The SIFT (Scale Invariant Feature Transform) Detector And .
The SIFT (Scale Invariant FeatureTransform) Detector and Descriptordeveloped by David LoweUniversity of British ColumbiaInitial paper 1999Newer journal paper 2004
Motivation The Harris operator is not invariant to scale and itsdescriptor was not invariant to rotation1. For better image matching, Lowe’s goal was todevelop an operator that is invariant to scale androtation. The operator he developed is both a detector and adescriptor and can be used for both image matchingand object recognition.1ButSchmidt and Mohr developed a rotation invariant descriptor for it in 1997.4/7/20062
Idea of SIFT Imagecontent is transformed into local featurecoordinates that are invariant to translation, rotation,scale, and other imaging parametersSIFT Features4/7/20063
Claimed Advantages of SIFT Locality: features are local, so robust to occlusionand clutter (no prior segmentation) Distinctiveness: individual features can bematched to a large database of objects Quantity: many features can be generated for evensmall objects Efficiency: close to real-time performance Extensibility: can easily be extended to wide rangeof differing feature types, with each addingrobustness4/7/20064
Overall Procedure at a High Level1. Scale-space extrema detectionSearch over multiple scales and image locations.2. Keypoint localizationFit a model to detrmine location and scale.Select keypoints based on a measure of stability.3. Orientation assignmentCompute best orientation(s) for each keypoint region.4. Keypoint descriptionUse local image gradients at selected scale and rotationto describe each keypoint region.4/7/20065
1. Scale-space extrema detection Goal: Identify locations and scales that can berepeatably assigned under different views of thesame scene or object.Method: search for stable features across multiplescales using a continuous function of scale.Prior work has shown that under a variety ofassumptions, the best function is a Gaussianfunction.The scale space of an image is a function L(x,y,σ)that is produced from the convolution of a Gaussiankernel (at different scales) with the input image.4/7/20066
Aside: Image PyramidsAnd so on.3rd level is derived from the2nd level according to the samefuntion2nd level is derived from theoriginal image according tosome functionBottom level is the original image.4/7/20067
Aside: Mean PyramidAnd so on.At 3rd level, each pixel is the meanof 4 pixels in the 2nd level.At 2nd level, each pixel is the meanof 4 pixels in the original image.meanBottom level is the original image.4/7/20068
Aside: Gaussian PyramidAt each level, image is smoothed andreduced in size.And so on.Apply Gaussian filterAt 2nd level, each pixel is the resultof applying a Gaussian mask tothe first level and then subsamplingto reduce the size.Bottom level is the original image.4/7/20069
Example: Subsampling with Gaussian pre-filteringG 1/8G 1/4Gaussian 1/24/7/200610
Lowe’s Scale-space extrema detection Scale-space function L Gaussian convolutionwhere σ is the width of the Gaussian. Laplacian of Gaussian kernel has been used in other workon scale invariance Difference of Gaussian kernel is a close approximate toscale-normalized Laplacian of Gaussian2 scales:σ and kσ4/7/200611
Scale-space extrema detection4/7/2006 Gaussian is an ad hocsolution of heatdiffusion equation Hence k is not necessarily verysmall in practice12
Lowe’s Pyramid Scheme Scale space is separated into octaves: Octave 1 uses scale σ Octave 2 uses scale 2σ etc. In each octave, the initial image is repeatedly convolvedwith Gaussians to produce a set of scale space images. Adjacent Gaussians are subtracted to produce the DOG After each octave, the Gaussian image is down-sampledby a factor of 2 to produce an image ¼ the size to startthe next level.4/7/200613
Lowe’s Pyramid Schemes 2 filtersσs 1 2(s 1)/sσ0.σi 2i/sσ0.σ2 22/sσ0σ1 21/sσ0σ04/7/2006s 3imagesincludingoriginalThe parameter s determines the number of images per octave.s 2differenceimages14
Key point localization Detect maxima andminima of difference-ofGaussian in scale spaceEach point is comparedto its 8 neighbors in thecurrent image and 9neighbors each in thescales above and below4/7/2006s 2 difference images.top and bottom ignored.s planes searched.es amRplelurBubtra ctSFor each max or min found,output is the location andthe scale.15
Scale-space extrema detection: experimental results over 32 imagesthat were synthetically transformed and noise added.% detectedaverage no. detected% correctly matchedaverage no. matchedStability ExpenseSampling in scale for efficiency How many scales should be used per octave? S ? 4/7/2006More scales evaluated, more keypoints foundS 3, stable keypoints increased tooS 3, stable keypoints decreasedS 3, maximum stable keypoints found16
