CS 2770: Intro to Computer Vision. Multiple Views. Prof. Adriana Kovashka University of Pittsburgh March 14, 2017

Transcription

1 CS 277: Intro to Computer Vision Multiple Views Prof. Adriana Kovashka Universit of Pittsburgh March 4, 27

2 Plan for toda Affine and projective image transformations Homographies and image mosaics Stereo vision Epipolar geometr

3 Wh multiple views? Structure and depth are inherentl ambiguous from single views. Multiple views help us to perceive 3d shape and depth. Kristen Grauman, images from Svetlana Lazebnik

4 Alignment problem We previousl discussed how to match features across images, of the same or different objects Now let s focus on the case of two images of the same object (e.g. i and i ) What transformation relates i and i? In alignment, we will fit the parameters of some transformation according to a set of matching feature pairs ( correspondences ). i i T Adapted from Kristen Grauman and Derek Hoiem

5 Two questions T Alignment: Given two images, what is the transformation between them? T Warping: Given a source image and a transformation, what does the transformed output look like? Kristen Grauman

6 Motivation: Image mosaics Kristen Grauman Image from

7 What are the correspondences? Min SSD = match? Compare content in local patches, find best matches. Simplest approach: Scan i with template formed from a point in i, and compute Euclidean distance (a.k.a. SSD) or normalized crosscorrelation between list of piel intensities in the patch. Kristen Grauman

8 What are the transformations? Eamples of transformations: translation rotation aspect affine perspective Alosha Efros

9 Parametric (global) warping T p = (,) p = (, ) Transformation T is a coordinate-changing machine: p = T(p) What does it mean that T is global? It is the same for an point p It can be described b just a few numbers (parameters) Let s represent T as a matri: p = Mp Alosha Efros M

10 Scaling Scaling a coordinate means multipling each of its components b a scalar Uniform scaling means this scalar is the same for all components: 2 Alosha Efros

11 Scaling Non-uniform scaling: different scalars per component X 2, Y.5 Alosha Efros

12 Scaling Scaling operation: Or, in matri form: b a q p n m scaling matri S Adapted from Alosha Efros b a

13 2D Linear transformations a c b d Onl linear 2D transformations can be represented with a 22 matri. Linear transformations are combinations of Scale, Rotation, Shear, and Mirror Alosha Efros

14 2D Rotate around (,)? * cos * sin * sin * cos cos sin sin cos 2D Shear? sh sh * * sh sh 2D Scaling? s s * * s s Modified from Alosha Efros Fig. from What transforms can we write w/ 22 matri?

15 What transforms can we write w/ 22 matri? 2D Mirror about Y ais? 2D Mirror over (,)? 2D Translation? t t CAN T DO! Alosha Efros

16 Homogeneous coordinates To convert to homogeneous coordinates: homogeneous image coordinates Converting from homogeneous coordinates Kristen Grauman

17 t t Translation t t t t t = 2 t = Homogeneous Coordinates Alosha Efros

18 2D affine transformations Affine transformations are combinations of Linear transformations, and Translations Maps lines to lines, parallel lines remain parallel w f e d c b a w Adapted from Alosha Efros

19 Fitting an affine transformation Assuming we know the correspondences, how do we get the transformation? ), ( i i ), ( i i t t m m m m i i i i i i i i i i t t m m m m Alosha Efros

20 Fitting an affine transformation How man matches (correspondence pairs) do we need to solve for the transformation parameters? Once we have solved for the parameters, how do we compute given? i i i i i i t t m m m m ), ( new new Adapted from Kristen Grauman ), ( new new

21 Projective transformations Projective transformations: Affine transformations, and Projective warps Parallel lines do not necessaril remain parallel w i h g f e d c b a w Kristen Grauman

22 Projective transformations A projective transformation is a mapping between an two projective planes with the same center of projection Also called Homograph PP2 PP * * * * * * * * * w w w H p p Adapted from Alosha Efros

23 Image mosaics: Goals... image from S. Seitz Obtain a wider angle view b combining multiple images. Kristen Grauman

24 Image mosaics: Camera setup Two images with camera rotation but no translation Adapted from Derek Hoiem Camera Center

25 Image mosaics: Man 2D views, one 3D object Steve Seitz mosaic plane The mosaic has a natural interpretation in 3D The images are reprojected onto a common plane The mosaic is formed on this plane Mosaic is a snthetic wide-angle camera

