Is SIFT scale invariant?

Jean-Michel Morel; Guoshen Yu

doi:10.3934/ipi.2011.5.115

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February 2011, 5(1): 115-136. doi: 10.3934/ipi.2011.5.115

Is SIFT scale invariant?

Jean-Michel Morel ^1, and Guoshen Yu ^2,

1.	CMLA, ENS Cachan, 61 avenue du Président Wilson, 94235 Cachan Cedex, France
2.	CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex, France

Received October 2010 Revised November 2010 Published February 2011

Abstract
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This note is devoted to a mathematical exploration of whether Lowe's Scale-Invariant Feature Transform (SIFT)[21], a very successful image matching method, is similarity invariant as claimed. It is proved that the method is scale invariant only if the initial image blurs are exactly guessed. Yet, even a large error on the initial blur is quickly attenuated by this multiscale method, when the scale of analysis increases. In consequence, its scale invariance is almost perfect. The mathematical arguments are given under the assumption that the Gaussian smoothing performed by SIFT gives an aliasing free sampling of the image evolution. The validity of this main assumption is confirmed by a rigorous experimental procedure, and by a mathematical proof. These results explain why SIFT outperforms all other image feature extraction methods when it comes to scale invariance.

Keywords: Shannon interpolation, sampling theory, Gaussian blur, scale invariance, aliasing., SIFT.

Mathematics Subject Classification: Primary: 68T10, 68T40; Secondary: 68T45, 93C8.

Citation: Jean-Michel Morel, Guoshen Yu. Is SIFT scale invariant?. Inverse Problems & Imaging, 2011, 5 (1) : 115-136. doi: 10.3934/ipi.2011.5.115

References:

