Camera Simulation

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell

Camera Simulation

ReferencesPhotography, B. London and J. Upton

Optics in Photography, R. Kingslake

The Camera, The Negative, The Print, A. Adams

Effect Cause

Field of view Film size, stops and pupils

Depth of field Aperture, focal length

Motion blur Shutter

Exposure Film speed, aperture, shutter


Topics

Ray tracing lenses

Focus

Field of view

Depth of focus / depth of field

Exposure

Lenses


Refraction

I

′I

sin sinn I n I′ ′=

Snell’s Law

N

n n′


Paraxial Approximation

0e ≈

Rays deviate only slightly from the axis

U

z

€

sin(U) ≈ U = u

tan(U) ≈ U = u

cos(U) ≈1


Incident Ray

I

U φ−

φ= −I U

Angles: ccw is positive; cw is negative

The sum of the interior angles is equal to the exterior angle.


Refracted Ray

− ′U′I

φ= −′ ′I U

φ− = + −′ ′( ) ( )I U

φ−


Derivation

Paraxial approximationφ φ= − ⇒ = −I U i uφ φ= − ⇒ = −′ ′ ′ ′I U i u


Derivation

Paraxial approximation

Snell’s Law= ⇒ =′ ′ ′ ′sin sinn I n I n i ni

φ φ= − ⇒ = −I U i uφ φ= − ⇒ = −′ ′ ′ ′I U i u

φ φ− = −′ ′( ) ( )n u n u


Ray Coordinates

−z z′

Ru

φ− =h

R

φ− − ′u

=−h

uz

− =′′

hu

z

h


Gauss’ Formula

Paraxial approximation to Snell’s Law

Ray coordinates

Thin lens equation

φ φ− = −′ ′( ) ( )n u n u

( ) ( )

( )

h h h hn n

z R z Rn n n n

z z R

′ − = −′

′ ′−= +

′

φ=−h

R=−′

′h

uz

=−h

uz

Holds for any height, any ray!


Vergence

Vergence

Thin lens equation

Surface Power equation

⎡ ⎤≡ ≈ =⎢ ⎥⎣ ⎦

1n nV diopters

r z m

V V P′= +

< 0V =0V > 0V

≡ −′1

( )P n nR

Diverging Converging


Lens-makers Formula

⎛ ⎞= − − =′ ⎜ ⎟⎝ ⎠1 2

1 1 1( )P n n

R R f

Converging Diverging

Refractive Power


Conjugate Points

To focus: move lens relative to backplaneHorizontal rays converge on focal point in the focal plane

1 1 1

z z f= +

′


Gauss’ Ray Tracing Construction

Parallel Ray

Focal RayChief Ray

Object Image


Ray Tracing: Finite Aperture

Focal Plane Back PlaneAperture Plane


Real Lens

Cutaway section of a Vivitar Series 1 90mm f/2.5 lensCover photo, Kingslake, Optics in Photography


Double Gauss

Radius (mm) Thick (mm) nd V-no aperture

58.950 7.520 1.670 47.1 50.4

169.660 0.240 50.4

38.550 8.050 1.670 47.1 46.0

81.540 6.550 1.699 30.1 46.0

25.500 11.410 36.0

9.000 34.2

-28.990 2.360 1.603 38.0 34.0

81.540 12.130 1.658 57.3 40.0

-40.770 0.380 40.0

874.130 6.440 1.717 48.0 40.0

-79.460 72.228 40.0

Data from W. Smith, Modern Lens Design, p 312


Ray Tracing Through Lenses

From Kolb, Mitchell and Hanrahan (1995)

200 mm telephoto

50 mm double-gauss

35 mm wide-angle

16 mm fisheye


Thick Lenses

Refraction occurs at the principal planes

Equivalent Lens

Field of View


Field of View

From London and Upton


Field of View



Field of View

Field of view

Types of lensesNormal 26º

Film diagonal focal length

Wide-angle 75-90ºNarrow-angle 10º

Redrawn from Kingslake, Optics in Photography

tan2

fov filmsize

f=


Perspective Transformation

Thin lens equation

Represent transformation as a 4x4 matrix

= + ⇒ =′+′

⇒ =′+

⇒ =′+

1 1 1 fzz

z z f z f

fxx

z f

fyy

z f

Depth of Field


Depth of Field



Circle of Confusion

d′

a

c d s z

a z z

′ ′ ′−= =

′ ′

s′sz z′

c

Circle of confusion proportional to the size of the aperture

Focal Plane Back Plane


Depth of Focus [Image Space]

