Mathematics in Independent Component Analysis

More documents

Recommendations

Info

198 Chapter 15. Neurocomputing, 69:1485-1501, 2006 Denoising Using Local Projective Subspace Methods P. Gruber, K. Stadlthanner, M. Böhm, F. J. Theis, E. W. Lang Abstract Institute of Biophysics, Neuro-and Bioinformatics Group University of Regensburg, 93040 Regensburg, Germany email: elmar.lang@biologie.uni-regensburg.de A. M. Tomé, A. R. Teixeira Dept. de Electrónica e Telecomunicações/IEETA Universidade de Aveiro, 3810 Aveiro, Portugal email: ana@ieeta.pt C. G. Puntonet, J. M. Gorriz Saéz Dep. Arqitectura y Técnologia de Computadores Universidad de Granada, 18371 Granada, Spain email: carlos@atc.ugr.es In this paper we present denoising algorithms for enhancing noisy signals based on Local ICA (LICA), Delayed AMUSE (dAMUSE) and Kernel PCA (KPCA). The algorithm LICA relies on applying ICA locally to clusters of signals embedded in a high dimensional feature space of delayed coordinates. The components resembling the signals can be detected by various criteria like estimators of kurtosis or the variance of autocorrelations depending on the statistical nature of the signal. The algorithm proposed can be applied favorably to the problem of denoising multidimensional data. Another projective subspace denoising method using delayed coordinates has been proposed recently with the algorithm dAMUSE. It combines the solution of blind source separation problems with denoising efforts in an elegant way and proofs to be very efficient and fast. Finally, KPCA represents a non-linear projective subspace method that is well suited for denoising also. Besides illustrative applications to toy examples and images, we provide an application of all algorithms considered to the analysis of protein NMR spectra. Preprint submitted to Elsevier Science 4 February 2005
Chapter 15. Neurocomputing, 69:1485-1501, 2006 199 1 Introduction The interpretation of recorded signals is often hampered by the presence of noise. This is especially true with biomedical signals which are buried in a large noise background most often. Statistical analysis tools like Principal Component Analysis (PCA), singular spectral analysis (SSA), Independent Component Analysis (ICA) etc. quickly degrade if the signals exhibit a low Signal to Noise Ratio (SNR). Furthermore due to their statistical nature, the application of such analysis tools can also lead to extracted signals with a larger SNR than the original ones as we will discuss below in case of Nuclear Magnetic Resonance (NMR) spectra. Hence in the signal processing community many denoising algorithms have been proposed [5,12,18,38] including algorithms based on local linear projective noise reduction. The idea is to project noisy signals in a high-dimensional space of delayed coordinates, called feature space henceforth. A similar strategy is used in SSA [20], [9] where a matrix composed of the data and their delayed versions is considered. Then, a Singular Value Decomposition (SVD) of the data matrix or a PCA of the related correlation matrix is computed. Noise contributions to the signals are then removed locally by projecting the data onto a subset of principal directions of the eigenvectors of the SVD or PCA analysis related with the deterministic signals. Modern multi-dimensional NMR spectroscopy is a very versatile tool for the determination of the native 3D structure of biomolecules in their natural aqueous environment [7,10]. Proton NMR is an indispensable contribution to this structure determination process but is hampered by the presence of the very intense water (H2O) proton signal. The latter causes severe baseline distortions and obscures weak signals lying under its skirts. It has been shown [26,29] that Blind Source Separation (BSS) techniques like ICA can contribute to the removal of the water artifact in proton NMR spectra. ICA techniques extract a set of signals out of a set of measured signals without knowing how the mixing process is carried out [2, 13]. Considering that the set of measured spectra X is a linear combination of a set of Independent Components (ICs) S, i.e. X = AS, the goal is to estimate the inverse of the mixing matrix A, using only the measured spectra, and then compute the ICs. Then the spectra are reconstructed using the mixing matrix A and those ICs contained in S which are not related with the water artifact. Unfortunately the statistical separation process in practice introduces additional noise not present in the original spectra. Hence denoising as a post-processing of the artifact-free spectra is necessary to achieve the highest possible SNR of the reconstructed spectra. It is important that the denoising does not change the spectral characteristics like integral peak intensities as the deduction of the 2