[1]	A. Agarwala, M. Agrawala, M. Cohen, D. Salesin and R. Szeliski, Photographing long scenes with multi-viewpoint panoramas, International Conference on Computer Graphics and Interactive Techniques, (2006), 853-861. Google Scholar
[2]	A. Baumberg, Reliable feature matching across widely separated views, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 1 (2000), 774-781. Google Scholar
[3]	M. Bennewitz, C. Stachniss, W. Burgard and S. Behnke, Metric localization with scale-Invariant visual features using a single perspective camera, European Robotics Symposium, (2006), 195. doi: 10.1007/11681120_16. Google Scholar
[4]	M. Brown and D. Lowe, Recognising panorama, in Proc. the 9th Int. Conf. Computer Vision, October, (2003), 1218-1225. Google Scholar
[5]	E. Y. Chang, EXTENT: Fusing context, content, and semantic ontology for photo annotation, Proceedings of the 2nd International Workshop on Computer Vision Meets Databases, (2005), 5-11. doi: 10.1145/1160939.1160945. Google Scholar
[6]	Q. Fan, K. Barnard, A. Amir, A. Efrat and M. Lin, Matching slides to presentation videos using SIFT and scene background matching, Proceedings of the 8th ACM International Workshop on Multimedia Information Retrieval, (2006), 239-248. Google Scholar
[7]	L. Février, "A Wide-baseline Matching Library for Zeno," Internship report, ENS, Paris, France, 2007, www.di.ens.fr/~fevrier/papers/2007-InternsipReportILM.pdf. Google Scholar
[8]	J. J. Foo and R. Sinha, Pruning SIFT for scalable near-duplicate image matching, Proceedings of the Eighteenth Conference on Australasian Database, 63 (2007), 63-71. Google Scholar
[9]	G. Fritz, C. Seifert, M. Kumar and L. Paletta, Building detection from mobile imagery using informative SIFT descriptors, Lecture Notes in Computer Science, (2005), 629-638. doi: 10.1007/11499145_64. Google Scholar
[10]	C. Gasquet and P. Witomski, "Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets," Springer Verlag, 1999. Google Scholar
[11]	I. Gordon and D. G. Lowe, What and where: 3D object recognition with accurate pose, Lecture Notes in Computer Science, 4170 (2006), 67. doi: 10.1007/11957959_4. Google Scholar
[12]	J. S. Hare and P. H. Lewis, Salient regions for query by image content, Image and Video Retrieval: Third International Conference, CIVR, (2004), 317-325. Google Scholar
[13]	C. Harris and M. Stephens, A combined corner and edge detector, Alvey Vision Conference, 15 (1988), 50. Google Scholar
[14]	T. Kadir, A. Zisserman and M. Brady, An affine invariant salient region detector, in European Conference on Computer Vision, (2004), 228-241. Google Scholar
[15]	Y. Ke and R. Sukthankar, PCA-SIFT: A more distinctive representation for local image descriptors, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 2 (2004), 506-513. Google Scholar
[16]	J. Kim, S. M. Seitz and M. Agrawala, Video-based document tracking: Unifying your physical and electronic desktops, Proc. of the 17th Annual ACM Symposium on User interface Software and Technology, 24 (2004), 99-107. Google Scholar
[17]	B. N. Lee, W. Y. Chen and E. Y. Chang, Fotofiti: Web service for photo management, Proceedings of the 14th Annual ACM International Conference on Multimedia, (2006), 485-486. doi: 10.1145/1180639.1180737. Google Scholar
[18]	H. Lejsek, F. H. Ásmundsson, B. T. Jónsson and L. Amsaleg, Scalability of local image descriptors: A comparative study, Proceedings of the 14th Annual ACM International Conference on Multimedia, (2006), 589-598. doi: 10.1145/1180639.1180760. Google Scholar
[19]	T. Lindeberg, Scale-space theory: A basic tool for analyzing structures at different scales, Journal of Applied Statistics, 21 (1994), 225-270. doi: 10.1080/757582976. Google Scholar
[20]	T. Lindeberg and J. Garding, Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D brightness structure, Proc. ECCV, (1994), 389-400. Google Scholar
[21]	D. G. Lowe, Distinctive image features from scale-invariant key points, International Journal of Computer Vision, 60 (2004), 91-110. doi: 10.1023/B:VISI.0000029664.99615.94. Google Scholar
[22]	J. Matas, O. Chum, M. Urban and T. Pajdla, Robust wide-baseline stereo from maximally stable extremal regions, Image and Vision Computing, 22 (2004), 761-767. doi: 10.1016/j.imavis.2004.02.006. Google Scholar
[23]	K. Mikolajczyk and C. Schmid, Indexing based on scale invariant interest points, Proc. ICCV, 1 (2001), 525-531. Google Scholar
[24]	K. Mikolajczyk and C. Schmid, An affine invariant interest point detector, Proc. ECCV, 1 (2002), 128-142. Google Scholar
