Lenses. Overview. Terminology. The pinhole camera. Pinhole camera Lenses Principles of operation Limitations

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Kristopher Stewart
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1 Overview Pinhole camera Principles of operation Limitations 1 Terminology The pinhole camera The first camera - camera obscura - known to Aristotle. In 3D, we can visualize the blur induced by the pinhole (a.k.a., aperture): Q: How would we reduce blur? 3 4

2 Shrinking the pinhole Shrinking the pinhole, cont d Q: What happens as we continue to shrink the aperture? 5 6 focus a bundle of rays to one point => can have larger aperture. A lens images a bundle of parallel rays to a focal point at a stance, f, beyond the plane of the lens. An aperture of ameter, D, restricts the extent of the bundle of refracted rays. Note: f is a function of the index of refraction of the lens. 7 8

3 Carnal points of a lens system For economical manufacture, lens surfaces are usually spherical. A spherical lens behaves ideally if we consider rays near the optical axis -- paraxial rays. Most cameras do not consist of a single thin lens. Rather, they contain multiple lenses, some thick. A system of lenses can be treated as a black box characterized by its carnal points. For a thin lens, we ignore lens thickness, and the paraxial approximation leads to the familiar Gaussian lens formula:!! + " $ "# =!!! " $ "# Magnification = 9 10 Carnal points of a lens system Carnal points of a lens system The focal points, principal points, and principal planes (well, surfaces actually) describe the paths of rays parallel to the optical axis. In a well-engineered lens system:! The principle planes are planar! The nodal and principal points are the same The nodal points describe the paths of rays that are not refracted, but are translated down the optical axis. The system still obeys Gauss s law, but all stances are now relative to the principal planes. The principal and nodal points are, together, called the carnal points. 11 1

4 Limitations of lens systems Depth of field exhibit a number of deviations from ideal. We ll consider a number of these deviations:! Depth of field! Primary (third order, Seidel) aberrations Distortion! Chromatic aberration! Flare! Vignetting Points that are not in the object plane will appear out of focus. The depth of field is a measure of how far from the object plane points can be before appearing too blurry Non-paraxial imaging Third order aberrations When we violate the paraxial assumption, we find that real imaging systems exhibit a number of imperfections. We can set up the geometry of a rotationally symmetric lens system in terms of an object, aperture, and image: The first set of non-ideal terms beyond perfect imaging and depth of field form the basis for the third order theory. Deviations from ideal optics are called the primary or Seidel aberrations: Smith 1996! Spherical aberration! Coma! Astigmatism! Petzval curvature! Distortion All of these aberrations can be reduced by stopping down the aperture, except stortion. We can then perform a Taylor series of the mapping from rays to image points: 15 16

5 Distortion Distortion Distortion follows the form (replacing h with r): r = a r+ a r + a r +! Sometimes this is re-written as: r = r ( a + a r + a r +! ) The effect is that non-raal lines curve out (barrel) or curve in (pin cushion). No stortion Pin cushion Barrel Chromatic aberration Flare Cause: Index of refraction varies with wavelength. Effect: Focus shifts with color, colored fringes on highlights Ways of improving: Achromatic designs Light rays refract and reflect at the interfaces between air and the lens. The stray light is not focused at the desired point in the image, resulting in ghosts or haziness, a phenomenon known as lens flare. 19 0

vignetting is best explained in raometric terms. A sensor responds to irraance (power per unit area) which is defined in terms of raance as:! optical vignetting!

6 Optical coatings Single vs. multiple coatings Optical coatings are tuned to cancel out reflections at certain angles and wavelengths. Burke 1996 Single coating Mutliple coatings 1 Vignetting Optical vignetting Light rays oblique to the lens will deliver less power per unit area (irraance) due to: Optical vignetting is best explained in raometric terms. A sensor responds to irraance (power per unit area) which is defined in terms of raance as:! optical vignetting! mechanical vignetting de = L cosθ dω E = L cosθ dω Result: darkening at the edges of the image. H For a given image plane and exit pupil: dω = dap r = cosθ da cos3 θ da = ( / cosθ ) L cos 4 θ da de = Thus: E!L 3 A cos 4! 4

Optical vignetting, cont d Mechanical vignetting We can rewrite this in terms of the ameter of the exit pupil: A π( D/) π D = = 4 Occlusion by apertures and lens extents results in mechanical

5 6 Lens design Bibilography Lens design is a complex optimization process that trades off:! achromatic aberrations! chromatic aberrations! field of view! aperture (speed)! stortions!

7 Optical vignetting, cont d Mechanical vignetting We can rewrite this in terms of the ameter of the exit pupil: A π( D/) π D = = 4 Occlusion by apertures and lens extents results in mechanical vignetting. In many cases, d o >> d i : = + f d d o = + 1 f d o 1 f f i As a result: D 4 E L cos θ f The term f/d is called the f-number. 5 6 Lens design Bibilography Lens design is a complex optimization process that trades off:! achromatic aberrations! chromatic aberrations! field of view! aperture (speed)! stortions! size, weight, cost, Burke, M.W. Image Acquisition: Handbook of Machine Vision Engineering. Volume 1. New York. Chapman and Hall, Goldberg, N. Camera Technology: The Dark Side of the Lens. Boston, Mass., Academic Press, Inc., 199. Hecht, E. Optics. Reang, Mass., Adson-Wesley Pub. Co., nd ed., Goldberg 199 Horn, B.K.P. Robot Vision. Cambridge, Mass., MIT Press, Smith, W., Modern Optical Engineering, McGraw Hill,

Image Formation and Capture. Acknowledgment: some figures by B. Curless, E. Hecht, W.J. Smith, B.K.P. Horn, and A. Theuwissen

Image Formation and Capture. Acknowledgment: some figures by B. Curless, E. Hecht, W.J. Smith, B.K.P. Horn, and A. Theuwissen Image Formation and Capture Acknowledgment: some figures by B. Curless, E. Hecht, W.J. Smith, B.K.P. Horn, and A. Theuwissen Image Formation and Capture Real world Optics Sensor Devices Sources of Error