computer vision no. 7: stereo part 2. today’s topics u projective geometry u camera projection...
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Computer Vision
No. 7: Stereo part 2
Today’s topics
Projective geometry Camera Projection Matrix F Matrix Multi View Shape Reconstruction
Elementary Projective Geometry
Topics
1. Projective Plane
2. Projective Space
3. Affine Space
4. Euclidean Space
Projective Plane (2D projective space)
In the Euclidean plane ),(: yxp
In the Projective plane )1,,(: yxp
Simply add a third coordinate of 1 at the end
Homogeneous coordinate
! Scaling is NOT important.
),,()1,,( yxyx Therefore the point (0, 0, 0) is disallowed in the projective plane.
)0,,( yx : Ideal points or Points at infinity
Description in Projective Plane
A line in the projective plane is described as…
A line through the point (x, y)…
0 cbyax
0),,()1,,( cbayx
),,( cbal
A line through a point
0 pllp tt
l p
Duality in Projective Plane
A point and a line ),,( cbal ),,( wyxp
The line through point and1p 2p 21 ppl
If point , and are on the same line1p 2p 3p
0det 321 ppp
The cross point of line and1l 2l 21 llp
If line , and are passing through the same point1l 2l 3l
0det 321 lll
Conics and Dual Conics
For symmetric 3 x 3 matrix …
All points which satisfy ….p
0pp tConic
Dual Conic
Therefore, for the matrix )( 1*
All lines which satisfy ….l
0* ll t
Transformation in Projective Plane
In the Projective plane, by 3 x 3 non-singular matrix T
A point is transformed intop p pTp
A line is transformed intol l
lTlplTpTTlpl ttttt )(0 1
A conic is…
1
11 )(0
TT
pTTppTTTTpppt
tttttt
Similarly a dual conic is…* tTT **
In the Euclidean space ),,(: ZYXP
In the Projective space )1,,,(: ZYXP
Homogeneous coordinate
! ),,,()1,,,( ZYXZYX
)0,,,( ZYX : Ideal points, Points at infinity
A plane in the projective space is described as…
),,,( DCBAL
A plane through a point 0 PLLP ttL ),,,( WZYXP
Projective Space (3D Projective space)
Quadrics and Dual Quadrics
For symmetric 4 x 4 matrix …All points which satisfy ….P
0PP tQuadric
Dual Quadric
Therefore, for the matrix )( 1*
All planes which satisfy ….L
0* LLt
Similarly a dual quadric is…* tTT **
A quadric is transformed by T into 1 TT t
N-Dimensional Affine Space
Among transformations T in the N-d projective space
0,0det,matrixsingular -non: 0
cANNA
c
bAT aat
aA
Affine Transformation
In N-D affine space… )0,( p
: point at infinity
The group of the points at infinity )1,0(
: plane at infinity
By an affine transformation
000
pAp
c
bA a
ta
a point at infinity
a point at infinity
N-Dimensional Euclidean Space
Among N-d affine transformations
0,,matrix orthogonal: 0
1
cAANNA
c
bAT e
teet
eE
Euclidean Transformation
AT
a group of points at infinity, which satisfy)}0,{( ep
0et
e pp
By an Euclidean transformation
0 et
eeet
et
eet
e pppAAppp
N=3: Absolute conic
N=2: Absolute point
3-Dimensional Euclidean Space
1,33: 0
RRR
c
bRT t
tE
By an Euclidean transformation
0
1
1
1
*E
Absolute Dual Quadric
***
0
0
00Ett
tEEEE
I
cb
RI
c
bRTT
The absolute dual quadric is invariant under Euclidean transformation!
