viral infection edition

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1 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Beginner s Guide to Image Processing or How to make vision-guided killbots step 1 viral infection edition Paul Pounds 8 April 2014 Universit of Queensland Sura Singh! 1

2 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 The Blorch Paul has contracted the Blorch Bad case of insides-fall-out sndrome Sura will guest lecture toda Paul s notes to Sura will take the form of Blorchnotes on the slide Blorch! 2

3 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 But first Some house keeping 3

4 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 House keeping Rubbish and drink bottles in the lab Warning 1. If ou see someone making/leaving a mess in the lab politel ecoriate them. Lab is No Food Zone. Eat outside plz kthbe. Blorchnote: Make sure the know we aren t kidding about this Progress Reviews are done (Almost) everone passed; some barel Wake-up call for a lot of people We are seeing lots of hacking ver little math 4

5 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 House keeping Group 10 is no more; Group 9 is now bigger Should have no advantage or disadvantage 5 On PCBs: Submit to Steve Wright for BEC batch runs If ou want to use a specific fab (eg. for fast turnaround) talk to ETSG the will allow ou to place the order ourself and get reimbursed Ask John Kolbach first. Blorchnote: If students are uncertain have them ask John

6 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Reminder: House keeping You must submit our designs to Doug for machining parts no later than week 7 No parts will be machined for ou after then You can machine our own parts but ou won t be able to go through the workshop 6

7 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Calendar at a glance Week Dates Lecture Reviews Demos Assessment submissions 1 3/3 7/3 Introduction You are here 2 10/3 15/3 Principles of Mechatronic Sstems design 3 17/3 21/3 Professional Engineering Topics 4 24/3 28/3 Your soldering is (probabl) terrible 5 31/3-4/3 No lecture Progress review 1 6 7/4 11/4 Image segmentation 7 14/4 18/4 B request Progress seminar 25% demo Design brief Break 21/4 25/4 Sooon 8 28/4 3/5 B request 9 5/5 9/5 B request Progress review 50% demo 10 12/5 16/ /5 23/5 75% demo Preliminar report 12 26/5 30/5 13 2/6 6/6 Closing lecture Final testing Final report and addendum 7

8 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Progress seminars Progress seminars are net week! First group-based assessment Gives ou presenting eperience and brings us up to date with our team s progress Sign up for session slots via Doodle poll Link to poll will be sent out via Blackboard announcement after the lecture (closes Frida) Blorchnote: Same proceedure as last time nothing scar 8

9 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Progress Seminar Group presentation 10 minutes per team Stand up and talk about our progress Each person talks for roughl equal time 9 Focus on progress not the requirements! We alread know what the project goal is. We know (roughl) what our approach is. Don t waste valuable time repeating them. Above all show evidence of our work. Blorchnote: The Blorch just loves evidence and justification

10 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 How to sign up: Progress seminars Have one and onl one member of our team nominate a time for our team on the When the sign up the must include their full name and team number. If the don t have both the slot will be cleared. If ou absolutel can t get a slot that works for all of our group me ASAP But this should never happen 10

11 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 FAQ Roundup Can we use off the shelf tires or do we have to make our own? Off the shelf tires are fine so long as ou aren t using a rubber wheel that is 100% tire. Can we use Lego shafts? Sure but wh would ou want to? Can we use Arduino/RaspPi/Raspduino? Yes but I won t respect ou in the morning. Wasn t there supposed to be a doodle poll for lecture topics? Yeah about that 11

12 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Lecture nominations Onl one person nominated a topic This presentation is therefore on that topic. Future topic polls will be conducted in class He let s do that now! 12 Blorchnote: Please take an informal poll of the students as to what topics the would like to hear about if no consensus let them know that coffee+q&a is often ver helpful too nice if the topics are eas to teach!

