Lecture 21: Cameras & Lenses II. Computer Graphics and Imaging UC Berkeley CS184/284A

Transcription

1 Lecture 21: Cameras & Lenses II Computer Graphics and Imaging UC Berkeley

2 Real Lens Designs Are Highly Complex [Apple] Topic o next lecture

3 Real Lens Elements Are Not Ideal Aberrations Real plano-convex lens (spherical surace shape). Lens does not converge rays to a point anywhere. More discussion next lecture

4 Today: Thin Lens Approximation

5 Ideal Thin Lens Focal Point Focal Point Credit: Karen Watson Focal Length Assume all parallel rays entering a lens pass through its ocal point.

6 Lens Focusing Conjugate Points Rays rom a point in object space intersect at a point in image space These are called conjugate points We create images ocused at a desired depth by placing a sensor at the conjugate distance Focusing involves changing the depth between the lens and sensor Object space Image space Question: what is the relationship between the position o a lens conjugate points?

7 Gauss Ray Diagrams

8 Gauss Ray Tracing Construction Parallel Ray Chie Ray Focal Ray Object Image

9 Gauss Ray Tracing Construction z o z i What is the relationship between conjugate depths z o,z i?

10 Gauss Ray Tracing Construction h o h o z i z o h i h i z o h o = h i h o = z i h i

11 Gauss Ray Tracing Construction z o h o = h i h o = z i h i h o h i = z o h o h i = z i z o = z i Object / image heights actor out - applies to all rays (z o )(z i )= 2 Newtonian Thin Lens Equation z o z i (z o + z i ) + 2 = 2 z o z i =(z o + z i ) 1 Gaussian Thin Lens Equation = 1 z i + 1 z o

12 The Thin Lens Equation z o z i 1 = 1 z i + 1 z o

13 Changing the Focus Distance 1 = 1 z i + 1 z o To ocus on objects at dierent distances, move the sensor Sensor relative to the lens For z i < z o the object is larger than the image At z i = z o we have 1:1 macro imaging For z i > z o the image is larger than the object (magniied) Can t ocus on objects closer than the lens ocal length

14 Magniication h o h i z o z i m = h i h o = z i z o

15 Magniication Example Focus at Ininity 1 = 1 z i + 1 z o m = z i z o I ocused on a distant mountain z o, so z i = sensor at ocal point magniication 0

16 Magniication Example Focus at 1:1 Macro 1 = 1 z i + 1 z o m = z i z o What coniguration do we need to achieve a magniication o 1 (i.e. image and object the same size, a.k.a. 1:1 macro)? Need z i = z o, so z i = z o = 2 - sensor at twice ocal length In 1:1 imaging, i the sensor is 36 mm wide, an object 36 mm wide will ill the rame

17 Thin Lens Demonstration

18 Thin Lens Demonstration Observations 3D image o object is: Compressed in depth or low magniication 1:1 in 3D or unit magniication Stretched in depth or high magniication

19 Lens Perorms a 3D Perspective Transorm Lenses transorm a 3D object to a 3D image; the sensor extracts a 2D slice rom that image As an object moves linearly (in Z), its image moves non-proportionally (in Z). And vice versa. As you change ocus o a camera, the image changes size!

20 Deocus Blur

21 Circle o Conusion

22 Circle o Conusion Further deocused point light Closer deocused point light

23 Circle o Conusion Deocus blur kernel or objects at this depth Deocus blur kernel or objects at this depth Size o blur kernel depends on depth rom ocal plane. Only see the blur kernel itsel i you have a point light. Why?

24 Circle o Conusion

25 Computing Circle o Conusion Diameter (C) z 0 s z s z o z i d! A C Object Focal Plane Image Sensor Plane Circle o conusion is proportional to the size o the aperture C A = d0 = z s z i z i z i

26 Deinition: F-Number (a.k.a. F-Stop) The F-Number o a lens is deined as the ocal length divided by the diameter o 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 -stop o 2 is sometimes written /2, relecting the act that the absolute aperture diameter (A) can be computed by dividing ocal length () by the relative aperture (N).

27 Example F-Stop Calculations D = 50 mm = 100 mm N = /D =2 D = 100 mm = 200 mm N = /D =2 D = 100 mm = 400 mm N = /D =4

28 Circle o Conusion is Inversely Proportional to F-Stop R. Berdan, canadiannaturephotographer.com C = A z s z i z i = N z s z i z i

29 Circle o Conusion Example 50mm /2 lens Full rame sensor (36x24mm) Focus: 1 meter Background: 10 meter Foreground: 0.3 meter A = 50mm/2 = 25mm 1 z s = 1/50 1/ mm 1 Background: z i = 1/50 1/10, mm C = A z s z i /z i =1.18mm 1 Foreground: z i = 1/50 1/ mm C = A z s z i /z i =3.07mm C = A z s z i z i ~65 pixels on HD TV ~169 pixels on HD TV

30 Circle o Conusion in Perspective Composition To maintain ield o view on subject, increase distance 16 mm (110 ) rom subject by same actor as ocal length 200 mm (12 ) (approx). What is the increase in background blur?

31 Circle o Conusion in Perspective Composition For subject at distance Z, 1 1 To maintain image size o subject when z s = 1 Z Distant background means z i =, changing zoom, increase distance C = N z s z i = N 1 1/ 1/Z =... rom subject z i by same actor as = N ocal length Z (approx). What is the increase in background blur? I we increase Z and by actor K, circle o conusion C also increases by K. (F-stop held constant)

32 Circle o Conusion in Perspective Composition 100mm, /4 138px 28mm, /4 40px From Paul van Walree, toothwalker.org/do.html As predicted, 100mm 28mm 138px 40px, but notice blur is constant relative to background object itsel!

33 Ray Tracing Ideal Thin Lenses

34 Examples o Renderings with Lens Focus Pharr and Humphreys

35 Ray Tracing or Deocus Blur (Thin Lens) x x x Sensor Subject plane z o z i Setup: Choose sensor size, lens ocal length and aperture size Choose depth o subject o interest z o Calculate corresponding depth o sensor z i rom thin lens equation (ocusing)

36 Ray Tracing or Deocus Blur (Thin Lens) x x x Sensor Subject plane z o z i To compute value o pixel at position x by Monte Carlo integration: Select random points x on lens plane Rays pass rom point x on image plane z i through points x on lens Each ray passes through conjugate point x on the plane o ocus z o Can determine x rom Gauss ray diagram So just trace ray rom x to x Estimate radiance on rays using path-tracing, and sum over all points x

37 Examples o Renderings with Lens Focus Pharr and Humphreys

38 Example o Rendering with Lens Focus Credit: Bertrand Benoit. Sweet Feast, [Blender /VRay]

39 Example o Rendering with Lens Focus Credit: Giuseppe Albergo. Colibri [Blender]

40 Acknowledgments Many thanks to Marc Levoy, who created many o these slides, and Pat Hanrahan. London, Stone, and Upton, Photography (9th ed.), Prentice Hall, Peterson, Understanding Exposure, AMPHOTO The Slow Mo Guys bobatkins.com Hari Subramanyan Canon EF Lens Work III