*E
Camera Projection
Topics
1. Pinhole Camera Model
2. Camera Parameters
3. Lens Distortion
Pinhole Camera
Pinhole Camera Modelgeometry
(X, Y, Z)
Image plane
X
Y
-Zx
y
(x, y)
Z
Y
Z
Xyx ,),(
Perspective projection
View point
(Optical center)
(sX, sY, sZ)
Pinhole Camera ModelThe projected point on the image plane (z = -1)
321
2
1
3
,)(
)(
)(
1rrrR
tPr
tPr
tPry
xt
t
t
The Rotation matrix of camera pose
The camera position t
R
In Projective Geometry
PtIRP
tIRtPR
tPr
tPr
tPr
tPry
x
p ttt
t
t
t
t
,
1,
)(
)(
)(
)(
1
1 3
2
1
3
Intrinsic parameters
yAn ideal image on the Image plane
x
u
v
θ An actual picture
u0
v0
(x, y)
(u, v)
pAy
x
vk
ukk
v
u
u v
uu
1100
sin0
cot
10
0
the Principal point
skew
Intrinsic parameters
yAn ideal image on the Image plane
x
u
v
θ An actual picture
u0
v0
(x, y)
(u, v)
the Principal point
skew
pAy
x
vfa
usf
v
u
u
1100
0
10
0
f : Focal Length
a : aspect ratio
Camera Projection Matrix
PtIARpAv
u
u t
,
1
Transformation of a point in the 3D world into a 2D picture…
tIARP tc
,Camera Matrix (3 x 4)
Camera calibration
11 parameters: rotation 3 + translation 3 + intrinsic 5
Estimate of these 11 parameters (= camera matrix)
Lens distortion
Short focal length Long focal length
Removal of lens distortion
Lens DistortionZ.Zhang (‘00) PAMI
R. Sagawa (‘05) IROS Non-parametric
F Matrix
Topics
1. Epipolar Geometry
2. E & F Matrix
3. Homography
4. Multi View Geometry
Correspondence Problem
Given two images, we want to solve a problem……
For a point in image 1, decide which point in image 2 it corresponds to.
?
plane 1
plane 2
Eye 1Eye 2
Epipolar Geometry
C1
C2
3D point Epipolar plane
EpipoleEpipole
Epipolar line
Epipolar line
Baseline
Essential Matrix
C1
C2
t
2p
1p
xRt p
tt
tt
tt
ptpt
0
0
0
12
13
23 2pRt
021 pRtp t
Essential matrix : E
2pR
Essential & Fundamental Matrix
1p
2p
021 pEp t Image planes
Pictures
1u2u
pApAu
0)(
)()(
21
21
11
21
211
1
21
uAEAu
uAEuA
pEp
tt
t
t
Fundamental matrix : F
Image 1 Image 2
Fundamental Matrix
image 1image 2
1u 2u
constraint
121
21 0
ARtAF
uFut
t Depends on the camera configuration!!
Fundamental Matrix PtIRApAu
PtIRApAu
t
t
222222
111111
,
,
tRAPRAPtIRApAu
PAPIIApAu
ttt
t
222222
11111
,
0,
tRAuARAu tt 21
1122
tuRAuA
,, 21
211
1 are on the same plane!Three vectors
021
21121
211
1
uRAtAuuRAtuA ttt
Epipolar Constraint
0
1
1,, 2
2
1121
v
u
FvuuFu t
tt lFvuFu 1111 1,,
Epipolar Constraint
0
1
1,, 2
2
1121
v
u
FvuuFu t
22
2
2
1
lv
u
FuF
Property of F Matrix
12
12
13
23
11
21
0
0
0
RA
tt
tt
tt
AARtAF tt
at most Rank 2
0det F
0
0
0
,0
3
2
1
e
e
e
e
FeF
There are some points which satisfye
Null Space of F Matrix (epipole)
1, uF for 021 eFu t
1u2e
All epipolar lines in image 2 pass through the point 2e
Null Space of F Matrix (epipole)
2u1e
Similarly, 2ufor 021 uFe t
21,ee : Epipoles
All epipolar lines in image 1 pass through the point 1e
Epipoles
Epipoles:• intersections of baseline with image planes• projection of the optical center in another image• the vanishing points of camera motion direction