[25]	K. Mikolajczyk and C. Schmid, A performance evaluation of local descriptors, in "International Conference on Computer Vision and Pattern Recognition," volume 2, (2003), 257-263. Google Scholar
[26]	K. Mikolajczyk and C. Schmid, Scale and affine invariant interest point detectors, International Journal of Computer Vision, 60 (2004), 63-86. doi: 10.1023/B:VISI.0000027790.02288.f2. Google Scholar
[27]	K. Mikolajczyk and C. Schmid, A performance evaluation of local descriptors, IEEE Trans. PAMI, (2005), 1615-1630. Google Scholar
[28]	K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir and L. V. Gool, A comparison of affine region detectors, International Journal of Computer Vision, 65 (2005), 43-72. doi: 10.1007/s11263-005-3848-x. Google Scholar
[29]	P. Monasse, Contrast invariant image registration, Proc. of the International Conf. on Acoustics, Speech and Signal Processing, Phoenix, Arizona, 6 (1999), 3221-3224. Google Scholar
[30]	P. Moreels and P. Perona, Common-frame model for object recognition, Neural Information Processing Systems, (2004), 953-960. Google Scholar
[31]	J. M. Morel and G. Yu, ASIFT: A new framework for fully affine invariant image comparison, SIAM Journal on Imaging Sciences, 2 (2009), 438-469. doi: 10.1137/080732730. Google Scholar
[32]	A. Murarka, J. Modayil and B. Kuipers, Building local safety maps for a wheelchair robot using vision and lasers, in "Proceedings of the The 3rd Canadian Conference on Computer and Robot Vision," IEEE Computer Society Washington, DC, USA, 2006. Google Scholar
[33]	P. Musé, F. Sur, F. Cao and Y. Gousseau, Unsupervised thresholds for shape matching, Proc. of the International Conference on Image Processing, 2 (2003), 647-650. Google Scholar
[34]	P. Musé, F. Sur, F. Cao, Y. Gousseau and J. M. Morel, An a contrario decision method for shape element recognition, International Journal of Computer Vision, 69 (2006), 295-315. doi: 10.1007/s11263-006-7546-0. Google Scholar
[35]	A. Negre, H. Tran, N. Gourier, D. Hall, A. Lux and J. L. Crowley, Comparative study of people detection in surveillance scenes, Structural, Syntactic and Statistical Pattern Recognition, Proceedings Lecture Notes in Computer Science, 4109 (2006), 100-108. doi: 10.1007/11815921_10. Google Scholar
[36]	D. Nister and H. Stewenius, Scalable recognition with a vocabulary tree, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, (2006), 2161-2168. Google Scholar
[37]	J. Rabin, Y. Gousseau and J. Delon, A statistical approach to the matching of local features, SIAM Journal on Imaging Sciences, 2 (2009), 931-958. doi: 10.1137/090751359. Google Scholar
[38]	F. Riggi, M. Toews and T. Arbel, Fundamental matrix estimation via TIP-transfer of invariant parameters, Proceedings of the 18th International Conference on Pattern Recognition (ICPR'06), 2 (2006), 21-24. Google Scholar
[39]	J. Ruiz-del Solar, P. Loncomilla and C. Devia, A new approach for fingerprint verification based on wide baseline matching using local interest points and descriptors, Lecture Notes in Computer Science, 4872 (2007), 586-599. doi: 10.1007/978-3-540-77129-6_51. Google Scholar
[40]	P. Scovanner, S. Ali and M. Shah, A 3-dimensional SIFT descriptor and its application to action recognition, Proceedings of the 15th International Conference on Multimedia, (2007), 357-360. doi: 10.1145/1291233.1291311. Google Scholar
[41]	C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal, 27 (1948), 623-656. Google Scholar
[42]	T. Tuytelaars and L. Van Gool, Matching widely separated views based on affine invariant regions, International Journal of Computer Vision, 59 (2004), 61-85. doi: 10.1023/B:VISI.0000020671.28016.e8. Google Scholar
[43]	L. Vacchetti, V. Lepetit and P. Fua, Stable real-time 3D tracking using online and offline information, IEEE Trans PAMI, (2004), 1385-1391. Google Scholar
[44]	M. Veloso, F. von Hundelshausen and P. E. Rybski, Learning visual object definitions by observing human activities, in "Proc. of the IEEE-RAS Int. Conf. on Humanoid Robots," (2005), 148-153. doi: 10.1109/ICHR.2005.1573560. Google Scholar
[45]	M. Vergauwen and L. Van Gool, Web-based 3D reconstruction service, Machine Vision and Applications, 17 (2005), 411-426. doi: 10.1007/s00138-006-0027-1. Google Scholar
[46]	K. Yanai, Image collector III: a web image-gathering system with bag-of-keypoints, Proc. of the 16th Int. Conf. on World Wide Web, (2007), 1295-1296. doi: 10.1145/1242572.1242816. Google Scholar

