Image formation
ECE 847:Digital Image Processing
Stan BirchfieldClemson University
Cameras
• First photograph due to Niepce
• Basic abstraction is the pinhole camera– lenses required to ensure image is not too
dark– various other abstractions can be applied
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Image formation overview
Image formation involves• geometry – path traveled by light• radiometry – optical energy flow• photometry – effectiveness of light to produce
“brightness” sensation in human visual system• colorimetry – physical specifications of light
stimuli that produce given color sensation• sensors – converting photons to digital form
Pinhole camera
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
Parallel lines meet: vanishing point
• each set of parallel lines (=direction) 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
Perspective projection
k
O
P
Q
j
ip
q
C
f
Properties of projection:• Points go to points• Lines go to lines• Planes go to whole image• Polygons go to polygons• Degenerate cases
– line through focal point to point
– plane through focal point to line
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Perspective projection (cont.)
Z
Y
w
vv
Z
X
w
uu
T
Z
Y
X
PI
w
v
u
p
ˆ
ˆ
:gnormalizinby scoordinate )(Euclidean imageRecover
0100
0010
0001
]0[
scoordinate e)(projectiv shomogeneou ofn nsformatioLinear tra
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Weak perspective 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
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
Weak perspective (cont.)
Pretend depth is constant (often OK !)
ˆ u X
Zr
ˆ v Y
Zr
Can also be written as a linear transformation :
u
v
w
1
Zr
1 0 0 0
0 1 0 0
0 0 0 Zr
X
Y
Z
T
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Orthographic projection
Let Z0=1:
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Pushbroom cameras
Pinhole too big - many directions are averaged, blurring the image
Pinhole too small- diffraction effects blur the image
Generally, pinhole cameras are dark, becausea very small set of raysfrom a particular pointhits the screen.
Pinhole size
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
The reason for lenses
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
The thin lens
z 1
z'
1
f
†
1
z'-
1
z=
1
f
focal points
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
Focusing
http://www.theimagingsource.com
Thick lens
• thick lens has 6 cardinal points:– two focal points (F1 and F2)
– two principal points (H1 and H2)
– two nodal points (N1 and N2)
• complex lens is formed by combining individual concave and convex lenses
http://physics.tamuk.edu/~suson/html/4323/thick.html
D. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
Complex lens
http://www.cambridgeincolour.com/tutorials/camera-lenses.htm
All but the simplest cameras contain lenses which are actually composed of several lens elements
Choosing a lens
• How to select focal length:– x=fX/Z– f=xZ/X
• Lens format should be >= CCD format to avoid optical flaws at the rim of the lens
http://www.theimagingsource.com/en/resources/whitepapers/download/choosinglenswp.en.pdf
Lenses – Practical issues
• standardized lens mount has two varieties:– C mount– CS mount
• CS mount lenses cannot be used with C mount cameras
http://www.theimagingsource.com/en/resources/whitepapers/download/choosinglenswp.en.pdf
Spherical aberration
perfect lens actual lens
On a real lens, even parallel rays are not focused perfectly
http://en.wikipedia.org/wiki/Spherical_aberration
Chromatic aberration
On a real lens, different wavelengths are not focused the same
http://en.wikipedia.org/wiki/Chromatic_aberration
Radial distortion
straight lines are curved:
uncorrected corrected
Radial distortion (cont.)
barrel distortion(more common)
pincushion distortion
http://en.wikipedia.org/wiki/Image_distortion
pincushion
barrel
http://foto.hut.fi/opetus/260/luennot/11/atkinson_6-11_radial_distortion_zoom_lenses.jpg
Two types:
Vignetting
vignetting – reduction of brightness at periphery of imageD. Forsyth, http://luthuli.cs.uiuc.edu/~daf/book/bookpages/slides.html
Normalized Image coordinates
P
Ou=X/Z = dimensionless !
1
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Pixel units
P
Ou=k f X/Z = in pixels !
[f] = m (in meters)[k] = pixels/m
f
Pixels are on a grid of a certain dimension
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Pixel coordinates
P
Ou=u0 + k f X/Z
f
We put the pixel coordinate origin on topleft
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Pixel coordinates in 2D
j
i(u0,v0)
(0.5,0.5) 640
480
(640.5,480.5)
u0 kfX
Z,v0 lf
Y
Z
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Summary: Intrinsic Calibration
33 Calibration Matrix K
p u
v
w
K[I 0]P
s u0
v0
1
1 0 0 0
0 1 0 0
0 0 1 0
X
Y
Z
T
Recover image (Euclidean) coordinates by normalizing:
ˆ u uw
X sY u0
Z
ˆ v v
w
Y v0
Z
skew
5 Degrees of Freedom !
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Camera Pose
In order to apply the camera model, objects in the scenemust be expressed in camera coordinates.