13 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Demos The first preliminar demonstration round runs net week Mark cap at 25 per cent. One demo attempt per team (in this round) If ou are read to test our car contact me now so I can arrange things. Bookings must be made b 5pm Frida. 13

14 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Back to business And now a special guest 14

15 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Special guest lecture Featuring 15

16 Perception / Computer Vision What s an object? METR 4202: Advanced Control & Robotics Dr Sura Singh Lecture # 7 September metr4202@itee.uq.edu.au School of Information Technolog and Electrical Engineering at the Universit of Queensland RoboticsCourseWare Contributor

17 Features Colour Corners Edges Lines Statistics on Edges: SIFT SURF METR 4202: Robotics 10 April

18 Features -- Colour Features Baer Patterns Fig: Ch. 10 Robotics Vision and Control METR 4202: Robotics April

19 Colour Spaces HSV YCrCb Gamma Corrected Luma (Y) + Chrominance BW Colour TVs : Just add the Chrominance γ Correction: CRTs γ= Source: Wikipedia HSV and YCrCb METR 4202: Robotics April

20 Edge Detection Cann edge detector: Fig: Ch. 10 Robotics Vision and Control METR 4202: Robotics 10 April

21 Edge Detection Cann edge detector: METR 4202: Robotics 10 April

22 Line Etraction and Segmentation Adopted from Williams Fitch and Singh MTRX 4700 METR 4202: Robotics 10 April

23 Line Formula Adopted from Williams Fitch and Singh MTRX 4700 METR 4202: Robotics 10 April

24 Line Estimation Least squares minimization of the line: Line Equation: Error in Fit: Solution: Adopted from Williams Fitch and Singh MTRX 4700 METR 4202: Robotics 10 April

25 Line Splitting / Segmentation What about corners? Split into multiple lines (via epectation maimization) 1. Epect (assume) a number of lines N (sa 3) 2. Find breakpoints b finding nearest neighbours upto a threshold or simpl at random (RANSAC) 3. How to know N? (Also RANSAC) Adopted from Williams Fitch and Singh MTRX 4700 METR 4202: Robotics 10 April

26 of a Point from a Line Segment d D Adopted from Williams Fitch and Singh MTRX 4700 METR 4202: Robotics April

27 Hough Transform Uses a voting mechanism Can be used for other lines and shapes (not just straight lines) METR 4202: Robotics 10 April

28 Hough Transform: Voting Space Count the number of lines that can go through a point and move it from the - plane to the a-b plane There is onl a one- infinite number (a line!) of solutions (not a two- infinite set a plane) METR 4202: Robotics 10 April

29 Hough Transform: Voting Space In practice the polar form is often used This avoids problems with lines that are nearl vertical METR 4202: Robotics 10 April

30 Hough Transform: Algorithm 1. Quantize the parameter space appropriatel. 2. Assume that each cell in the parameter space is an accumulator. Initialize all cells to zero. 3. For each point () in the (visual & range) image space increment b 1 each of the accumulators that satisf the equation. 4. Maima in the accumulator arra correspond to the parameters of model instances. METR 4202: Robotics 10 April

31 Line Detection Hough Lines [1] A line in an image can be epressed as two variables: Cartesian coordinate sstem: mb Polar coordinate sstem: r θ avoids problems with vert. lines =m+b For each point ( 1 1 ) we can write: Each pair (rθ) represents a line that passes through ( 1 1 ) See also OpenCV documentation (cv::houghlines) METR 4202: Robotics 10 April

32 Line Detection Hough Lines [2] Thus a given point gives a sinusoid Repeating for all points on the image See also OpenCV documentation (cv::houghlines) METR 4202: Robotics 10 April

33 Line Detection Hough Lines [3] Thus a given point gives a sinusoid Repeating for all points on the image NOTE that an intersection of sinusoids represents (a point) represents a line in which piel points la. Thus a line can be detected b finding the number of Intersections between curves See also OpenCV documentation (cv::houghlines) METR 4202: Robotics 10 April

34 Stereo: Epipolar geometr Match features along epipolar lines epipolar line epipolar plane viewing ra Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

35 Stereo: epipolar geometr for two images (or images with collinear camera centers) can find epipolar lines epipolar lines are the projection of the pencil of planes passing through the centers Rectification: warping the input images (perspective transformation) so that epipolar lines are horizontal Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

36 Rectification Project each image onto same plane which is parallel to the epipole Resample lines (and shear/stretch) to place lines in correspondence and minimize distortion [Zhang and Loop MSR-TR-99-21] Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

37 Rectification BAD! Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics April

38 Rectification GOOD! Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics April

39 Matching criteria Raw piel values (correlation) Band-pass filtered images [Jones & Malik 92] Corner like features [Zhang ] Edges [man people ] Gradients [Seitz 89; Scharstein 94] Rank statistics [Zabih & Woodfill 94] Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

40 Finding correspondences Appl feature matching criterion (e.g. correlation or Lucas-Kanade) at all piels simultaneousl Search onl over epipolar lines (man fewer candidate positions) Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

