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Image Matching Using GeneralizedScale-Space Interest PointsTony LindebergstarSchool of Computer Science and CommunicationKTH Royal Institute of Technology, Stockholm, SwedenAbstract. The performance of matching and object recognition meth-ods based on interest points depends on both the properties of theunderlying interest points and the associated image descriptors. Thispaper demonstrates the advantages of using generalized scale-space in-terest point detectors when computing image descriptors for image-basedmatching.These generalized scale-space interest pointsarebased on link-ing of image features over scale and scale selection by weighted averagingalong feature trajectories over scale and allow for a higher ratio of correctmatches anda lower ratio of false matchescompared topreviously knowninterest point detectors within the same class. Specifically, it is shownhow a significant increase in matching performance can be obtained inrelation to the underlying interest point detectors in the SIFT and theSURF operators. We propose that these generalized scale-space interestpoints when accompanied by associated scale-invariant image descriptorsshould allow for better performance of interest point based methods forimage-based matching, object recognition and related vision tasks.Keywords: interest points, scale selection, scale linking, matching, ob-ject recognition, feature detection, scale invariance, scale space.1 IntroductionA common approach to image-based matching consists of detecting interestpoints with associated image descriptors from image data and then establishinga correspondence between the image descriptors. Specifically, the SIFT opera-tor 1 and the SURF operator 2 have been demonstrated to be highly usefulfor this purpose with many successful applications, including object recognition,3-D object and scene modelling, video tracking, gesture recognition, panoramastitching as well as robot localization and mapping.In the SIFT operator, the intitial detection of interest points is based ondierences-of-Gaussians from which local extrema over space and scale are com-puted. Such points are referred to as scale-space extrema. The dierence ofGaussian operator can be seen as an approximation of the Laplacian operator,starThe support from the Swedish Research Council (contract 2010-4766), the RoyalSwedish Academy of Sciences and the Knut and Alice Wallenberg Foundation isgratefully acknowledged.A. Kuijper et al. (Eds.): SSVM 2013, LNCS 7893, pp. 355367, 2013.c Springer-Verlag Berlin Heidelberg 2013356 T. Lindebergand it follows from general results in 3 that the scale-space extrema of theLaplacian have scale-invariant properties that can be used for normalizing localimage patches or image descriptors with respect to scaling transformations. TheSURF operator is on the other hand based on initial detection of image featuresthat can be seen as approximations of the determinant of the Hessian operatorwith the underlying Gaussian derivatives replaced by an approximation in termsof Haar wavelets. From the general results in 3 it follows that scale-space ex-tremaofthedeterminantofthe Hessiandoalsoleadtoscale-invariantbehaviour,which can be used for explaining the good performance of the SIFT and SURFoperators under scaling transformations.The subject of this article is to show how the performance of image matchingcan be improved by using a generalized framework for detecting interest pointsfrom scale-space features involving (i) new Hessian feature strength measuresat a fixed scale, (ii) linking of image features over scale into feature trajectoriesto allow for a better selection of significant image features and (iii) scale selec-tion by weighted averaging along feature trajectories to allow for more robustscale estimates. By replacing the interest points in the regular SIFT and SURFoperators by generalized scale-space interest points to be described below, itis possible to define new scale-invariant image descriptors that lead to bettermatching performance compared to the performance obtained by correspondinginterest point detection mechanisms as used in the SIFT and SURF operators.2 Generalized Scale-Space Interest PointsBasic requirements on the interest points on which image matching is to beperformed are that they should (i) have a clear, preferably mathematically well-founded, definition, (ii) have a