High-Dimensional Range profile geometrical visualization and performance estimation of classification of radar targets via a Gaussian mixture model

28/08/2013
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High-Dimensional Range profile geometrical visualization and performance estimation of classification of radar targets via a Gaussian mixture model

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High-Dimensional Range profile geometrical visualization and performance estimation of classification of radar targets via a Gaussian mixture model Thomas Boulay1,2 , Ali Mohammad-Djafari1 , Nicolas Gac1 , Julien Lagoutte2 1 Laboratoire des Signaux et Syst`emes (L2S, UMR 8506 CNRS - SUPELEC - Univ Paris Sud 11) Sup´elec, Plateau de Moulon, F-91192 Gif-sur-Yvette, FRANCE. 2 THALES AIR SYSTEMS, voie Pierre Gilles de Gennes, 91470 Limours en Hurepoix, FRANCE Emails: thomas.boulay@lss.supelec.fr, djafari@lss.supelec.fr, nicolas.gac@lss.supelec.fr, julien.lagoutte@thalesgroup.com GSI 2013, MINES ParisTech, Paris, France' & $ % Non Cooperative Target Recognition (NCTR) Œ Formation of range profile Example of range profile ' & $ % Objectives  • Visualize high-dimensional range profiles in 2D space • Estimate graphically and analytically classification performances ' & $ % Non Linear Dimension Reduction (NLDR) Ž From high-dimensional space to low-dimensional space xT,i → yT,i xT,i ∈ RM → yT,i ∈ R2 Stochastic Neighbor Embedding (SNE) pj|i = exp −||xT,i − xT,j||2 /(2σ2 i ) k=i exp −||xT,i − xT,k||2/(2σ2 i ) qj|i = exp −||yT,i − yT,j||2 k=i exp −||yT,i − yT,k||2 t-Distributed Stochastic Neighbor Embedding (t-SNE) qij = 1 + ||yT,i − yT,j||2 −1 k,l,k=l 1 + ||yT,k − yT,l||2 −1 ' & $ % Mixture Models for NCTR application  p(zT,i|ZT ) = K k=1 πk T p(zT,i|θk T ) In low-dimensional space zT,i = yT,i, in high-dimensional space zT,i = xT,i Gaussian Mixture Models (GMM) p(zT,i|θk T ) = N(zT,i|θk T ), θk T = {mk T , Ck T } mk T = 1 Nk T Nk T i=1 zk T,i Ck T (p, q) = 1 Nk T − 1 Nk T i=1 (zk T,i(p) − mk T (p)) (zk T,i(q) − mk T (q)) Decision Rule if pi ≥ TG, the class i is granted → Granted Matrix (GrM) if TD ≤ pi < TG, the class i is dubious → Dubious Matrix (DuM) if pi < TD, the class i is denied → Denied Matrix (DeM) ' & $ % Probability map in low-dimensional space  t-SNE 2D visualization of data for class 1, 2 and 3 (row), for SNR = 30dB, SNR = 20dB and SNR = 15dB (column). ' & $ % Decision matrix in low-dimensional space ‘ GrM DuM DeM PPPPPPPPPPPP Test Class 1 2 3 1 2 3 1 2 3 SNR=30dB Class 1 99.1 0.9 0 0 0 0 0.9 99.1 100 Class 2 0 100 0 0 0 0 100 0 100 Class 3 0 0 100 0 0 0 100 100 0 SNR=20dB Class 1 99.1 0.9 0 0 0 0 0.9 99.1 100 Class 2 0 98.5 0 0.9 1.5 0 98.5 0 100 Class 3 0 0 100 0 0 0 100 100 0 SNR=15dB Class 1 95.6 0.9 0 3.5 1.1 0 0.9 98 100 Class 2 0.6 94.4 0 5 4.7 0 94.4 0.9 100 Class 3 0 0 100 0 0 0 100 100 0 ' & $ % Decision matrix in high-dimensional space ’ GrM DuM DeM PPPPPPPPPPPP Test Class 1 2 3 1 2 3 1 2 3 SNR=30dB Class 1 100 0 0 0 0 0 0 100 100 Class 2 0 100 0 0 0 0 100 0 100 Class 3 0 0 100 0 0 0 100 100 0 SNR=20dB Class 1 99.4 0.6 0 0 0.3 0 0.6 99.1 100 Class 2 0.3 99.7 0 0 0.3 0 99.7 0 100 Class 3 0 0 100 0 0 0 100 100 0 SNR=15dB Class 1 97.7 2.3 0 0.3 0 0 2 97.7 100 Class 2 0.3 99.7 0 0 0 0 99.7 0.3 100 Class 3 0 0 100 0 0 0 100 100 0 ' & $ % Perspectives “ • Generalize this method for any kind of algorithm • Evaluate for each kind of algorithm, the critical dimension from which the performances begin to degrade • Determine the best appropriate algorithm from data