Interest Point Detection. Lecture-4
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1 nterest Point Detection Lectre-4
2 Contents Harris Corner Detector Sm o Sqares Dierences (SSD Corrleation Talor Series Eigen Vectors and Eigen Vales nariance and co-ariance
3 What is an interest point Epressie tetre The point at which the direction o the bondar o object changes abrptl ntersection point between two or more edge segments
4 What is an interest point Epressie tetre The point at which the direction o the bondar o object changes abrptl ntersection point between two or more edge segments
5 Snthetic & Real nterest Points Corners are indicated in red
6 Properties o nterest Point Detectors Detect all (or most tre interest points No alse interest points Well localized. Robst with respect to noise. Eicient detection
7 Possible Approaches to Corner Detection Based on brightness o images Usall image deriaties Based on bondar etraction First step edge detection Cratre analsis o edges
8 Where can we se it? Atomate object tracking Point matching or compting disparit Stereo calibration Estimation o ndamental matri Motion based segmentation Recognition 3D object reconstrction Robot naigation mage retrieal and indeing
9 Correspondence across iews Correspondence: matching points patches edges or regions across images Slide Credit: James Has
10 Eample: estimating ndamental matri that corresponds two iews Slide rom Silio Saarese
11 Eample: strctre rom motion Slide Credit: James Has
12 This class: interest points Sppose o hae to click on some point go awa and come back ater deorm the image and click on the same points again. Which points wold o choose? original Slide Credit: James Has deormed
13 Oeriew o Kepoint Matching 1. Find a set o distinctie kepoints A 1 A 2 A 3 2. Deine a region arond each kepoint A d( A B T B K. Graman B. Leibe 3. Etract and normalize the region content 4. Compte a local descriptor rom the normalized region 5. Match local descriptors
14 Goals or Kepoints Detect points that are repeatable and distinctie
15 Ke trade os A 1 A 2 A 3 Detection o interest points More Repeatable Robst detection Precise localization More Points Robst to occlsion Works with less tetre Description o patches More Distinctie Minimize wrong matches Slide Credit: James Has More Fleible Robst to epected ariations Maimize correct matches
16 nariant Local Featres mage content is transormed into local eatre coordinates that are inariant to translation rotation scale and other imaging parameters Featres Descriptors
17 Choosing interest points Where wold o tell or riend to meet o? Slide Credit: James Has
18 Featre etraction: Corners 9300 Harris Corners Pkw Charlotte NC Slides rom Rick Szeliski Setlana Lazebnik and Kristin Graman
19 Local eatres: main components 1 Detection: denti the interest points 2 Description :Etract eatre ector descriptor srronding each interest point. (1 [ (1 1 1 d ] 3 Matching: Determine correspondence between descriptors in two iews (2 [ (2 2 1 d ] Kristen Graman
20 Goal: interest operator repeatabilit We want to detect (at least some o the same points in both images. No chance to ind tre matches! Yet we hae to be able to rn the detection procedre independentl per image. Kristen Graman
21 Goal: descriptor distinctieness We want to be able to reliabl determine which point goes with which.? Kristen Graman Mst proide some inariance to geometric and photometric dierences between the two iews.