well-defined position in image space, (iii) havelocal image structures around the interest point that are rich in informationcontent such that the interest points carry important information to later stagesand (iv) be stable under local and global deformations of the image domain,including perspective image deformations and illumination variations such thatthe interest points can be reliably computed with a high degree of repeatability.2.1 Dierential Entities for Detecting Scale-Space Interest PointsAs basis for performing local image measurements on a two-dimensional imagef, we will consider a scale-space representation 410L(x,y; t)=integraldisplay(u,v)IR2f(xu,y v)g(u,v; t)dudv (1)generated by convolution with Gaussian kernels g(x,y; t)=12te(x2+y2)/2tofincreasing width, where the variance t is referred to as the scale parameter, andwith scale-normalized derivatives with = 1 defined according to a = t/2xand = t/2y3. To detect interest points within this scale-space framework,we will consider:Image Matching Using Generalized Scale-Space Interest Points 357(i) either the scale-normalized Laplacian operator 32normL = t(Lxx+Lyy)(2)or the scale-normalized determinant of the Hessian 3detHnormL = t2(LxxLyyL2xy), (3)(ii) either of the following dierential analogues/extensions of the Harris oper-ator 11 proposed in 12, 13; the unsigned Hessian feature strength mea-sure ID1,normL =braceleftbiggt2(detHLk trace2HL)ifdetHLk trace2HL00ohrwis(4)or the signed Hessian feature strength measure ID1,normL =t2(detHLk trace2HL)ifdetHLk trace2HL0t2(detHL+k trace2HL)ifdetHL+k trace2HL0) 0.7484 0.7994 0.7512 0.7574 0.7498 0.7784detHnormL (D1L0) 0.7721 0.8225 0.7635 0.7932 0.7678 0.8079detHnormL (D1L0) 0.7691 0.8163 0.7602 0.7841 0.7647 0.8002D1,normL 0.7719 0.8280 0.7596 0.7977 0.7658 0.8128D1,normL 0.7698 0.8241 0.7578 0.7916 0.7638 0.8079D2,normL (D1L0) 0.7203 0.8187 0.7111 0.7776 0.7157 0.7981D2,normL (D1L0) 0.7204 0.8261 0.7113 0.7766 0.7159 0.8014Harris-Laplace 0.7002 0.7855 0.7046 0.7535 0.7024 0.7695Harris-detHessian 0.7406 0.7608 0.7561 0.7319 0.7406 0.7463Eciency: SURF-like image descriptorscaling foreshortening averageInterest points extr link extr link extr link2normL (D1L0) 0.7424 0.7832 0.7280 0.7140 0.7352 0.7486detHnormL (D1L0) 0.7656 0.8072 0.7402 0.7504 0.7529 0.7788detHnormL (D1L0) 0.7628 0.8015 0.7372 0.7430 0.7500 0.7723D1,normL 0.7661 0.8126 0.7354 0.7537 0.7507 0.7831D1,normL 0.7640 0.8081 0.7334 0.7478 0.7487 0.7779D2,normL (D1L0) 0.7157 0.8014 0.6870 0.7284 0.7013 0.7649D2,normL (D1L0) 0.7158 0.8100 0.6873 0.7328 0.7015 0.7714Harris-Laplace 0.6948 0.7620 0.6724 0.6944 0.6836 0.7282Harris-detHessian 0.7345 0.7381 0.7192 0.6705 0.7268 0.7043the Harris operator 11: (i) the Harris-Laplace operator 19 based on spatialextrema of the Harris measure and scale selection from local extrema over scaleofthescale-normalizedLaplacian,(ii)ascale-linkedversionoftheHarris-Laplaceoperator with scale selection by weighted averaging over feature trajectories ofHarrisfeatures12,and(iii-iv)twoHarris-detHessianoperatorsanalogoustotheHarris-Laplace operators, with the dierence that scale selection is performedbased on the scale-normalized determinant of the Hessian instead of the scale-normalized Laplacian 12.Image Matching Using Generalized Scale-Space Interest Points 363Table 2. The five best combinations of interest points and image descriptors amongthe 229 = 36 combinations considered in this experimental evaluation as rankedon the ratio of interest points that lead to correct matches. For comparison, resultsare also shown for the SIFT descriptor based on scale-space extrema of the Laplacian,the SIFT or SURF descriptors based on scale-space extrema of the determinant of theHessian and the SIFT descriptor based on Harris-Laplace interest points.Interest points and image descriptors ranked on matching eciencyInterest points Scale selection Descriptor EciencyD1,normL link SIFT 0.8128D1,normL link SIFT 0.8079detHnormL (D1L0) link SIFT 0.8079D2,normL (D1L0) link SIFT 0.8014detHnormL (D1L0) link SIFT 0.8002.detHnormL (D1L0) extr SIFT 0.7721detHnormL (D1L0) extr SURF 0.76562normL (D1L0) extr SIFT 0.7484Harris-Laplace extr SIFT 0.7002The experiments are based on detecting the N = 800 strongest interest pointsextracted from the first image, regarded as reference image for the homography.To obtain an approximate uniform density of interest points under scaling trans-formations, an adapted