Chapter2 Image Formation
Reading: Szeliski, Chapter 2
What are we tuned to?
The visual system is tuned to process structures
typically found in the world.
What is a natural image?
The structure of ambient light
What is a natural image?
What is a natural image?
The visual system seems to be tuned to a set of images:
What is a natural image?
The visual system seems to be tuned to a set of images:
Did you saw this image?
The visual system seems to be tuned to a set of images:
Demo inspired from D. Field
6 images
http://www.alexfito.com/
http://www.rowland.harvard.edu/images/ModPurp.jpg
Not all these images are the result of sampling a real-world plenoptic function
• Proposition 1. The primary task of early vision is to deliver a small set of useful
measurements about each observable location in the plenoptic function.
• Proposition 2. The elemental operations of early vision involve the measurement
of local change along various directions within the plenoptic function.
• Goal: to transform the image into other representations (rather than pixel values)
that makes scene information more explicit
Cavanagh, Perception 95
What are “visual features”
Shape, color, texture, etc
2.1 Photometric Image Formation
• Discrete color or intensity values
• Where do these value come from?
– Geometry, projection
– Camera optics, sensor properties
– Lighting, surface properties
Images as Functions
Images as Functions
• We can think of an image as a function, f, from R2 to R:
– f( x, y ) gives the intensity at position ( x, y )
– Realistically, we expect the image only to be defined over a
rectangle, with a finite range:
• f: [a,b]x[c,d]  [0,1]
• A color image is just three functions pasted together.
We can write this as a “vector-valued” function:
 r ( x, y ) 
f ( x, y )   g ( x, y ) 



 b ( x, y ) 

Images as functions
What is a digital image?
• We usually work with digital (discrete) images:
– Sample the 2D space on a regular grid
– Quantize each sample (round to nearest integer)
• If our samples are D apart, we can write this as:
•
f[i ,j] = Quantize{ f(i D, j D) }
• The image can now be represented as a matrix of integer
values
Photometric Image Formation
Perspective projection
Lens optics
Light scattering
Bayer color filter array
Photometric Image Formation
2.2 Lighting
• Point light source
– Single location (small light bulb)
– Infinity: the sun --directional light
• Area light source
– A finite rectangular area emitting light equally in all
directions
• Environment map
• Light direction to color mapping
2.2.2 Reflectance and Shading
many models for reflectance and shading
BRDF: Bidirectional Reflectance Distribution Function
BRDF
• BRDF is reciprocal
• For isotropic surface,
no preferred directions for light transport
BRDF
• Light existing a surface point:
Foreshortening factor
Diffuse Reflection
• Also called Lambertian or matte reflection
– Light is scattered uniformly in all directions, i.e.
– BRDF is constant:
Think about the inverse problem
Specular Reflection
• Depends on the direction of outgoing light
• Mirror surface:
– Specular reflection direction
vr
cos( s )  (vr  si )
Specular Reflection
• Amount of light
– Phone model
– Micro-facet model
– Larger , , more specular surface with
hightlights ; Smaller, softer gloss
Phone Shading Model
• Diffuse
• Specular
• Ambient:
– Does not depend on surface orientation
– Color and both ambient illumination
the object
and
Phone Shading Model
Lr (vr ;  )
 ka ( ) La ( )
 kd ( ) Li ( )(vi  ni ) 
i
 ks ( ) Li ( )(vr  si ) ke
i
The recent advent of
programmable pixel shaders
makes the use of more
complex models feasible.
Example
Realistic Rendering
Ioannis Gkioulekas, et al, Siggraph’13
The recent advent of programmable pixel shaders makes the use of more complex
models feasible.
Optics
• Lens, sensor
• Ideal pinhole camera
• More complex: focus, exposure, vignetting,
aberation,…,
Thin lens model
Thin lens: low, equal curvature on both sides
Optical axis
Thin lens model
object
Focus plane
Thin lens model
object
Circle of confusion
Pinhole camera
Pinhole Camera Model
object
Pinhole Camera Model
object
Pinhole Camera Model
object
2.3 3D to 2D Projection
book: pp32-60
3D view of world
perspective
• 3D perspective: the most commonly used
projection in computer vision and computer graphics
Pinhole Camera Model
object
xf
u
z
yf
v
z
Pinhole Camera Model
object
Pinhole Camera Model
• Using homogeneous (projective) coordinate
–
xf
u
z
yf
v
z
f
u 
 fx / z  
 

