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

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2 Camera Simulation Effect Cause Field of view Film size, focal length Depth of field Aperture, focal length Exposure Film speed, aperture, shutter Motion blur Shutter References Photography, B. London and J. Upton Optics in Photography, R. Kingslake The Camera, The Negative, The Print, A. Adams

3 Cameras (5D integral) R L( x, ω, t) P( x) S( t) cosθ da dω dt = T Ω A Motion Blur Depth of Field Cook, Porter, Carpenter, 1984 Mitchell, 1991

4 Why is Area Better than Solid Angle? Solid Angle 100 shadow rays Area 100 shadow rays

5 Topics Lenses Depth of field Exposure

6 Lenses

7 Thin Lens Demonstration

8 Lenses Focus! Rays from a point in object space intersect at a point in image space These are conjugate points Points at infinity converge at the focal point or focus Points on different object planes focus on different image planes The position of the point of focus can be changed by moving the lens

9 Gauss Ray Tracing Construction Parallel Ray Focal Point Focal Ray Chief Ray Object Image

10 Snell s Law Î ˆN θ i ϕi ϕt θ t ˆT ϕ = ϕ ± π t i n sinθ = n sinθ i i t t n i Nˆ Iˆ = n Nˆ Tˆ t

11 Law of Refraction Î ˆN Tˆ = µ Iˆ + γ Nˆ θ i ˆ ˆ ˆ T = 1 = µ + γ + 2µγ I N µ = n / n i t θ t ˆT { ( ( ) 2 )} γ = µ Iˆ Nˆ ± µ Iˆ Nˆ 1 2 Total internal reflection: ( I N) µ (1 ˆ ˆ ) < 0 { 2 2 } = µ cosθ ± 1 µ sin θ = µ cosθ ± cosθ = µ cosθ cosθ i i i t t i 1 2 γ = µ 1

12 Optical Manhole Total internal reflection n w = 4 3 From Livingston and Lynch

13 Refraction at a Spherical Lens n n! R n : index of refraction

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

15 Angle of Refraction Opposite angles are equals

16 Snell s Law I n n! I!

17 Snell s Law I n n! I!

18 Paraxial Approximation: Small Angles I n n! I! Rays deviate only slightly from the axis

19 Paraxial Approximation: Small Angles I n n! I!

20 Angles to Slopes u h R z!

21 Gauss Formula Paraxial approximation to Snell s Law Ray coordinates h h h h n! ( ) = n( ) z! R z R n! n ( n! n) = + z! z R Holds for any height, any ray!

22 Perspective Transformation How does z transform? fz = + z" = z" z f z + f fx x" = z + f fy y" = z + f Represent lens transformation as a 4x4 matrix

23 Ray Tracing: Finite Aperture 1.Pick a point on image plane x 2.Pick a point on the lens u 3.Transform x to x; form ray (u,x-u) x u x! Focal Plane Aperture Plane Image Plane

24 Real Lens Cutaway section of a Vivitar Series 1 90mm f/2.5 lens Cover photo, Kingslake, Optics in Photography

25 Double Gauss Data from W. Smith, Modern Lens Design, p 312 Radius Thick nd V-no aperture

26 Ray Tracing Through Lenses 200 mm telephoto 35 mm wide-angle 50 mm double-gauss 16 mm fisheye From Kolb, Mitchell and Hanrahan (1995)

27 Depth of Field

28 Depth of Field From London and Upton

29 Circle of Confusion s z z! s! d! a c Focal Plane Back Plane Circle of confusion proportional to the size of the aperture c d! s! z! = = a z! z!

30 Resolving Power Diffraction limit c f = 1.22 λ = µ m=0.040 mm a [ ] 35mm film (Leica standard) c = 0.025mm CCD/CMOS pixel aperture c = mm (Nikon D1)

31 Exposure

32 Image Irradiance a f π % a & = = = ' ( 4 f Ω ) * 2 E L cosθ dω Lπ sin θ L 2

33 Uniform Disk Source Geometric Derivation Algebraic Derivation r Ω! = cosα 2π 1 0 cosθ dφ d cosθ h sinα 2 Ω! = π sin α α = = = 2π π π r cos 2 sin 2 2 α 2 r + h θ 2 2 cosα 1

34 Relative Aperture or F-Stop a f a = f N F-Number and exposure: E π 4 1 N = L 2 Fstops: stop doubles exposure

35 Camera Exposure Exposure H = E T Exposure overdetermined Aperture: f-stop - 1 stop doubles H Decreases depth of field Shutter: Doubling the open time doubles H Increases motion blur

36 Aperture vs Shutter Constant Exposure f/16 1/8s f/4 1/125s f/2 1/500s From London and Upton CS348b Lecture 7 Pat Hanrahan, Spring 2016

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