26 How to stitch together panorama (mosaic)? Basic Procedure Take a sequence of images from the same position Rotate the camera about its optical center Compute the homograph (transformation) between first and second image Transform the second image to overlap with the first Blend the two together to create a mosaic (If there are more images, repeat) Adapted from Steve Seitz

27 Computing the homograph,, 2, 2, 2 2,, n n n n To compute the homograph given pairs of corresponding points in the images, we need to set up an equation where the parameters of H are the unknowns Kristen Grauman

28 Computing the homograph Assume we have four matched points: How do we compute homograph H? w w w p h h h h h h h h h H h h h h h h h h h h Adapted from Derek Hoiem, Kristen Grauman p =Hp h A i h g f e d c b a w w w h h h h h h h h h Using = w / w, = w / w: Can set scale factor h 9 =. So, there are 8 unknowns. Need at least 8 eqs, but the more the better p

29 How to stitch together panorama (mosaic)? Basic Procedure Take a sequence of images from the same position Rotate the camera about its optical center Compute the homograph (transformation) between first and second image Transform the second image to overlap with the first Blend the two together to create a mosaic (If there are more images, repeat) Adapted from Steve Seitz

30 * * * * * * * * * w w w H p p w w w w,,, To appl a given homograph H Compute p = Hp (regular matri multipl) Convert p from homogeneous to image coordinates Modified from Kristen Grauman Transforming the second image Image canvas Image 2

31 Transforming the second image Image 2 Image canvas H(,) f(,) g(, ) Forward warping: Send each piel f(,) to its corresponding location (, ) = H(,) in the right image Modified from Alosha Efros

32 Transforming the second image H(,) f(,) g(, ) Forward warping: Send each piel f(,) to its corresponding location (, ) = H(,) in the right image Q: what if piel lands between two piels? A: distribute color among neighboring piels (, ) Alosha Efros

33 Transforming the second image Image 2 Image canvas H - (,) f(,) g(, ) Inverse warping: Get each piel g(, ) from its corresponding location (,) = H - (, ) in the left image Modified from Alosha Efros

34 Transforming the second image H - (,) f(,) g(, ) Inverse warping: Get each piel g(, ) from its corresponding location (,) = H - (, ) in the left image Q: what if piel comes from between two piels? A: interpolate color value from neighbors Alosha Efros

35 Homograph eample: Image rectification p p To unwarp (rectif) an image solve for homograph H given p and p : p =Hp Derek Hoiem

36 Summar of affine/projective transforms Write 2d transformations as matri-vector multiplication (including translation when we use homogeneous coordinates) Fitting transformations: solve for unknown parameters given corresponding points from two views linear, affine, projective (homograph) Mosaics: uses homograph and image warping to merge views taken from same center of projection Perform image warping (forward, inverse) Adapted from Kristen Grauman

37 Net topic: Stereo vision Homograph: Same camera center, but camera rotates Stereo vision: Camera center is not the same (we have multiple cameras) Epipolar geometr Relates cameras from two positions/cameras Stereo depth estimation Recover depth from disparities between two images Adapted from Derek Hoiem

38 Stereo photograph and stereo viewers Take two pictures of the same subject from two slightl different viewpoints and displa so that each ee sees onl one of the images. Invented b Sir Charles Wheatstone, 838 Image from fisher-price.com Kristen Grauman

39 Stereo photograph and stereo viewers Kristen Grauman

40 Depth from stereo for computers Two cameras, simultaneous views Single moving camera and static scene Kristen Grauman

41 Depth from stereo Goal: recover depth b finding image coordinate that corresponds to X X z f f C Baseline C B Derek Hoiem

42 Depth from stereo Goal: recover depth b finding image coordinate that corresponds to Sub-Problems. Calibration: How do we recover the relation of the cameras (if not alread known)? 2. Correspondence: How do we search for the matching point? X Derek Hoiem

43 Geometr for a simple stereo sstem Assume parallel optical aes, known camera parameters (i.e., calibrated cameras). What is epression for Z? Similar triangles (p l, P, p r ) and (O l, P, O r ): T l Z f r T Z Depth is inversel proportional to disparit. depth disparit Z f T r l Adapted from Kristen Grauman

44 Depth from disparit We have two images taken from cameras with different intrinsic and etrinsic parameters. How do we match a point in the first image to a point in the second? image I(,) Disparit map D(,) image I (, ) If we could find the corresponding points in two images, we could estimate relative depth Kristen Grauman

