응용통계연구 (The Korean Journal of Applied Statistics)

  • 제37권6호
  • /
  • Pages.751-768
  • /
  • 2024
  • /
  • 1225-066X(pISSN)
  • /
  • 2383-5818(eISSN)
한국통계학회 (The Korean Statistical Society)
권호 표지
DOI QR코드

DOI QR Code

서포트 벡터 머신을 활용한 일래스틱 평면 형태데이터의 선형공간 속 분류 연구

Support vector machine for elastic planar shape on the linearized space

  • 투고 : 2024.06.20
  • 심사 : 2024.09.01
  • 발행 : 2024.12.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

본 논문에서는 컴퓨터 비전, 의학 이미징과 같은 다양한 응용 분야에서 활용되는 형태데이터에 대하여 서포트 벡터 머신 기반의 분류 모형을 제안하고 다른 통계적 모형들과의 분류 성능을 비교한다. 형태를 함수형 데이터로 표현했을 때, 위치이동, 크기조절, 회전, 재매개변수화와 같은 변동 요인에 불변하는 형태 거리를 가지고 분석하기 위하여 최근 활발히 연구되어지고 있는 일래스틱 형태 분석을 기반으로 한다. 이 분석 틀은 형태를 표현하는 곡선을 제곱근속도함수로 변환하여 곡선의 본래 공간인 리마니안 다양체를 단위 초구로 재구성할 수 있다. 초구 위에 변환된 표본 형태데이터의 평균을 중심으로 탄젠트 공간을 만들고, 그 위로 사영시킨 유클리디안 벡터를 통해 서포트 벡터 머신 방법들로 분류한다. 폰 미제스-피셔 혼합분포를 이용하여 생성한 모의실험 형태데이터와 조류 형태를 분석하는 실제 데이터를 통해 제안한 서포트 벡터 머신 방법과 다른 통계적 분류 모형들을 적용하고 그 성능을 비교한다.

In this paper, we consider a classification model based on support vector machines (SVM) for shape data, which is utilized in various application areas such as computer vision, medical imaging, and so on. When shape is represented as a function, we need a shape distance invariant to translation, scaling, rotation, and reparameterization. We adopt the elastic shape analysis framework based on the square-root velocity function (SRVF) representation. The framework enables us to analyze shape data on a unit hypersphere instead of a Riemannian manifold, the original representation space. The data could be even linearized using a tangent space at the mean of the transformed sample shapes. We apply the SVM to the tangent Euclidean vectors after projection. We design simulation studies for shape classification by generating planar curves from a mixture of von Mises-Fisher distributions. We analyze real data of algal shapes, and compare its performance with other statistical classification methods.

키워드

과제정보

이 성과는 정부(과학기술정보통신부)의 재원으로 한국연구재단의 지원을 받아 수행된 연구임 (RS-2022-00167077). 이 성과는 한국수자원공사(K-water)의 지원을 받아 수행된 연구임.

