Last Two Lectures Panoramic Image Stitching
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Last Two Lectures
Panoramic Image Stitching
Feature Detection and Matching
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TodayMore on Mosaic
Projective Geometry
Single View Modeling
Vermeer’s Music Lesson Reconstructions by Criminisi et al.
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Image Alignment
Feature Detection and MatchingCylinder:Translation2 DoF
Plane:Homography8 DoF
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Plane perspective mosaics
– 8-parameter generalization of affine motion• works for pure rotation or planar surfaces
– Limitations:• local minima • slow convergence
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Revisit Homography
x
(Xc,Yc,Zc)
xcf
ZYX
yfxf
yx
c
c
1000
0~
11
1
ZYX
yfxf
yx
c
c
R100
00
~1
2
2
211 ~ xxKRK
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Estimate f from H?
x
(Xc,Yc,Zc)
xcf
ZYX
ff
yyxx
c
c
1000000
~1
1
1
1
1
ZYX
ff
yyxx
c
c
R1000000
~1
2
2
2
2
211
12 ~)( xxRKK
H
{
1
222
1
1
11
2
***
//
~
ffjfifh
fgedfcba
HKKR
??, 21 ff
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The drifting problem
• Error accumulation– small errors accumulate over time
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Bundle AdjustmentAssociate each image i with iK iREach image i has features ijp
Trying to minimize total matching residuals
),(
211~) and all(mi j
mjmmiiijiifE pKRRKpR
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Rotations
• How do we represent rotation matrices?
1. Axis / angle (n,θ)R = I + sinθ [n] + (1- cosθ) [n]
2 (Rodriguez Formula), with [n] be the cross product matrix.
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Incremental rotation update
1. Small angle approximationΔR = I + sinθ [n] + (1- cosθ) [n]
2
≈ I +θ [n] = I+[ω] linear in ω= θn
2. Update original R matrixR ← R ΔR
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Recognizing Panoramas
[Brown & Lowe, ICCV’03]
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Finding the panoramas
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Finding the panoramas
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Algorithm overview
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Algorithm overview
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Algorithm overview
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Algorithm overview
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Algorithm overview
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Algorithm overview
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Finding the panoramas
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Finding the panoramas
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Algorithm overview
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Algorithm overview
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Algorithm overview
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Get you own copy!
[Brown & Lowe, ICCV 2003][Brown, Szeliski, Winder, CVPR’05]
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How well does this work?
Test on 100s of examples…
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How well does this work?
Test on 100s of examples…
…still too many failures (5-10%)for consumer application
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Matching Mistakes: False Positive
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Matching Mistakes: False Positive
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Matching Mistakes: False Negative
• Moving objects: large areas of disagreement
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Matching Mistakes
• Accidental alignment– repeated / similar regions
• Failed alignments– moving objects / parallax– low overlap– “feature-less” regions
• No 100% reliable algorithm?
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How can we fix these?
• Tune the feature detector• Tune the feature matcher (cost metric)• Tune the RANSAC stage (motion model)• Tune the verification stage• Use “higher-level” knowledge
– e.g., typical camera motions
• → Sounds like a big “learning” problem– Need a large training/test data set (panoramas)
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Enough of images!
We want more from the image
We want real 3D scene
walk-throughs:Camera rotationCamera
translation
on to 3D…
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So, what can we do here?
• Model the scene as a set of planes!
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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 does a line in the image correspond to in
projective space?
• A line is a plane of rays through origin– all rays (x,y,z) satisfying: ax + by + cz = 0
zyx
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– 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• can switch the meanings of points and lines to get another
formula
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
(a,b,0)y
xz image plane
• Corresponds to a line in the image (finite coordinates)
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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 postmultiplication 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 N is also represented by a 4-vector• Points and planes are dual in 4D: N P=0
– Projective transformations• Represented by 4x4 matrices T: P’ = TP, N’
= N T-1
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3D to 2D: “perspective” projection
• Matrix Projection: ΠPp
1************
ZYX
wwywx
What is not preserved under perspective projection?
What IS preserved?
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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 v– The ray from C through v is parallel to the lines– An image may have more than one vanishing point
• in fact every pixel is a potential 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
• also called vanishing line– 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
• also called vanishing line– 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 P
V
DPP t 0
0/1///
1Z
Y
X
ZZ
YY
XX
ZZ
YY
XX
t DDD
t
tDtPDtPDtP
tDPtDPtDP
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
• points higher than C project above 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
1
2
3
4
55.4
2.83.3
Camera height
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q1
Computing vanishing points (from lines)
• Intersect p1q1 with p2q2
v
p1
p2
q2
Least squares version• Better to use more than two lines and compute the “closest” point of
intersection• See notes by Bob Collins for one good way of doing this:
– http://www-2.cs.cmu.edu/~ph/869/www/notes/vanishing.txt
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C
Measuring height without a ruler
ground plane
Compute Z from image measurements• Need more than vanishing points to do this
Z
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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
1423
2413
PPPPPPPP
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
1i
i
i
i ZYX
P
3421
2431
PPPPPPPP
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vZ
rt
b
tvbrrvbt
Z
Z
image cross ratio
Measuring height
B (bottom of object)
T (top of object)
R (reference point)
ground plane
HC
TBRRBT
scene cross ratio
1ZYX
P
1yx
pscene points represented as image points as
RH
RH
R
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Measuring height
RH
vz
r
b
t
RH
Z
Z
tvbrrvbt
image cross ratio
H
b0
t0
vvx vy
vanishing line (horizon)
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Measuring height vz
r
b
t0
vx vy
vanishing line (horizon)
v
t0
m0
What if the point on the ground plane b0 is not known?• Here the guy is standing on the box, height of box is known• Use one side of the box to help find b0 as shown above
b0
t1
b1
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Computing (X,Y,Z) coordinates
• Okay, we know how to compute height (Z coords)– how can we compute X, Y?
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Camera calibration• Goal: estimate the camera parameters
– Version 1: solve for projection matrix
ΠXx
1************
ZYX
wwywx
• Version 2: solve for camera parameters separately– intrinsics (focal length, principle point, pixel size)– extrinsics (rotation angles, translation)– radial distortion
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Vanishing points and projection matrix
************
Π 4321 ππππ
1π 2π 3π 4π
T00011 Ππ = vx (X vanishing point)
Z3Y2 , similarly, vπvπ
origin worldof projection10004 TΠπ
ovvvΠ ZYXNot So Fast! We only know v’s up to a scale factor
ovvvΠ ZYX cba• Can fully specify by providing 3 reference points
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3D Modeling from a photograph
https://research.microsoft.com/vision/cambridge/3d/3dart.htm