Mathematics in Independent Component Analysis

More documents

Recommendations

Info

294 Chapter 21. Proc. BIOMED 2005, pages 209-212 1.5 1 0.5 0 −0.5 −1 −1.5 −2 4 2 0 −2 −4 −3 Figure 3. Data set with 120 samples after 3-dimensional PCA projection (91% of the data was retained). The dots mark the 60 samples representing cells, the x’s mark the 60 non-cell data points. The two circles indicate clusters of a k-means application with a search for two clusters. Obviously, k-means nicely differentiates between the cell and the non-cell components. A perceptron with output dimension 1 consists of a single neuron only, so the output function y can be written as y(x) = θ(w ⊤ x + w0) with weight w ∈ R n , n input dimension, w0 ∈ R the bias and as activation function θ, the Heaviside function (θ(x) = 0 for x < 0 and θ(x) = 1 for x ≥ 0). Often, the bias w0 is added as additional weight to w with fixed input 1. Learning in a perceptron means minimizing the error energy function shown above. This can be performed for example by gradient descent with respect to w and w0. This induces the well known delta-rule for the weight update −2 −1 ∆w = η(y(x) − t)x, where η denotes a chosen learning rate parameter, y(x) the output of the neural network at sample x and t the observation of input x. It is easy to see that a perceptron separates the data linearly with the boundary hyperplane given by {x ∈ R n |w ⊤ x + w0 = 0}. For the cell classifier, we use a single-unit perceptron with linear activation function in order to get a measure for the certainty of cell/non-cell classification. Application of delta-learning to the 5-dimensional data set from above gives excellent performance after already 4 epochs of batch learning. The final performance error (variance of perceptron estimation error of the training set) after 55 epochs was 0.0038 which confirms the good performance as well as the linearity of the classification problem. This was further shown, when we used a two-layered network 0 1 2 3 4 with 5 hidden neurons in order to test for nonlinearities in the data set. After only 10 epochs, the error was already very small, and it could finally be diminished to 3 · 10 −19 . Still the performance of the perceptron is more than enough for classification. 4 Confidence map 4.1 Generation The cell classifier from above only has to be trained once. Given such a cell classifier, section pictures can now be analyzed as follows: A pixelwise scan of the image gives an image patch with center location at the scan point; to this image patch the cell classifier is then applied to give a probability of whether a cell is at the given location or not. This altogether (after image extension at the boundaries) yields a probability distribution over the whole image which is called confidence map. Each point of the confidence map is a value in [0, 1] stating how probable it is that a cell is depicted at the specified location. In practice a pixelwise scan is too expensive in terms of calculation time, so instead a grid value say 5 for 20 × 20-patches is introduced, and the picture is scanned only every 5-th pixel. This yields a rasterization of the original confidence map, which is still good enough to detect cells. Figure 4 shows the rasterized confidence map of a section part. The maxima of the confidence map correspond to the cell locations; small but non-zero values in the confidence map typically depict misclassifications that can be avoided by thresholding. 4.2 Evaluation After the confidence map has been generated, it can be evaluated by simple maxima analysis. However as seen in figure 4, maxima not always correspond to cell positions, so thresholding in the confidence map is first applied. Values of 0.5 to 0.8 yield good results in experiments. Furthermore, the cell classifier could have responded to one cell when applied to image patches with large overlap. Therefore after a maximum has been detected, adjacent points in the confidence map are also set to zero within a given radius (15 to 18 were good values for 20 × 20 image patches). Iterative application of this algorithm then gave the final cell positions and hence the image segmentation. 5 Results In practice we used perceptron learning after preprocessing with PCA and also ICA [5] [6] in order to help the learning algorithm with linearly separated data. The patch size was chosen to be 20 × 20, a thresholding of 0.8 was applied in the confidence map and the