C1
C2
e1e2
Examples
Examples
Examples
Computing F Matrix (8-point algorithm)021 uFu t
0
1
1 2
2
333231
232221
131211
11
v
u
FFF
FFF
FFF
vu
01
33
32
31
23
22
21
13
12
11
112212122121
F
F
F
F
F
F
F
F
F
vuvvvvuuuvuu
1& F
Computing F Matrix
Epipolar pencil by linear solution (due to noise and error)
Computing F MatrixSingular value decomposition (SVD)
321
3
2
1
00
00
00
tVUF
2rank F Without noise, σ3 must be 0
modification
tVUF
000
00
00
2
1
Computing F Matrix
tVUF
3
2
1
00
00
00
tVUF
000
00
00
2
1
Homography
ij
d
n
P
In Camera 1
Camera 1
Camera 2
PAPIAu
111 0
1Pnd t
The distance between camera 1 and the plane
In Camera 2
121222 Pd
ntIRAtPRAPtIRAu
tttt
Homography and F Matrix
ij
d
n
P
Camera 1
Camera 2
11
122 uAd
ntIRAu
tt
12 uHu
0111
112
121
ttAtAAd
ntIRARAtAFH t
ttt
0)( 111
111
111
AttAAtAAtAFHFH tttttt
Planer homography mosaicing
Planer homography mosaicing
camera
Image plane’
Image rectification
Camera 1 Camera 2
Rectification
Fusiello (MVA 2000) Pollefeys
Constraints of 3 images
L
1l2l
3l
13
12
11
1
l
l
l
l
23
22
21
2
l
l
l
l
33
32
31
3
l
l
l
l
Line – line – line
13
332
12
322
11
312
llTl
llTl
llTl
t
t
t
Tensor TrifocaliT
3 x 3 matrix
Constraints of 3 imagesL
1l2l
3l
Line – line – line
132i
ijkkj lTll
Tensor description
L
2l
3l
Point – line – line
1u 0321 ijkkji Tllu
13
12
11
1
u
u
u
u
Constraints of 3 images
Point – point – line
ri
pkkjprji Tluu 0321
rsi
pqkqskjprji Tuuu 0321
L
2u
3l
1u
Point – point – point
2u
3u
1u
otherwise
123 ofn permutatio oddan is
123 ofn permutatioeven an is
0
1
1
rst
rst
rst
2-Views Geometry
2u1u
X
matrix 43 scalar,
ii
ii
i
P
XPu
00
02
122
11
X
uP
uP
M : (3 x 2) x (4 + 2) matrix
0detif M 0
2
1
X
2-Views Geometry
2u1u
X
matrix 43 scalar,
ii
ii
i
P
XPu
00
02
122
11
X
uP
uP
M : (3 x 2) x (4 + 2) matrix
0det M 021 ijji Fuu
In tensor form
3-Views Geometry
2u
3u
1u
X
matrix 43 scalar,
ii
ii
i
P
XPu
00
0
00
0
0
3
2
1
33
22
11
X
uP
uP
uP
M : (3 x 3) x (4 + 3) matrix
0det M rsi
pqkqskjprji Tuuu 0321
4-Views Geometry
2u
3u
1u
X
00
0
0
000
00
00
00
4
3
2
1
44
33
22
11
X
uP
uP
uP
uP
0det M
wxyzpqrslszlkrykjqxjipwi Quuuu 04321
4u
tensorlquadrifocapqrsQ
Camera Calibration
Topics
1. 3D Ref.
2. 2D Ref.
3. 1D Ref.
4. Self Calibration (Kruppa equations)
The meaning of camera calibration
tIARP tc
,
0
1
1
1
*E
Absolute Dual Quadric
*E
ttt
ttcEc AARA
t
IItIARPP
0**
Camera Calibration = to estimate the image of the absolute quadric
*
Camera Calibration (3D Ref.)
11 34
24
14
333231
232221
131211
Z
Y
X
P
P
P
PPP
PPP
PPP
v
u
known
tcec
t PPAA **
tIARP tc
,
Homography ( 2D-2D mapping )
11 34
24
14
333231
232221
131211
Z
Y
X
P
P
P
PPP
PPP
PPP
v
u
ij
d
(x, y)
1100
1
y
xdji
Z
Y
X
xHy
x
MPv
u
u c
11
(u, v)
Homography (3 x 3)
the origin
Camera Calibration (2D Ref.)
Without loss of generality…
0,
tIR
0
IAPc
djiAdji
AMPH c
1000
Z.Zhang (‘00) PAMI
Camera Calibration (2D Ref.)
