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UCSD CSE 252C - Viewpoint Invariant Region Detection

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Viewpoint Invariant Region Detectiona review by Erik Murphy-Chutorian “Scale & Affine Invariant Interest Point Detectors”(Mikolajczyk & Schmid)and“Features for Recognition: Viewpoint Invariance for Non-Planar Scenes”(Vedaldi & Soatto)Invariance to ViewThe 2D projection of a 3D scene differs greatly in appearance due to the position and orientation of the cameraCannot assume all transformations are uniform scaling or rotation.Important for Object Recognition and Wide-baseline stereo.Viewpoint Invariant RegionsScale & Affine Invariant Interest Point Detectors 71where the scalars σIand σDare the integration anddifferentiation scales respectively. We can then derivethe following relation:"I,R= A"I,LAT= σI!AM−1LAT"= σI(A−TMLA−1)−1= σIM−1R(8)"D,R= A"D,LAT= σD!AM−1LAT"= σD(A−TMLA−1)−1= σDM−1RThis shows that imposing the conditions, defined inEq. (7) leads to the relations 8, under the assumptionthat the points are related by an affine transformationand the matrices are computed for corresponding scalesσIand σD. We can now invert the problem and supposethat we have two points related by an unknown affinetransformation. If we estimate the matrices "Rand"Lsuch that the matrices verify conditions 7 and 8,then relation 6 will be true. This property enables thetransformation parameters to be expressed directly bythe matrix components. The affine transformation canthen be defined by:A = M−1/2RR M1/2Lwhere R is an orthogonal matrix which represents anarbitrary rotation or mirror transformation. In the nextsection we present an iterative algorithm for estimat-ing the matrices "Rand "L. The affine transformationcan be estimated up to a rotation between two cor-responding points without any prior knowledge aboutthis transformation. Furthermore, the matrices MLandMR, computed under conditions 7 and 8, determinecorresponding regions defined by xTMx = 1. If theneighborhood of points xRand xLare normalized bytransformations x"R= M1/2RxRand x"L= M1/2LxL, re-spectively, the normalized regions are related by a sim-ple rotation x"L= Rx"R(Baumberg, 2000; Garding andLindeberg, 1994).xR= AxL= M−1/2RR M1/2LxL,M1/2RxR= R M1/2LxL(9)The matrices M"Land M"Rin the normalized frames areequal to a pure rotation matrix (see Fig. 4). In otherwords, the intensity patterns in the normalized framesare isotropic in terms of the second moment matrix.Isotropy Measure. The second moment matrix canalso be interpreted as an isotropy measure. WithoutFigure 4. Diagram illustrating the affine normalization based on thesecond moment matrices. Image coordinates are transformed withmatrices M−1/2Land M−1/2R. The transformed images are related byan orthogonal transformation.loss of generality we suppose that a local anisotropicstructure is an affine transformed isotropic structure.To compensate for the affine deformation, we have tofind the transformation that projects the anisotropic pat-tern to the isotropic one. Note that rotation preservesthe isotropy of an image patch, therefore, the affinedeformation of an isotropic structure can be deter-mined up to a rotation factor. This rotation can be reco-vered by methods based on the gradient orientation(Lowe, 1999; Mikolajczyk, 2002). The local isotropycan be measured by the eigenvalues of the second mo-ment matrix µ(x, σI, σD). If the eigenvalues are equalwe consider the point isotropic. To obtain a normalizedmeasure we use the eigenvalue ratio:Q =λmin(µ)λmax(µ)(10)The value of Q varies in the range [0 . . . 1] with 1 fora perfect isotropic structure. This measure can give aslightly different response for different scales as thematrix µ is computed for a given integration and dif-ferentiation scale. These scales should be selected in-dependently of the image resolution. The scale selec-tion technique (see Section 2.1) gives the possibilityto determine the integration scale related to the lo-cal image structure. The differentiation and integrationscales can be related by a constant factor s, σD= sσI.For obvious reasons the differentiation scale should al-ways be smaller than the integration scale. The factorScale & Affine Invariant Interest Point Detectors 71where the scalars σIand σDare the integration anddifferentiation scales respectively. We can then derivethe following relation:"I,R= A"I,LAT= σI!AM−1LAT"= σI(A−TMLA−1)−1= σIM−1R(8)"D,R= A"D,LAT= σD!AM−1LAT"= σD(A−TMLA−1)−1= σDM−1RThis shows that imposing the conditions, defined inEq. (7) leads to the relations 8, under the assumptionthat the points are related by an affine transformationand the matrices are computed for corresponding scalesσIand σD. We can now invert the problem and supposethat we have two points related by an unknown affinetransformation. If we estimate the matrices "Rand"Lsuch that the matrices verify conditions 7 and 8,then relation 6 will be true. This property enables thetransformation parameters to be expressed directly bythe matrix components. The affine transformation canthen be defined by:A = M−1/2RR M1/2Lwhere R is an orthogonal matrix which represents anarbitrary rotation or mirror transformation. In the nextsection we present an iterative algorithm for estimat-ing the matrices "Rand "L. The affine transformationcan be estimated up to a rotation between two cor-responding points without any prior knowledge aboutthis transformation. Furthermore, the matrices MLandMR, computed under conditions 7 and 8, determinecorresponding regions defined by xTMx = 1. If theneighborhood of points xRand xLare normalized bytransformations x"R= M1/2RxRand x"L= M1/2LxL, re-spectively, the normalized regions are related by a sim-ple rotation x"L= Rx"R(Baumberg, 2000; Garding andLindeberg, 1994).xR= AxL= M−1/2RR M1/2LxL,M1/2RxR= R M1/2LxL(9)The matrices M"Land M"Rin the normalized frames areequal to a pure rotation matrix (see Fig. 4). In otherwords, the intensity patterns in the normalized framesare isotropic in terms of the second moment matrix.Isotropy Measure. The second moment matrix canalso be interpreted as an isotropy measure. WithoutFigure 4. Diagram illustrating the affine normalization based on thesecond moment matrices. Image coordinates are transformed withmatrices M−1/2Land M−1/2R. The transformed images are related byan orthogonal transformation.loss of generality we suppose that a local anisotropicstructure is an affine transformed isotropic structure.To compensate for the affine deformation, we have tofind the


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