Chapter 21. Proc. BIOMED 2005, pages 209-212 295 1 0.5 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 Figure 4. The plot shows the confidence map with grid value 5 of the image part shown above. cut-out radius for cell detection in the confidence map was 18 pixels. In figure 1 an automatically segmented picture is shown. We see good performances of the counting algorithm. So far we only compared cell-numbers of sections, counted by the algorithm and by an expert. We get calculation errors of about 5%. In further experiments, we also want to compare cell positions, detected by the algorithm and by an expert. 6 Conclusion We have presented a framework for brain section image segmentation and analysis. The feature detector, here cell classifier, was first trained on a given sample set using a neural network. This detector was then applied by scanning over the image to get a confidence map. Maxima analysis yields the cell locations. Experiments showed good performance of the classifier, however larger tests will have to be performed. In future work, various problems will have to be dealt with. First of all the scanning performance should be increased in order to be able to use smaller grid values, which could significantly increase the classification rate. This could be done for example by using some kind of hierarchical neural network like a cellular neural network, see [9]. In typical brain section images, some cells not directly 15 20 25 30 35 40 45 50 lying in the focus plane occur in a blurred fashion. In order to count those without counting them twice in two section images with different focus planes, a three-dimensional cell classifier could be trained for fixed focus plane distances. A different approach for accounting for non-focused cells would be to simply allow ’overcounting’, and then to reduce doubles in the segmented images according to location. This seems suitable given the fact that cells do not vary greatly in size. Finally, sections typically span more than one microscope image. In order to count cells of the whole section some way of not counting cells twice in both image parts has to be devised. This could be done for example by using techniques from image fusion. Furthermore, often not the whole image but only parts of the section are to be counted; so far this choice of the ’region of interest’ is done manually. We hope to automate this in the future by finding separating features of these regions. 7 Acknowledgements F.J.T. and E.W.L. gratefully acknowledges financial support by the DFG 1 and the BMBF 2 . References [1] J. Altman and G. Das. Autoradiographic and histological evidence of postnatal hippocampal neurogenesis in rats. J. Comp. Neurol., 124(3):319–335, 1965. [2] H. Cameron, C. Woolley, B. McEwen, and E. Gould. Differentiation of newly born neurons and glia in the dentate gyrus of the adult rat. Neuroscience, 56(2):336–344, 1993. [3] F. Dolbeare. Bromodeoxyuridine: a diagnostic tool in biology and medicine, part i: Historical perspectives, histochemical methods and cell kinetics. Histochem. J., 27(5):339–369, 1995. [4] S. Haykin. Neural networks. Macmillan College Publishing Company, 1994. [5] J. Hérault and C. Jutten. Space or time adaptive signal processing by neural network models. In J.S. Denker, editor, Neural Networks for Computing. Proceedings of the AIP Conference, pages 206–211, New York, 1986. American Institute of Physics. [6] A. Hyvärinen, J.Karhunen, and E.Oja. Independent Component Analysis. John Wiley & Sons, 2001. [7] H.G. Kuhn, H. Dickinson-Anson, and F. Gage. Neurogenesis in the dentate gyrus of the adult rat: age-related decrease of neuronal progenitor proliferation. J. Neurosci., 16(6):2027– 2033, 1996. [8] H.G. Kuhn, T. Palmer, and E. Fuchs. Adult neurogenesis: a compensatory mechanism for neuronal damage. Eur. Arch. Psychiatry Clin. Neurosci., 251(4):152–158, 2001. [9] T.W. Nattkemper, H. Ritter, and W. Schubert. A neural classifier enabling high-throughput topological analysis of lymphocytes in tissue sections. IEEE Trans. ITB, 5:138–149, 2001. 1 graduate college ‘nonlinear dynamics’ 2 project ‘ModKog’
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:
Page 158 and 159:
Page 160 and 161:
Page 162 and 163:
Page 164 and 165:
156 Chapter 10. IEEE TNN 16(4):992-
Page 166 and 167:
Page 168 and 169:
Page 170 and 171:
162 Chapter 11. EURASIP JASP, 2007
Page 172 and 173:
Page 174 and 175:
Page 176 and 177:
Page 178 and 179:
Page 180 and 181:
Page 182 and 183:
Page 184 and 185:
176 Chapter 12. LNCS 3195:718-725,
Page 186 and 187:
178 Chapter 12. LNCS 3195:718-725,
Page 188 and 189:
180 Chapter 12. LNCS 3195:718-725,
Page 190 and 191:
182 Chapter 12. LNCS 3195:718-725,
Page 192 and 193:
184 Chapter 12. LNCS 3195:718-725,
Page 194 and 195:
186 Chapter 13. Proc. EUSIPCO 2005
Page 196 and 197:
Page 198 and 199:
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 206 and 207:
198 Chapter 15. Neurocomputing, 69:
Page 208 and 209:
Page 210 and 211:
Page 212 and 213:
Page 214 and 215:
Page 216 and 217:
Page 218 and 219:
Page 220 and 221:
Page 222 and 223:
Page 224 and 225:
Page 226 and 227:
Page 228 and 229:
Page 230 and 231:
Page 232 and 233:
Page 234 and 235:
Page 236 and 237:
Page 238 and 239:
230 Chapter 16. Proc. ICA 2006, pag
Page 240 and 241:
Page 242 and 243:
Page 244 and 245:
Page 246 and 247:
Page 248 and 249:
240 Chapter 17. IEEE SPL 13(2):96-9
Page 250 and 251:
242 Chapter 17. IEEE SPL 13(2):96-9
Page 252 and 253: 244 Chapter 17. IEEE SPL 13(2):96-9
Page 254 and 255: 246 Chapter 18. Proc. EUSIPCO 2006
Page 260 and 261: 252 Chapter 19. LNCS 3195:977-984,
Page 270 and 271: 262 Chapter 20. Signal Processing 8
Page 300 and 301: 292 Chapter 21. Proc. BIOMED 2005,
Page 304 and 305: 296 Chapter 21. Proc. BIOMED 2005,
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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?