Lens Equation
Ren Ng
The Thin Lens Equation
zo
f
f
(zo
z o zi
f )(zi
f
=
zi
2
ff) = f
2
zo zi )f + f = f
(zo +
f
f
2
zi
zo zi = (zo + zi )f
1
1
1
=
+
f
zi
zo
CS184/284A
Ren Ng
f )(zi
f) = f
+ zChanging
i )f + f = f
2
2
the Focus Distance
zo zi = (zo + zi )f
1
1
1
=
+
f
zi
zo
Sensor
• To focus on objects at different
distances, move the sensor
relative to the lens
• For zi < zo the object is larger
than the image
• At zi = zo we have 1:1 macro
imaging
• For zi > zo the image is larger
than the object
• Can’t focus on objects closer
than the lens’ focal length
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Ren Ng
Magnification
ho
hi
zo
zi
hi
zi
m=
=
ho
zo
CS184/284A
Ren Ng
)(zi
f) = f
2
2
2
Magnification
Example
–
Focus
at
Infinity
)f
+
f
=
f
i
zo zi = (zo + zi )f
1
1
1
=
+
f
zi
zo
zi
m=
zo
If focused on a distant mountain
•
•
•
zo ≈ ∞, so zi = f
sensor at focal point
magnification ≈ 0
CS184/284A
Ren Ng
(zo
f )(zi
f) = f
2
Magnification
zo zi Example
(zo + zi )f +–f Focus
= f at 1:1 Macro
2
2
zo zi = (zo + zi )f
1
1
1
=
+
f
zi
zo
zi
m=
zo
What configuration do we need to achieve a magnification of 1
(i.e. image and object the same size, a.k.a. 1:1 macro)?
•
•
Need zi = zo, so zi = zo = 2f —- sensor at twice focal length
In 1:1 imaging, if the sensor is 36 mm wide, an object 36 mm
wide will fill the frame
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Ren Ng
Thin Lens Demonstration
http://graphics.stanford.edu/courses/cs178-10/applets/gaussian.html
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Ren Ng
Thin Lens Demonstration Observations
3D image of object is:
•
Compressed in depth for
low magnification
•
1:1 in 3D for unit
magnification
•
Stretched in depth for high
magnification
CS184/284A
Ren Ng
Lens Performs a 3D Perspective Transform
•
Lenses transform a 3D object to a 3D image; the
sensor extracts a 2D slice from that image
•
As an object moves linearly (in Z), its image moves non-proportionally (in Z). And vice versa.
•
As you refocus a camera, the image changes size !
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Ren Ng
Defocus Blur
Circle of Confusion
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Ren Ng
Circle of Confusion
Further defocused object
Closer defocused object
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Ren Ng
Computing Circle of Confusion Diameter (C)
0
zs
zs
zo
zi
d!
A
Object
Focal Plane
Circle of confusion is proportional
to the size of the aperture
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C
Image Sensor Plane
0
C
d
|zs zi |
=
=
A
zi
zi
Ren Ng
Circle of Confusion is Inversely Proportional to F-Stop
R. Berdan, canadiannaturephotographer.com
C=A
CS184/284A
|zs
zi
zi |
f |zs zi |
=
N
zi
Ren Ng
Definition: F-Number (a.k.a. F-Stop)
•
The F-Number of a lens is defined as the focal length divided
by the diameter of the aperture
•
Common F-stops on real lenses: 1.4, 2, 2.8, 4.0, 5.6, 8, 11,
16, 22, 32
•
•
1 stop doubles exposure
An f-stop of 2 is sometimes written f/2, reflecting the fact
that the absolute aperture diameter (A) can be computed by
dividing focal length (f) by the relative aperture (N).