number Nprime= N/s2of interest points is searched for(i) within the subwindow of the reference image that is mapped to the interiorof the transformed image and (ii) in the transformed image, with s denotingrelative scaling factor between the two images.This procedureis repeatedforallpairsof imageswithin thegroupsofdistancevariations or viewing variations respectively, implying up to 55 image pairs forthe scaling transformations and 6 image pairs for the foreshortening transforma-tions, i.e. up to 61 matching experiments for each one of the 12 posters, thus upto 732 experiments for each one of 29 interest point detectors.As can be seen from the results of matching SIFT- or SURF-like image de-scriptors in Table 1, the interest point detectors based on scale linking and withscale selection by weighted averaging along feature trajectories generally lead tosignificantly higher eciency rates compared to the corresponding interest pointdetectors based on scale selection from local extrema over scale. Specifically, thehighest eciency rates are obtained with the scale linked version of the unsignedHessian feature strength measure D1,normL, followed by scale-linked versions ofthe unsigned signed Hessian feature strength measureD1,normL and the deter-minant of the Hessian operator detHnormL with complementary thresholdingon D1,normL0.364 T. LindebergScaling variations Foreshortening variations0 0.5 1 1.5 2 2.5 30.550.60.650.70.750.80.852Ldet H L (D1L 0)det H L (D1L 0)D1LD1LD2L (D1L 0)HarrisLaplace2L (extr)20 25 30 35 40 450.550.60.650.70.750.80.852Ldet H L (D1L 0)det H L (D1L 0)D1LD1LD2L (D1L 0)HarrisLaplace2L (extr)Fig.3. Graphs showing how the matching eciency depends upon (left) the amountof scaling s 1.25,6.0 for scaling transformations (with log2s on the horizontalaxis) and (right) the dierence in viewing angle 22.5,45 for the foreshorteningtransformations for interest point matching based on SIFT-like image descriptors.Corresponding experimental results that cannot be included here because oflack of space show that the lowestand thus the best 1-precisionscore is obtainedwith the determinant of the Hessian operator detHnormL with complementarythresholding onD1,normL0, followed by the determinant of the Hessian op-erator detHnormL with complementary thresholding on D1,normL0.Among the more traditional feature detectors based on scale selection fromlocal extrema over scale, we can also note that the determinant of the Hes-sian operator detHnormL performs significantly better than both the Laplacianoperator 2normL and the Harris-Laplace operator. We can also note that theHarris-Laplace operator can be improved by either scale linking or by replacingscale selection based on the scale-normalized Laplacian by scale selection basedon the scale-normalized determinant of the Hessian.When comparing the results obtained for SIFT-like and SURF-like image de-scriptors, we can see that the SIFT-like image descriptors lead to both highereciency rates and lower 1-precision scores than the SURF-like image descrip-tors. This qualitative relationship holds over all types of interest point detectors.In this respect, the pure image descriptor in the SIFT operator is clearly betterthan the pure image descriptor in the SURF operator. Specifically, more reliableimage matches can be obtained by replacing pure image descriptor in the SURFoperator by the pure image descriptor in the SIFT operator.Table 2 lists the five best combinations of interest point detectors and imagedescriptorsinthisevaluationasrankedontheireciencyvalues.Forcomparison,the results of our corresponding analogues of the SIFT operator with interestpoint detection from scale-space extrema of the Laplacian and our analogue ofthe SURF operator based on scale-space extrema of the determinant of the Hes-sian are also shown. As can be seen from this ranking, the best combinationsImage Matching Using Generalized Scale-Space Interest Points 365of generalized points with SIFT-like image descriptors perform significantly bet-ter than the corresponding analogues of regular SIFT or regular SIFT based onscale-space extrema of the Laplacian or the determinant of the Hessian.Figure 3 shows graphs of how the eciency rate depends upon the amount ofscaling for the scaling transformationsand the dierence in viewing anglefor theforeshortening transformations. As can be seen from these graphs, the interestpoint detectors detHnormL, D1,normL