22 Some patches can be localized or matched with higher accrac than others.
23 Some patches can be localized or matched with higher accrac than others.
24 Harris nterest Point Detector Cited b 8636
25 Harris Corner Detector Corner point can be recognized in a window Shiting a window in an direction shold gie a large change in intensit C.Harris M.Stephens. A Combined Corner and Edge Detector. 1988
26 Basic dea lat region: no change in all directions edge : no change along the edge direction corner : signiicant change in all directions
27 Apertre Problem
28 Apertre Problem
29 Correlation k l l k h l k h Kernel mage h h 1 h 2 h 3 h 4 h 5 h 6 h 7 h 8 h 9 h * h h h h h h h h h h
30 Correlation k lhk l h k l Cross correlation k l k l k l Ato correlation
31 minimize minimize Correlation Vs SSD SSD SSD maimize maimize k l hk l 2 k l k l k l k l k l Sm o Sqares Dierence 2 k l 2hk l k l hk l k l k l SSD 2h k l 2hk l k l SSD k l hk l k l Correlatio n k l These terms do not depend on correlation 2
32 Mathematics o Harris Detector Change o intensit or the shit ( E( Ato-correlation w( window nction [ ( shited intensit ( ] intensit 2
33 Ato-Correlation
34 Brook Talor ( His marriage in 1721 with Miss Brdges o Wallington Srre led to an estrangement rom his ather which ended in 1723 ater her death in giing birth to a son who also died he married this time with his ather's approal Sabetta Sawbridge o Olantigh Kent who also died in childbirth in 1730 ; in this case howeer his daghter Elizabeth sried. Talor was elected a ellow o the Roal Societ earl in 1712 and in the same ear sat on the committee or adjdicating the claims o Sir saac Newton and Gottried Leibniz abot Calcls.
35 Talor Series ƒ( Can be represented at point a in terms o its deriaties a a a a 3! ( 2! ( ( ( ( 3 2
36 Talor Series ( ( ( ( ( ( : at ( ( Epress
37 Talor Series Talor Series o right side ( ( ( ( ( t dt t t d d t t ( ( dt t d d t dt d d t 0 t 0 dt dt dt d dt d t 0 0 t Optical Flow Constrain Eqation
38 Mathematics o Harris Detector E 2 ] [ ( E 2 ( E ( E ( E 2 intensit shited intensit ] ( ( [ ( E 2 intensit shited intensit ] ( ( [ ( Talor Series M E ( M
39 Mathematics o Harris Detector E( is an eqation o an ellipse. Let 1 and 2 be eigenales o M M E ( M ( 2 ( 1
40 Eigen Vectors and Eigen Vales The eigen ector o a matri A is a special ector with the ollowing propert Where is called eigen ale To ind eigen ales o a matri A irst ind the roots o: Then sole the ollowing linear sstem or each eigen ale to ind corresponding eigen ector
41 Eample Eigen Vales Eigen Vectors
42 Eigen Vales
43 Eigen Vectors
44 MATLAB Fction [ectorcalec]=eig(c;
45 Mathematics o Harris Detector Classiication o image points sing eigenales o M: 2 Edge 2 >> 1 Corner 1 and 2 are large 1 ~ 2 ; E increases in all directions 1 and 2 are small; E is almost constant in all directions Flat region Edge 1 >> 2 1
46 Mathematics o Harris Detector Measre o cornerness in terms o 1 2 M SDS 1 D R det D ktrace D 2 R k
47 Mathematics o Harris Detector R depends onl on eigenales o M 2 Edge R < 0 Corner R is large or a corner R > 0 R is negatie with large magnitde or an edge R is small or a lat region Flat Edge R small R < 0 1