 0
w  v   z  fy / z  
1 
1
 0
 

 
0
0
f
0
0
0 0  x 
 
0 0   y 
1 0  z 
 
0 1  1 
wx = Kp
Camera Intrinsics
• Imperfect camera
• image sensor
f
K   0
 0
•
•
•
•
s
af
0
cx 
c y 
1 
e.g., (cx , c y )  (W / 2, H / 2)
s: possible skew between sensor axes
a: aspect ratio
(cx , c y ) : optical center
Five intrinsic parameters
F : focal length
Camera Intrinsics
•
•
•
•
Focal length
Actual focal length, e.g. 18~55mm,
Conventional sensor width: 35 mm
Digital Image: integer values, [0,W) x [0,H)
Field of view
Focal length
Sensor width
Extrinsic Parameters
• World Coordinate system to Camera
Coordinate system
pc  [R | t]p w
j
Extrinsic parameters
i
k
i
O j
C
k
Cw  R 1t
Extrinsic Parameters
with
wx = Kp
wx  K[R | t]p w
M  K[R | t]
Camera Matrix
2.3 Digital Camera
• Process chart
2.3 Digital Camera
• Process chart
2.3.2 Color
• Light from different parts of the spectrum is
somehow integrated into discrete RGB color
values
wx  K[R | t]p w
2.3.2 Color
• Primary and Secondary Colors
• Additive colors (projector, monitor)
• Subtractive colors (printing, printing)
CIE color matching
• Commission Internationale d’Eclairage (CIE)
• Color matching experiments
pure colors to the
R=700.0nm,
G=546.1nm, and
B=435.8nm
XYZ Color Space
Y=1 for pure R (1,0,0)
XYZ Color Space
• Y=1 for (1,1,1)
XYZ Color Space
• Chromaticity coordinates
• Yxy (luminance plus the two most distinctive
chrominance components)
Chromaticity Diagram
L*a*b* Color Space
• Human visual system is roughly logarithmic
• Differences in luminance or chrominance are
more perceptually uniform
• Non-linear mapping from XYZ to L*a*b* space
L*a*b* Color Space
Color Cameras
• Spectral response function
• Make sure to generate the standard color values
HDTV, new monitors, new standard ITU-R BT.709
Color Filter Arrays
• Separate sensors for three primary colors
Bayer RGB pattern: (a) color filter array layout; (b) interpolated pixel values
Bayer Pattern, 1976
• Green filters are twice as many as red and
blue filters
• Human visual system is much more sensitive
to high frequency detail in luminance than
chrominance
• Luminance is mostly determined by green
value
Color Balance
• Move the white point of a given image closer
to pure white (R=G=B)
– Multiply RGB values by a different factor
– Color twist, general 3x3 transform matrix
– Exercise 2.9 (optional)
Gamma
• CRT Monitor: non-linear relationship between
the voltage and the resulting brightness is
determined by gamma
  2.2
• Pre-map the sensed luminance Y through an
inverse gamma
1

 0.45
Gamma Compensation
Noise added during transmission or quantization will be reduced in
the darker regions of the signal where it was more visible
Other Color Spaces
• XYZ, RGB for spectral content of color signals
• Others for image coding and computer
graphics
– YUV, YCrCb, HSV
YUV Color Space
• YUV for video transmission
– Luma
– Two lower frequency chroma channels
YCrCb Color Space
• Closely related to YUV
• Different scale factor to fit within the 8-bit
range for digital signals
• Useful for careful image de-blocking, et al.
HSV Color Space
• Hue: direction around a
color wheel
• Saturation: scaled
distance from the diagonal
• Value: mean or maximum
color value
More suitable for color picking
Color Ratios
• Suitable for algorithms that only affect the
value/luminance and not saturation or hue
• After processing, scale rgb back by the color
ratio Ynew/Yold
Color FAQ, http://www.poynton.com/ColorFAQ.html
2.3.3 Compression
• Converting signal into YCbCr (or related
variant)
• Compress the luminance signal with higher
fidelity than the chrominance signal