Image Formation in Man and Machines. Image formation - 1

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1 Image Formation in Man and Machines Image ormation - 1

2 Overview v Pinhole camera v Reraction o light v Thin-lens equation v Optical power and accommodation v Image irradiance and scene radiance v Human eye v Geometry o perspective imaging Image ormation - 2

3 Lens-less Imaging Systems - Pinhole Optics v Pinhole optics projects images without lens Scene Pinhole Image with ininite depth o ield v Smaller the pinhole better the ocus Blurred Image less the light energy rom any single point v Good or tracking solar eclipses Image ormation - 3

4 Diraction v Two disadvantages to pinhole systems Low light collecting power diraction v Diraction θ Light bends as it passes by the edge o a narrow aperture v Human vision at high light levels, pupil (aperture) is small and blurring is due to diraction at low light levels, pupil is open and blurring is due to lens imperections λ D D Image ormation - 4

5 Diraction and pinhole optics Image ormation - 5

6 Lenses Collect More Light v With a lens, diverging rays rom a scene point are converged back to an image point Scene Image plane Lens Center o projection Image ormation - 6

7 Reraction: Snell s law relected ray v I θ is the angle o incidence and θ is the angle o reraction then n sinθ n'sinθ where n and n are the reractive indices o the two media v Reractive index is the ratio o speed o light in a vacuum to speed o light in the medium ' θ reracted ray incident ray reracted ray Reractive indices glass water air Image ormation - 7

8 Lens Equations Image ormation - 8

9 Thin-Lens Equation v Thin-lens equation relates the distance between the scene point being viewed and the lens to the distance between the lens and the point s image (where the rays rom that point are brought into ocus by the lens) Let M be a point being viewed u u p is the distance o M rom the lens along the optical axis The thin lens ocuses all the rays rom M onto the same point, the image point m at distance q rom the lens. M Q q- H F s S O h p q m Image ormation - 9

10 Thin-Lens Equation 1 p + 1 q 1 v m can be determined by intersecting two known rays MQ is parallel to the optical axis, so it must be reracted to pass through F. MO passes through the lens center, so it is not bent. v Note two pairs o similar triangles MSO and Osm (yellow) OQF and Fsm (green) Image ormation - 10 H p S M H h q p H + h p + q H h H + q q h Q O q - F s h q m Divide 2 equations: p p + q q 1 p + q p q

11 Thin-Lens Equation 1 p + 1 q 1 v Notice that the distance behind the lens, q, at which a point, M, is brought into ocus depends on p, the distance o that point rom the lens amiliar to us rom rotating the ocus ring o any camera M Q q- H F s S O h p q m Image ormation - 11

12 Thin-Lens Equation 1 p + 1 q 1 M M Q S S O F m p q m v As p gets large, q approaches v As q approaches, p approaches ininity Image ormation - 12

13 Optical Power and Accommodation v Optical power o a lens - how strongly the lens bends the incoming rays Short ocal length lens bends rays signiicantly It images a point source at ininity (large p) at distance behind the lens. The smaller, the more the rays must be bent to bring them into ocus sooner. Optical power is 1/, with measured in meters. The unit is called the diopter Human vision: when viewing araway objects the distance rom the lens to the retina is 0.017m. So the optical power o the eye is 58.8 diopters Image ormation - 13

14 M Accommodation v How does the human eye bring nearby points into ocus on the retina? by increasing the power o the lens muscles attached to the lens change its shape to change the lens power accommodation: adjusting the ocal length o the lens bringing points that are nearby into ocus causes araway points to go out o ocus depth-o-ield: range o distances in ocus M S S O m m p F F q Image ormation - 14

15 Accommodation Image ormation - 15

16 Accommodation Sources at > 1 meter are imaged at same distance Sources closer than 1 m are imaged at dierent distances Image ormation - 16

17 Accommodation M v Physical cameras - mechanically change the distance between the lens and the image plane M F S S O m m p q Image ormation - 17

18 Image ormation - 18 Pixel Brightness and Scene Brightness 2 2 ) cos / ( cos cos / ( cos α θ α α Z da da ) 2 cos cos Z da da θ α θ α π cos cos Z D LdA dp θ α π cos cos Z D da da L da dp E D da Ω θ α L D E α π 4 2 cos 4 da dp LdA Ωcosθ Z

19 Image Irradiance and Scene Radiance E 2 π 4 4 v Image irradiance E is proportional to scene radiance v Brighter scene points produce brighter pixels D cos v Image irradiance is proportional to inverse o square o -number (/D), is larger or small -number α L Image ormation - 19

20 Retina Image ormation - 20 Light

21 Light Retina v Limitations o human vision Blood vessels and other cells in ront o photoreceptors shadows cast on photoreceptors non-uniorm brightness Image ormation - 21

22 Photoreceptor Mosaics v The retina is covered with a mosaic o photoreceptors v Two dierent types o photoreceptors Image ormation - 22 Rods - approximately 100,000,000 Cones - approximately 5,000,000 v Rods Sensitive to low levels o light: scotopic light levels v Cones Sensitive to higher levels o light: photopic light levels v Mesopic light levels - both rods and cones active

23 Photoreceptor Mosaics v (A) Cones in the Fovea. Several neurons per cone v (B) Cones and Rods in the periphery v Rods are small but several rods per neuron Image ormation - 23

24 Photoreceptor Mosaics v Fovea is area o highest concentration o photoreceptors ovea contains no rods, just cones approximately 50,000 cones in the ovea cannot see dim light sources (like stars) when we look straight at them! v TV camera photoreceptor mosaics nearly square mosaic o approximately 800X640 elements or complete ield o view Image ormation - 24