djiAH
11 333231
232221
131211
y
x
HHH
HHH
HHH
v
u
known
0
1
ji
jjiit
tt
0
1
t
tttt AA
Camera Calibration (2D Ref. Results)
Camera Calibration (1D Ref.)Z.Zhang (‘04) PAMI
1P
3P
2P1u
3u
2u
213 )1( PPP
On the line in 3D
2133 )1( uuPPu c
22111 ,0 PAuPAPIAu
1
1,1)1(
121
122
11
212
122 uuAAuuuAuAPPL tt
:const. 1P :fix
Camera Calibration (1D Ref. Results)
Self Calibration
In most cases, there are NOT any reference objects…
be able to calibrate the camera by unknown scenes
Homography at infinity
P
Camera 1
Camera 2
Considering P on the infinity plane …
)0,,,()0,( ZYXPP t
0222 ZYXPP t
and on the absolute quadric…
PRPtIRp tt
0 PPPRRPpp tttt
In camera 2 coordinate system
The image of the absolute quadric
P
Camera 1
Camera 2
PAu
11
The image of the absolute quadric at the infinity plane
0111
11
111
uu
uAAuPPt
ttt
011*
111 ueue t
01*
11 ll t
The Kruppa equations
P
Camera 1
Camera 2
1
011
*111
11*
111
ueeu
ueue
tt
t
02*
22 ll t
tt lFu 21
01*
21 uFFu tt
tt FFee *21
*11
2Given the F Matrix…
References
Multiple View GeometryHartley and Zisserman, Cambridge
The Geometry of Multiple ImagesFaugeras and Luong, MIT
Three-Dimensional Computer VisionFaugeras, MIT
Algebraic Projective GeometrySemple and Kneebone, Oxford
Multiple View Shape Reconstruction
Topics
•Basic Principle•Surface Approach
Multi-baseline stereo•Volumetric Approach
Volume intersection method•Combinatorial Approach
Photo hull
•Extended Algorithm•Level Set Approach•Graph-cut Approach
Basic Principle #1
Surface Approach
Surface Approach Try to reconstruct a surface of an object
– Find depth d that minimizes some matching cost function F (d) defined on N views
Camera 1
Camera 2Camera N
)(dF
d
d
Matching Cost Function
2
11,
),(),()(
wcc
cNc
ybd
fxIyxIdF
)),(()),((
)),(),,(()(
1
1
1, ybdf
xIVaryxIVar
ybdf
xIyxICovdF
cc
cc
cNc
(b) SSSD (sum of sum. of squared difference)
(a) SSAD (sum. of sum of absolute difference)
w
cccNc
ybd
fxIyxIdF ),(),()( 1
1,
(c) SNCC (sum of normalized cross correlation)
Camera 1 (reference) Camera 2 Camera N
Epipolar line
2bd
fNb
d
f
Why Multiple Views? Binocular stereo is sometimes ambiguous
e.x. F(d) = SAD (sum. of absolute difference)
w
ybd
fxIyxI ),(),( 21
d
Camera 1 Camera 2
)(dF
d
Which is the right match?
b
Multi-baseline Stereo
T. Kanade, M. Okutomi, and T. Nakahara, "A Multiple-Baseline Stereo Method," Proc. ARPA Image Understanding Workshop, pp. 409-426, 1992.
Camera 1(reference)
Camera 2
d
Camera 3 Camera N
d
SAD2
d
SSAD
d
SAD3
d
SADN
Unique minima
cSAD
Video-rate Stereo Vision Hardware implementation is easy
S. Kimura, T. Kanade, H. Kano, A. Yoshida, E. Kawamura, K. Oda, “CMU video-rate stereo machine", Proc. of Mobile Mapping Symposium, 1995
Image capture
Apply LoG filter
SSAD computationDisparity computation
Block matchingLUT instead of rectificationSAD computation
2
22
22
22
4 21
1),(
yx
eyx
yxLoG
Real-time Robot Vision
9 cameras stereo
Real-time Robot Vision
Model-based 3D Object Tracking
Virtualized Reality ™ 51 video cameras surrounding the scene
– Recorded to VCR
– Off-line 3D modeling
T. Kanade, P. Rander, P. J. Narayanan, “Virtualized Reality: Constructing Virtual Worlds from Real Scenes”, IEEE MultiMedia, vol.4, no.1, pp.34-47, 1997
Large scene from overlapped dense stereo images
Input images
3D video
Basic Principles #2
Volumetric Approach
Volumetric Approach Try to reconstruct a volume that contains an object
– Detect silhouette in each image– Backproject each silhouette– Intersect backprojected volumes
Binary ImagesBinary Images
Volume Intersection
Reconstruction contains the true scene– But is generally not the same
Visual Hull Minimum convex hull of an object
⇒ A volume obtained from infinite number of surrounding cameras
Visual hullTrue shape
Reconstructed
volume
Concave area is never recovered
Bound of the reconstruction
Voxel Algorithm for Volume Intersection
Find voxel that is projected inside silhouette in all the images– O(NM3), for N images, M3 voxels
– Don’t have to search 2M3 possible scenes!