andD1,normL that possess ane covari-ancepropertiesorapproximationsthereof12,13do alsohavethe bestmatchingproperties under foreshortening transformations. Specifically, the generalized in-terest point detectors based on scale linking perform significantly better thanscale-space extrema of the Laplacian or the determinant of the Hessian as wellas better than the Harris-Laplace operator.5 Summary and ConclusionsWe have presented a set of extensions of the SIFT and SURF operators, byreplacing the underlying interest point detectors used for computing the SIFTor SURF descriptors by a family of generalized scale-space interest points.These generalized scale-space interest points are based on (i) new dieren-tial entities for interest point detection at a fixed scale in terms of new Hessianfeature strength measures, (ii) linking of image structures into feature trajec-tories over scale and (ii) performing scale selection by weighted averaging ofscale-normalized feature responses along these feature trajectories 12.The generalized scale-space interest points are all scale-invariant in the sensethat (i) the interest points are preserved under scaling transformation and that(ii) the detection scales obtained from the scale selection step are transformed inascalecovariantway.Thereby,thedetectionscalecanbeusedfordefiningalocalscale normalized reference frame around the interest point, which means thatimage descriptors that are defined relative to such a scale-normalized referenceframe will also be scale invariant.By complementing these generalized scale-space interest points with local im-agedescriptorsdefinedinaconceptuallysimilarwayasthepureimagedescriptorparts in SIFT or SURF, while being based on image measurements in terms ofGaussianderivativesinstead ofimagepyramids orHaar wavelets,wehaveshownthat the generalized interest points with their associated scale-invariant imagedescriptors lead to a higher ratio of correct matches and a lower ratio of falsematches compared to corresponding results obtained with interest point detec-tors based on more traditional scale-space extrema of the Laplacian, scale-spaceextrema of the determinant of the Hessian or the Harris-Laplace operator.Intheliterature,therehasbeensomedebateconcerningwhichoneoftheSIFTor SURF descriptors leads to the best performance. In our experimental evalua-tions, we have throughtout found that our SIFT-like image descriptor based onGaussian derivatives generally performs much better than our SURF-like imagedescriptor, also expressed in terms of Gaussian derivatives. In this respect, thepure image descriptor in the SIFT operator can be seen as significantly betterthan the pure image descriptor in the SURF operator.366 T. LindebergConcerning the underlying interest points, we have on the other hand foundthat the determinant of the Hessian operator to generally perform significantlybetter than the Laplacian operator, both for scale selection based on scale-spaceextrema and scale selection based on weighted averaging of featureresponsesalongfeaturetrajectoriesobtainedbyscalelinking.Sincethedierence-of-Gaussians interest point detector in the regular SIFT operator can be seenas an approximation of the scale-normalized Laplacian, we can therefore regardthe underlying interest point detector in the SURF operator as significantlybetter than the interest point detector in the SIFT operator. Specifically, wecould expect a significant increase in the performance of SIFT by just replacingthe scale-space extrema of the dierence-of-Gaussians operator by scale-spaceextrema of the determinant of the Hessian.In addition, our experimental evaluations show that further improvements arepossiblebyreplacingthe interestpointsobtainedfromscale-spaceextremaintheSIFT and SURF operators by generalized scale-space interest points obtainedby scale linking, with the best results obtained with the Hessian feature strengthmeasures D1,normL andD1,normL followed by the determinant of the HessiandetHnormL and the Hessian feature strength measureD2,normL.References1. Lowe, D.: Distinctive image features from scale-invariant keypoints. Int. J. Comp.Vis. 60, 91110 (2004)2. Bay, H., Ess, A., Tuytelaars, T.: van Gool: Speeded up robust features (SURF).CVIU 110, 346359 (2008)3. Lindeberg, T.: Feature detection with automatic scale selection. Int. J. Comp.Vis. 30, 77116 (1998)4. Witkin, A.P.:
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