How to achieve this goal? (1) Cameras

Transcription

1 How to achieve this goal? (1) Cameras History, progression and comparisons of different Cameras and optics. Geometry, Linear Algebra

2 Images Image from Chris Jaynes, U. Kentucky Discrete vs. Continuous I[x,y,t] array I(x,y,t) function Derivatives, Convolutions

3 Image formation Today s lecture was largely created by Cornelia Fermuller, at the University of Maryland.

4 Image Formation Vision infers world properties form images. How do images depend on these properties? Two key elements Geometry Radiometry We consider only simple models of these

5 Camera Obscura Camera = Latin for room Obscura = Latin for dark "When images of illuminated objects... penetrate through a small hole into a very dark room... you will see [on the opposite wall] these objects in their proper form and color, reduced in size... in a reversed position, owing to the intersection of the rays". Da Vinci (Russell Naughton)

6 DaVinci was not the first: Chinese philosopher Mo-Ti (5th century BC) formally recorded the creation of an inverted image formed by light rays passing through a pinhole into a darkened room. He called this darkened room a "collecting place" or the "locked treasure room." Aristotle ( BC) viewed crescent shape of partial solar eclipse through holes in a sieve or gaps in leaves of a tree. Alhazen of Basra (arabian scholar, 10 th century) had a portable tent room for solar observation.

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8 Used to observe eclipses (eg., Bacon, ) By artists (eg., Vermeer). (hard to draw correct perspective!).

9 Jetty at Margate England, (Jack and Beverly Wilgus)

10 Cameras First photograph due to Niepce Camera shown is from 1826.

11 Pinhole cameras Abstract camera model - box with a small hole in it Pinhole cameras work in practice (Forsyth & Ponce)

12 Properties of pinhole cameras: Distant objects are smaller B C (Forsyth & Ponce)

13 Parallel lines meet Common to draw image plane in front of the focal point. Moving the image plane merely scales the image. (Forsyth & Ponce)

14 Vanishing points Each set of parallel lines meets at a different point The vanishing point for this direction Sets of parallel lines on the same plane lead to collinear vanishing points. The line is called the horizon for that plane Parallel planes have the same horizon too

15 Properties of Projection Invariants Points project to points Lines project to lines Planes project to the whole image or a half image Degenerate cases Line through focal point projects to a point. Plane through focal point projects to line Plane perpendicular to image plane projects to part of the image (with horizon). Angles are not preserved Parallel lines may intersect.

16 Take out paper and pencil

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19 Camera Geometry 3d 2d transformation: perspective projection center of projection object focal length image plane

20 Camera Geometry 3d 2d transformation: perspective projection center of projection object image focal length image plane Each point is projected along a ray through the center of projection. x? y?

21 Perspective Projection center of projection image object image plane focal length Save mental gyrations by placing the image plane in front of the COP.

22 The equation of projection (Forsyth & Ponce)

23 The equation of projection Cartesian coordinates: We have, by similar triangles, that Ignore the third coordinate, and get ) ', ' ( ),, ( z y f z x f z y x z y f y z x f x ' ' ' ' = = '), ', ' ( ),, ( f z y f z x f z y x

24 Orthographic projection x' = x y' = y

25 Weak perspective (scaled Issue orthographic projection) perspective effects, but not over the scale of individual objects collect points into a group at about the same depth, then divide each point by the depth of its group (Forsyth & Ponce)

26 The Equation of Weak Perspective ( x, y, z) s( x, y) s is constant for all points. Parallel lines no longer converge, they remain parallel.

27 Pros and Cons of These Models Weak perspective much simpler math. Accurate when object is small and distant (satellite pictures of Russian shipyards) Most useful for recognition. (satellite pictures of Russian shipyards) Pinhole perspective much more accurate for scenes. Used in structure from motion. When accuracy really matters, must model real cameras.

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29 Cameras with Lenses (Forsyth & Ponce)

30 Interaction of light with matter Absorption Scattering Refraction Reflection Other effects: Diffraction: deviation of straight propagation in the presence of obstacles Fluorescence:absorbtion of light of a given wavelength by a fluorescent molecule causes reemission at another wavelength

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32 Refraction n1, n2: indexes of refraction

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34 Thin lens projection model More Accuracy

35 ( v u Similar triangles <P F S >,<ROF > and <PSF><QOF> ( v f )( u f ) = f 1 + z' v 1 z u 1 = f 2

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37 Assumptions for thin lens equation Lens surfaces are spherical Incoming light rays make a small angle with the optical axis The lens thickness is small compared to the radii of curvature The refractive index is the same for the media on both sides of the lens

38 No place is the point completely in focus.

39 More Accuracy Real cameras don t create exactly a pinhole projection... Radial Distortion

40 Distortion from spherical lens aspheric (like non-linear)

41 Other aberrations Astigmatism: unevenness of the cornea (reason for hard contact lenses!). Distortion : different areas of lens have different focal length Coma : point not on optical axis is depicted as asymmetrical comet-shaped blob Chromatic aberration Astigmatism, front of back of cornea.

42 Recap can. coords: (x,y) y u image coordinates v (X,Y,Z) in canonical coords x object coordinates canonical coordinates z f x = f X Z 2d - 3d y = f Y Z What s left? You know where on the image plane Each point projects to, but which pixel is that?

43 2d - 2d transforms Canonical coordinates u Pixel coordinates v y p p x C p = (p x,p y ) P p = (p u,p v ) How many degrees of freedom are there in a rigid transformation from one 2d coordinate system to another?

44 3d - 3d transforms Canonical coordinates Object coordinates y x z p z y x p C p = (X C,Y C,Z C ) O p = (X O,Y O,Z O ) How many DOF are there in a rigid transformation from one 3d coordinate system to another?

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