WorldCoordinates
CameraCoordinates
Calibration target looks tilted from cameraviewpoint. This can be explained as adifference in coordinate systems.
wc T
y
x
z
z
x
y
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rigid Body Transformations
• Need a way to specify the six degrees-of-freedom of a rigid body.
• Why are there 6 DOF?
A rigid body is acollection of pointswhose positionsrelative to eachother can’t change
Fix one point,three DOF
Fix second point,two more DOF(must maintaindistance constraint)
Third point addsone more DOF,for rotationaround line
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Notations
• Superscript references coordinate frame• AP is coordinates of P in frame A• BP is coordinates of P in frame B
• Example:
A P
A xA yA z
OP A x iA A y jA A z kA
kA
jA
iA
OA
PF. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Translation
kA
jA
iA
kB
jB
iB
OB
OA
B PAPBOA
P
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Translation
• Using homogeneous coordinates, translation can be expressed as a matrix multiplication.
• Translation is commutative
B A BAP P O
1 0 1 1
B B AAP I O P
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rotation
A B
A BA A A B B B
A B
x x
OP i j k y i j k y
z z
77777777777777
B B AAP R P
BA R
means describing frame A inThe coordinate system of frame B
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rotation
. . .
. . .
. . .
A B A B A BBA A B A B A B
A B A B A B
R
i i j i k i
i j j j k j
i k j k k k
B B BA A A i j k
Orthogonal matrix!
A TB
A TB
A TB
i
j
k
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Example: Rotation about z axis
What is the rotation matrix?
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rotation in homogeneous coordinates
• Using homogeneous coordinates, rotation can be expressed as a matrix multiplication.
• Rotation is not communicative
B B AAP R P
0
1 0 1 1
B B AAP R P
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rigid transformations
B B A BA AP R P O
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Rigid transformations (con’t)
• Unified treatment using homogeneous coordinates.
1 0
1 0 1 0 1 1
1 1
B B B AA A
B B AA A
T
P O R P
R O P
0
1 1
B ABA
P PT
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Projective Camera Matrix
CameraCalibrationProjectionExtrinsics
p u
v
w
K[I 0]TP
s u0
v0
1
1 0 0 0
0 1 0 0
0 0 1 0
R t
0 1
X
Y
Z
T
K R t P MP
5+6 DOF = 11 !
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Projective Camera Matrix
T
Z
Y
X
mmmm
mmmm
mmmm
w
v
u
MPPtRKp
34333231
24232221
14131211
5+6 DOF = 11 !
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Columns & Rows of M
4321
3
2
1
mmmm
m
m
m
M
m2P=0
i
ii
i
ii
Pm
Pmv
Pm
Pmu
3
2
3
1
O
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Effect of Illumination
Light source strength and direction has a dramatic impact on distribution of brightness in the image (e.g. shadows, highlights, etc.)
(Subject 8 from the Yale face database due to P. Belhumeur et. al.)
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Image formation
• Light source emits photons
• Absorbed, transmitted, scattered
• fluorescenceCamerasource
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Surfaces receives and emits
• Incident light from lightfield
• Act as a light source
• How much light ?
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Irradiance
• Irradiance – amount of light falling on a surface patch
• symbol=E, units = W/m2
dA
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Radiosity
• power leaving a point per area
• symbol=B, units = W/m2
dA
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Light = Directional
• Light emitted varies w. direction
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Steradians (Solid Angle)
• 3D analogue of 2D angle
A
R
angle L
R, circle 2 radians
solid angle A
R2, sphere 4 steradians
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Steradians (cont’d)
1 m2 subtends a solid angle of 1 steradian (sr).
Sphere 4, hemisphere 2 steradians.
