# Stereo and Projective Structure from Motion

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Stereo and Projective Structure from Motion
04/13/10 Stereo and Projective Structure from Motion Computer Vision CS 543 / ECE 549 University of Illinois Derek Hoiem 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 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

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
Assume we have matched points x x’ with outliers Homography (No Translation) Fundamental Matrix (Translation)

Recap: Automatically Relating Projections
Assume we have matched points x x’ with outliers Homography (No Translation) Fundamental Matrix (Translation) Correspondence Relation Normalize image coordinates RANSAC with 4 points De-normalize:

Recap: Automatically Relating Projections
Assume we have matched points x x’ with outliers Homography (No Translation) Fundamental Matrix (Translation) Correspondence Relation Normalize image coordinates RANSAC with 4 points De-normalize: Correspondence Relation Normalize image coordinates RANSAC with 8 points Enforce by SVD De-normalize:

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]; See HZ p

Recap Fundamental matrix song

Triangulation: Linear Solution
X x x' Generally, rays Cx and C’x’ will not exactly intersect Can solve via SVD, finding a least squares solution to a system of equations Further reading: HZ p

Triangulation: Linear Solution
Given P, P’, x, x’ Precondition points and projection matrices Create matrix A [U, S, V] = svd(A) X = V(:, end) Pros and Cons Works for any number of corresponding images Not projectively invariant Code:

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

Projective SFM: Two-camera case
Compute fundamental matrix F between the two views First camera matrix: [I|0] Second camera matrix: [M|t] Then e is the epipole (FTe = 0), A = –[e×]F F&P sec

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 points cameras

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 points cameras

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 points cameras

Bundle adjustment Non-linear method for refining structure and motion
Minimizing reprojection error Xj P1Xj x3j x1j P3Xj P2Xj x2j P1 P3 P2

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: For a moving camera, 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

Based on Photo Tourism by Noah Snavely, Steve Seitz, and Rick Szeliski
Photosynth.net Based on Photo Tourism by Noah Snavely, Steve Seitz, and Rick Szeliski

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 image image 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

Essential matrix for parallel images
Epipolar constraint: R = I t = (T, 0, 0) x x’ t

Special case of fundamental matrix
Epipolar constraint: R = I t = (T, 0, 0) x x’ t The y-coordinates of corresponding points are the same!

Depth from disparity f x x’ Baseline B z O O’ X
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’)

Correspondence search
Left Right scanline Matching cost disparity 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

Correspondence search
Left Right scanline SSD

Correspondence search
Left Right scanline Norm. corr

Effect of window size Smaller window Larger window More detail
More noise Larger window Smoother disparity maps Less detail smaller window: more detail, more noise bigger window: less noise, more detail

Failures of correspondence search
Occlusions, repetition Textureless surfaces Non-Lambertian surfaces, specularities

Results with window search
Data Window-based matching Ground truth

How can we improve window-based matching?
So far, matches are independent for each point What constraints or priors can we add?

How can we improve window-based matching?
The similarity constraint is local (each reference window is matched independently) Need to enforce non-local correspondence constraints or priors

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
D W1(i ) W2(i+D(i )) D(i ) 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

Stereo matching as energy minimization
D W1(i ) W2(i+D(i )) D(i ) Probabilistic interpretation: we want to find a Maximum A Posteriori (MAP) estimate of disparity image D:

Many of these constraints can be encoded in an energy function and solved using graph cuts
Ground truth Y. Boykov, O. Veksler, and R. Zabih, Fast Approximate Energy Minimization via Graph Cuts, PAMI 2001 For the latest and greatest:

“Shortest paths” for scan-line stereo
Left image Right image Right occlusion s q Left occlusion t p correspondence Can be implemented with dynamic programming Ohta & Kanade ’85, Cox et al. ‘96 Slide credit: Y. Boykov

Coherent stereo on 2D grid
Scanline stereo generates streaking artifacts Can’t use dynamic programming to find spatially coherent disparities/ correspondences on a 2D grid

Stereo matching as energy minimization
Note: the above formulation does not treat the two images symmetrically, does not enforce uniqueness, and does not take occlusions into account It is possible to come up with an energy that does all these things, but it’s a bit more complex Defined over all possible sets of matches, not over all disparity maps with respect to the first image Includes an occlusion term The smoothness term looks different and more complicated V. Kolmogorov and R. Zabih, Computing Visual Correspondence with Occlusions using Graph Cuts, ICCV 2001

Summary Recap of epipolar geometry Recovering structure
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

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