Projection
Readings
Nalwa 2.1

Projection
Readings
Nalwa 2.1

Müller-Lyer Illusion

Image formation
Let’s design a camera
Idea 1:  put a piece of film in front of an object
Do we get a reasonable image?

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?

Camera Obscura
The first camera
Known to Aristotle
How does the aperture size affect the image?

Shrinking the aperture
Why not make the aperture as small as possible?

Shrinking the aperture

Adding a lens
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
Changing the shape of the lens changes this distance

Lenses
A lens focuses parallel rays onto a single focal point
focal point at a distance f beyond the plane of the lens
f is a function of the shape and index of refraction of the lens
Aperture of diameter D restricts the range of rays
aperture may be on either side of the lens
Lenses are typically spherical (easier to produce)

Thin lenses
Thin lens equation:
Any object point satisfying this equation is in focus
What is the shape of the focus region?
How can we change the focus region?
Thin lens applet:  http://www.phy.ntnu.edu.tw/java/Lens/lens_e.html  (by Fu-Kwun Hwang )

Depth of field
Changing the aperture size affects depth of field
A smaller aperture increases the range in which the object is approximately in focus

The eye
The human eye is a camera
Iris - colored annulus with radial muscles
Pupil - the hole (aperture) whose size is controlled by the iris
What’s the “film”?

Digital camera
A digital camera replaces film with a sensor array
Each cell in the array is a Charge Coupled Device
light-sensitive diode that converts photons to electrons
other variants exist:  CMOS is becoming more popular
http://electronics.howstuffworks.com/digital-camera.htm

Issues with digital cameras
Noise
big difference between consumer vs. SLR-style cameras
low light is where you most notice noise
Compression
creates artifacts except in uncompressed formats (tiff, raw)
Color
color fringing artifacts from Bayer patterns
Blooming
charge overflowing into neighboring pixels
In-camera processing
oversharpening can produce halos
Interlaced vs. progressive scan video
even/odd rows from different exposures
Are more megapixels better?
requires higher quality lens
noise issues
Stabilization
compensate for camera shake (mechanical vs. electronic)

Modeling projection
The coordinate system
We will use the pin-hole model as an approximation
Put the optical center (Center Of Projection) at the origin
Put the image plane (Projection Plane) in front of the COP
Why?
The camera looks down the negative z axis
we need this if we want right-handed-coordinates

Modeling projection
Projection equations
Compute intersection with PP of ray from (x,y,z) to COP
Derived using similar triangles (on board)

Homogeneous coordinates
Is this a linear transformation?

Perspective Projection
Projection is a matrix multiply using homogeneous coordinates:

Perspective Projection
How does scaling the projection matrix change the transformation?

Orthographic projection
Special case of perspective projection
Distance from the COP to the PP is infinite
Good approximation for telephoto optics
Also called “parallel projection”:  (x, y, z) → (x, y)
What’s the projection matrix?

Other types of projection
Scaled orthographic
Also called “weak perspective”
Affine projection
Also called “paraperspective”

Camera parameters

Distortion
Radial distortion of the image
Caused by imperfect lenses
Deviations are most noticeable for rays that pass through the edge of the lens

Correcting radial distortion

Distortion

Modeling distortion
To model lens distortion
Use above projection operation instead of standard projection matrix multiplication

Other types of lenses