41 Image registration (revisited) How do we determine correspondences? block matching or SSD (sum squared differences) d is the disparit (horizontal motion) How big should the neighborhood be? Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

42 Neighborhood size Smaller neighborhood: more details Larger neighborhood: fewer isolated mistakes w = 3 w = 20 Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

43 Stereo: certaint modeling Compute certaint map from correlations input depth map certaint map METR 4202: Robotics 10 April

44 Plane Sweep Stereo Sweep famil of planes through volume projective re-sampling of (XYZ) input image composite virtual camera each plane defines an image composite homograph METR 4202: Robotics 10 April

45 Plane sweep stereo Re-order (piel / disparit) evaluation loops for ever piel for ever disparit compute cost for ever disparit for ever piel compute cost METR 4202: Robotics 10 April

46 Stereo matching framework For ever disparit compute raw matching costs Wh use a robust function? occlusions other outliers Can also use alternative match criteria METR 4202: Robotics 10 April

47 Stereo matching framework Aggregate costs spatiall Here we are using a bo filter (efficient moving average implementation) Can also use weighted average [non-linear] diffusion METR 4202: Robotics 10 April

48 Stereo matching framework Choose winning disparit at each piel Interpolate to sub-piel accurac E(d) d* d METR 4202: Robotics 10 April

49 Traditional Stereo Matching Advantages: gives detailed surface estimates fast algorithms based on moving averages sub-piel disparit estimates and confidence Limitations: narrow baseline nois estimates fails in tetureless areas gets confused near occlusion boundaries METR 4202: Robotics 10 April

50 Stereo with Non-Linear Diffusion Problem with traditional approach: gets confused near discontinuities New approach: use iterative (non-linear) aggregation to obtain better estimate provabl equivalent to mean-field estimate of Markov Random Field METR 4202: Robotics 10 April

51 Feature-based stereo Match corner (interest) points Interpolate complete solution Slide from Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

52 Cool Robotics Share D. Wedge The Fundamental Matri Song METR 4202: Robotics 10 April

53 Edge Detection Cann edge detector: Pepsi Sequence: Image Data: and Szeliski CS223B-L9 See also: Use of Temporal information to aid segmentation: METR 4202: Robotics 10 April

54 Wh etract features? Object detection Robot Navigation Scene Recognition Steps: Etract Features Match Features Adopted drom S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

55 Wh etract features? [2] Panorama stitching Step 3: Align images (see: Hartle & Zisserman Multiple View Geometr) Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

56 Characteristics of good features Repeatabilit The same feature can be found in several images despite geometric and photometric transformations Salienc Each feature is distinctive Compactness and efficienc Man fewer features than image piels Localit A feature occupies a relativel small area of the image; robust to clutter and occlusion Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

57 Finding Corners Ke propert: in the region around a corner image gradient has two or more dominant directions Corners are repeatable and distinctive C.Harris and M.Stephens. "A Combined Corner and Edge Detector. Proceedings of the 4th Alve Vision Conference: pages Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

58 Corner Detection: Basic Idea Look through a window Shifting a window in an direction should give a large change in intensit flat region: no change in all directions Source: A. Efros edge : no change along the edge direction corner : significant change in all directions METR 4202: Robotics 10 April

59 Corner Detection: Mathematics Change in appearance of window w() for the shift [uv]: E( u v) w( ) I( u v) I( ) 2 I( ) E(u v) E(32) Adopted from S. Lazebnik Gang Hua (CS 558) w( ) METR 4202: Robotics 10 April

60 Corner Detection: Mathematics Change in appearance of window w() for the shift [uv]: E( u v) w( ) I( u v) I( ) 2 I( ) E(u v) E(00) Adopted from S. Lazebnik Gang Hua (CS 558) w( ) METR 4202: Robotics 10 April

61 Corner Detection: Mathematics Change in appearance of window w() for the shift [uv]: E( u v) w( ) I( u v) I( ) 2 Window function Shifted intensit Intensit Window function w() = or Adopted from S. Lazebnik Gang Hua (CS 558) 1 in window 0 outside Gaussian Source: 10 April R Szeliski - METR 4202: Robotics 61

62 Corner Detection: Mathematics Change in appearance of window w() for the shift [uv]: E( u v) w( ) I( u v) I( ) 2 We want to find out how this function behaves for small shifts E(u v) Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