Octree representation
Acceleration via Octree
Find voxel that is projected inside silhouette in all the images– Multi-resolution approach– Voxels that project to only background pixels in any image are carved– Voxels that project to only foreground pixels in all images remain– Ambiguous voxels are subdivided
Resolution 1Resolution 1
Acceleration via Octree
Find voxel that is projected inside silhouette in all the images– Multi-resolution approach– Voxels that project to only background pixels in any image are carved– Voxels that project to only foreground pixels in all images remain– Ambiguous voxels are subdivided
Resolution 2Resolution 2
Acceleration via Octree
Find voxel that is projected inside silhouette in all the images– Multi-resolution approach– Voxels that project to only background pixels in any image are carved– Voxels that project to only foreground pixels in all images remain– Ambiguous voxels are subdivided
Resolution 3Resolution 3
Results Reconstruction from 4 synthetic images
Silhouette images
16x16x16 voxel space 32x32x32 voxel space 64x64x64 voxel space
How do we get silhouette images?
Dependent on the purpose of the application
– Interactive Foreground Separation » Building static 3d model from small number of images
– Statistical Background Subtraction » Motion capture system in controlled environment from large
number of image sequence (multiple video streams)
Interactive Foreground Separation
Use of an interactive segmentation program
Y. Li, J. Sun, C.K. Tang, H.Y. Shum, “Lazy snapping,” ACM Trans. on Graphics Vol.23,No. 3, pp.303—308, 2004.
Foreground region Background region
Separation Wrong segmentation
Background region
Separation
See “Example of Graph-cut” in the lecture 6
Building Static 3d Model Volume intersection from 8 images
– No concavities
•Brightness component
•Chromaticity component
Statistical Background Subtraction
i
i
2)min(arg iiii EE
|||| iii IECD
iE iI
Background image Image of interest
1 i
CDiCD if then pixel Ii is foreground
else if then pixel Ii is shadow
else pixel Ii is shadow
Choice of Threshold Learn statistics of brightness and chromaticity component for
each pixel
1 2 CDBrightness distribution Chromaticity distribution
P % P %P is around 99%
Motion Capture System in Controlled Environment
Input image Foreground & shadow Silhouette image
Basic Principles #3
Combinatorial Approach
Combinatorial Approach1. Start with an initial volume
Arbitrary rectangular shape Result of volume intersection
2. The volume is iteratively carved to be getting closer to the true shape Considering photo-consictency 2 approaches: Voxel coloring and Space-carving
Carve an inconsistentsurface voxel
Photo-consistency
Color variance of the imaged voxel among the cameras
If color variance > thresholdthe voxel is carved
If color variance < thresholdthe voxel is preserved
Voxel of interest
Voxel of interest
True surface
Voxel Coloring Approach
Input ImagesInput Images
(Calibrated)(Calibrated)
Goal: Goal: Assign RGBA values to voxels in VAssign RGBA values to voxels in Vphoto-consistentphoto-consistent with images with images
Discrete formulation
1. Choose voxel1. Choose voxel2. Project and correlate2. Project and correlate3. Color if photo-consistent3. Color if photo-consistent
Visibility Problem
In what order do we have to choose a voxel?
In which images is each voxel visible?
The Global Visibility Problem
Inverse Visibility known images
Unknown SceneUnknown Scene
Which points are visible in which images?
Known SceneKnown Scene
Forward Visibility Unknown images
LayersLayers
Solution: Depth Ordering
SceneScene
TraversalTraversal
Condition:Condition:
voxel space and group of cameras must be completely separatedvoxel space and group of cameras must be completely separated
Visit occluders first
Space-carving Algorithm
General case
Image 1 Image N
…...
