Lenses
Real-time Rendering of Physically Based Optical Effect in Theory and Practice
SIGGRAPH 2015 Course

Masaki Kawase
Silicon Studio, Corp.
masa@siliconstudio.co.jp

Various Bokeh from Photographs

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Contents• Aberrations and Corrections• Residual Aberrations and Bokeh Characteristics• Phenomena of Multiple-Lens Systems• Conclusion3

Aberrations and Corrections

Optical Aberrations• Actual lenses do not image ideally– Imperfect focus– Image distortion– Color dispersion– And more …

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Major Aberrations• Monochromatic aberrations– Occur even with single-wavelength rays
– Also known as Seidel's five aberrations• Chromatic aberrations– Caused by dispersion• The separation of visible light into its different colors• Different refractive indices in multi-wavelength rays

– Caused with multi-wavelength rays but:
• Occurs as blur in monochrome film
• Does not occur in color film with single-wavelength rays
– Such as Sodium-vapor Lamps

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Monochromatic and Chromatic Aberrations• Monochromatic aberrations (Seidel's five aberrations)– Spherical Aberration (SA)– Coma– Field Curvature– Astigmatism– Distortion• Chromatic aberrations (CA)– Lateral Chromatic Aberration (CA of Magnification)– Longitudinal Chromatic Aberration (Axial CA)
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Details of Important Aberrations
Which Affect Bokeh

Spherical Aberration• The focal length deviation of rays parallel to the optical axis
• The aberration is caused by a spherical lens
– Spherical surfaces are:• Not ideal for lenses• Commonly used due to the high manufacturability

9Principle of Spherical Aberration
Focal length of each incident height
Incident height
Optical axis

Paraxial focal length• The farther the rays are from the optical axis,
the closer they intersect the optical axis10

Spherical Lens Bokeh

Optical axis

Spot diagrams

Front bokeh (sharp-edged)

Circle ofleast confusion(a.k.a. COLC)
Back bokeh (soft-edged)
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Corrections for Spherical Aberration• Doublet lens
– Pair of convex and concave lenses– Concave lens aberration cancels convex lens one– Cannot cancel perfectly• Triplet lens
– An additional lens to doublet– Still not perfect, but much better• Aspherical lens
– Surface is close to ideal– Expensive to make– Perfectly remove spherical aberration

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Example of Doublet Lens Correction
Focal point of each incident height

Optical axis

Circle of least confusion plane• More complicated bokeh than spherical

Front bokeh

Back bokeh

13Comparison

Spherical lens

Doublet lensSharper focus
Flatter bokeh
14Spherical Aberration Charts(Longitudinal Aberration Diagrams)
Y: Incident height

Y: Incident height
X: Amount of aberration

Spherical lens

Doublet lens

• Y-axis : Incident height (independent variable)• X-axis : Amount of spherical aberration (dependent variable)
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Spherical Aberration Charts(Longitudinal Aberration Diagrams)

Incident height
Without correction

DoubletTriplet

Aspherical

Spherical

Doublet

TripletAspherical16

Diagrams and Bokeh• Closer to vertical line,
better correction– Sharper focus– Flatter bokeh

Spherical

Doublet

TripletAspherical

Front bokeh

Back bokeh
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Axial Chromatic Aberration• Differences of ray wavelengths cause aberration• Refractive indices differ by wavelengths

DrBob, https://en.wikipedia.org/wiki/File:Chromatic_aberration_lens_diagram.svg
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Principle of Axial CA
Incident height

Focal length on each wavelength19Bokeh of Axial CASpot diagrams of marginal rays

Front bokehBack bokeh
20Effects of Axial CA• Front bokeh shows red fringe• Back bokeh shows blue fringe• Relatively larger fringe around the focal point

Front bokeh

Back bokeh

Out-of-focus images made by a magnifier
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Correction of Axial Chromatic Aberration• Achromatic lens
– Correction with doublet or triplet etc.
• Coupling of different dispersion property lenses• Focusable multi-wavelength rays on a single point• Cannot correct perfectly on all wavelengths

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Achromatic Lens
• Achromatic lens (Achromat)– Achromatic doublet etc.
– Focusable two wavelength rays on thesame point• e.g. red and blue

• Apochromatic lens (APO)– Apochromatic triplet etc.
– Generally focusable three wavelength rays
• e.g. red, green and blue

Apochromatic triplet
DrBob, https://commons.wikimedia.org/wiki/File:Lens6b-en.svg
Egmason, https://commons.wikimedia.org/wiki/File:Apochromat_2.svg
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Example of Achromatic Doublet Correction
Focal plane on each wavelength• Cannot match all color focal points– If you match red and blue, you cannot match others (green etc.)24

