Structure from Motion - PDF Free Download

/8/ Structure from Motion Computer Vision CS 43, Brown James Hays Many slides adapted from Derek Hoiem, Lana Lazebnik, Silvio Saverese, Steve Seitz, and Martial Hebert

This class: structure from motion Recap of epipolar geometry Depth from two views Affine structure from motion

Recap: Epipoles Point 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 C

Recap: Fundamental Matri Fundamental matri maps from a point in one image to a line in the other If and correspond to the same 3d point X:

Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R,t R 2,t 2 R 3,t 3 Camera Camera 2 Camera 3?? Slide credit: Noah Snavely

Structure from motion ambiguity If we scale the entire scene by some factor k and, at the same time, scale the camera matrices by the factor of /k, the projections of the scene points in the image remain eactly the same: PX k P( kx) It is impossible to recover the absolute scale of the scene!

How do we know the scale of image content?

Structure from motion ambiguity If we scale the entire scene by some factor k and, at the same time, scale the camera matrices by the factor of /k, the projections of the scene points in the image remain eactly the same More generally: if we transform the scene using a transformation Q and apply the inverse transformation to the camera matrices, then the images do not change PX PQ - QX

Projective structure from motion Given: m images of n fied 3D points ij = P i X j, i =,, m, j =,, n Problem: estimate m projection matrices P i and n 3D points X j from the mn corresponding points ij With no calibration info, cameras and points can only be recovered up to a 44 projective transformation Q: X QX, P PQ - We can solve for structure and motion when 2mn >= m +3n 5 For two cameras, at least 7 points are needed

Types of ambiguity Projective 5dof A T v t v Preserves intersection and tangency Affine 2dof A T 0 t Preserves parallellism, volume ratios Similarity 7dof s R T 0 t Preserves angles, ratios of length Euclidean 6dof R T 0 t Preserves angles, lengths With no constraints on the camera calibration matri or on the scene, we get a projective reconstruction Need additional information to upgrade the reconstruction to affine, similarity, or Euclidean

Projective ambiguity Q p A T v t v PX PQ - Q X P P

Projective ambiguity

Affine ambiguity Affine Q A A T 0 t PX PQ - Q X A A

Affine ambiguity

Similarity ambiguity Q s sr T 0 t PX PQ -Q X S S

Similarity ambiguity

Bundle adjustment Non-linear method for refining structure and motion Minimizing reprojection error m E( P, X) D n i j 2, P X ij i j X j P X j j 3j P P 2 X j 2j P 3 X j P 2 P 3

Photo synth Noah Snavely, Steven M. Seitz, Richard Szeliski, "Photo tourism: Eploring photo collections in 3D," SIGGRAPH 2006 http://photosynth.net/

Structure from motion Let s start with affine cameras (the math is easier) center at infinity

Affine structure from motion Affine projection is a linear mapping + translation in inhomogeneous coordinates y a a 2 a a 2 22 a a 3 23 X t Y t Z y AX t a 2 a X Projection of world origin. We are given corresponding 2D points () in several frames 2. We want to estimate the 3D points (X) and the affine parameters of each camera (A)

Affine structure from motion Centering: subtract the centroid of the image points For simplicity, assume that the origin of the world coordinate system is at the centroid of the 3D points After centering, each normalized point ij is related to the 3D point X i by j i n k k j i n k i k i i j i n k ik ij ij n n n A X X X A b A X b A X j i ij X A

Suppose we know 3D points and affine camera parameters then, we can compute the observed 2d positions of each point mn m m n n n m X X X A A A 2 2 22 2 2 2 2 Camera Parameters (2m3) 3D Points (3n) 2D Image Points (2mn)

What if we instead observe corresponding 2d image points? Can we recover the camera parameters and 3d points? cameras (2 m) points (n) n m mn m m n n X X X A A A D 2 2 2 2 22 2 2? What rank is the matri of 2D points?

Factorizing the measurement matri AX Source: M. Hebert

Factorizing the measurement matri Singular value decomposition of D: Source: M. Hebert

Factorizing the measurement matri Obtaining a factorization from SVD: Source: M. Hebert

Factorizing the measurement matri Obtaining a factorization from SVD: This decomposition minimizes D-MS 2 Source: M. Hebert

Affine ambiguity A ~ X ~ S The decomposition is not unique. We get the same D by using any 3 3 matri C and applying the transformations A AC, X C - X That is because we have only an affine transformation and we have not enforced any Euclidean constraints (like forcing the image aes to be perpendicular, for eample) Source: M. Hebert

Eliminating the affine ambiguity Orthographic: image aes are perpendicular and scale is a a 2 = 0 a 2 a a 2 = a 2 2 = X This translates into 3m equations in L = CC T : A i L A T i = Id, i =,, m Solve for L Recover C from L by Cholesky decomposition: L = CC T Update M and S: M = MC, S = C - S Source: M. Hebert

Algorithm summary Given: m images and n tracked features ij For each image i, center the feature coordinates Construct a 2m n measurement matri D: Column j contains the projection of point j in all views Row i contains one coordinate of the projections of all the n points in image i Factorize D: Compute SVD: D = U W V T Create U 3 by taking the first 3 columns of U Create V 3 by taking the first 3 columns of V Create W 3 by taking the upper left 3 3 block of W Create the motion (affine) and shape (3D) matrices: A = U 3 W 3 ½ and X = W 3 ½ V 3 T Eliminate affine ambiguity Source: M. Hebert

Dealing with missing data So far, we have assumed that all points are visible in all views In reality, the measurement matri typically looks something like this: cameras points One solution: solve using a dense submatri of visible points Iteratively add new cameras

A nice short eplanation Class notes from Lischinksi and Gruber http://www.cs.huji.ac.il/~csip/sfm.pdf