NewDevelopmentsinBiomedicalEngineering32 val of 1s, 85% of the stimuli at 800Hz and randomly presented 15% deviant tones at 560Hz. The subject was sitting in a chair and was asked to press a button every time he heard the deviant target tone. The sampling rate of the EEG was 500 Hz. From the recordings, channel Cz was selected for analysis, after bandpass filtering in the range 1-40Hz. Average responses from the two conditions are shown in Figure 2 (Section 2). For investigation of the single trial variability of the P300 peak, EEG epochs from -100 ms to 600 ms relative to the stimulus onset of each deviant stimulus were here used. The model was designed as in section 7.1 but now for the slower P300 wave the selection f c = 10Hz was made. The application of the empirical rule (27) gave in this case k = 15. Kalman smoother estimates were computed with the selection σ 2 ω = 9, with respect to the expected faster variability of the potential. In Figure 5 (I) there are presented the EP measurements in the original stimulus order (trial-by- trial). In the same figure (II) the obtained estimates based on the measurements (I) are shown. Clearly, in the estimates, the dynamic variability of the P300 peak potential is revealed, sug- gesting that it cannot be considered as occurring at fixed latency from the stimuli presentation. At the same image (II), the estimated latency is also plotted as a function of the consecutive trial t. The latency of the peak was estimated from the Kalman smoother estimates based on the maximum value within the time interval 250-370ms after the presentation of the stimuli. The estimated time-varying latency of the P300 peak was then used to order the single-trial measurements. The sorted single-trials (condition-by-condition) are shown at Figure 5 (III). The shorted latency estimates are plotted again over the image plot. This plot clearly demon- strates that the latency estimates obtained with Kalman smoother are of acceptable accuracy. Finally, the algorithm was also applied to the sorted measurements (III). The value σ 2 ω = 4 was selected and new point estimates for the latency were obtained as before. Kalman smoother estimates and the new latency estimates are plotted in Figure 5 (IV). The linear trend of the sorted potentials allows the use of even smaller value for state-noise variance parameter (Georgiadis et al., 2005b), thus reducing even more the noise without reducing the variability of the peak. The last obtained estimates of the latencies were plotted over the original non sorted measurements (I). The similarities between the estimated latency fluctuations in (I) and (II) underline the robustness of the method. 8. Conclusion and Future Directions EP research has to deal with several inherent difficulties. Traditional analysis is based on aver- aged data often by forming extra grand averages of different populations. Thus, trial-to-trial variability and individual subject characteristics are largely ignored (Fell, 2007). Therefore, the study of isolated components retrieved by averages might be misleading, or at least it is a simplification of the reality. For example, habituation may occur and the responses could be different from the beginning to the end of the recording session. Furthermore, cognitive potentials exhibit rich latency and amplitude variability that traditional research based on av- eraging is not able to exploit for studying complex cognitive processes. Latency variability could be used, for instance, for studying perceptual changes, quantifying stimulus classifica- tion speed or task difficulty. In this chapter, state-space modeling for single-trial estimation of EPs was presented in its general form based on Bayesian estimation theory. This formulation enables the selection of different models for dynamical estimation. In general, the applicability of the proposed Fig. 5. Single-trial EP latency variability. State-spacemodelingforsingle-trialevokedpotentialestimation 33 val of 1s, 85% of the stimuli at 800Hz and randomly presented 15% deviant tones at 560Hz. The subject was sitting in a chair and was asked to press a button every time he heard the deviant target tone. The sampling rate of the EEG was 500 Hz. From the recordings, channel Cz was selected for analysis, after bandpass filtering in the range 1-40Hz. Average responses from the two conditions are shown in Figure 2 (Section 