Stereo and Projective Structure from Motion
Computer VisionCS 543 / ECE 549
University of Illinois
Derek Hoiem
04/13/10
Many slides adapted from Lana Lazebnik, Silvio Saverese, Steve Seitz
This class
• Recap of epipolar geometry
• Recovering structure– Generally, how can we estimate 3D positions for
matched points in two images? (triangulation)– If we have a moving camera, how can we recover
3D points? (projective structure from motion)– If we have a calibrated stereo pair, how can we
get dense depth estimates? (stereo fusion)
Basic Questions
• Why can’t we get depth if the camera doesn’t translate?
• Why can’t we get a nice panorama if the camera does translate?
Recap: Epipoles
C
• Point x in left image corresponds to epipolar line l’ in right image
• Epipolar line passes through the epipole (the intersection of the cameras’ baseline with the image plane
C
Recap: Fundamental Matrix• Fundamental matrix maps from a point in one
image to a line in the other
• If x and x’ correspond to the same 3d point X:
Recap: Automatically Relating Projections
Homography (No Translation) Fundamental Matrix (Translation)
Assume we have matched points x x’ with outliers
Recap: Automatically Relating Projections
Homography (No Translation)
• Correspondence Relation
1. Normalize image coordinates
2. RANSAC with 4 points
3. De-normalize:
Assume we have matched points x x’ with outliers
0HxxHxx ''
Txx ~ xTx ~
THTH~1
Fundamental Matrix (Translation)
Recap: Automatically Relating Projections
Homography (No Translation) Fundamental Matrix (Translation)
• Correspondence Relation
1. Normalize image coordinates
2. RANSAC with 8 points3. Enforce by
SVD4. De-normalize:
• Correspondence Relation
1. Normalize image coordinates
2. RANSAC with 4 points
3. De-normalize:
Assume we have matched points x x’ with outliers
0HxxHxx ''
Txx ~ xTx ~
THTH~1
Txx ~ xTx ~
TFTF~1
0~
det F
0 Fxx T
Recap• We can get projection matrices P and P’ up to a
projective ambiguity
• Code:function P = vgg_P_from_F(F)[U,S,V] = svd(F);e = U(:,3);P = [-vgg_contreps(e)*F e];
0IP | e|FeP 0 Fe T
See HZ p. 255-256
Recap• Fundamental matrix song
Triangulation: Linear Solution
• Generally, rays Cx and C’x’ will not exactly intersect
• Can solve via SVD, finding a least squares solution to a system of equations
X
x x'
0 XPx 0PXx
0AX
TT
TT
TT
TT
v
u
v
u
23
13
23
13
pp
pp
pp
pp
A
Further reading: HZ p. 312-313
Triangulation: Linear SolutionGiven P, P’, x, x’1. Precondition points and projection
matrices2. Create matrix A3. [U, S, V] = svd(A)4. X = V(:, end)
Pros and Cons• Works for any number of
corresponding images• Not projectively invariant
1
v
u
x
1
v
u
x
T
T
T
3
2
1
p
p
p
P
TT
TT
TT
TT
v
u
v
u
23
13
23
13
pp
pp
pp
pp
A
T
T
T
3
2
1
p
p
p
P
Code: http://www.robots.ox.ac.uk/~vgg/hzbook/code/vgg_multiview/vgg_X_from_xP_lin.m
Triangulation: Non-linear Solution• Minimize projected error while satisfying
xTFx=0
• Solution is a 6-degree polynomial of t, minimizing
Further reading: HZ p. 318
Projective structure from motion• Given: m images of n fixed 3D points
• xij = Pi Xj , i = 1,… , m, j = 1, … , n • Problem: estimate m projection matrices Pi and n 3D points Xj from
the mn corresponding points xij
x1j
x2j
x3j
Xj
P1
P2
P3
Slides from Lana Lazebnik
Projective structure from motion• Given: m images of n fixed 3D points
• xij = Pi Xj , i = 1,… , m, j = 1, … , n • Problem: estimate m projection matrices Pi
and n 3D points Xj from the mn corresponding points xij
• With no calibration info, cameras and points can only be recovered up to a 4x4 projective transformation Q:
• X → QX, P → PQ-1
• We can solve for structure and motion when • 2mn >= 11m +3n – 15
• For two cameras, at least 7 points are needed
Sequential structure from motion•Initialize motion from two images using fundamental matrix
•Initialize structure by triangulation
•For each additional view:– Determine projection matrix of
new camera using all the known 3D points that are visible in its image – calibration ca
mer
as
points
Sequential structure from motion•Initialize motion from two images using fundamental matrix
•Initialize structure by triangulation
•For each additional view:– Determine projection matrix of new
camera using all the known 3D points that are visible in its image – calibration
– Refine and extend structure: compute new 3D points, re-optimize existing points that are also seen by this camera – triangulation
cam
eras
points
Sequential structure from motion•Initialize motion from two images using fundamental matrix
•Initialize structure by triangulation
•For each additional view:– Determine projection matrix of new
camera using all the known 3D points that are visible in its image – calibration
– Refine and extend structure: compute new 3D points, re-optimize existing points that are also seen by this camera – triangulation
•Refine structure and motion: bundle adjustment
cam
eras
points
Bundle adjustment• Non-linear method for refining structure and motion• Minimizing reprojection error
2
1 1
,),(
m
i
n
jjiijDE XPxXP
x1j
x2j
x3j
Xj
P1
P2
P3
P1Xj
P2XjP3Xj
Self-calibration• Self-calibration (auto-calibration) is the process
of determining intrinsic camera parameters directly from uncalibrated images
• For example, when the images are acquired by a single moving camera, we can use the constraint that the intrinsic parameter matrix remains fixed for all the images– Compute initial projective reconstruction and find 3D
projective transformation matrix Q such that all camera matrices are in the form Pi = K [Ri | ti]
• Can use constraints on the form of the calibration matrix: zero skew
Summary so far• From two images, we can:
– Recover fundamental matrix F– Recover canonical cameras P and P’ from F– Estimate 3d position values X for corresponding
points x and x’
• For a moving camera, we can:– Initialize by computing F, P, X for two images– Sequentially add new images, computing new P,
refining X, and adding points– Auto-calibrate assuming fixed calibration matrix to
upgrade to similarity transform
Photo synth
Noah Snavely, Steven M. Seitz, Richard Szeliski, "Photo tourism: Exploring photo collections in 3D," SIGGRAPH 2006
http://photosynth.net/
3D from multiple images
Building Rome in a Day: Agarwal et al. 2009
Plug: Steve Seitz Talk
• Steve Seitz will talk about “Reconstructing the World from Photos on the Internet”– Monday, April 26th, 4pm in Siebel Center
Special case: Dense binocular stereo• Fuse a calibrated binocular stereo pair to
produce a depth imageimage 1 image 2
Dense depth map
Many of these slides adapted from Steve Seitz and Lana Lazebnik
Basic stereo matching algorithm
• For each pixel in the first image– Find corresponding epipolar line in the right image– Examine all pixels on the epipolar line and pick the best match– Triangulate the matches to get depth information
• Simplest case: epipolar lines are scanlines– When does this happen?
