announcements
DESCRIPTION
Announcements. Email list – let me know if you have NOT been getting mail Assignment hand in procedure web page at least 3 panoramas: (1) sample sequence, (2) w/Kaidan, (3) handheld hand procedure will be posted online Readings for Monday (via web site) Seminar on Thursday, Jan 18 - PowerPoint PPT PresentationTRANSCRIPT
• Email list – let me know if you have NOT been getting mail• Assignment hand in procedure
– web page
– at least 3 panoramas:» (1) sample sequence, (2) w/Kaidan, (3) handheld
– hand procedure will be posted online
• Readings for Monday (via web site)• Seminar on Thursday, Jan 18
Ramin Zabih:
“Energy Minimization for Computer Vision via Graph Cuts “
• Correction to radial distortion term
Announcements
Today
Single View Modeling
Projective Geometry for Vision/Graphics
on Monday (1/22)• camera pose estimation and calibration• bundle adjustment• nonlinear optimization
3D Modeling from a Photograph
Shape from X
Shape from X• Shape from shading• Shape from texture• Shape from focus• Others…
Make strong assumptions – limited applicability
Lee & Kau 91 (from survey by Zhang et al., 1999)
Model-Based Techniques
Blanz & Vetter, Siggraph 1999
Model-based reconstruction• Acquire a prototype face, or database of prototypes• Find best fit of prototype(s) to new image
User Input + Projective Geometry
• Horry et al., “Tour Into the Picture”, SIGGRAPH 96
• Criminisi et al., “Single View Metrology”, ICCV 1999
• Shum et al., CVPR 98
• Images above by Jing Xiao, CMU
Projective Geometry for Vision & Graphics
Uses of projective geometry• Drawing• Measurements• Mathematics for projection• Undistorting images• Focus of expansion• Camera pose estimation, match move• Object recognition via invariants
Today: single-view projective geometry• Projective representation• Point-line duality• Vanishing points/lines• Homographies• The Cross-Ratio
Later: multi-view geometry
1 2 3 4
1
2
3
4
Measurements on Planes
Approach: unwarp then measure
Planar Projective Transformations
Perspective projection of a plane• lots of names for this:
– homography, texture-map, colineation, planar projective map
• Easily modeled using homogeneous coordinates
1***
***
***
'
'
y
x
s
sy
sx
H pp’
To apply a homography H• compute p’ = Hp• p’’ = p’/s normalize by dividing by third component
Image Rectification
To unwarp (rectify) an image• solve for H given p’’ and p• solve equations of the form: sp’’ = Hp
– linear in unknowns: s and coefficients of H
– need at least 4 points
(0,0,0)
The Projective Plane
Why do we need homogeneous coordinates?• represent points at infinity, homographies, perspective projection,
multi-view relationships
What is the geometric intuition?• a point in the image is a ray in projective space
(sx,sy,s)
• Each point (x,y) on the plane is represented by a ray (sx,sy,s)– all points on the ray are equivalent: (x, y, 1) (sx, sy, s)
image plane
(x,y,1)y
xz
Projective Lines
What is a line in projective space?
• A line is a plane of rays through origin
• all rays (x,y,z) satisfying: ax + by + cz = 0
z
y
x
cba0 :notationvectorin
• A line is also represented as a homogeneous 3-vector ll p
l
Point and Line Duality• A line l is a homogeneous 3-vector (a ray)• It is to every point (ray) p on the line: l p=0
p1p2
• What is the intersection of two lines l1 and l2 ?
• p is to l1 and l2 p = l1 l2
• Points and lines are dual in projective space• every property of points also applies to lines
l1
l2
p
• What is the line l spanned by rays p1 and p2 ?