Depth of focus Equal circles of confusion

Two planes: near and far

−′ ′ ′= =

′ ′f f

f f

d s zc

a z zc

a

fz

s

nz

c

′s

′fz

′nz′fd

−′ ′ ′= =

′ ′n n

n n

d z sc

a z z

′nd




−′ ′ ′ ⎛ ⎞= = ⇒ = +⎜ ⎟⎝ ⎠′ ′ ′ ′1 1

1f f

f f f

d s zc c

a z z z s ac

a

fz

s

nz

c

′s

′fz

′nz′fd

−′ ′ ′ ⎛ ⎞= = ⇒ = −⎜ ⎟⎝ ⎠′ ′ ′ ′1 1

1n n

n n n

d z sc c

a z z z s a

′nd




⎛ ⎞= +⎜ ⎟⎝ ⎠′ ′1 1

1f

c

z s a

c

′s′fz

′nz

⎛ ⎞= −⎜ ⎟⎝ ⎠′ ′1 1

1n

c

z s a

+ =′ ′ ′

1 1 12

f nz z s

− =′ ′ ′

1 1 2 1

f n

c

z z a s


Depth of Field [Object Space]

Depth of field Equal circles of confusion

+ =1 1 1

2n fz z s

⎛ ⎞− = − ≈⎜ ⎟⎝ ⎠

1 1 2 1 1 2 1

n f

c c

z z a f s a f

= +′

1 1 1

f fz z f= +

′1 1 1

n nz z fc

fz

nz

c

= +′

1 1 1

s s f


Hyperfocal Distance

c

fz

nz

cWhen

→ ⇒ = =∞,2n f

Hs H z z

H is the hyperfocal distance

+ =1 1 1

2n fz z s

€

N ≡f

a

€

1

zn

−1

z f

=2c

a

1

f= 2

cN

f 2≡ 2

1

H


Depth of Field Scale


Factors Affecting DOFFrom http://www.kodak.com/global/en/consumer/pictureTaking/cameraCare/cameCar6.shtml

= 2

1 cN

H f


Resolving Power

Diffraction limit

35mm film (Leica standard)

CCD/CMOS pixel aperture

[ ]λ μ= = × ×1.22 1.22 64 .500 m=0.040mmf

ca

=0.025mmc

=0.0116mm (Nikon D1)c

Exposure


Image Irradiance

f

2

2cos sin4

aE L d L L

f

πθ ω π θΩ

⎛ ⎞= = = ⎜ ⎟

⎝ ⎠∫

a


Image Irradiancen̂

θ

€

φ

€

φ

€

ω

€

ω =π4

a2 cosφ

(z /cosφ)2=

π

4

a

z

⎛

⎝ ⎜

⎞

⎠ ⎟2

cos3 φSolid angle subtended by thelens, as seen by the patch A

€

ω

€

δP =LΩ δA cosθ = LδAπ

4

a

z

⎛

⎝ ⎜

⎞

⎠ ⎟2

cos3 φcosθPower from patch Athrough the lens

€

E =δP

δI=L

δA

δI

π

4

a

z

⎛

⎝ ⎜

⎞

⎠ ⎟2

cos3 φcosθIrradiance at thesensor element

€

δA

€

δI

€

a

€

f

€

z


Image Irradiance

€

ˆ n

θ

€

φ

€

φ

€

δAcosθ

(z /cosφ)2=

δI cosφ

( f /cosφ)2

€

δA

δI=

cosφ

cosθ

z

f

⎛

⎝ ⎜

⎞

⎠ ⎟

2€

δI

€

δA

€

z

€

a

€

f


Image Irradiance

€

E =δP

δI=L

δA

δI

π

4

a

z

⎛

⎝ ⎜

⎞

⎠ ⎟2

cos3 φcosθ

€

E = Lπ

4

a

f

⎛

⎝ ⎜

⎞

⎠ ⎟

2

cos4 φ

€

δA

δI=

cosφ

cosθ

z

f

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

E = Lπ

4

a

f

⎛

⎝ ⎜

⎞

⎠ ⎟

2

On axis


Relative Aperture or F-Stop

F-Number and exposure:

Fstops: 1.4 2 2.8 4.0 5.6 8 11 16 22 32 45 641 stop doubles exposure

f

π= 2

1

4E L

N

=f

aN

a


Camera Exposure

Exposure

Exposure overdeterminedAperture: f-stop - 1 stop doubles H

Decreases depth of field

Shutter: Doubling the open time doubles HIncreases motion blur

H E T= ×


Aperture vs Shutter


f/161/8s

f/41/125s

f/21/500s


High Dynamic Range

Sixteen photographs of the Stanford Memorial Church taken at 1-stop increments from 30s to 1/1000s.From Debevec and Malik, High dynamic range photographs.


Simulated Photograph

Adaptive histogram With glare, contrast, blur


Camera Simulation

Sensor response

Lens

Shutter

Scene radiance

λ ω ω λ λ ω λΩ Λ

= •′ ′ ′ ′ ′ ′ ′∫∫∫∫r r

( , ) ( , , ) ( ( , , ), , ) ( )A T

R P x S x t L T x t dA x d dt d

ω ω λ= ′ ′( , ) ( , , )x T x

λ′( , )P x

( , , , )L x tω λω′ ′( , , )S x t

ω λ′ ′( , , , )L x tω λ( , , , )L x tA

Ω

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Camera Simulation

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