Page 1 and 2:
Statistical machine learning of bio
Page 3:
to Jakob
Page 6 and 7:
vi Preface
Page 8 and 9:
viii CONTENTS 3 Signal Processing 8
Page 11 and 12:
Chapter 1 Statistical machine learn
Page 13 and 14:
1.1. Introduction 5 auditory cortex
Page 15 and 16:
1.2. Uniqueness issues in independe
Page 17 and 18:
1.2. Uniqueness issues in independe
Page 19 and 20:
S U M M A R Y T his �le contains
Page 21 and 22:
1.3. Dependent component analysis 1
Page 23 and 24:
Table 1.1: BSS algorithms based on
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
1.4. Sparseness 25 1.4 Sparseness O
Page 35 and 36:
1.4. Sparseness 27 Theorem 1.4.3 (S
Page 37 and 38:
1.4. Sparseness 29 R 3 R 3 A BSRA R
Page 39 and 40:
1.4. Sparseness 31 Sparse projectio
Page 41 and 42:
1.5. Machine learning for data prep
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
1.6. Applications to biomedical dat
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
1.7. Outlook 51 noise data signal i
Page 61:
Part II Papers 53
Page 64 and 65:
56 Chapter 2. Neural Computation 16
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
82 Chapter 3. Signal Processing 84(
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
90 Chapter 4. Neurocomputing 64:223
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
104 Chapter 5. IEICE TF E87-A(9):23
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
114 Chapter 6. LNCS 3195:726-733, 2
Page 124 and 125:
116 Chapter 6. LNCS 3195:726-733, 2
Page 126 and 127:
118 Chapter 6. LNCS 3195:726-733, 2
Page 128 and 129:
120 Chapter 6. LNCS 3195:726-733, 2
Page 130 and 131:
122 Chapter 6. LNCS 3195:726-733, 2
Page 132 and 133:
124 Chapter 7. Proc. ISCAS 2005, pa
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
130 Chapter 8. Proc. NIPS 2006 Towa
Page 140 and 141:
132 Chapter 8. Proc. NIPS 2006 1.3
Page 142 and 143:
134 Chapter 8. Proc. NIPS 2006 stru
Page 144 and 145:
136 Chapter 8. Proc. NIPS 2006 5.5
Page 146 and 147:
138 Chapter 8. Proc. NIPS 2006
Page 148 and 149:
140 Chapter 9. Neurocomputing (in p
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
Page 156 and 157: 148 Chapter 9. Neurocomputing (in p
Page 164 and 165: 156 Chapter 10. IEEE TNN 16(4):992-
Page 170 and 171: 162 Chapter 11. EURASIP JASP, 2007
Page 184 and 185: 176 Chapter 12. LNCS 3195:718-725,
Page 194 and 195: 186 Chapter 13. Proc. EUSIPCO 2005
Page 200 and 201: 192 Chapter 14. Proc. ICASSP 2006 S
Page 202 and 203: 194 Chapter 14. Proc. ICASSP 2006 A
Page 204 and 205: 196 Chapter 14. Proc. ICASSP 2006
Page 208 and 209: 200 Chapter 15. Neurocomputing, 69:
Page 238 and 239: 230 Chapter 16. Proc. ICA 2006, pag
Page 248 and 249: 240 Chapter 17. IEEE SPL 13(2):96-9
Page 256 and 257:
248 Chapter 18. Proc. EUSIPCO 2006
Page 258 and 259:
250 Chapter 18. Proc. EUSIPCO 2006
Page 260 and 261:
252 Chapter 19. LNCS 3195:977-984,
Page 262 and 263:
254 Chapter 19. LNCS 3195:977-984,
Page 264 and 265:
256 Chapter 19. LNCS 3195:977-984,
Page 266 and 267:
258 Chapter 19. LNCS 3195:977-984,
Page 268 and 269:
260 Chapter 19. LNCS 3195:977-984,
Page 270 and 271:
262 Chapter 20. Signal Processing 8
Page 272 and 273:
Page 274 and 275:
Page 276 and 277:
Page 278 and 279:
Page 280 and 281:
Page 282 and 283:
Page 284 and 285:
Page 286 and 287:
Page 288 and 289:
Page 290 and 291:
Page 292 and 293:
Page 294 and 295:
Page 296 and 297:
Page 298 and 299:
Page 300 and 301:
292 Chapter 21. Proc. BIOMED 2005,
Page 302 and 303:
Page 304 and 305:
Page 306 and 307:
298 BIBLIOGRAPHY Barber, C., Dobkin
Page 308 and 309:
300 BIBLIOGRAPHY Georgiev, P. (2001
Page 310 and 311:
302 BIBLIOGRAPHY Keck, I., Theis, F
Page 312 and 313:
304 BIBLIOGRAPHY Schießl, I., Sch
Page 314 and 315:
306 BIBLIOGRAPHY Theis, F. and Inou
show all

Mathematics in Independent Component Analysis

Create successful ePaper yourself

Delete template?

Save as template?