63 Corner Detection: Mathematics Change in appearance of window w() for the shift [uv]: E( u v) w( ) I( u v) I( ) We want to find out how this function behaves for small shifts E( u v) E(00) Eu (00) [ u v] E (00) v 1 E [ u v] 2 E uu uv (00) (00) E E uv vv (00) (00) u v 2 Local quadratic approimation of E(uv) in the neighborhood of (00) is given b the second-order Talor epansion: Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

64 Corner Detection: Mathematics v u E E E E v u E E v u E v u E vv uv uv uu v u (00) (00) (00) (00) ] [ 2 1 (00) (00) ] [ (00) ) ( 2 ( ) ( ) ( ) ( ) E u v w I u v I Second-order Talor epansion of E(uv) about (00): ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2 ) ( ) ( ) ( 2 ) ( ) ( ) ( ) ( ) ( 2 ) ( v u I I v u I w v u I v u I w v u E v u I I v u I w v u I v u I w v u E v u I I v u I w v u E uv uu u 10 April METR 4202: Robotics 64 Adopted from S. Lazebnik Gang Hua (CS 558)

65 Corner Detection: Mathematics v u I w I I w I I w I w v u v u E 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ] [ ) ( 2 ( ) ( ) ( ) ( ) E u v w I u v I Second-order Talor epansion of E(uv) about (00): ) ( ) ( ) ( 2 (00) ) ( ) ( ) ( 2 (00) ) ( ) ( ) ( 2 (00) 0 (00) 0 (00) 0 (00) I I w E I I w E I I w E E E E uv vv uu v u 10 April METR 4202: Robotics 65 Adopted from S. Lazebnik Gang Hua (CS 558)

66 Harris detector: Steps Compute Gaussian derivatives at each piel Compute second moment matri M in a Gaussian window around each piel Compute corner response function R Threshold R Find local maima of response function (nonmaimum suppression) C.Harris and M.Stephens. A Combined Corner and Edge Detector. Proceedings of the 4th Alve Vision Conference: pages Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

67 Harris Detector: Steps Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics April

68 Harris Detector: Steps Compute corner response R Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics April

69 Harris Detector: Steps Find points with large corner response: R>threshold Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics April

70 Harris Detector: Steps Take onl the points of local maima of R Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics April

71 Harris Detector: Steps Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics April

72 Invariance and covariance We want corner locations to be invariant to photometric transformations and covariant to geometric transformations Invariance: image is transformed and corner locations do not change Covariance: if we have two transformed versions of the same image features should be detected in corresponding locations Adopted from S. Lazebnik Gang Hua (CS 558) METR 4202: Robotics 10 April

73 RANdom SAmple Consensus 1. Repeatedl select a small (minimal) subset of correspondences 2. Estimate a solution (in this case a the line) 3. Count the number of inliers e <Θ (for LMS estimate med( e ) 4. Pick the best subset of inliers 5. Find a complete least-squares solution Related to least median squares See also: MAPSAC (Maimum A Posteriori SAmple Consensus) From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 September

74 Cool Robotics Share (this week) D. Wedge The RANSAC Song METR 4202: Robotics 10 April

75 Scale Invariant Feature Transform Basic idea: Take 1616 square window around detected feature Compute edge orientation (angle of the gradient - 90 ) for each piel Throw out weak edges (threshold gradient magnitude) Create histogram of surviving edge orientations 0 2 angle histogram Adapted from slide b David Lowe METR 4202: Robotics 10 April

76 SIFT descriptor Full version Divide the 1616 window into a 44 grid of cells (22 case shown below) Compute an orientation histogram for each cell 16 cells * 8 orientations = 128 dimensional descriptor Adapted from slide b David Lowe METR 4202: Robotics 10 April

77 Properties of SIFT Etraordinaril robust matching technique Can handle changes in viewpoint Up to about 60 degree out of plane rotation Can handle significant changes in illumination Sometimes even da vs. night (below) Fast and efficient can run in real time Lots of code available From David Lowe and Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

78 Cool Robotics Share Source: Youtube: Wired How the Tesla Model S is Made METR 4202: Robotics 10 April

79 Feature matching Given a feature in I 1 how to find the best match in I 2? 1. Define distance function that compares two descriptors 2. Test all the features in I 2 find the one with min distance From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

80 Feature distance How to define the difference between two features f 1 f 2? Simple approach is SSD(f 1 f 2 ) sum of square differences between entries of the two descriptors can give good scores to ver ambiguous (bad) matches f 1 f 2 I 1 I 2 From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