2. Keypoint localization Detailed keypoint determination Sub-pixel and sub-scale location scaledetermination Ratio of principal curvature to reject edges andflats (like detecting corners)4/7/200617
Keypoint localization Once a keypoint candidate is found, perform adetailed fit to nearby data to determine location, scale, and ratio of principal curvaturesIn initial work keypoints were found at location andscale of a central sample point.In newer work, they fit a 3D quadratic function toimprove interpolation accuracy.The Hessian matrix was used to eliminate edgeresponses.4/7/200618
Eliminating the Edge Response Reject flats: 0.03Reject edges:Let α be the eigenvalue withlarger magnitude and β the smaller.Let r α/β.So α rβr 10What does this look like? 4/7/2006(r 1)2/r is at amin when the2 eigenvaluesare equal.19
3. Orientation assignment Create histogram oflocal gradient directionsat selected scaleAssign canonicalorientation at peak ofsmoothed histogramEach key specifiesstable 2D coordinates(x, y, scale,orientation)If 2 major orientations, use both.4/7/200620
Keypoint localization with orientation233x189832initial keypoints729keypoints aftergradient threshold4/7/2006536keypoints afterratio threshold21
4. Keypoint Descriptors At this point, each keypoint has locationscaleorientationNext is to compute a descriptor for the localimage region about each keypoint that is 4/7/2006highly distinctiveinvariant as possible to variations such aschanges in viewpoint and illumination22
Normalization Rotate the window to standard orientation Scale the window size based on the scale atwhich the point was found.4/7/200623
Lowe’s Keypoint Descriptor use the normalized circular region about thekeypointcompute gradient magnitude and orientation at eachpoint in the regionweight them by a Gaussian window overlaid on thecirclecreate an orientation histogram over the 4 X 4subregions of the window4 X 4 descriptors over 16 X 16 sample array wereused in practice. 4 X 4 times 8 directions gives avector of 128 values.4/7/200624
Lowe’s Keypoint Descriptor(shown with 2 X 2 descriptors over 8 X 8) Invariant to other changes (Complex Cell)In experiments, 4x4 arrays of 8 bin histogram is used,a total of 128 features for one keypoint4/7/200625
Harris-Laplacian1Find local maximum of: Harris cornerdetector in space(image coordinates) Laplacian in scalescaley Harris x DoG scale SIFT (Lowe)2Find local maximumyof:– Difference of Gaussiansin space and scale DoG Laplacian Scale Invariant Detectorsx1 K.Mikolajczyk,C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20012 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. IJCV 20044/7/200626
Scale Invariant Detectors Experimental evaluation of detectorsw.r.t. scale changeRepeatability rate:# correspondences# possible correspondencesK.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20014/7/200627
Schmid’s Comparison with Harris-Laplacian Affine-invariant comparison Translation-invariant – local features: both OKRotation-invariant Harris-Laplacian SIFT OrientationShear-invariant Harris-Laplacian EigenvaluesSIFT PCANoWithin 50 degree of viewpoint, SIFT is better thanHL, after 70 degree, HL is better.4/7/200628
Comparison with Harris-Laplacian Computational time: 4/7/2006SIFT uses few floating point calculationHL uses iterative calculation which costs muchmore29
Using SIFT for Matching “Objects”4/7/200630
4/7/200631
Uses for SIFT Feature points are used also for: 4/7/2006Image alignment (homography, fundamentalmatrix)3D reconstructionMotion trackingObject recognitionIndexing and database retrievalRobot navigation other32
(image coordinates) Laplacian in scale scale x y Harris Laplacian SIFT (Lowe)2 Find local maximum of: – Difference of Gaussians in space and scale scale x y DoG DoG 1 K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 2001 2 D.Lowe. “
Lowe, D. “Distinctive image features from scale-invariant keypoints” International Journal of Computer Vision, 60, 2 (2004), pp. 91-110 Pele, Ofir. SIFT: Scale Invariant Feature Transform. Sift.ppt Lee, David. Object Recognition from Local Scale-Invariant Features (SIFT). O319.S
Lowe, D. “Distinctive image features from scale-invariant keypoints” International Journal of Computer Vision, 60, 2 (2004), pp. 91-110 Pele, Ofir. SIFT: Scale Invariant Feature Transform. Sift.ppt Lee, David. Object Recognition from Local Scale-
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