Example of Achromatic Doublet Bokeh

Spot diagram of marginal rays

Front bokeh

Back bokeh
25Comparison

Axial chromatic aberration

Residual chromatic aberration
a.k.a. secondary spectrum
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Correction by Achromatic Doublet• Doublet also corrects spherical aberration• Combination bokeh of each character– Residual aberration of spherical aberration• Soft / Sharp edge
• Dark center / sharp peak

– Residual aberration of
axial chromatic aberration＊• Concentric colored circles＝• ⇒Complicated gradation
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Diagrams and Bokeh with Multiple Wavelengths

Spherical lens
without correction

Doublet lens
28Corrected Bokeh from Aberrations• Correction by achromatic doublets– Widely used– Typical correction example• Soft purple fringe on front bokeh• Sharp green fringe on back bokeh

Front bokeh in photographs

Back bokeh in photographs
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Front Bokeh with Purple Fringe

Front bokeh in photographs
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Back Bokeh with Green Fringe

Back bokeh in photographs
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Is Residual Aberration Visible or Not?
Large

Strongly visible

Larger bokeh

Slightly out of focus
with a large aperture

Aperture size

Large out of focus
with a small apertureSmallLess visible

Smaller bokehFocus

BlurryOut-of-focus amount32

Is Residual Aberration Visible or Not? (Cont’d)• Strongly visible– Slightly out of focus with a large aperture• Less visible– Large out of focus with a small aperture

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Residual Aberrations andBokeh Characteristics

Bokeh Characteristics• Bokeh Characteristics vary by:– Aberrations– Residual aberrations• Different corrections make different characteristics• Residual aberrations are essentially undesired– But they are characteristics of real photos

35Various Bokeh from Photographs36Phenomena of Multiple-Lens Systems

Multiple-Lens Systems• Actual optical system is composed of multiple lenses
in order to:– Correct aberrations– Zoom– Reduce focus breathing– Others

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Multiple-Lens vs. Single-Lens• More complex aberrations• Various bokeh characteristics• Different focus breathing• Variable maximum aperture• Optical Vignetting• And more …

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Focus Breathing• Focus breathing– FOV varies when focusing• Types of focus breathing– Single Lens
• Focusing by shifting lens or sensor– Focal length is constant and independent of focus distance• At close focus, FOV becomes narrower– In spite of constant focal length» Extend image distance (between lens and sensor)» While the F-number is the same, the effective F-number is larger (darker)– Multiple-lens system• Breathing varies by the focusing mechanism
40Focal Length, Sensor Size and FOV• Field of view is often explained as…
– Depends on the ratio of sensor size and focal length• fov = atan(h / 2f) * 2• f = h / (tan(fov / 2) * 2)

ffovD• fov : field of view
• h : sensor size

do=∞hdi = f• Not accurate– Accurate only when focusing on infinite distance41

Accurate FOV Calculation• Field of view
– Depends on the ratio of sensor size and image distance• fov = atan(h / 2di) * 2• di = h / (tan(fov / 2) * 2)

– Effective calculation only when a lens exists• fov = atan(h (do - f) / 2do f) * 2• f = (do h / 2) / (tan(fov / 2) * do + h / 2)

ffovDh• Effective F-number– Fe = di / D– Effective calculation only when a lens exists

do <∞• Fe = (1 + M) F
• Fe = (di / f) F

• Focus distance is also required in order to calculate correctly– If the focal length is constant, FOV becomes narrower with finite focus

di > fOptical magnification ‘M’
M = di / do
M = f / (do – f) = di / f - 1

Focusing Mechanisms• All-Group Focusing / Film-Back Focusing– Same mechanism as single-lens system– Used in old lenses– FOV becomes narrower when close focus• An Effective F-number becomes decreased• Front-Group Focusing– Used in old lenses– Usually FOV becomes narrower when close focus• An Effective F-number becomes decreased• Inner (Internal) / Rear Focusing––––

a.k.a. IF / RF
Used in recent zoom lenses
Usually FOV becomes wider when close focus (less expensive lenses)
No-breathing focus (relatively expensive lenses)• An Effective F-number is constant43

Focusing Mechanism and Breathing Examples
Inner Focusing (expensive lens)
No breathing
Focal length becomes shrunk

All-Group / Film-Back Focusing
FOV becomes narrower
ffovf

fov

h

Infinite focus
do=∞

hdo=∞

di =fffov

di =fffovhClose focus
dodihdodi44

Focusing Mechanism and Breathing Examples
Shift sensor to backward 2f (or shift lens)
Focal length is constantEffective F-number becomes darker
f=3

fov

Image distance is fixed
Focal length is shrunk to 66.7%
FOV and Effective F-number are constant
f=3

fov

h

Infinite focus
do=∞

fov

h
do=∞

di=3=ff=3

fov

di=3=f

f=2

hFocus on 2fdo=6=2f
M = di / do = 1.0

di=6=2fh
do=6=3f
di=3=1.5f
M = di / do = 0.5

45Focusing Mechanism and Breathing Examples
Typical Inner Focusing
FOV becomes wider
Focal length is largely shrunk