81 Feature distance How to define the difference between two features f 1 f 2? Better approach: ratio distance = SSD(f 1 f 2 ) / SSD(f 1 f 2 ) f 2 is best SSD match to f 1 in I 2 f 2 is 2 nd best SSD match to f 1 in I 2 gives small values for ambiguous matches f 1 f f ' 2 2 I 1 I 2 From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

82 Evaluating the results How can we measure the performance of a feature matcher? feature distance From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

83 True/false positives true match 200 false match The distance threshold affects performance feature distance True positives = # of detected matches that are correct Suppose we want to maimize these how to choose threshold? False positives = # of detected matches that are incorrect Suppose we want to minimize these how to choose threshold? From Szeliski Computer Vision: Algorithms and Applications

84 Camera calibration Determine camera parameters from known 3D points or calibration object(s) internal or intrinsic parameters such as focal length optical center aspect ratio: what kind of camera? eternal or etrinsic (pose) parameters: where is the camera? How can we do this? From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

85 Camera calibration approaches Possible approaches: linear regression (least squares) non-linear optimization vanishing points multiple planar patterns panoramas (rotational motion) From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

86 Image formation equations (X c Y c Z c ) f u c u From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

87 Calibration matri Is this form of K good enough? non-square piels (digital video) skew radial distortion From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

88 Levenberg-Marquardt Iterative non-linear least squares [Press 92] Linearize measurement equations Substitute into log-likelihood equation: quadratic cost function in Dm From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

89 Levenberg-Marquardt What if it doesn t converge? Multipl diagonal b (1 + l) increase l until it does Halve the step size Dm (m favorite) Use line search Other ideas? Uncertaint analsis: covariance S = A-1 Is maimum likelihood the best idea? How to start in vicinit of global minimum? From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

90 Camera matri calibration Advantages: ver simple to formulate and solve can recover K [R t] from M using QR decomposition [Golub & VanLoan 96] Disadvantages: doesn't compute internal parameters more unknowns than true degrees of freedom need a separate camera matri for each new view From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

91 Multi-plane calibration Use several images of planar target held at unknown orientations [Zhang 99] Compute plane homographies Solve for K-TK-1 from Hk s 1plane if onl f unknown 2 planes if (fucvc) unknown 3+ planes for full K Code available from Zhang and OpenCV From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

92 Rotational motion Use pure rotation (large scene) to estimate f estimate f from pairwise homographies re-estimate f from 360º gap optimize over all {KRj} parameters [Stein 95; Hartle 97; Shum & Szeliski 00; Kang & Weiss 99] f=510 f=468 Most accurate wa to get f short of surveing distant points From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

93 SFM: Structure from Motion (& Cool Robotics Share (this week)) METR 4202: Robotics 10 April

94 Structure [from] Motion Given a set of feature tracks estimate the 3D structure and 3D (camera) motion. Assumption: orthographic projection Tracks: (u fp v fp ) f: frame p: point Subtract out mean 2D position i f : rotation s p : position From Szeliski Computer Vision: Algorithms and Applications

95 Structure from motion How man points do we need to match? 2 frames: (Rt): 5 dof + 3n point locations 4n point measurements n 5 k frames: 6(k 1)-1 + 3n 2kn alwas want to use man more From Szeliski Computer Vision: Algorithms and Applications METR 4202: Robotics 10 April

96 Measurement equations Measurement equations u fp = i T f s p i f : rotation s p : position v fp = j T f s p Stack them up W = R S R = (i 1 i F j 1 j F ) T S = (s 1 s P ) From Szeliski Computer Vision: Algorithms and Applications

97 Factorization W = R 2F 3 S 3 P SVD W = U Λ V Λ must be rank 3 W = (U Λ 1/2 )(Λ 1/2 V) = U V Make R orthogonal R = QU S = Q -1 V i ft Q T Qi f = 1 From Szeliski Computer Vision: Algorithms and Applications

98 Results Look at paper figures From Szeliski Computer Vision: Algorithms and Applications

99 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Questions? 106

100 METR4810 Mechatronics Team Project 2 Paul Pounds 8 April 2014 Tune-in net time for <Class voted topic here>! or When we switch to Q&A we can get coffee. Fun fact: Mathematical models predict the Blorch will overwhelm all of humanit some time in