참고문헌

  1. Amit Y, Grenander U, and PiccioniM(1991). Structural image restoration through deformable templates, Journal of the American Statistical Association, 86, 376-387.
  2. Bookstein FL (1986). Size and shape spaces for landmark data in two dimensions, Statistical Science, 1, 181-222.
  3. Boser BE, Guyon IM, and Vapnik VN (1992). A training algorithm for optimal margin classifiers, In Proceedings of the Fifth Annual Workshop on Computational Learning Theory, Pittsburgh, PA, 144-152.
  4. Cho H, DeMeo B, Peng J, and Berger B (2019a). Large-margin classification in hyperbolic space, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics, 89, 1832-1840.
  5. Cho MH, Asiaee A, and Kurtek S (2019b). Elastic statistical shape analysis of biological structures with case studies: a tutorial, Bulletin of Mathematical Biology, 81, 2052-2073.
  6. Cho MH, Kurtek S, and Bharath K (2022). Tangent functional canonical correlation analysis for densities and shapes, with applications to multimodal imaging data, Journal of Multivariate Analysis, 189, 104870.
  7. Cho MH, Kurtek S, and MacEachern SN (2021). Aggregated pairwise classification of elastic planar shapes, The Annals of Applied Statistics, 15, 619-637.
  8. Cortes C and Vapnik V (1995). Support-vector networks, Machine Learning, 20, 273-297.
  9. Dryden IL and Mardia KV (2016). Statistical Shape Analysis: With Applications in R, John Wiley & Sons Ltd, Chichester, UK.
  10. Hastie T, Tibshirani R, and Friedman J (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, New York.
  11. Heo TY, Lee JM, Woo MH, Lee H, and Cho MH (2024a). Logistic regression models for elastic shape of curves based on tangent representations, Journal of the Korean Statistical Society, 53, 416-434.
  12. Heo TY, Kim J, and Cho MH (2024b). Classification of algae in watersheds using elastic shape, Communications for Statistical Applications and Methods, 31, 309-322.
  13. HsuCWand Lin CJ (2002). A comparison of methods for multiclass support vector machines, IEEE Transactions on Neural Networks, 13, 415-425.
  14. Joshi SH, Klassen E, Srivastava A, and Jermyn I (2007). A novel representation for Riemannian analysis of elastic curves in ℝn, In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, Minneapolis, MN, 1-7.
  15. Kendall DG (1984). Shape manifolds, Procrustean metrics, and complex projective spaces, Bulletin of the London Mathematical Society, 16, 81-121.
  16. Klassen E, Srivastava A, Mio M, and Joshi SH (2004). Analysis of planar shapes using geodesic paths on shape spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 26, 372-383.
  17. Knerr S, Personnaz L, and Dreyfus G (1990). Single-layer learning revisited: a stepwise procedure for building and training a neural network, In Proceedings of Neurocomputing: Algorithms, Architectures and Applications, Paris, 41-50.
  18. Kurtek S and Bharath K (2015). Bayesian sensitivity analysis with the fisher-rao metric, Biometrika, 102, 601-616.
  19. Kurtek, S, Srivastava A, Klassen E, and Ding Z (2012). Statistical modeling of curves using shapes and related features, Journal of the American Statistical Association, 107, 1152-1165.
  20. Kurtek S, Su J, Grimm C, Vaughan M, Sowell R, and Srivastava A (2013). Statistical analysis of manual segmentations of structures in medical images, Computer Vision and Image Understanding, 117, 1036-1050.
  21. Lang S (2012). Fundamentals of Differential Geometry, Springer, New York.
  22. Loosli G, Canu S, and Ong CS (2015). Learning SVM in krein spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 1204-1216.
  23. Mardia K, Kent J, and Walder A (1991). Statistical shape models in image analysis, In Proceedings of the 23rd Symposium on the Interface, Seattle, 550-557.
  24. Mardia KV and Jupp PE (2009). Directional Statistics, John Wiley & Sons Ltd, Chichester, UK.
  25. Michor PW and Mumford D (2005). Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms, Documenta Mathematica, 10, 217-245.
  26. Mio W, Srivastava A, and Joshi S (2007). On shape of plane elastic curves, International Journal of Computer Vision, 73, 307-324.
  27. Pal S, Woods RP, Panjiyar S, Sowell E, Narr KL, and Joshi SH (2017). A Riemannian framework for linear and quadratic discriminant analysis on the tangent space of shapes, In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition Workshops, 47-55.
  28. Park J, Lee H, Park CY, Hasan S, Heo TY, and Lee WH (2019). Algal morphological identification in watersheds for drinking water supply using neural architecture search for convolutional neural network,Water, 11, 1338.
  29. Rossi F and Villa N (2006). Support vector machine for functional data classification, Neurocomputing, 69, 730-742.
  30. Sharon E and Mumford D (2006). 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70, 55-75.
  31. Srivastava A, Klassen E, Joshi SH, and Jermyn IH (2010). Shape analysis of elastic curves in euclidean spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 33, 1415-1428.
  32. Srivastava A and Klassen EP (2016). Functional and Shape Data Analysis, Springer, New York.
  33. Younes L (1998). Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58, 565-586.