ImageFormation图像构造

Thomas Wiegand o 3 / 42
Fundamentals of Image Formation
The Simplest Optics: The Pinhole
Add a barrier between the scene and the image plane: Block off most of the light Produces image on photosensitive surface (or screen) Opening is known as aperture
Total time that light requires from point A to point B √ b2 + (c − x)2 a2 + x2 + t= v1 v2 Determining the position x for which t is minimized by setting d t=0 dx yields v1 √ x = a2 + x2 v2 b2 c−x + (c − x)2
Thomas Wiegand o 6 / 42
Fundamentals of Image Formation
Ambiguity of Projection
Thomas Wiegand
o 7 / 42
Fundamentals of Image Formation
Geometric Blur due to Pinhole Size
Thomas Wiegand
Refraction index n n= c speed of light in vacuum = v speed of light in medium
o 12 / 42
Fundamentals of Image Formation
Snell’s Law as Special Case of Fermat’s Principle
Focusing Parallel Light to a Point
From Fermat’s principle, it follows: The time tBC that light requires for the path B → C must be equal to the time tAC that light requires for the path A → C : tBC = tAC n · d + f = n · (d − x) + (x + f )2 + y 2 where n denotes the refraction index of the lens (refraction index of air is n0 ≈ 1) Parallel light is only focused to a point, if the lens has the shape of a hyperbola x+
Thomas Wiegand o 10 / 42
Fundamentals of Image Formation
Pinhole Photography
Properties of Pinhole Photography Straight lines remain straight Infinite depth of field Serious diffraction blur Requires very long exposure times
Thomas Wiegand o 5 / 42
Fundamentals of Image Formation
Perspective Projection by Pinhole Camera
Perspective projection formulas x = −f · X Z and y = −f · Y Z
Thomas Wiegand
[Hecht]
o 15 / 42
Fundamentals of Image Formation
Paraxial Approximation for Thin Spherical Lenses
Lensmaker’s formula
Paraxial rays and thin lenses (d ≈ 0) 1 1 + = (n − 1) so si 1 1 + R1 R2
[Juergen Kollmorgen]
Thomas Wiegand o 11 / 42
Fundamentals of Image Formation
Refraction at the Surface Between Two Media
Snell’s law of refraction v1 n2 sin θ1 = = sin θ2 v2 n1 Fermat’s principle: The path taken between two points by a ray of light is the path that can be traversed in the least time Can be used to derive Snell’s law
Thomas Wiegand o 2 / 42
Fundamentals of Image Formation
Why Do We Need Optics?
Without optics: Each point on the sensor (or photographic film) would record the integral of all light arriving from every point of a scene All sensor points would record similar colors (slight differences, because irradiance depends on angle of incidence)
Thomas Wiegand o 4 / 42
Fundamentals of Image Formation
Camera Obscura
First mention of the principle Chinese philosopher Mozi (470 BC – 391 BC) Greek philosopher Aristotle (384 BC – 322 BC)
Yields focal length f for so → ∞ 1 = (n − 1) f 1 1 + R1 R2
Gaussian lens formula
Follows from lensmaker’s formula Valid for thin lenses (d ≈ 0)
Thomas Wiegand
f n+1
2
−
n2
y2 f2 = −1 (n + 1)2
o 14 / 42
Fundamentals of Image Formation
Spherical Lenses
Hyperbolic lenses
Optimal for focusing monochromatic light Expensive to manufacture Still suffer from chromatic aberration
Diffraction Property of wave propagation Inherent to the wave character of light Diffraction blur increases as the aperture decreases Can we improve the sharpness of the image on a sensor? Diffraction blur basically only depends on the aperture Diffraction cannot be reduced for given aperture Geometric blur for large apertures can be significantly reduced by using lenses
∆ y ≈ 1 µm · f d
Sharpest images for
dopt ≈ 1mm · z·f (z + f ) · 1m
[Hecht]
Thomas Wiegand
o 9 / 42
Fundamentals of Image Formation
ห้องสมุดไป่ตู้
Diffraction and Image Sharpness
Fundamentals of Image Formation
Fundamentals of Image Formation
Thomas Wiegand
o 1 / 42
Fundamentals of Image Formation
Image Formation Using Optics
[Cross section of Nikon D4, Photokina 2012]
Projection from 3-d space to 2-d image plane Ambiguity between object size (X and Y ) and object distance Z Lines in 3-d space are mapped to lines on 2-d image plane
point C can be freely moved on surface
Using the definition of sine yields Snell’s law of refraction sin θ1 sin θ2 = v1 v2