GENERAL ANNOUNCEMENTS: - stnicholasoca.org … · Web viewGENERAL ANNOUNCEMENTS: - stnicholasoca.org
Announcements
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Transcript of Announcements
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• 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
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Today
Single View Modeling
Projective Geometry for Vision/Graphics
on Monday (1/22)• camera pose estimation and calibration• bundle adjustment• nonlinear optimization
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3D Modeling from a Photograph
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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)
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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
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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
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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
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Measurements on Planes
Approach: unwarp then measure
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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
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'
'
y
x
s
sy
sx
H pp’
To apply a homography H• compute p’ = Hp• p’’ = p’/s normalize by dividing by third component
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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
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(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
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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
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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
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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
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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
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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
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3D to 2D: “Perspective” Projection
Matrix Projection: ΠPp
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Z
Y
X
s
sy
sx
It’s useful to decompose into T R project A
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0100
0010
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projectionintrinsics orientation position
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Projection Models
Orthographic
Weak Perspective
Affine
Perspective
Projective
1000
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Π
1000yzyx
xzyx
tjjj
tiii
Π
tRΠ
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Π
1000yzyx
xzyx
tjjj
tiii
fΠ
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Properties of Projection
Preserves• Lines and conics• Incidence• Invariants (cross-ratio)
Does not preserve• Lengths• Angles• Parallelism
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Vanishing Points
Vanishing point• projection of a point at infinity
image plane
cameracenter
ground plane
vanishing point
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Vanishing Points (2D)
image plane
cameracenter
line on ground plane
vanishing point
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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
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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
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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
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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
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ZZ
YY
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YY
XX
t D
D
D
t
t
DtP
DtP
DtP
tDP
tDP
tDP
PP
ΠPv
P0
D
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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
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Fun With Vanishing Points
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Perspective Cues
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Perspective Cues
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Perspective Cues
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Comparing Heights
VanishingVanishingPointPoint
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Measuring Height
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Camera height
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C
Measuring Height Without a Ruler
ground plane
Compute Y from image measurements• Need more than vanishing points to do this
Y
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The Cross Ratio
A Projective Invariant• Something that does not change under projective
transformations (including perspective projection)
P1
P2
P3
P4
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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
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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
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• 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
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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
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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)
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3D Modeling from a Photograph
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3D Modeling from a Photograph
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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
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