References Photography, B. London and J. Upton Optics in Photography, R. Kingslake The Camera, The Negative, The Print, A. Adams

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1 Page 1 Camera Simulation Eect Cause Field o view Depth o ield Motion blur Exposure Film size, stops and pupils Aperture, ocal length Shutter Film speed, aperture, shutter Reerences Photography, B. London and J. Upton Optics in Photography, R. Kingslake The Camera, The Negative, The Print, A. Adams Topics Ray tracing lenses Focus Field o view Depth o ocus / depth o ield Exposure

2 Page 2 Lenses Reraction Snell s Law n sin I = nsin I N I n n I

3 Page 3 Paraxial Approximation sinu u e 0 tanu u U Rays deviate only slightly rom the axis z Incident Ray Angles: ccw is positive; cw is negative I U φ I = U φ The sum o the interior angles is equal to the exterior angle.

4 Page 4 Reracted Ray φ I U I = U φ ( φ) = I + ( U ) Derivation Paraxial approximation I= U φ i= u φ I = U φ i = u φ

5 Page 5 Derivation Paraxial approximation I= U φ i= u φ I = U φ i = u φ Snell s Law n sin I = nsin I ni = ni n ( u φ) = n( u φ) Ray Coordinates u h = z u = h z φ = h R u h R φ u z z

6 Page 6 Gauss Formula Paraxial approximation to Snell s Law n ( u φ) = n( u φ) Ray coordinates h u = z φ = h R Thin lens equation h h h h n ( ) = n( ) z R z R n n ( n n) = + z z R u = h z Holds or any height, any ray! Vergence Diverging Converging V < 0 V = 0 V > 0 Vergence Thin lens equation Surace Power equation n n 1 V = diopters r z m V = V + P 1 P ( n n) R

7 Page 7 Lens-makers Formula Reractive Power P = ( n n) = R R 1 2 Converging Diverging Conjugate Points = + z z To ocus: move lens relative to backplane Horizontal rays converge on ocal point in the ocal plane

8 Page 8 Gauss Ray Tracing Construction Parallel Ray Focal Ray Chie Ray Object Image Ray Tracing: Finite Aperture Focal Plane Aperture Plane Back Plane

9 Page 9 Real Lens Cutaway section o a Vivitar Series 1 90mm /2.5 lens Cover photo, Kingslake, Optics in Photography Double Gauss Data rom W. Smith, Modern Lens Design, p 312 Radius (mm) Thick (mm) n d V-no aperture

10 Page 10 Ray Tracing Through Lenses 200 mm telephoto 35 mm wide-angle 50 mm double-gauss 16 mm isheye From Kolb, Mitchell and Hanrahan (1995) Thick Lenses Equivalent Lens Reraction occurs at the principal planes

11 Page 11 Field o View Field o View From London and Upton

12 Page 12 Field o View From London and Upton Field o View Field o view Redrawn rom Kingslake, Optics in Photography ov tan 2 = ilmsize Types o lenses Normal 26º Film diagonal ocal length Wide-angle 75-90º Narrow-angle 10º

13 Page 13 Perspective Transormation Thin lens equation z = + z = z z z+ x x = z + y y = z + Represent transormation as a 4x4 matrix Depth o Field

14 Page 14 Depth o Field From London and Upton Circle o Conusion s z z s d a c Focal Plane Back Plane Circle o conusion proportional to the size o the aperture c d s z = = a z z

15 Page 15 Depth o Focus [Image Space] Depth o ocus Equal circles o conusion s a s Two planes: near and ar z z c d s z = = a z z c d z n z n c dn zn s = = a z z n n c d n Depth o Focus [Image Space] Depth o ocus Equal circles o conusion s a s z z c d s z 1 1 c = = = 1 + a z z z s a c d z n z n c dn zn s 1 1 c = = = 1 a z z z s a n n n c d n

16 Page 16 Depth o Focus [Image Space] Depth o ocus Equal circles o conusion 1 1 c = 1 + z s a 1 1 c = 1 z s a n = 2 1 z z s n z z n s c 1 1 = 2 c 1 z z a s n Depth o Field [Object Space] Depth o ield Equal circles o conusion 1 = s s 1 = z z n n 1 = z z z c = 2 z z s n z n 1 1 2c 1 1 2c 1 = z z a s a n c

17 Page 17 Hyperocal Distance = 2 z z s n 1 1 2c 1 cn 1 = = z z a H n a N z c When H s H zn =, z = 2 z n c H is the hyperocal distance Depth o Field Scale

18 Page 18 Factors Aecting DOF From 1 cn = H 2 Resolving Power Diraction limit c = 1.22 λ = µ m=0.040 mm a [ ] 35mm ilm (Leica standard) c = 0.025mm CCD/CMOS pixel aperture c = mm (Nikon D1)

19 Page 19 Exposure Image Irradiance a π a = θ ω = π θ = 4 Ω 2 E Lcos d L sin L 2

20 Page 20 Relative Aperture or F-Stop a a = N F-Number and exposure: E π 1 L 4 N = 2 Fstops: stop doubles exposure Camera Exposure Exposure H = E T Exposure overdetermined Aperture: -stop - 1 stop doubles H Decreases depth o ield Shutter: Doubling the open time doubles H Increases motion blur

21 Page 21 Aperture vs Shutter /16 1/8s /4 1/125s From London and Upton /2 1/500s High Dynamic Range Sixteen photographs o the Stanord Memorial Church taken at 1-stop increments rom 30s to 1/1000s. From Debevec and Malik, High dynamic range photographs.

22 Page 22 Simulated Photograph Adaptive histogram With glare, contrast, blur Camera Simulation L(, xω,, t λ) Ω A L( x, ω, t, λ) R= P( x, λ) S( x, ω, t) LT ( ( x, ω, λ), t, λ) da( x ) dω dtdλ AΩT Λ Sensor response Lens Shutter Scene radiance Px (, λ) (, xω) = T( x, ω, λ) Sx (, ω, t) Lx (, ω,, tλ)