Search for


TEXT SIZE
search for




 

서포트 벡터 머신을 활용한 일래스틱 평면 형태데이터의 선형공간 속 분류 연구
Support vector machine for elastic planar shape on the linearized space
Korean J Appl Stat 2024;37(6):751-768
Published online December 31, 2024
© 2024 The Korean Statistical Society.

우명훈a, 이형석a, 이준명a, 조민호1,a
Myung Hun Wooa, Hyeongseok Leea, Joon Myoung Leea, Min Ho Cho1,a

aDepartment of Statistics, Inha University
1Corresponding author: Department of Statistics, Inha University, 100 Inha-ro, Michuhol-gu, Incheon 22212, Korea.
E-mail: mcho@inha.ac.kr
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (RS-2022-00167077). This study was supported by K-water (Korea Water Resources Corportion).
Received June 20, 2024; Revised August 17, 2024; Accepted September 1, 2024.
Abstract
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.
주요어 : 서포트 벡터 머신, 일래스틱 형태, 제곱근속도함수, 탄젠트 공간, 폰 미제스-피셔 혼합분포
Keywords : elastic shape, square-root velocity function, support vector machines, tangent space, von Mises-Fisher distributions
References
  1. Amit Y, Grenander U, and PiccioniM(1991). Structural image restoration through deformable templates, Journal of the American Statistical Association, 86, 376-387.
    CrossRef
  2. Bookstein FL (1986). Size and shape spaces for landmark data in two dimensions, Statistical Science, 1, 181-222.
    CrossRef
  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.
    CrossRef
  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.
    Pubmed KoreaMed CrossRef
  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.
    Pubmed KoreaMed CrossRef
  7. Cho MH, Kurtek S, and MacEachern SN (2021). Aggregated pairwise classification of elastic planar shapes, The Annals of Applied Statistics, 15, 619-637.
    CrossRef
  8. Cortes C and Vapnik V (1995). Support-vector networks, Machine Learning, 20, 273-297.
    CrossRef
  9. Dryden IL and Mardia KV (2016). Statistical Shape Analysis: With Applications in R, John Wiley & Sons Ltd, Chichester, UK.
    CrossRef
  10. Hastie T, Tibshirani R, and Friedman J (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer, New York.
    CrossRef
  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.
    CrossRef
  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.
    CrossRef
  13. HsuCW and Lin CJ (2002). A comparison of methods for multiclass support vector machines, IEEE Transactions on Neural Networks, 13, 415-425.
    Pubmed CrossRef
  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.
    CrossRef
  15. Kendall DG (1984). Shape manifolds, Procrustean metrics, and complex projective spaces, Bulletin of the London Mathematical Society, 16, 81-121.
    CrossRef
  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.
    Pubmed CrossRef
  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.
    CrossRef
  18. Kurtek S and Bharath K (2015). Bayesian sensitivity analysis with the fisher-rao metric, Biometrika, 102, 601- 616.
    CrossRef
  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.
    CrossRef
  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.
    CrossRef
  21. Lang S (2012). Fundamentals of Differential Geometry, Springer, New York.
  22. Loosli G, Canu S, and Ong CS (2015). Learning SVM in kre˘ın spaces, IEEE Transactions on Pattern Analysis and Machine Intelligence, 38, 1204-1216.
    Pubmed CrossRef
  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.
    CrossRef
  26. Mio W, Srivastava A, and Joshi S (2007). On shape of plane elastic curves, International Journal of Computer Vision, 73, 307-324.
    CrossRef
  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.
    Pubmed KoreaMed CrossRef
  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.
    CrossRef
  29. Rossi F and Villa N (2006). Support vector machine for functional data classification, Neurocomputing, 69, 730- 742.
    CrossRef
  30. Sharon E and Mumford D (2006). 2D-shape analysis using conformal mapping, International Journal of Computer Vision, 70, 55-75.
    CrossRef
  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.
    Pubmed CrossRef
  32. Srivastava A and Klassen EP (2016). Functional and Shape Data Analysis, Springer, New York.
    CrossRef
  33. Younes L (1998). Computable elastic distances between shapes, SIAM Journal on Applied Mathematics, 58, 565-586.
    CrossRef
December 2024, 37 (6)
Full Text(PDF) Free