– Initialize to a volume V containing the true scene
– Repeat until convergence
– Choose a voxel on the current surface
– Carve if not photo-consistent– Project to visible input images
Output of Space-carving: Photo hull
The Photo Hull is the UNION of all photo-consistent scenes in V
– Tightest possible bound on the true scene
True SceneTrue Scene
VV
Photo HullPhoto Hull
VV
Algorithm finds a surface consistent with the input images, but not necessarily the actual surface
Typical Error in Photo hull
actual surface
reconstructed surface
Cusping artifact Floating voxel artifact
Space-carving Implementation Multi-pass plane sweep
– Efficient, easy to implement
Sweep in –z direction Sweep in -x direction Sweep in -y direction
Repeat
Results
Details are recovered
Volume intersection Space carving Space carvingwith textures
Excessively carved
Input image
Using 16 images300k polygons
Extended Algorithm #1
Level Set Approach
Level Set Approach Define problem in 1 higher dimension
– An additional dimension, time t, is added φ(p(t),t). [Osher98]» P(t) is a point in the space
» Φ(p(t),t) is a signed distance function
– Surface, zero level set φ=0, moves along its normal direction
– Front propagation is guided by Partial Differential Equation (PDE).
– Cross-correlation is used as a constraint in PDE. [Faugeras98]
0)),(( 11 ttp
0)0),0(( p
True shape
How to move the level set surface
Define a velocity field, F, that specifies surface points move in time– In the case of stereo, F is a function of the matching cost on p
(t) in multi-baseline stereo
– F decreases to 0 around the true surface
Build an initial value for the level set function, φ(p(0),0), based on the initial surface
Adjust φ over time– Surface at time t is defined by φ(p(t),t)=0
Level Set Formulation Constraint: level set value of a point p(t) on the surface must alwa
ys 0
By the chain rule
Since F supplies the speed in the outward normal direction
Hence evaluation equation for φ is
0)),(( ttp
0)(')),(( tpttpt
/ where,)(' nFntp
0 Ft
Essentially a gradient-descent local optimization problem
Front Propagation by PDE Front Propagation can be efficiently calculated in level
set framework– Fast marching method
– Narrow band method
Results
Can handle topology change
Extended Algorithm #2
Graph-cut Approach
GoalCalibrated images of Lambertian scene
3D model of scene
Volumetric Graph-cuts Energy function to minimize
][][][ SESESE volsurf dAxSE Ssurf )(][
Outer surface (Visual hull)
Inner surface (at constant offset from outer surface)
dVSE SVvol )(][ Volume between outer surface and S
(to avoid over-smoothing)
Reconstructed surface (S)
Integral of photo-consistency over S(S should be photo-consistent))(x Photo-consistency
Designing a Graph
1. Assign a node for each voxel
2. Connect neighboring nodes– Weight is set as ρ(x)
3. Connect outer nodes to source– Weight is infinite
4. Connect inner nodes to sink– Weight is infinite
Outer surface
Inner surface
Sink
Source
)(x
Graph-cut algorithm minimizes the energy
Designing a Graph
1. Connect all the nodes to source– Weight is λ
Outer surface
Inner surface
Sink
Source
A cut that separates source and sink
is equal to
][][][ SESESE volsurf
Cut
Results
Volume intersection
Volumetric Graph-cut
Summary Multi-baseline stereo for accurate depth map
– Good for real-time applications, dense depth map is obtained directly
Voxel-based reconstruction– Volume intersection for conservative estimate– 2 background subtraction methods for different applications– Concavities can be recovered by considering Photo-consistency
Extended algorithm– 2 approaches to add smoothness constraint– Level set approach solves local optimization problem
» can handle topology change
– Graph-cut approach solves global optimization problem» cannot handle topology change
Volume intersection– Martin & Aggarwal, “Volumetric description of objects from multiple views”, Tran
s. Pattern Analysis and Machine Intelligence, 5(2), 1991, pp. 150-158.– Szeliski, “Rapid Octree Construction from Image Sequences”, Computer Vision, Gr
aphics, and Image Processing: Image Understanding, 58(1), 1993, pp. 23-32.
Voxel coloring and Space-carving– Seitz & Dyer, “Photorealistic Scene Reconstruction by Voxel Coloring”, Proc. Comp
uter Vision and Pattern Recognition (CVPR), 1997, pp. 1067-1073.– Kutulakos & Seitz, “A Theory of Shape by Space Carving”, Proc. ICCV, 1998, pp.
307-314.
Extended algorithm– Faugeras & Keriven, “Variational principles, surface evolution, PDE's, level set met
hods and the stereo problem", IEEE Trans. on Image Processing, 7(3), 1998, pp. 336-344.
– G. Vogiatzis and P. H. S. Torr and R. Cipolla, “Multi-View Stereo via Volumetric Graph-Cuts”, Proc. CVPR, 2005, pp.391-398.
References
Next Week
6/7 Motion understanding– Prof. Ikeuchi
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