2). For investigation of the single trial variability of the P300 peak, EEG epochs from -100 ms to 600 ms relative to the stimulus onset of each deviant stimulus were here used. The model was designed as in section 7.1 but now for the slower P300 wave the selection f c = 10Hz was made. The application of the empirical rule (27) gave in this case k = 15. Kalman smoother estimates were computed with the selection σ 2 ω = 9, with respect to the expected faster variability of the potential. In Figure 5 (I) there are presented the EP measurements in the original stimulus order (trial-by- trial). In the same figure (II) the obtained estimates based on the measurements (I) are shown. Clearly, in the estimates, the dynamic variability of the P300 peak potential is revealed, sug- gesting that it cannot be considered as occurring at fixed latency from the stimuli presentation. At the same image (II), the estimated latency is also plotted as a function of the consecutive trial t. The latency of the peak was estimated from the Kalman smoother estimates based on the maximum value within the time interval 250-370ms after the presentation of the stimuli. The estimated time-varying latency of the P300 peak was then used to order the single-trial measurements. The sorted single-trials (condition-by-condition) are shown at Figure 5 (III). The shorted latency estimates are plotted again over the image plot. This plot clearly demon- strates that the latency estimates obtained with Kalman smoother are of acceptable accuracy. Finally, the algorithm was also applied to the sorted measurements (III). The value σ 2 ω = 4 was selected and new point estimates for the latency were obtained as before. Kalman smoother estimates and the new latency estimates are plotted in Figure 5 (IV). The linear trend of the sorted potentials allows the use of even smaller value for state-noise variance parameter (Georgiadis et al., 2005b), thus reducing even more the noise without reducing the variability of the peak. The last obtained estimates of the latencies were plotted over the original non sorted measurements (I). The similarities between the estimated latency fluctuations in (I) and (II) underline the robustness of the method. 8. Conclusion and Future Directions EP research has to deal with several inherent difficulties. Traditional analysis is based on aver- aged data often by forming extra grand averages of different populations. Thus, trial-to-trial variability and individual subject characteristics are largely ignored (Fell, 2007). Therefore, the study of isolated components retrieved by averages might be misleading, or at least it is a simplification of the reality. For example, habituation may occur and the responses could be different from the beginning to the end of the recording session. Furthermore, cognitive potentials exhibit rich latency and amplitude variability that traditional research based on av- eraging is not able to exploit for studying complex cognitive processes. Latency variability could be used, for instance, for studying perceptual changes, quantifying stimulus classifica- tion speed or task difficulty. In this chapter, state-space modeling for single-trial estimation of EPs was presented in its general form based on Bayesian estimation theory. This formulation enables the selection of different models for dynamical estimation. In general, the applicability of the proposed Fig. 5. Single-trial EP latency variability. NewDevelopmentsinBiomedicalEngineering34 methodology primarily relates on the assumption of hidden dynamic variability from trial-to- trial or from condition-to-condition. A practical method for designing an observation model was also presented and its capability to reveal meaningful amplitude and latency fluctuations in EP measurements was demonstrated. In the approach, optimal estimates for the states are obtained with Kalman filter and smoother algorithms. When all the measurements are available (batch processing) Kalman smoother should be used. EPs also contain rich spatial information that can be used for describing brain dynamics (Makeig et al., 2004; Ranta-aho et al., 2003). In this study, this important issue was not dis- cussed and emphasis was given on optimal estimation of some temporal EP characteristics. Future development of the presented methodology involves the extension of the approach to multichannel and multimodal data sets, for instance, simultaneously measured EEG/ERP and fMRI/BOLD signals (Debener et al., 2006), for the study of dynamic changes of the central nervous system. Acknowledgments The authors acknowledge financial support from the Academy of Finland (project numbers: 123579, 1.1.2008-31.12.2011, and 126873, 1.1.2009-31.12.2011). 