25 The Human Eye v Limitations o human vision the image is upside-down! high resolution vision only in the ovea u only one small ovea in man u other animals (birds, cheetas) have dierent oveal organizations blind spot Image ormation - 25

26 v Blind Spot Close let eye v Look steadily at white cross v Move head slowly toward and away rom igure v At a particular head position, the white disk disappears completely rom view. Image ormation - 26

27 Cones and color v There are three dierent types o cones they dier in their sensitivity to dierent wavelengths o light Image ormation - 27

28 Color Cameras v Two types o color cameras Single CCD array u in ront o each CCD element is a ilter - red, green or blue u color values at each pixel are obtained by hardware interpolation subject to artiacts lower intensity quality than a monochromatic camera 3 CCD arrays packed together, each sensitive to dierent wavelengths o light Image ormation - 28

29 3-CCD Camera Image ormation - 29

30 Cones, CCD s and space v How much o the world does a cone see? measured in terms o visual angle the eye lens collects light over a total ield o view o about 100 o each cone collects light over a visual angle o about 8.5 x 10-3 degrees (about 30 seconds) v How much o the world does a single camera CCD see example: 30 o lens 30/500 gives about 6 x10-2 degrees per CCD Eye s acuity is 10 times higher. Image ormation - 30

31 Perspective Imaging - Pinhole Camera Model Image center Image plane y C x Line o sight z s Scene point (x s, y s, z s ) z Image point (x i, y i, -) pinhole at the center o projection v The point on the image plane that corresponds to a particular point in the scene is ound by ollowing the line that passes through the scene point and the center o projection Image ormation - 31

32 Perspective ImagingCentral Projection v Line o sight to a point in the scene is the line through the center o projection to that point v Image plane is parallel to the x-y plane distance to image plane is -ocal length this inverts the image move the image plane in ront o the center o projection Image ormation - 32

33 Perspective Imaging Image point (x i, y i, -) v Fundamental equations or perspective projection onto a plane Image ormation - 33 Image plane center o projection y C x Image point (x i, y i, ) x y i i Scene point (x s, y s, z s ) x z s s y z s s z

34 Field o View v As gets smaller, image becomes more wide angle (more world points project onto the inite image plane) v As gets larger, image becomes more telescopic (smaller part o the world projects onto the inite image plane) Field o view Image ormation - 34

35 Vanishing Points and Lines v We are looking at a scene on a plane v To each point M on the scene plane we associate an image point m v The line, v, o the image plane that belongs to the plane through C parallel to the scene plane is called the vanishing line, or the horizon line C v m m M Image plane Scene plane Image ormation - 35

36 Vanishing Points and Lines v The images o the points on rail L 1 belong to a plane P 1 deined by C and L 1 The lines o sight or each point o the track L 1 lie on this plane v The image o rail L 2 belongs to a plane P 2 deined by C and track v v v v v v The image l 1 o L 1 belongs to intersection o plane P 1 and image plane The image l 2 o L 2 belongs to intersection o plane P 1 and image plane Planes P 1 and P 2 intersect along line CV parallel to L 1 and L 2 Point V belongs to P 1, P 2 and image plane. Thereore, l 1 and l 2 intersect in V V is image o a scene point o L 1 and a scene point o L 2. Since CV is parallel to L 1 and L 2, these points are at ininity. Image ormation - 36 C V L m 1 l 2 V v l 1 Image plane Scene plane L 2

37 From Perspective Projection to Orthographic Projection v Select origin o coordinate system at image center v World coordinates are independent o ocal length v β 1/ v When β 0, orthographic projection Image ormation - 37 C x y i i O (x i, y i, 0) y x s + y + s z z x s s Scene point (x s, y s, z s ) x y z s β s β z z s s

38 Orthographic Projection C O (x i, y i, 0) y x Scene point (x s, y s, z s ) z 1/ 0 Image plane x y i i x s y s Image ormation - 38

39 Depth o Field and -number v Depth o ield is smaller or small -number M F S m Image ormation - 39

40 Lens Imperections v Lens imperections might cause rays not to intersect at a point Deviations in shape rom the ideal lens Material imperections that might cause the reractive index to vary within the lens Image ormation - 40

41 Reraction o Color v Why does the prism separate the light into its spectral components? Prism bends dierent wavelengths o light by dierent amounts u Reractive index is a unction o wavelength u Shorter wavelengths are reracted more strongly than longer wavelengths Image ormation - 41

42 Chromatic Aberration v Chromatic aberration Dierent wavelengths o light rom the same point source are ocused at dierent distances behind the lens When incident light is a mixture o wavelengths, we can observe a chromatic ringe at edges Accommodation can bring any wavelength into good ocus, but not all simultaneously Human visual system has other mechanisms or reducing chromatic aberration Color cameras have similar problems Image ormation - 42

43 Chromatic Aberration Image ormation - 43

44 Compound Eyes Many (small) animals have compound eyes - each photoreceptor has its own lens - images seen by these eyes are equally sharp in all directions -examples: lies and other insects But these eyes do not scale well biologically Image ormation - 44

45 Reerences v Foundations o Vision, Brian Wandell, Sinauer Associates, Sunderland MA, 1995 v Introductory Techniques or 3-D Computer Vision, E. Trucco and A. Verri, Prentice-Hall, pp v Computer Vision, Ballard & Brown, Prentice- Hall, pp , pp v Robot Vision, B.K.P. Horn, MIT Press, pp v A Guided Tour o Computer Vision, V. Nalwa, AT&T Press, pp Image ormation - 45