CS184/284A
Ren Ng
Example F-Stop Calculations
D = 50 mm
f = 100 mm
N = f /D = 2
D = 100 mm
f = 200 mm
N = f /D = 2
D = 100 mm
f = 400 mm
N = f /D = 4
CS184/284A
Ren Ng
Circle of Confusion is Inversely Proportional to F-Stop
R. Berdan, canadiannaturephotographer.com
C=A
CS184/284A
|zs
zi
zi |
f |zs zi |
=
N
zi
Ren Ng
Circle of Confusion – Example
50mm f/2 lens
Full frame sensor (36x24mm)
Focus: 1 meter
Background: 10 meter
Foreground: 0.3 meter
|zs zi |
f |zs zi |
A = 50mm/2 = 25mm
C=A
=
1
zi
N
zi
zs =
⇡ 52.63mm
1/50 1/1000
1
Background: zi =
⇡ 50.25mm
1/50 1/10,000
C = A|zs zi |/zi = 1.18mm
~65 pixels on HD TV
1
Foreground: zi =
⇡ 55.56mm
1/50 1/300
C = A|zs
CS184/284A
zi |/zi = 3.07mm
~167 pixels on HD TV
Ren Ng
Circle of Confusion in Perspective Composition
•
•
To maintain field of
view on subject, increase distance
from subject
by same factor as
focal length
(approx).
16 mm (110°)
200 mm (12°)
What is the increase
in background blur?
CS184/284A
Ren Ng
Circle of Confusion in Perspective Composition
•
•
To maintain field of
view on subject, increase distance
from subject
by same factor as
focal length
(approx).
What is the increase
in background blur?
CS184/284A
For subject at distance Z,
1
1
1
=
zs
f
Z
Distant background means zi = f,
f |zs zi |
f
C=
=
N
zi
N
f f
=
NZ f
•
1
1/f
1/Z
f
f
= ...
If we increase Z and f by factor K,
circle of confusion C also increases
by K. (F-stop held constant)
Ren Ng
Circle of Confusion in Perspective Composition
100mm, f/4
138px
40px
28mm, f/4
From Paul van Walree, toothwalker.org/dof.html
100mm
138px
⇡
As predicted,
, but notice blur is constant relative to background object itself!
28mm
40px
CS184/284A
Ren Ng
Ray Tracing Ideal Thin Lenses
Examples of Renderings with Lens Focus
Pharr and Humphreys
CS184/284A
Ren Ng
Ray Tracing for Defocus Blur (Thin Lens)
x’’
x’
x’’’
Sensor
Subject plane
zo
zi
Setup:
• Choose sensor size, lens focal length and aperture size
• Choose depth of subject of interest zo
• Calculate corresponding depth of sensor zi from thin lens
equation (focusing)
CS184/284A
Ren Ng
Ray Tracing for Defocus Blur (Thin Lens)
x’’
x’
x’’’
Sensor
Subject plane
zo
zi
To compute value of pixel at position x’ by Monte Carlo integration:
• Select random points x’’ on lens plane
• Rays pass from point x’ on image plane zi through points x’’ on lens
• Each ray passes through conjugate point x’’’’ on the plane of focus zo
• Can determine x’’’ from Gauss’ ray diagram
• So just trace ray from x’’ to x’’’
• Estimate radiance on rays using path-tracing, and sum over all points x’’
CS184/284A
Ren Ng
Examples of Renderings with Lens Focus
Pharr and Humphreys
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Ren Ng
Example of Rendering with Lens Focus
Credit: Bertrand Benoit. “Sweet Feast,” 2009. [Blender /VRay]
Example of Rendering with Lens Focus
Credit: Giuseppe Albergo. “Colibri” [Blender]
Acknowledgments
Many thanks to Marc Levoy, who created many of these
slides, and Pat Hanrahan.
•
London, Stone, and Upton, Photography (9th ed.),
Prentice Hall, 2008.
•
•
•
•
•
Peterson, Understanding Exposure, AMPHOTO 1990.
The Slow Mo Guys
bobatkins.com
Hari Subramanyan
Canon EF Lens Work III
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Ren Ng