Simplest Case: Parallel images• Image planes of cameras are
parallel to each other and to the baseline
• Camera centers are at same height
• Focal lengths are the same
Simplest Case: Parallel images• Image planes of cameras are
parallel to each other and to the baseline
• Camera centers are at same height
• Focal lengths are the same• Then, epipolar lines fall along
the horizontal scan lines of the images
Special case of fundamental matrix
RtExExT ][,0
00
00
000
][
T
TRtE
Epipolar constraint:
vTTv
vT
Tvuv
u
T
Tvu
0
0
10
100
00
000
1
R = I t = (T, 0, 0)
The y-coordinates of corresponding points are the same!
t
x
x’
Depth from disparity
f
x x’
BaselineB
z
O O’
X
f
z
fBxxdisparity
Disparity is inversely proportional to depth!
Stereo image rectification
Stereo image rectification
• Reproject image planes onto a common plane parallel to the line between optical centers
• Pixel motion is horizontal after this transformation
• Two homographies (3x3 transform), one for each input image reprojection
C. Loop and Z. Zhang. Computing Rectifying Homographies for Stereo Vision. IEEE Conf. Computer Vision and Pattern Recognition, 1999.
Rectification example
Basic stereo matching algorithm
• If necessary, rectify the two stereo images to transform epipolar lines into scanlines
• For each pixel x in the first image– Find corresponding epipolar scanline in the right image– Examine all pixels on the scanline and pick the best match x’– Compute disparity x-x’ and set depth(x) = 1/(x-x’)
Matching cost
disparity
Left Right
scanline
Correspondence search
• Slide a window along the right scanline and compare contents of that window with the reference window in the left image
• Matching cost: SSD or normalized correlation
Left Right
scanline
Correspondence search
SSD
Left Right
scanline
Correspondence search
Norm. corr
Effect of window size
– Smaller window+ More detail• More noise
– Larger window+ Smoother disparity maps• Less detail
W = 3 W = 20
Failures of correspondence search
Textureless surfaces Occlusions, repetition
Non-Lambertian surfaces, specularities
Results with window search
Window-based matching Ground truth
Data
How can we improve window-based matching?
• So far, matches are independent for each point
• What constraints or priors can we add?
Stereo constraints/priors• Uniqueness
– For any point in one image, there should be at most one matching point in the other image
Stereo constraints/priors• Uniqueness
– For any point in one image, there should be at most one matching point in the other image
• Ordering– Corresponding points should be in the same order in
both views
Stereo constraints/priors• Uniqueness
– For any point in one image, there should be at most one matching point in the other image
• Ordering– Corresponding points should be in the same order in
both views
Ordering constraint doesn’t hold
Non-local constraints• Uniqueness
– For any point in one image, there should be at most one matching point in the other image
• Ordering– Corresponding points should be in the same order in both
views• Smoothness
– We expect disparity values to change slowly (for the most part)
Stereo matching as energy minimization
I1 I2 D
• Energy functions of this form can be minimized using graph cuts
Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001
W1(i ) W2(i+D(i )) D(i )
)(),,( smooth21data DEDIIEE
ji
jDiDE,neighbors
smooth )()( 2
21data ))(()( i
iDiWiWE
Many of these constraints can be encoded in an energy function and solved using graph cuts
Graph cuts Ground truth
For the latest and greatest: http://www.middlebury.edu/stereo/
Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001
Summary
• Recap of epipolar geometry– Epipoles are intersection of baseline with image planes– Matching point in second image is on a line passing through its
epipole– Fundamental matrix maps from a point in one image to an epipole in
the other– Can recover canonical camera matrices from F (with projective
ambiguity)
• Recovering structure– Triangulation to recover 3D position of two matched points in images
with known projection matrices– Sequential algorithm to recover structure from a moving camera,
followed by auto-calibration by assuming fixed K– Get depth from stereo pair by aligning via homography and searching
across scanlines to match; Depth is inverse to disparity.
Next class
• KLT tracking
• Elegant SFM method using tracked points, assuming orthographic projection
• Optical flow