Imputation of Multiple Missing Values by Normal Mixture Model under Markov Random Field: Application to Imputation of Pixel Values of Color Image

마코프 랜덤 필드 하에서 정규혼합모형에 의한 다중 결측값 대체기법: 색조영상 결측 화소값 대체에 응용

Kim, Seung-Gu

  • Received : 20091000
  • Accepted : 20091000
  • Published : 2009.11.30


There very many approaches to impute missing values in the iid. case. However, it is hardly found the imputation techniques in the Markov random field(MRF) case. In this paper, we show that the imputation under MRF is just to impute by fitting the normal mixture model(NMM) under several practical assumptions. Our multivariate normal mixture model based approaches under MRF is applied to impute the missing pixel values of 3-variate (R, G, B) color image, providing a technique to smooth the imputed values.


Multiple missing values;imputation;Markov random field;EM algorithm;color image


  1. Besag, J. (1975). Statistical analysis of non-lattice data, The Statistician, 24, 179–195
  2. Besag, J. (1986). On the statistical analysis of dirty pictures (with discussion), Journal of the Royal Statistical Society B, 48, 259–302
  3. Blanchet, J. and Vignes, M. (2009). A model-based approach to gene clustering with missing observation reconstruction in a Markov random field framework, Journal of Computational Biology, 16, 475–486
  4. Dass, S. C. and Nair, V. N. (2003). Edge detection, spatial smoothing, and image reconstruction with partially observed multivariate data, Journal of the American Statistical Association, 98, 77–89
  5. Hunt, L. and Jorgensen, M. (2003). Mixture model clustering mixed data with missing information, Computation Statistics & Data Analysis, 41, 429–440
  6. Kalton, G. and Kasprzyk, D. (1986). The treatment of missing survey data, Survey Methodology, 12, 1–16
  7. Kim, D., Lee, Y. and Oh, H. S. (2006). Hierarchical likelihood-based wavelet method for denoising signals with missing data, IEEE Signal Processing Letters
  8. Little, J. and Rubin, D. (2002). Statistical Analysis of Missing Data, Wiely, New York
  9. Marker, D. A., Judkins, D. R. and Winglee, M. (2002). Large-Scale Imputation for Complex Surveys, In: Groves, R. M. and Eltinge, E. L.
  10. McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models, Wiely, New York
  11. Ogawa, T., Haseyama, M. and Kitajima, H. (2006). Restoration of Missing Intensity of Still Images by Using Optical Flows, System and Computers in Japan, 37, 1786–1795
  12. Owen, A. (1986). Contribution to the discussion of paper by B.D.Ripley, Canadian Journal of Statistics, 14, 106–110
  13. Qian and Titterington. (1992). Stochastic relaxations and EM algorithms for Markov random fields, Journal of Statistical Computation and Simulation, 40, 55–69
  14. Zio, M. D., Guarnera, U. and Luzi, O. (2007). Imputation through finite Gaussian mixture models, Computation Statistics & Data Analysis, 51, 5305–5316