48
49 Compte corner response
50 Find points with large corner response: R> threshold Find points with large corner response: R> threshold
51 Take onl the points o local maima o R piel ale is greater than its neighbors then it is a local maima.
52
53 Other Version o Harris Detectors R 1 2 Triggs R det( D trace( D Szeliski (Harmonic mean R 1 Shi-Tomasi
54 2 1
55 Mathematics o Harris Detector Change o intensit or the shit ( E( Ato-correlation w( window nction [ ( shited intensit ( ] intensit 2 Window nctions UNFORM GAUSSAN
56 Mathematics o Harris Detector Change o intensit or the shit ( E( Ato-correlation w( window nction [ ( shited intensit ( ] intensit 2 Window nctions UNFORM GAUSSAN
57 Mathematics o Harris Detector E 2 ] [ ( w( E 2 ( w( E ( w( E ( w( E 2 intensit shited intensit ] ( ( [ ( window nction w( E 2 intensit shited intensit ] ( ( [ ( nctionwindoww(talor Series M E ( M
58 Corner Detection: Mathematics Change in appearance o window w( or the shit []: 2 E ( w ( ( ( ( E( E(32 w( Slide Credit: James Has
59 Corner Detection: Mathematics Change in appearance o window w( or the shit []: 2 E ( w ( ( ( ( E( E(00 w( Slide Credit: James Has
60 Corner Detection: Mathematics Change in appearance o window w( or the shit []: 2 E ( w ( ( ( Window nction Shited intensit ntensit Window nction w( = or 1 in window 0 otside Gassian Sorce: R. Szeliski
61 Corner Detection: Mathematics Change in appearance o window w( or the shit []: 2 E ( w ( ( ( We want to ind ot how this nction behaes or small shits E( Slide Credit: James Has
62 Corner Detection: Mathematics The qadratic approimation simpliies to E ( [ ] M where M is a second moment matri compted rom image deriaties: M 2 w( 2 M Slide Credit: James Has
63 w M ( Corners as distinctie interest points 2 2 matri o image deriaties (aeraged in neighborhood o a point. Notation: Slide Credit: James Has
64 The srace E( is locall approimated b a qadratic orm. Let s tr to nderstand its shape. nterpreting the second moment matri M E ] [ ( w M 2 2 ( Slide Credit: James Has
65 ( w M First consider the ais-aligned case (gradients are either horizontal or ertical either λ is close to 0 then this is not a corner so look or locations where both are large. nterpreting the second moment matri Slide Credit: James Has
66 nterpreting the second moment matri Consider a horizontal slice o E( : [ ] M const This is the eqation o an ellipse. Slide Credit: James Has
67 nterpreting the second moment matri Consider a horizontal slice o E( : [ ] M const This is the eqation o an ellipse. Diagonalization o M: M R R 0 2 The ais lengths o the ellipse are determined b the eigenales and the orientation is determined b R direction o the astest change direction o the slowest change ( ma -1/2 ( min -1/2
68 Visalization o second moment matrices Slide Credit: James Has
69 Algorithm
![Harris Detector [Harris88]](/docs-images/87/95744635/images/70-1.jpg "Second moment matri 2 ( D ( D")
![det[ ( ] [trace( ( ] g D 2 2](/docs-images/87/95744635/images/70-7.jpg "2 2 2 ( g( [ g( ] [ g( g( D ]")
70 Harris Detector [Harris88] Second moment matri 2 ( D ( D ( D g( 2 ( D ( D 1. mage deriaties (optionall blr irst det M trace M Sqare o deriaties 3. Gassian ilter g( 2 2 g( 2 g( 2 g( 4. Cornerness nction both eigenales are strong 2 har det[ ( ] [trace( ( ] g D ( g( [ g( ] [ g( g( D ] 2 5. Non-maima sppression Slide Credit: James Has 70 har
71 nariance and coariance We want corner locations to be inariant to photometric transormations and coariant to geometric transormations nariance: image is transormed and corner locations do not change Coariance: i we hae two transormed ersions o the same image eatres shold be detected in corresponding locations Slide Credit: James Has
72 Aine intensit change Onl deriaties are sed => inariance to intensit shit + b ntensit scaling: a a + b R threshold R (image coordinate (image coordinate Partiall inariant to aine intensit change Slide Credit: James Has
73 mage translation Deriaties and window nction are shit-inariant Corner location is coariant w.r.t. translation Slide Credit: James Has
74 mage rotation Second moment ellipse rotates bt its shape (i.e. eigenales remains the same Corner location is coariant w.r.t. rotation Slide Credit: James Has
75 Scaling Corner All points will be classiied as edges Corner location is not coariant to scaling! Slide Credit: James Has
76 Algorithm
77 Reading Material Section Featre Detectors Richard Szeliski "Compter Vision: Algorithms and Application Springer.
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