## Lenses

Lenses focus a bundle of rays to one point => can have larger aperture. Hecht 19 . 87. 8. Lenses. A lens images a bundle of parallel rays to a focal point at a.

Overview

Lenses Pinhole camera Lenses • Principles of operation • Limitations

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Terminology

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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?

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Hecht 1987

Shrinking the pinhole, cont’d

Hecht 1987

Shrinking the pinhole

Q: What happens as we continue to shrink the aperture?

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Lenses

Lenses 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 distance, f, beyond the plane of the lens.

Hecht 1987

Lenses

An aperture of diameter, D, restricts the extent of the bundle of refracted rays. Note: f is a function of the index of refraction of the lens.

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Lenses

Cardinal points of a lens system

For economical manufacture, lens surfaces are usually spherical.

Most cameras do not consist of a single thin lens. Rather, they contain multiple lenses, some thick.

A spherical lens behaves ideally if we consider rays near the optical axis -- “paraxial rays.”

A system of lenses can be treated as a “black box” characterized by its cardinal points.

For a “thin” lens, we ignore lens thickness, and the paraxial approximation leads to the familiar Gaussian lens formula:

! ! ! + = "\$ "# !

Hecht 1987

!

"\$

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Magnification = 9

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Cardinal points of a lens system

Cardinal 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 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 distances are now relative to the principal planes.

! The principle planes are planar ! The nodal and principal points are the same

The principal and nodal points are, together, called the cardinal points.

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Limitations of lens systems

Depth of field

Lenses exhibit a number of deviations from ideal. We’ll consider a number of these deviations:

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.”

! Depth of field ! Primary (third order, Seidel) aberrations • Distortion ! Chromatic aberration ! Flare ! Vignetting

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Non-paraxial imaging

Third order aberrations

When we violate the paraxial assumption, we find that real imaging systems exhibit a number of imperfections.

The first set of non-ideal terms beyond perfect imaging and depth of field form the basis for the third order theory.

We can set up the geometry of a rotationally symmetric lens system in terms of an object, aperture, and image:

Smith 1996

Deviations from ideal optics are called the primary or Seidel aberrations: ! ! ! ! !

Spherical aberration Coma Astigmatism Petzval curvature Distortion

All of these aberrations can be reduced by stopping down the aperture, except distortion. We can then perform a Taylor series of the mapping from rays to image points:

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Distortion

Distortion

Distortion follows the form (replacing h with r):

r ′ = a1r + a3r 3 + a5r 5 +! Sometimes this is re-written as:

r ′ = r ⋅ ( a1 + a3r 2 + a5r 4 +!)

Hecht 1987

Hecht 1987

The effect is that non-radial lines curve out (barrel) or curve in (pin cushion).

No distortion

Pin cushion

Barrel

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Chromatic aberration

Flare

Cause:

Light rays refract and reflect at the interfaces between air and the lens.

Index of refraction varies with wavelength.

The “stray” light is not focused at the desired point in the image, resulting in ghosts or haziness, a phenomenon known as lens flare.

Effect: Focus shifts with color, colored fringes on highlights Ways of improving:

Hecht 1987

Achromatic designs

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Optical coatings

Single vs. multiple coatings

Hecht 1987

Optical coatings are tuned to cancel out reflections at certain angles and wavelengths.

Burke 1996

Single coating

Mutliple coatings

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Vignetting

Optical vignetting

Light rays oblique to the lens will deliver less power per unit area (irradiance) due to:

Optical vignetting is best explained in radiometric terms. A sensor responds to irradiance (power per unit area) which is defined in terms of radiance 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

2

=

cosθ dA cos3 θ dA = 2 2 (di / cosθ ) di

L cos 4 θ dA dE = 2 di

Thus:

E !L

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A cos 4 ! 2 di

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Optical vignetting, cont’d

Mechanical vignetting

We can rewrite this in terms of the diameter of the exit pupil:

Occlusion by apertures and lens extents results in mechanical vignetting.

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Hecht 1987

A π (D / 2)2 π  D  = =   di2 di2 4  di  In many cases, do >> di:

1 1 1 = + f do di di d i = +1 f do di ≈ 1 ⇒ di ≈ f f As a result: 2

D E ≈ L   cos 4 θ f  The term f/D is called the f-number.

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Lens design

Bibilography

Lens design is a complex optimization process that trades off:

Burke, M.W. Image Acquisition: Handbook of Machine Vision Engineering. Volume 1. New York. Chapman and Hall, 1996.

achromatic aberrations chromatic aberrations field of view aperture (speed) distortions size, weight, cost, …

Goldberg, N. Camera Technology: The Dark Side of the Lens. Boston, Mass., Academic Press, Inc., 1992. Hecht, E. Optics. Reading, Mass., Addison-Wesley Pub. Co., 2nd ed., 1987. Horn, B.K.P. Robot Vision. Cambridge, Mass., MIT Press, 1986. Goldberg 1992

! ! ! ! ! !

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Smith, W., Modern Optical Engineering, McGraw Hill, 1996.

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