Solid angle of a small planar patch of area d A at a distance R :
R
A
d dAcos
R2
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Polar Coordinates
Any direction from patch can be expressed as two angles ,.
d d(d sin)
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Intensity
• Intensity – amount of light emitted from a point per steradian
• symbol=I, units = W/sr
intensity I
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
Irradiance and Intensity
dA
Example : point light source :
intensity I /4
watts
irradiance E d /dA Id /dA I /r2
intensity I
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Radiance
• Radiance – amount of light passing through an area dA and
• symbol=L, units = W x m-2 x sr-1
radiance L(P,)
P
dA
d
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Radiance is important
• Response of camera/eye is proportional to radiance
• Pixel values
• Constant along a ray
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lightfield = Gibson optic array !
• 5DOF: Position = 3DOF, 2 DOF for direction
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lightfield Sampler
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lightfield Sample
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lambertian Emitters
• Lambertian = constant radiance• More photons emitted straight up• Oblique: see fewer photons, but area looks smaller• Same brightness !• Total power is proportional to wedge area• “Cosine law”• Sun approximates Lambertian: Different angle, same
brightness• Moon should be less bright at edges, as gets less light from
sun.• Reflects more light at grazing angles than a Lambertian
reflector
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Radiance Emitted/Reflected
• Radiance – amount of light emitted from a surface patch per steradian per area
• foreshortened !
dA
radiance L(P,,)
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Calculating Radiosity
constant radiance L
radiosity B Lcos d Lcos sindd L
radiance L(P,,)
If reflected light is not dependent on angle, then can integrate over angle: radiosity is an approximate radiometric unit
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Example: Sun
• Power= 3.91 1026 W
• Surface Area:6.07 1018 m2
• Power = Radiance . Area . • L = 2.05 107 W/m2.sr
Example from P. Dutre SIGGRAPH tutorialF. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Irradiance (again)
• Integrate incoming radiance over hemisphere
i
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Example: Sun
irradiance E L dA cosd
Area
(2.05107W/m2.sr)(1m2)(6.710 5 sr)
1373.5W
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
BRDF
L
E
oi
Bidirectional Reflectance Distribution Function (BRDF) :
(o,o;i,i) outgoing radiance L
incident irradiance E
Symmetric in incoming and outgoing directions – this is the Helmholtz reciprocity principle
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
BRDF Example
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lambertian surfaces and albedo
• For some surfaces, the DHR is independent of illumination direction too– cotton cloth, carpets,
matte paper, matte paints, etc.
• For such surfaces, radiance leaving the surface is independent of angle
• Called Lambertian surfaces (same Lambert) or ideal diffuse surfaces
• Use radiosity as a unit to describe light leaving the surface
• DHR is often called diffuse reflectance, or albedo
• for a Lambertian surface, BRDF is independent of angle, too.
• Useful fact:
brdf d
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Specular surfaces
• Another important class of surfaces is specular, or mirror-like.– radiation arriving along a
direction leaves along the specular direction
– reflect about normal– some fraction is absorbed,
some reflected– on real surfaces, energy
usually goes into a lobe of directions
– can write a BRDF, but requires the use of funny functions
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Lambertian + specular
• Widespread model– all surfaces are Lambertian plus specular component
• Advantages– easy to manipulate
– very often quite close true
• Disadvantages– some surfaces are not
• e.g. underside of CD’s, feathers of many birds, blue spots on many marine crustaceans and fish, most rough surfaces, oil films (skin!), wet surfaces
– Generally, very little advantage in modelling behaviour of light at a surface in more detail -- it is quite difficult to understand behaviour of L+S surfaces
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Radiometry vs. Photometry
http://www.optics.arizona.edu/Palmer/rpfaq/rpfaq.htm
Name Unit Symbol Name Unit Symbolradiant flux watt (W) luminous flux lumen (lm)
irradiance/radiosity W/m2 E/M illuminance lm/m2 = lux (lx) Evradiant intensity W/sr I luminous intensity lm/sr = candela (cd) Iv
radiance W/m2-sr L luminance lm/m2-sr = cd/m2 = nit Lv
RADIOMETRIC QUANTITIES PHOTOMETRIC QUANTITIES
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Sensors
• CCD vs. CMOS
• Types of CCDs: linear, interline, full-frame, frame-transfer
• Bayer filters
• progressive scan vs. interlacing
• NTSC vs. PAL vs. SECAM
• framegrabbers
• blooming
F. Dellaert, http://www.cc.gatech.edu/~dellaert/vision/html/materials.html
Bayer color filter
http://en.wikipedia.org/wiki/Bayer_filter