All-Group / Film-Back Focusing
FOV becomes narrower
f

fov

f

fov

h

Infinite focus
do=∞

h
do=∞

di =fffov

fov

di =ffhh

Close focus
do

dido

di46

Focusing Mechanism and Breathing Examples
Shift sensor to backward 2f (or shift lens)
Focal length is constantEffective F-number becomes darkerf=3fov

Focal length is shrunk to 40% in this case
Image distance is also shrunk to 50%
FOV becomes wider
f=3

fov

h

Infinite focus
do=∞

fov

h
do= ∞

di=3=ffov

f=3

do=6=2f
M = di / do = 1.0

di=6=2f

f=1.2

hh

Focus on 2f

di=3=fdo=6=5f

di=1.5=1.25f
M = di / do = 0.2547

Variable Aperture Zoom Lenses

Wide (12mm)
Maximum aperture is f/2.8

Narrow (60mm)
Maximum aperture is f/4.048

Effective Aperture Diameter ‘D’• Diameter of “Entrance Pupil”– Virtual image of the aperture as seen from the front– NOT a physical aperture diameter

Effective aperturediameter49

Zooming Varies Virtual Image Diameter• To keep the exposure, narrower FOV requires larger diameter
D=f/F

Φ15mm

Φ~4.2mm

Zoom lens12-60mm F2.8-4.0
Wide (12mm) f/2.8
D = 12/2.8 = ~4.2mm

Narrow (60mm) f/4
D = 60/4 = 15mm
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Zoom Lens Types• Fixed Aperture Zoom Lens
– Minimum F-number is constant over the entire zoom range– Effective diameter is proportional to focal length (D = f / F)

• Variable Aperture Zoom Lens
– Minimum F-number becomes larger as the FOV becomes narrower– Effective diameter is not proportional to focal length*Note that the “Minimum F-number”
means the “Maximum Aperture”
Wide (12mm)
f/2.8

Narrow (60mm)
f/4.051

Examples of Zoom Lens Products•

OLYMPUS D.ZUIKO (4/3“)–––•Fixed aperture

17-55mm F2.8
18-140mm F3.5-5.6
55-200mm F4.0-5.6

Fixed aperture

CANON EF (35mm)–––•

17-55mm F2.8
18-135mm F3.5-5.6
55-250mm F4.0-5.6

DX NIKKOR (APS-C)–––•

Fixed aperture

CANON EF-S (APS-C)–––•

14-42mm F3.5-5.6
12-60mm F2.8-4.0
35-100mm F2.0

24-70mm F2.8
70-200mm F2.8
100-400mm F4.5-5.6

Fixed aperture
Fixed aperture

FX NIKKOR (35mm)–––

24-70mm F2.8
70-200mm F2.8
80-400mm F4.5-5.6

Fixed aperture
Fixed aperture52

Tendency of Actual Lenses• Lower magnification zoom
• More expensive “Brighter lens”

Minimum F-numbervaries a little• Higher magnification zoom
• Less expensive “Darker lens”

Minimum F-numbervaries a lot

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Conclusion

Conclusion• Actual lenses have various aberrations– Many solutions correct aberrations– Aberrations cannot be completely corrected• Residual aberrations give bokeh its character• Bokeh is rich in variety– Different corrections show different representations– Color fringes and gradation vary between front and back bokeh– Conspicuousness: smaller out-of-focus > larger out of focus
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Conclusion (cont’d)• Actual optical system is composed of multiple lenses
in order to:– Correct aberrations– Zoom– Reduce focus breathing• Many phenomena do not conform to single lens rules– Different focus breathing– Different zooming aperture varying

by different mechanisms

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References
••
•
••
•

Kawase, M. “Camera, Optics Theory and Post Effects for Renderists.” Computer Entertainment Developers
Conference, 2007.
Kawase, M. “Optics Knowledge to Achieve Attractive Images.” Computer Entertainment Developers
Conference, 2010.
Trávník, J. “On Bokeh.” Jakub Trávník's resources. http://jtra.cz/stuff/essays/bokeh/index.html
安藤幸司 『光と光の記録「レンズ編」』 AnfoWorld http://www.anfoworld.com/LensMF.html
吉田正太郎（1997）『カメラマンのための写真レンズの科学』地人書館.
永田信一（2002）『図解 レンズがわかる本』日本実業出版社.

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