• l is to p1 and p2 l = p1 p2
• l is the plane normal
Ideal points and lines
Ideal point (“point at infinity”)• p (x, y, 0) – parallel to image plane• It has infinite image coordinates
(sx,sy,0)y
xz image plane
Ideal line• l (a, b, 0) – parallel to image plane• Corresponds to a line in the image (finite coordinates)
(sx,sy,0)y
xz image plane
Homographies of Points and Lines
Computed by 3x3 matrix multiplication• To transform a point: p’ = Hp• To transform a line: lp=0 l’p’=0
– 0 = lp = lH-1Hp = lH-1p’ l’ = lH-1
– lines are transformed by premultiplication of H-1
3D Projective Geometry
These concepts generalize naturally to 3D• Homogeneous coordinates
– Projective 3D points have four coords: P = (X,Y,Z,W)
• Duality– A plane L is also represented by a 4-vector
– Points and planes are dual in 3D: L P=0
• Projective transformations– Represented by 4x4 matrices T: P’ = TP, L’ = L T-1
However• Can’t use cross-products in 4D. We need new tools
– Grassman-Cayley Algebra» generalization of cross product, allows interactions between
points, lines, and planes via “meet” and “join” operators
– Won’t get into this stuff today
3D to 2D: “Perspective” Projection
Matrix Projection: ΠPp
1****
****
****
Z
Y
X
s
sy
sx
It’s useful to decompose into T R project A
11
0100
0010
0001
100
0
0
31
1333
31
1333
x
xx
x
xxyy
xx
f
ts
ts
00
0 TIRΠ
projectionintrinsics orientation position
Projection Models
Orthographic
Weak Perspective
Affine
Perspective
Projective
1000
****
****
Π
1000yzyx
xzyx
tjjj
tiii
Π
tRΠ
****
****
****
Π
1000yzyx
xzyx
tjjj
tiii
fΠ
Properties of Projection
Preserves• Lines and conics• Incidence• Invariants (cross-ratio)
Does not preserve• Lengths• Angles• Parallelism
Vanishing Points
Vanishing point• projection of a point at infinity
image plane
cameracenter
ground plane
vanishing point
Vanishing Points (2D)
image plane
cameracenter
line on ground plane
vanishing point
Vanishing Points
Properties• Any two parallel lines have the same vanishing point• The ray from C through v point is parallel to the lines• An image may have more than one vanishing point
image plane
cameracenter
C
line on ground plane
vanishing point V
line on ground plane
Vanishing Lines
Multiple Vanishing Points• Any set of parallel lines on the plane define a vanishing point• The union of all of these vanishing points is the horizon line
– called vanishing line in the Criminisi paper• Note that different planes define different vanishing lines
v1 v2
Vanishing Lines
Multiple Vanishing Points• Any set of parallel lines on the plane define a vanishing point• The union of all of these vanishing points is the horizon line
– called vanishing line in the Criminisi paper• Note that different planes define different vanishing lines
Computing Vanishing Points
Properties• P is a point at infinity, v is its projection
• They depend only on line direction
• Parallel lines P0 + tD, P1 + tD intersect at X
V
DPP t 0
0/1
/
/
/
1Z
Y
X
ZZ
YY
XX
ZZ
YY
XX
t D
D
D
t
t
DtP
DtP
DtP
tDP
tDP
tDP
PP
ΠPv
P0
D
Computing Vanishing Lines
Properties• l is intersection of horizontal plane through C with image plane
• Compute l from two sets of parallel lines on ground plane
• All points at same height as C project to l
• Provides way of comparing height of objects in the scene
ground plane
lC
Fun With Vanishing Points
Perspective Cues
Perspective Cues
Perspective Cues
Comparing Heights
VanishingVanishingPointPoint
Measuring Height
1
2
3
4
55.4
2.8
3.3
Camera height
C
Measuring Height Without a Ruler
ground plane
Compute Y from image measurements• Need more than vanishing points to do this
Y
The Cross Ratio
A Projective Invariant• Something that does not change under projective
transformations (including perspective projection)
P1
P2
P3
P4
1423
2413
PPPP
PPPP
The Cross-Ratio of 4 Collinear Points
Can permute the point ordering• 4! = 24 different orders (but only 6 distinct values)
This is the fundamental invariant of projective geometry• likely that all other invariants derived from cross-ratio
vZ
vL
t
b
YZL
LZ
bvvt
vvbtImage Cross Ratio
Measuring Height
B (bottom of object)
T (top of object)
L (height of camera on the this line)
ground plane
YC
YY
LTBLT
LBT 1Scene Cross Ratio
• Eliminating and yields
• Can calculate given a known height in scenetvbl
tb
Y
TY
t
C
Measuring Height
reference plane TZX 10
TZYX 1
Algebraic Derivation• •
lvvΠb ZXT ZbXaZX 10
lvvvΠt ZYXT ZYbXaZYX 1
b
Y
So Far
Can measure• Positions within a plane• Height
– More generally—distance between any two parallel planes
These capabilities provide sufficient tools for single-view modeling
To do:• Compute camera projection matrix
Vanishing Points and Projection Matrix
4321 ππππΠ
lvvvΠ ZYX
Not So Fast! We only know v’s up to a scale factor
lvvvΠ ZYX ba
T00011 Ππ
Z3Y2 , similarly, vπvπ
origin worldof projection10004 TΠπ
lvv
vvπ thiscall choose toconvenient 4
YX
YX
• Can fully specify by providing 3 reference lengths
= vx (X vanishing point)
3D Modeling from a Photograph
3D Modeling from a Photograph
Conics
Conic (= quadratic curve)• Defined by matrix equation
• Can also be defined using the cross-ratio of lines• “Preserved” under projective transformations
– Homography of a circle is a…
– 3D version: quadric surfaces are preserved
• Other interesting properties (Sec. 23.6 in Mundy/Zisserman)– Tangency
– Bipolar lines
– Silhouettes
0
1***
***
***
1
y
x
yx