Lecture 13: Cameras and Geometry
Lecture 12: Cameras and Geometry
CAP 5415 Fall 2010 The midterm
• What does the response of a derivative filter tell me about whether there is an edge or not? Things aren't working
• Did you look at the filters? • Why not? • Normalize the filters How do we see the world?
Let’s design a camera – Idea 1: put a piece of film in front of an object – Do we get a reasonable image?
Slide by Steve Seitz Pinhole camera
Add a barrier to block off most of the rays – This reduces blurring – The opening known as the aperture – How does this transform the image?
Slide by Steve Seitz Pinhole camera model
Pinhole model: – Captures pencil of rays – all rays through a single point – The point is called Center of Projection (COP) – The image is formed on the Image Plane – Effective focal length f is distance from COP to Image Plane Slide by Steve Seitz A little bit of history on building cameras
Camera Obscura • Latin for “Dark Box” • Dark room with a pinhole in wall • Projects image onto wall • Allows artists to get perspective right
Image from Wikipedia Camera Obscura Camera Obscura, Gemma Frisius, 1558
The first camera – Known to Aristotle
– Depth of the room is the effective focal length Camera Obscura
• Can also be a box We’ll use the pinhole camera model to describe image formation
Notice how the image is inverted
(Image from Slides by Forsyth) Projection Effects
Pinhole
• Height of objects depends on the distance from the pinhole (O)
(Image from Slides by Forsyth) Projection Effects: Horizon Line • Consider two parallel lines that lie in a plane (Π) • Will converge to a point on the horizon line(H)
Pinhole
(Image from Slides by Forsyth) • Observe this next time you are driving on a flat road Vanishing points • Each set of parallel • Good ways to spot lines (=direction) faked images meets at a different – scale and perspective point don’t work – The vanishing point for – vanishing points this direction behave badly • Sets of parallel lines – supermarket tabloids on the same plane are a great source. lead to collinear vanishing points. – The line is called the horizon for that plane
(From Slides by Forsyth) The equation of projection
(Image from Slides by Forsyth) The equation of projection
We know: so
(Image from Slides by Forsyth) Lenses
• Why Lenses? • For an ideal pinhole, only one ray of light reaches each point – Very Dim Image
• Why not make pinhole bigger? Why not make pinhole bigger? • Only one point can generate rays that strike a particular point on the image plane
Why not make pinhole bigger? • Now add an aperture
Pinhole too big - many directions are averaged, blurring the image
Pinhole too small- diffraction effects blur the image
Generally, pinhole cameras are dark, because a very small set of rays from a particular point hits the screen.
(From Slides by Forsyth) Lenses
• The lens focuses multiple rays coming from the same point
(Image from Slides by Forsyth) Thin Lens Equation
Focus and Defocus
A lens focuses light onto the film – There is a specific distance at which objects are “in focus” • other points project to a “circle of confusion” in the image
– How can we change focus distance?
Slide by Steve Seitz More on Lenses
Canon EF 28-135mm f/3.5-5.6 IS USM Standard Zoom Lens for Canon SLR Cameras
28-135mm is the focal length
i o
P
P’
f
Diagram by Shree Nayar
What's f/3.5-5.6?
Canon EF 28-135mm f/3.5-5.6 IS USM Standard Zoom Lens for Canon SLR Cameras
f-number
• f is the focal length • D is the diameter of the pupil or aperture
• f/2 is the same as N=2 • f/16 is the same as N=16
• Which has the bigger aperture? What's f/3.5-5.6?
• This is the widest possible aperture
Canon EF 28-135mm f/3.5-5.6 IS USM Standard Zoom Lens for Canon SLR Cameras
Why should I adjust the aperture? • Big aperture means more light, shorter exposure time • Also affects sharpness and depth of field
Here, the rays are focused on the image plane
Now, look at a point that is farther way
Circle of Confusion
It grows as you move farther away
Circle of Confusion
Circle of Confusion • Spot caused by a point that is not in focus
• You decide the tolerable limits
(Diagram from Wikipedia) Aperture also causes blurring • Go back to pinhole camera model • Only one point can generate rays that strike a particular point on the image plane
Aperture also causes blurring • Now add an aperture
Depth of Field • Increasing the aperture diameter increases the size of the circle of confusion
f/22 f/5.6
Diffraction • When light passes through a small aperture the rays begin to interfere with each other • For a perfectly circular aperture this leads to the airy disc pattern
Image from http://www.cambridgeincolour.com/tutorials/diffraction-photography.htm# This leads to a loss of sharpness
f/8 f/11 f/16
f/22 From http://www.cambridgeincolour.com/tutorials/diffraction-photography.htm# After Light Strikes the sensor
• Engineering problem: – I have sensor that records the amount of light at different pixels – How do I get a color image instead of a black and white image?
Solutions
• Three sensors • One sensor with a color mask – Each pixel records one wavelength • A common pattern for the mask is the Bayer pattern:
Mosaicing
• So, if I took a • My sensor would picture of this edge record this image
Demosaicing
• I have 1 color at each pixel • I need three • Easy solution: Interpolate
+
Problem! This smooths across the edge • Because the different pixels are used to red and green, the smoothing may be different
+
Result: Color Fringing
Color Fringing
(Results from Brainard et al) Fast Solution
• The fringing occurs when the correlation between the color channels is incorrectly estimated • One measure of this correlation is the color difference • Can fix errors using median filtering
Simple Demosaicing Algorithm (Freeman) • Use linear interpolation to get first estimate • Compute difference images between color channels • Median filter these difference images • Use filtered difference images to reconstruct
(Slide by Freeman)