show all references

References:

[1]	A. Agarwala, M. Agrawala, M. Cohen, D. Salesin and R. Szeliski, Photographing long scenes with multi-viewpoint panoramas, International Conference on Computer Graphics and Interactive Techniques, (2006), 853-861. Google Scholar
[2]	A. Baumberg, Reliable feature matching across widely separated views, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 1 (2000), 774-781. Google Scholar
[3]	M. Bennewitz, C. Stachniss, W. Burgard and S. Behnke, Metric localization with scale-Invariant visual features using a single perspective camera, European Robotics Symposium, (2006), 195. doi: 10.1007/11681120_16. Google Scholar
[4]	M. Brown and D. Lowe, Recognising panorama, in Proc. the 9th Int. Conf. Computer Vision, October, (2003), 1218-1225. Google Scholar
[5]	E. Y. Chang, EXTENT: Fusing context, content, and semantic ontology for photo annotation, Proceedings of the 2nd International Workshop on Computer Vision Meets Databases, (2005), 5-11. doi: 10.1145/1160939.1160945. Google Scholar
[6]	Q. Fan, K. Barnard, A. Amir, A. Efrat and M. Lin, Matching slides to presentation videos using SIFT and scene background matching, Proceedings of the 8th ACM International Workshop on Multimedia Information Retrieval, (2006), 239-248. Google Scholar
[7]	L. Février, "A Wide-baseline Matching Library for Zeno," Internship report, ENS, Paris, France, 2007, www.di.ens.fr/~fevrier/papers/2007-InternsipReportILM.pdf. Google Scholar
[8]	J. J. Foo and R. Sinha, Pruning SIFT for scalable near-duplicate image matching, Proceedings of the Eighteenth Conference on Australasian Database, 63 (2007), 63-71. Google Scholar
[9]	G. Fritz, C. Seifert, M. Kumar and L. Paletta, Building detection from mobile imagery using informative SIFT descriptors, Lecture Notes in Computer Science, (2005), 629-638. doi: 10.1007/11499145_64. Google Scholar
[10]	C. Gasquet and P. Witomski, "Fourier Analysis and Applications: Filtering, Numerical Computation, Wavelets," Springer Verlag, 1999. Google Scholar
[11]	I. Gordon and D. G. Lowe, What and where: 3D object recognition with accurate pose, Lecture Notes in Computer Science, 4170 (2006), 67. doi: 10.1007/11957959_4. Google Scholar
[12]	J. S. Hare and P. H. Lewis, Salient regions for query by image content, Image and Video Retrieval: Third International Conference, CIVR, (2004), 317-325. Google Scholar
[13]	C. Harris and M. Stephens, A combined corner and edge detector, Alvey Vision Conference, 15 (1988), 50. Google Scholar
[14]	T. Kadir, A. Zisserman and M. Brady, An affine invariant salient region detector, in European Conference on Computer Vision, (2004), 228-241. Google Scholar
[15]	Y. Ke and R. Sukthankar, PCA-SIFT: A more distinctive representation for local image descriptors, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, 2 (2004), 506-513. Google Scholar
[16]	J. Kim, S. M. Seitz and M. Agrawala, Video-based document tracking: Unifying your physical and electronic desktops, Proc. of the 17th Annual ACM Symposium on User interface Software and Technology, 24 (2004), 99-107. Google Scholar
[17]	B. N. Lee, W. Y. Chen and E. Y. Chang, Fotofiti: Web service for photo management, Proceedings of the 14th Annual ACM International Conference on Multimedia, (2006), 485-486. doi: 10.1145/1180639.1180737. Google Scholar
[18]	H. Lejsek, F. H. Ásmundsson, B. T. Jónsson and L. Amsaleg, Scalability of local image descriptors: A comparative study, Proceedings of the 14th Annual ACM International Conference on Multimedia, (2006), 589-598. doi: 10.1145/1180639.1180760. Google Scholar
[19]	T. Lindeberg, Scale-space theory: A basic tool for analyzing structures at different scales, Journal of Applied Statistics, 21 (1994), 225-270. doi: 10.1080/757582976. Google Scholar
[20]	T. Lindeberg and J. Garding, Shape-adapted smoothing in estimation of 3-D depth cues from affine distortions of local 2-D brightness structure, Proc. ECCV, (1994), 389-400. Google Scholar
[21]	D. G. Lowe, Distinctive image features from scale-invariant key points, International Journal of Computer Vision, 60 (2004), 91-110. doi: 10.1023/B:VISI.0000029664.99615.94. Google Scholar
[22]	J. Matas, O. Chum, M. Urban and T. Pajdla, Robust wide-baseline stereo from maximally stable extremal regions, Image and Vision Computing, 22 (2004), 761-767. doi: 10.1016/j.imavis.2004.02.006. Google Scholar
[23]	K. Mikolajczyk and C. Schmid, Indexing based on scale invariant interest points, Proc. ICCV, 1 (2001), 525-531. Google Scholar
[24]	K. Mikolajczyk and C. Schmid, An affine invariant interest point detector, Proc. ECCV, 1 (2002), 128-142. Google Scholar
