Structure from motion

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1 Structure from motion

2 Structure from motion Given a set of corresponding points in two or more images, compute the camera parameters and the 3D point coordinates?? R 1,t 1 R 2,t R 2 3,t 3 Camera 1 Camera 2 Camera 3?? Slide credit: Noah Snavely

3 Structure from motion Given: m images of n fixed 3D points λ ij x ij = P i X j, i = 1,, m, j = 1,, n Problem: estimate m projection matrices P i and n 3D points X j from the mn correspondences x ij X j x 1j x 3j P 1 x 2j P 2 P 3

4 Outline Reconstruction ambiguities Affine structure from motion Factorization Projective structure from motion Bundle adjustment Modern structure from motion pipeline

5 Is SfM always uniquely solvable? Necker cube Source: N. Snavely

6 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 1/k, the projections of the scene points in the image remain exactly the same: x = PX = 1 k P ( kx) It is impossible to recover the absolute scale of the scene!

7 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 1/k, the projections of the scene points in the image remain exactly 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: x = PX = ( PQ -1)( QX)

8 Types of ambiguity Projective 15dof A T v t v Preserves intersection and tangency Affine 12dof A T 0 t 1 Preserves parallellism, volume ratios Similarity 7dof s R T 0 t 1 Preserves angles, ratios of length Euclidean 6dof R T 0 t 1 Preserves angles, lengths With no constraints on the camera calibration matrix or on the scene, we get a projective reconstruction Need additional information to upgrade the reconstruction to affine, similarity, or Euclidean

9 Projective ambiguity With no constraints on the camera calibration matrix or on the scene, we can reconstruct up to a projective ambiguity Q p = A T v t v x = PX = ( PQ -1 )( Q X) P P

10 Projective ambiguity

11 Affine ambiguity If we impose parallelism constraints, we can get a reconstruction up to an affine ambiguity Affine Q A = A T 0 t 1 x = PX = ( PQ -1 )( Q X) A A

12 Affine ambiguity

13 Similarity ambiguity A reconstruction that obeys orthogonality constraints on camera parameters and/or scene Q s = sr T 0 t 1 x = PX = ( PQ -1)( Q X) S S

14 Similarity ambiguity

15 Affine structure from motion Let s start with affine or weak perspective cameras (the math is easier) center at infinity

16 Recall: Orthographic Projection Image World Projection along the z direction

17 Affine cameras Orthographic Projection Parallel Projection

18 Affine cameras A general affine camera combines the effects of an affine transformation of the 3D space, orthographic projection, and an affine transformation of the image: Affine projection is a linear mapping + translation in non-homogeneous coordinates = = = affine] [ affine] 3 [ b A P b a a a b a a a x X a 1 a 2 b AX x + = + = = b b Z Y X a a a a a a y x Projection of world origin

19 Affine structure from motion Given: m images of n fixed 3D points: x ij = A i X j + b i, i = 1,, m, j = 1,, n Problem: use the mn correspondences x ij to estimate m projection matrices A i and translation vectors b i, and n points X j The reconstruction is defined up to an arbitrary affine transformation Q (12 degrees of freedom): A 1 0 b 1 A 0 b 1 Q, X Q 1 We have 2mn knowns and 8m + 3n unknowns (minus 12 dof for affine ambiguity) Thus, we must have 2mn >= 8m + 3n 12 For two views, we need four point correspondences X 1

20 Affine structure from motion Centering: subtract the centroid of the image points in each view xˆ ij = x ij 1 n n k = 1 x ik = A X i j + b i 1 n n k= 1 ( A X + b ) i k i = A i X j 1 n n k= 1 X k = A Xˆ i j For simplicity, set the origin of the world coordinate system to the centroid of the 3D points After centering, each normalized 2D point is related to the 3D point X j by x ˆ = A ij i X j

21 Affine structure from motion D Let s create a 2m n data (measurement) matrix: = xˆ xˆ xˆ m1 xˆ xˆ xˆ m2!! "! xˆ xˆ xˆ 1n 2n mn cameras (2 m) points (n) C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method. IJCV, 9(2): , November 1992.

22 Affine structure from motion Let s create a 2m n data (measurement) matrix: [ ] n m mn m m n n X X X A A A x x x x x x x x x D! "! #!! ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = = cameras (2 m 3) points (3 n) The measurement matrix D = MS must have rank 3! C. Tomasi and T. Kanade. Shape and motion from image streams under orthography: A factorization method. IJCV, 9(2): , November 1992.

23 Factorizing the measurement matrix Source: M. Hebert

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

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

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

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

28 Affine ambiguity The decomposition is not unique. We get the same D by using any 3 3 matrix C and applying the transformations M MC, S C -1 S That is because we have only an affine transformation and we have not enforced any Euclidean constraints (like forcing the image axes to be perpendicular, for example) Source: M. Hebert

29 Eliminating the affine ambiguity Transform each projection matrix A to another matrix AC to get orthographic projection Image axes are perpendicular and scale is 1 a 2 a 1 x a 1 a 2 = 0 a 1 2 = a 2 2 = 1 X This translates into 3m equations: (A i C)(A i C) T = A i (CC T )A i = Id, i = 1,, m Solve for L = CC T Recover C from L by Cholesky decomposition: L = CC T Update M and S: M = MC, S = C -1 S Source: M. Hebert

