Feature extraction: Corners Harris Corners Pkwy, Charlotte, NC

Transcription

1 Featre etraction: Corners 9300 Harris Corners Pkw Charlotte NC

2 Wh etract featres? Motiation: panorama stitching We hae two images how do we combine them?

3 Wh etract featres? Motiation: panorama stitching We hae two images how do we combine them? Step 1: etract featres Step : match featres

4 Wh etract featres? Motiation: panorama stitching We hae two images how do we combine them? Step 1: etract featres Step : match featres Step 3: align images

5 Characteristics of good featres Repeatabilit The same featre can be fond in seeral images despite geometric and photometric transformations Salienc ach featre is distinctie Compactness and efficienc Man fewer featres than image piels Localit A featre occpies a relatiel small area of the image; robst to cltter and occlsion

6 Applications Featre points are sed for: mage alignment 3D reconstrction Motion tracking Robot naigation ndeing and database retrieal Object recognition

7 Finding Corners Ke propert: in the region arond a corner image gradient has two or more dominant directions Corners are repeatable and distinctie CH i dmst h "A C bi d C d d D t t C.Harris and M.Stephens. "A Combined Corner and dge Detector. Proceedings of the 4th Ale Vision Conference: pages

8 Corner Detection: Basic dea We shold easil recognize the point b looking throgh a small window Shifting a window in an direction shold gie a large change in intensit Sorce: A. fros flat region: no change in all directions edge : no change along the edge direction corner : significant change in all directions

9 Corner Detection: Mathematics Change in appearance of window w for the shift []: [ ] = w 3 w

10 Corner Detection: Mathematics Change in appearance of window w for the shift []: [ ] = w 00 w

11 Corner Detection: Mathematics Change in appearance of window w for the shift []: [ ] = w Window fnction Shifted intensit ntensit Window fnction w = or 1 in window 0 otside Gassian Sorce: R. Szeliski

12 Corner Detection: Mathematics Change in appearance of window w for the shift []: [ ] = w We want to find ot how this fnction behaes for small shifts

13 Corner Detection: Mathematics Change in appearance of window w for the shift []: [ ] = w We want to find ot how this fnction behaes for small shifts Local qadratic approimation of in the neighborhood of 00 0 isgienbthesecond second-order order Talor epansion: 00 [ ] [ ]

14 Corner Detection: Mathematics [ ] w = Second-order Talor epansion of abot 00: ] [ 1 00 ] [ ] [ 00 ] [ 00 [ ] w = w = [ ] w w = [ ] w

15 Corner Detection: Mathematics [ ] w = Second-order Talor epansion of abot 00: ] [ 1 00 ] [ = ] [ 00 ] [ = = w w = = 00 w =

16 Corner Detection: Mathematics [ ] w = Second-order Talor epansion of abot 00: w w ] [ w w ] [ 0 00 = = = w w = = 00 w =

17 Corner Detection: Mathematics The qadratic approimation simplifies to [ ] M where M is a second moment matri compted from image deriaties: es M = w M

18 nterpreting the second moment matri The srface is locall approimated b a qadratic form. Let s tr to nderstand its shape. M ] [ = w M = w M

19 nterpreting the second moment matri First consider the ais-aligned case gradients are either horizontal or ertical 0 λ gradients are either horizontal or ertical = = λ λ w M f either λ is close to 0 then this is not a corner so look for locations where both are large.

20 nterpreting the second moment matri Consider a horizontal slice of : [ ] M = const This is the eqation of an ellipse.

21 nterpreting the second moment matri Consider a horizontal slice of : [ ] M = const This is the eqation of an ellipse. Diagonalization of M: M λ = R R 0 λ The ais lengths of the ellipse are determined b the eigenales and the orientation is determined b R direction of the fastest change direction of the slowest change λ ma -1/ λ -1/ min

22 Visalization of second moment matrices

23 Visalization of second moment matrices

24 nterpreting the eigenales Classification of image points sing eigenales of M: λ dge λ >> λ 1 Corner λ 1 and λ are large λ 1 ~ λ ; increases in all directions λ 1 and λ are small; is almost constant in all directions Flat region dge λ 1 >> λ λ 1

25 Corner response fnction R = det M M α trace = λ1λ α λ1 λ α: constant t to dge R < 0 Corner R>0 R small Flat region dge R < 0

26 Harris detector: Steps 1. Compte Gassian deriaties at each piel. Compte second moment matri M in a Gassian window arond each piel 3. Compte corner response fnction R 4. Threshold R 5. Find local maima of response fnction nonmaimm sppression CH i d MSt h A C bi d C d d D t t C.Harris and M.Stephens. A Combined Corner and dge Detector. Proceedings of the 4th Ale Vision Conference: pages

27 Harris Detector: Steps

28 Harris Detector: Steps Compte corner response R

29 Harris Detector: Steps Find points with large corner response: R>threshold

30 Harris Detector: Steps Tk Take onl the points it of flocal lmaima of frr

31 Harris Detector: Steps

32 nariance and coariance We want corner locations to be inariant to photometric transformations and coariant to geometric transformations nariance: image is transformed and corner locations do not change Coariance: if we hae two transformed ersions of the same image featres shold be detected in corresponding locations

33 Affine intensit change a b Onl deriaties are sed => inariance to intensit shift b ntensit scaling: a R threshold R image coordinate image coordinate Partiall inariant to affine intensit change

34 mage translation Deriaties and window fnction are shift-inariant Corner location is coariant w.r.t. translation

35 mage rotation Second moment ellipse rotates bt its shape i.e. eigenales remains the same Corner location is coariant w.r.t. rotation

36 Scaling Corner All points will be classified as edges Corner location is not coariant to scaling!