45 Stereo correspondence constraints Given p in left image, where can corresponding point p be? Kristen Grauman

46 Epipolar constraint Geometr of two views constrains where the corresponding piel for some image point in the first view must occur in the second view. It must be on the line where () the plane connecting the world point and optical centers, and (2) the image plane, intersect. Potential matches for p have to lie on the corresponding line l. Potential matches for p have to lie on the corresponding line l. Adapted from Kristen Grauman, Derek Hoiem

47 Epipolar geometr: notation P p p Adapted from Derek Hoiem Baseline line connecting the two camera centers Epipoles = intersections of baseline with image planes = projections of the other camera center Epipolar Plane plane containing baseline Epipolar Lines - intersections of epipolar plane with image planes (alwas come in corresponding pairs)

48 Epipolar constraint The epipolar constraint is useful because it reduces the correspondence problem to a D search along an epipolar line. Kristen Grauman, image from Andrew Zisserman

49 Stereo geometr, with calibrated cameras If the stereo rig is calibrated, we know : how to rotate and translate camera reference frame to get to camera reference frame 2. Rotation: 33 matri R; translation: 3 vector T. Kristen Grauman

50 Stereo geometr, with calibrated cameras If the stereo rig is calibrated, we know : how to rotate and translate camera reference frame to get to camera reference frame 2. Adapted from Kristen Grauman X RX T

51 An aside: cross product Vector cross product takes two vectors and returns a third vector that s perpendicular to both inputs. So here, c is perpendicular to both a and b, which means the dot product =. Kristen Grauman

52 From geometr to algebra Kristen Grauman X RX T TX Normal to the plane TRX TRX TT X Cross-product of vector with itself is. T X XT RX

53 Aside: Matri form of cross product Can be epressed as a matri multiplication. c b b b a a a a a a b a a a a a a a a Kristen Grauman

54 X X Essential matri T RX [T ] RX Let E [T ] R XEX X T EX E is called the essential matri, and it relates corresponding image points between both cameras, given the rotation and translation. Before we said: If we observe a point in one image, its position in other image is constrained to lie on line defined b above. It turns out that: E T is the epipolar line l through in the second image, corresponding to. E is the epipolar line l through in the first image, corresponding to. Adapted from Kristen Grauman, Derek Hoiem

55 Essential matri eample: parallel cameras R I p [,, f ] T E [ d,,] [ T ]R d d p [,, f ] p Ep For the parallel cameras, image of an point must lie on same horizontal line in each image plane. Kristen Grauman

56 image I(,) Disparit map D(,) image I (, ) (, )=(+D(,),) Adapted from Kristen Grauman

57 Basic stereo matching algorithm For each piel in the first image Find corresponding epipolar scanline in the right image Search along epipolar line and pick the best match Slide a window along the right scanline and compute Euclidean distance between contents of that window with the reference window in the left image; take the window corresponding to the minimum as the match Compute disparit - and set depth() = f*t/(- ) Adapted from Derek Hoiem

58 Results with window search Data Left image Right image Window-based matching Window-based matching Ground truth Ground truth Derek Hoiem

59 Summar of stereo vision Epipolar geometr Epipoles are intersection of baseline with image planes Matching point in second image is on a line passing through its epipole Epipolar constraint limits where points from one view will be imaged in the other, which makes search for correspondences quicker Essential matri E maps from a point in one image to a line (its epipolar line) in the other Stereo depth estimation Find corresponding points along epipolar scanline Estimate disparit (depth is inverse to disparit) Adapted from Kristen Grauman and Derek Hoiem

60 How can we improve? Uniqueness For an 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 epect disparit values to change slowl (for the most part) Man of these constraints can be encoded in an energ function and solved using graph cuts Derek Hoiem

61 Before Graph cuts Ground truth Y. Bokov, O. Veksler, and R. Zabih, Fast Approimate Energ Minimization via Graph Cuts, PAMI 2 For the latest and greatest: Derek Hoiem

62 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 2D points ij X j j 3j P 2j Svetlana Lazebnik P 2 P 3

63 Photo tourism Noah Snavel, Steven M. Seitz, Richard Szeliski, "Photo tourism: Eploring photo collections in 3D," SIGGRAPH 26

64 3D from multiple images Sameer Agarwala, Noah Snavel, Ian Simon, Steven M. Seitz, Richard Szeliski, "Building Rome in a Da," ICCV 29