30 Reconstruction results C. Tomasi and T. Kanade, Shape and motion from image streams under orthography: A factorization method, IJCV 1992

31 Algorithm summary Given: m images and n features x ij For each image i, center the feature coordinates Construct a 2m n measurement matrix 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 and shape matrices: M = U 3 W 3 ½ and S = W 3 ½ V 3 T (or M = U 3 and S = W 3 V 3T ) Eliminate affine ambiguity Source: M. Hebert

32 Dealing with missing data So far, we have assumed that all points are visible in all views In reality, the measurement matrix typically looks something like this: cameras points Possible solution: decompose matrix into dense subblocks, factorize each sub-block, and fuse the results Finding dense maximal sub-blocks of the matrix is NPcomplete (equivalent to finding maximal cliques in a graph)

33 Dealing with missing data Incremental bilinear refinement (1) Perform factorization on a dense sub-block (2) Solve for a new 3D point visible by at least two known cameras (triangulation) (3) Solve for a new camera that sees at least three known 3D points (calibration) F. Rothganger, S. Lazebnik, C. Schmid, and J. Ponce. Segmenting, Modeling, and Matching Video Clips Containing Multiple Moving Objects. PAMI 2007.

34 Projective structure from motion Given: m images of n fixed 3D points λ ij x ij = P i X j, i = 1,, m, j = 1,, n Problem: estimate m projection matrices P i and n 3D points X j from the mn correspondences x ij X j x 1j x 3j P 1 x 2j P 2 P 3

35 Projective structure from motion Given: m images of n fixed 3D points λ ij x ij = P i X j, i = 1,, m, j = 1,, n Problem: estimate m projection matrices P i and n 3D points X j from the mn correspondences x ij 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

36 Projective SFM: Two-camera case Compute fundamental matrix F between the two views First camera matrix: [I 0] Second camera matrix: [A b] Then b is the epipole (F T b = 0), A = [b ]F F&P sec

37 Incremental structure from motion Initialize motion from two images using fundamental matrix Initialize structure by triangulation points For each additional view: Determine projection matrix of new camera using all the known 3D points that are visible in its image calibration cameras

38 Incremental structure from motion Initialize motion from two images using fundamental matrix Initialize structure by triangulation points For each additional view: Determine projection matrix of new camera using all the known 3D points that are visible in its image calibration cameras Refine and extend structure: compute new 3D points, re-optimize existing points that are also seen by this camera triangulation

39 Incremental structure from motion Initialize motion from two images using fundamental matrix Initialize structure by triangulation points For each additional view: Determine projection matrix of new camera using all the known 3D points that are visible in its image calibration cameras 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

40 Bundle adjustment Non-linear method for refining structure and motion Minimize reprojection error m n w ij i=1 j=1 x ij 1 λ ij P i X j 2 X j visibility flag: is point j visible in view i? P 1 X j P 1 x 1j P 2 X j x 2j P 3 X j x 3j P 2 P 3

41 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 P i = K [R i t i ] Can use constraints on the form of the calibration matrix: zero skew Can use vanishing points

42 Representative SFM pipeline N. Snavely, S. Seitz, and R. Szeliski, Photo tourism: Exploring photo collections in 3D, SIGGRAPH 2006.

43 Feature detection Detect SIFT features Source: N. Snavely

44 Feature detection Detect SIFT features Source: N. Snavely

45 Feature matching Match features between each pair of images Source: N. Snavely

46 Feature matching Use RANSAC to estimate fundamental matrix between each pair Source: N. Snavely

47 Feature matching Use RANSAC to estimate fundamental matrix between each pair Image source

48 Feature matching Use RANSAC to estimate fundamental matrix between each pair Source: N. Snavely

49 Image connectivity graph (graph layout produced using the Graphviz toolkit: Source: N. Snavely

50 Incremental SFM Pick a pair of images with lots of inliers (and preferably, good EXIF data) Initialize intrinsic parameters (focal length, principal point) from EXIF Estimate extrinsic parameters (R and t) using five-point algorithm Use triangulation to initialize model points While remaining images exist Find an image with many feature matches with images in the model Run RANSAC on feature matches to register new image to model Triangulate new points Perform bundle adjustment to re-optimize everything

51 The devil is in the details Handling degenerate configurations (e.g., homographies) Eliminating outliers Dealing with repetitions and symmetries Handling multiple connected components Closing loops Making the whole thing efficient! See, e.g., Towards Linear-Time Incremental Structure from Motion

52 SFM software Bundler OpenSfM OpenMVG VisualSFM See also Wikipedia s list of toolboxes

53 Review: Structure from motion Ambiguity Affine structure from motion Factorization Dealing with missing data Incremental structure from motion Projective structure from motion Bundle adjustment Modern structure from motion pipeline

54 Summary: 3D geometric vision

55 Summary: 3D geometric vision Single-view geometry The pinhole camera model Variation: orthographic projection The perspective projection matrix Intrinsic and extrinsic parameters Calibration Single-view metrology, calibration using vanishing points Multiple-view geometry Triangulation The epipolar constraint Essential matrix and fundamental matrix Stereo Binocular, multi-view Structure from motion Reconstruction ambiguity Affine SFM Projective SFM

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