Lecture #7

1. Lecture #7 What makes a good eye? Part I. Resolution Land and Nilsson first half of chapter 3 2/14/13

2. Two key features of an eye • Resolution (today) • Precision with which eye can resolve or distinguish a spatial image • Sensitivity (next time) • How much light can be detected by the eye

3. Eye design • Optical devices are subject to laws of physics and optimized by the engineers who design them • Eyes are subject to laws of physics and optimized by evolution

4. Camera - transfers object image to “film”

5. Image resolution depends on pixel density

6. Vertebrate eye - single lens which transfers image to retina

7. Spatial resolution depends on cone spacing Wolfe et al Sensory Perception fig 2.9

8. Human retinal mosaic in fovea JW temporal nasal AN nasal Huge variation from person to person in distribution of cones and in M/L cone ratios! Roorda and Williams 1999

9. Optimizing resolution • Receptor size and spacing • Lens focal length • Optical imperfections

10. You Are Here

11. v v

12. v

13. Low Resolution High Resolution Why does this even exist?

15. Resolution - seeing one point of light receptor

16. Reasons that points of light might end up as blurred points of light • Lens does not focus perfectly • Aberration = imperfection • Cornea, lens or humours scatter some of light • Diffraction • When light goes through an aperture, it gets diffracted

17. Resolution

18. Optimal receptor spacing to resolve objects • Need enough photoreceptors to define boundaries between objects

19. Optimal receptor spacing • If receptors are too big (spaced too coarsely) they can’t resolve fine detail

20. Want optimal receptor spacing • No reason to have more receptors than this as they won’t resolve the images any better

21. Refraction: Light bends at interface Bigger incident angle = more bending Less dense More dense

22. Lens focal length, f • Focal length is distance at which parallel light is focused f

23. Lens focal length • Determined by curvature of lens surface Lens radius of curvature Curvature as if cut lens out of surface of sphere with that radius

24. Lens focal length • Determined by curvature of lens surface Radius of curvature of sphere that matches lens’s curvature

25. Lens focal length is determined by curvature of lens surfaces Large radius of curvature = large focal length Small radius of curvature = small focal length

26. Calculate focal length from lens makers equation

27. Lens focal length • Determined by curvature of lens surface For thin lens

28. Lens focal length How can we make focal length longer? If lens is symmetric, R1=-R2=R If lens is flat on one side, R2=∞

29. PhET lens simulator

30. Physics lens equation f Distance of image and object in relation to lens focal length Object Image dO di

31. So di = f Physics lens equation If dO, object distance is very large dO>>di O I dO di

32. Principal rays and nodal point dOdi

33. Note: both cornea and lens focus the light so retina is located at combined focal length of lens and cornea Retina

34. Eye resolution Determine resolution based on how well can resolve a periodic pattern or grating One period of grating

35. Quantifying resolution • Minimum resolvable grating will match receptor spacing • If receptor spacing is s then grating covers 2s on retina • Angle = 2s / f Angle f 2s

36. Angle s f One cycle of grating covers two photoreceptors • Angle = 2s / f • So can resolve one cycle if it covers no less than that angle • Max resolvable spatial frequency • v = 1 / angle of one grating cycle = f / 2s

37. Angles grating retina θ Tanθ= 2s / f But for small θ tanθ=θ So θ= 2s/f 2s f θ is in radians!

38. Angle θ s f Interreceptor angle • Angle between two adjacent photoreceptors • θ= s / f

39. One cycle of grating covers two photoreceptors • To cover full cycle of grating the angle is 2θ= 2s / f • Maximum resolvable spatial frequency, v • v = 1/2θ= f/ 2s Angle 2θ s f

40. Δρ f d Minimum receptor acceptance angle • Another angle: • Δρ minimum receptor acceptance angle • f eye focal length = distance to retina • d receptor diameter • Resolution =1/ Δρ= f/d In general s ≈ d and 2s/f ≈ 2d/f

41. Δρ Angle 2θ=2s/f s f f d Two ways to characterize eye resolution • Maximum resolvable spatial frequency, v = f/2s Inter-receptor spacing, Δρ = d/f Grating resolution = f / 2s Receptor resol = 1/Δρ= f/d = f /s

42. Some typical spatial frequencies - Vertebrate eyes

43. Some typical spatial frequencies - Vertebrate eyes Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad

44. Some typical spatial frequencies - Vertebrate eyes Δρ= 0.0036º * π / 180 º = 6.28 x 10-5 rad v = 1 / 2Δρ = 1 / (2*6.28 x 10-5) = 7960 cycles / rad

45. WHAT ABOUT ME?

46. How can we modify an eye to get high acuity night vision?

47. Chicken Owl

48. Image size relative to object size: Similar triangles • f is focal length of the eye • O = object size • I = image size • U=distance to object • O / U= I / f f

49. Image size relative to object size: Similar triangles O • O / U= I / f • f is focal length of the eye • O = object size • I = image size • U=distance to object Magnification = I / O = f / U U f I

50. PhET lens simulator