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CHAPTER 8 OPTICS
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Assume that the viewable area on the image plane is a rectangle with height h and width w centered about the principal point The corners of the rectangle and the location of the center of projection define four planes (top, bottom, left and right) that bound the view volume The near and far planes are perpendicular to the image plane at distances determined by the depth of field By convention, the plane normals point to the outside of the view volume The six bounding planes can be calculated from the position and orien tation of the camera, the dimensions of the imaging element, and the lens optics The plane coordinates are in the same coordinate system used to define the position and orientation of the camera, typically the absolute co ordinate system for the scene The bounding planes can be used to determine if all scene points that should be within the view of the camera are inside the view volume The position and orientation of the camera, the nominal focus, the aperture diameter, and the focal length can be adjusted to change the view volume until all scene points that should be viewable are within the view volume The minumum aperture diameter is constrained by the ambient illumination and the shutter speed necessary to minimize motion blur
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Exposure
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The amount of light collected by the camera depends on the intensity of light falling on the image plane (the image irradiance) and the duration of the exposure (shutter speed):
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Exposure is in units of joules (energy) per square meter while image irradi ance is in units of watts (power) per square meter Multiplying power by time gives energy, so the units work out correctly The F-number or F -stop is proportional to the ratio of the focal length f to the diameter d of the aperture: F-number =
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The lens aperture on cameras is designated in units of F-number because image intensity (or film exposure for constant shutter speed) is the same for
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EXERCISES
different lenses at the same F -stop: F-number
(813) for constant shutter speed t (814)
In other words, the F-stop is the aperture diameter provided in a system of units that factors out the different light gathering capabilities of lenses with different focal lengths The F-number on camera lenses is marked with numbers that are multi ples of V2 because doubling the aperture area is equivalent to increasing the aperture diameter by V2, 2 x aperture area
V2d ~
l4d
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The F-stops are numbered 28,4,56, 8, 11, and so on, so each F-stop changes the aperture diameter by a factor of 14 and increases the amount of light on the image plane by a factor of 2
Further Reading
There are several good books on optics, but most texts give short coverage to radiometry An excellent text on optics which includes some coverage of radiometry and the optics of the eye is a paperback book by Young [259J The vision text by Horn [109J contains a good discussion of radiometry and imaging Horn was the pioneer in the use of optics and radiometry in machine vision The text by Beynon and Lamb [31J contains a chapter on CCD image sensors
Exercises
81 The geometry of the view volume was formulated in Section 84 As sume that the camera position and orientation are given in camera coordinates, so that the position is (0,0,0) , the view vector is (0,0,1), and the up vector is (0, 1,0)
CHAPTER 8 OPTICS
a Calculate the position of the corners of the view rectangle on the image plane in terms of the position and orientation of the camera and the height and width of the imaging element h Calculate the formulas for the coefficients of the top bottom left, and right bounding planes in terms of the camera position and orientation and the height and width of the imaging element c Calculate the formulas for the coefficients of the near and far planes in terms of the focal length, aperture diameter , and nomi nal distance to a scene point d Generalize these formulas to the camera position and orientation in absolute coordinates e Outline an iterative algorithm for adjusting the camera position and orientation, nominal focus , aperture diameter, and focal length to accommodate a given list of scene points that should be in the view volume
82 Consider a moving object with position (Px,Py,Pz) and velocity vector (vx, v y, v z ) in camera coordinates It is desirable to adjust the shutter speed so that the blur due to motion is reduced to an acceptable level, defined for this exercise to be the minimum distance between imaging elements a Calculate the projected velocity vector on the image plane assum ing perspective projection and camera constant f h Calculate the minimum acceptable shutter speed c Suppose that there are no moving objects in the scene, but the camera moves with velocity vector (v x , v y, v z ) Repeat the calcu lation for the minimum shutter speed The result will depend on the distance to points in the scene d Suppose that the camera motion is due to vibration, modeled as movement in a uniformly distributed direction with vector mag nitude v Repeat the calculation for minimum shutter speed in terms of the maximum magnitude of the velocity vector due to vibration
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