[25]	K. Mikolajczyk and C. Schmid, A performance evaluation of local descriptors, in "International Conference on Computer Vision and Pattern Recognition," volume 2, (2003), 257-263. Google Scholar
[26]	K. Mikolajczyk and C. Schmid, Scale and affine invariant interest point detectors, International Journal of Computer Vision, 60 (2004), 63-86. doi: 10.1023/B:VISI.0000027790.02288.f2. Google Scholar
[27]	K. Mikolajczyk and C. Schmid, A performance evaluation of local descriptors, IEEE Trans. PAMI, (2005), 1615-1630. Google Scholar
[28]	K. Mikolajczyk, T. Tuytelaars, C. Schmid, A. Zisserman, J. Matas, F. Schaffalitzky, T. Kadir and L. V. Gool, A comparison of affine region detectors, International Journal of Computer Vision, 65 (2005), 43-72. doi: 10.1007/s11263-005-3848-x. Google Scholar
[29]	P. Monasse, Contrast invariant image registration, Proc. of the International Conf. on Acoustics, Speech and Signal Processing, Phoenix, Arizona, 6 (1999), 3221-3224. Google Scholar
[30]	P. Moreels and P. Perona, Common-frame model for object recognition, Neural Information Processing Systems, (2004), 953-960. Google Scholar
[31]	J. M. Morel and G. Yu, ASIFT: A new framework for fully affine invariant image comparison, SIAM Journal on Imaging Sciences, 2 (2009), 438-469. doi: 10.1137/080732730. Google Scholar
[32]	A. Murarka, J. Modayil and B. Kuipers, Building local safety maps for a wheelchair robot using vision and lasers, in "Proceedings of the The 3rd Canadian Conference on Computer and Robot Vision," IEEE Computer Society Washington, DC, USA, 2006. Google Scholar
[33]	P. Musé, F. Sur, F. Cao and Y. Gousseau, Unsupervised thresholds for shape matching, Proc. of the International Conference on Image Processing, 2 (2003), 647-650. Google Scholar
[34]	P. Musé, F. Sur, F. Cao, Y. Gousseau and J. M. Morel, An a contrario decision method for shape element recognition, International Journal of Computer Vision, 69 (2006), 295-315. doi: 10.1007/s11263-006-7546-0. Google Scholar
[35]	A. Negre, H. Tran, N. Gourier, D. Hall, A. Lux and J. L. Crowley, Comparative study of people detection in surveillance scenes, Structural, Syntactic and Statistical Pattern Recognition, Proceedings Lecture Notes in Computer Science, 4109 (2006), 100-108. doi: 10.1007/11815921_10. Google Scholar
[36]	D. Nister and H. Stewenius, Scalable recognition with a vocabulary tree, Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition, (2006), 2161-2168. Google Scholar
[37]	J. Rabin, Y. Gousseau and J. Delon, A statistical approach to the matching of local features, SIAM Journal on Imaging Sciences, 2 (2009), 931-958. doi: 10.1137/090751359. Google Scholar
[38]	F. Riggi, M. Toews and T. Arbel, Fundamental matrix estimation via TIP-transfer of invariant parameters, Proceedings of the 18th International Conference on Pattern Recognition (ICPR'06), 2 (2006), 21-24. Google Scholar
[39]	J. Ruiz-del Solar, P. Loncomilla and C. Devia, A new approach for fingerprint verification based on wide baseline matching using local interest points and descriptors, Lecture Notes in Computer Science, 4872 (2007), 586-599. doi: 10.1007/978-3-540-77129-6_51. Google Scholar
[40]	P. Scovanner, S. Ali and M. Shah, A 3-dimensional SIFT descriptor and its application to action recognition, Proceedings of the 15th International Conference on Multimedia, (2007), 357-360. doi: 10.1145/1291233.1291311. Google Scholar
[41]	C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal, 27 (1948), 623-656. Google Scholar
[42]	T. Tuytelaars and L. Van Gool, Matching widely separated views based on affine invariant regions, International Journal of Computer Vision, 59 (2004), 61-85. doi: 10.1023/B:VISI.0000020671.28016.e8. Google Scholar
[43]	L. Vacchetti, V. Lepetit and P. Fua, Stable real-time 3D tracking using online and offline information, IEEE Trans PAMI, (2004), 1385-1391. Google Scholar
[44]	M. Veloso, F. von Hundelshausen and P. E. Rybski, Learning visual object definitions by observing human activities, in "Proc. of the IEEE-RAS Int. Conf. on Humanoid Robots," (2005), 148-153. doi: 10.1109/ICHR.2005.1573560. Google Scholar
[45]	M. Vergauwen and L. Van Gool, Web-based 3D reconstruction service, Machine Vision and Applications, 17 (2005), 411-426. doi: 10.1007/s00138-006-0027-1. Google Scholar
[46]	K. Yanai, Image collector III: a web image-gathering system with bag-of-keypoints, Proc. of the 16th Int. Conf. on World Wide Web, (2007), 1295-1296. doi: 10.1145/1242572.1242816. Google Scholar

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Is SIFT scale invariant?

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