9. References Cerutti, S., Bersani, V., Carrara, A. & Liberati, D. (1987). Analysis of visual evoked potentials through Wiener filtering applied to a small number of sweeps, Journal of Biomedical Engineering 9(1): 3–12. Debener, S., Ullsperger, M., Siegel, M. & Engel, A. (2006). Single-trial EEG-fMRI reveals the dynamics of cognitive function, Trends in Cognitive Sciences 10(2): 558–63. Delorme, A. & Makeig, S. (2004). EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis, Journal of Neuroscience Methods 134(1): 9–21. Doncarli, C., Goering, L. & Guiheneuc, P. (1992). Adaptive smoothing of evoked potentials, Signal Processing 28(1): 63–76. Fell, J. (2007). Cognitive neurophysiology: Beyond averaging, NeuroImage 37: 1069–1027. Georgiadis, S. (2007). State-Space Modeling and Bayesian Methods for Evoked Potential Estimation, PhD thesis, Kuopio University Publications C. Natural and Environmental Sciences 213. (available: http://bsamig.uku.fi/). Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2005a). Recursive mean square estimators for single-trial event related potentials, Proc. Finnish Signal Processing Sym- posium - FINSIG’05, Kuopio, Finland. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2005b). Single-trial dynamical estimation of event related potentials: a Kalman filter based approach, IEEE Transac- tions on Biomedical Engineering 52(8): 1397–1406. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2007). A subspace method for dynamical estimation of evoked potentials, Computational Intelligence and Neuroscience 2007: Article ID 61916, 11 pages. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2008). Tracking single-trial evoked potential changes with Kalman filtering and smoothing, 30th Annual Inter- national Conference of the IEEE Engineering in Medicine and Biology Society, Vancouver, Canada, pp. 157–160. Holm, A., Ranta-aho, P., Sallinen, M., Karjalainen, P. & Müller, K. (2006). Relationship of P300 single trial responses with reaction time and preceding stimulus sequence, Interna- tional Journal of Psychophysiology 61(2): 244–252. Intriligator, J. & Polich, J. (1994). On the relationship between background EEG and the P300 event-related potential, Biological Psychology 37(3): 207–218. Jansen, B., Agarwal, G., Hegde, A. & Boutros, N. (2003). Phase synchronization of the ongoing EEG and auditory EP generation, Clinical Neurophysiology 114(1): 79–85. Kaipio, J. & Somersalo, E. (2005). Statistical and Computational Inverse Problems, Applied Math- ematical Sciences, Springer. Kalman, R. (1960). A new approach to linear filtering and prediction problems, Transactions of the ASME, Journal of Basic Engineering 82: 35–45. Karjalainen, P., Kaipio, J., Koistinen, A. & Vauhkonen, M. (1999). Subspace regularization method for the single trial estimation of evoked potentials, IEEE Transactions on Biomedical Engineering 46(7): 849–860. Knuth, K., Shah, A., Truccolo, W., Ding, M., Bressler, S. & Schroeder, C. (2006). Differentially variable component analysis (dVCA): Identifying multiple evoked components us- ing trial-to-trial variability, Journal of Neurophysiology 95(5): 3257–3276. Li, R., Principe, J., Bradley, M. & Ferrari, V. (2009). A spatiotemporal filtering methodology for single-trial ERP component estimation, IEEE Transactions on Biomedical Engineering 56(1): 83–92. Makeig, S., Debener, S. & Delorme, A. (2004). Mining event-related brain dynamics, Trends in Cognitive Science 8(5): 204–210. Makeig, S., Westerfield, M., Jung, T P., Enghoff, S., Townsend, J., Courchesne, E. & Sejnowski, T. (2002). Dynamic brain sources of visual evoked responses, Science 295: 690–694. Mäkinen, V., Tiitinen, H. & May, P. (2005). Auditory even-related responses are generated independently of ongoing brain activity, NeuroImage 24(4): 961–968. Malmivuo, J. & Plonsey, R. (1995). Bioelectromagnetism, Oxford university press, New York. Niedermeyer, E. & da Silva, F. L. (eds) (1999). Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, 4th edn, Williams and Wilkins. Qiu, W., Chang, C., Lie, W., Poon, P., Lam, F., Hamernik, R., Wei, G. & Chan, F. (2006). Real- time data-reusing adaptive learning of a radial basis function network for tracking evoked potentials, IEEE Transanctions on Biomedical Engineering 53(2): 226–237. Quiroga, R. Q. & Garcia, H. (2003). Single-trial evoked potentials with wavelet denoising, Clinical Neurophysiology 114: 376–390. Ranta-aho, P., Koistinen, A., Ollikainen, J., Kaipio, J., Partanen, J. & Karjalainen, P. (2003). Single-trial estimation of multichannel evoked-potential measurements, IEEE Trans- actions on Biomedical Engineering 50(2): 189–196. Rauch, H., Tung, F. & Striebel, C. (1965). Maximum likelihood estimates of linear dynamic systems, AIAA Journal 3: 1445–1450. Sorenson, H. (1980). Parameter Estimation, Principles and Problems, Vol. 9 of Control and Systems Theory, Marcel Dekker Inc., New York. Thakor, N., Vaz, C., McPherson, R. & Hanley, D. F. (1991). Adaptive Fourier series modeling of time-varying evoked potentials: Study of human somatosensory evoked response to etomidate anesthetic, Electroencephalography and Clinical Neurophysiology 80(2): 108– 118. State-spacemodelingforsingle-trialevokedpotentialestimation 35 methodology primarily relates on the assumption of hidden dynamic variability from trial-to- trial or from condition-to-condition. A practical method for designing an observation model was also presented and its capability to reveal meaningful amplitude and latency fluctuations in EP measurements was demonstrated. In the approach, optimal estimates for the states are obtained with Kalman filter and smoother algorithms. When all the measurements are available (batch processing) Kalman smoother should be used. EPs also contain rich spatial information that can be used for describing brain dynamics (Makeig et al., 2004; Ranta-aho et al., 2003). In this study, this important issue was not dis- cussed and emphasis was given on optimal estimation of some temporal EP characteristics. Future development of the presented methodology involves the extension of the approach to multichannel and multimodal data sets, for instance, simultaneously measured EEG/ERP and fMRI/BOLD signals (Debener et al., 2006), for the study of dynamic changes of the central nervous system. Acknowledgments The authors acknowledge financial support from the Academy of Finland (project numbers: 123579, 1.1.2008-31.12.2011, and 126873, 1.1.2009-31.12.2011). 9. References Cerutti, S., Bersani, V., Carrara, A. & Liberati, D. (1987). Analysis of visual evoked potentials through Wiener filtering applied to a small number of sweeps, Journal of Biomedical Engineering 9(1): 3–12. Debener, S., Ullsperger, M., Siegel, M. & Engel, A. (2006). Single-trial EEG-fMRI reveals the dynamics of cognitive function, Trends in Cognitive Sciences 10(2): 558–63. Delorme, A. & Makeig, S. (2004). EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis, Journal of Neuroscience Methods 134(1): 9–21. Doncarli, C., Goering, L. & Guiheneuc, P. (1992). Adaptive smoothing of evoked potentials, Signal Processing 28(1): 63–76. Fell, J. (2007). Cognitive neurophysiology: Beyond averaging, NeuroImage 37: 1069–1027. Georgiadis, S. (2007). State-Space Modeling and Bayesian Methods for Evoked Potential Estimation, PhD thesis, Kuopio University Publications C. Natural and Environmental Sciences 213. (available: http://bsamig.uku.fi/). Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2005a). Recursive mean square estimators for single-trial event related potentials, Proc. Finnish Signal Processing Sym- posium - FINSIG’05, Kuopio, Finland. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2005b). Single-trial dynamical estimation of event related potentials: a Kalman filter based approach, IEEE Transac- tions on Biomedical Engineering 52(8): 1397–1406. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2007). A subspace method for dynamical estimation of evoked potentials, Computational Intelligence and Neuroscience 2007: Article ID 61916, 11 pages. Georgiadis, S., Ranta-aho, P., Tarvainen, M. & Karjalainen, P. (2008). Tracking single-trial evoked potential changes with Kalman filtering and smoothing, 30th Annual Inter- national Conference of the IEEE Engineering in Medicine and Biology Society, Vancouver, Canada, pp. 157–160. Holm, A., Ranta-aho, P., Sallinen, M., Karjalainen, P. & Müller, K. (2006). Relationship of P300 single trial responses with reaction time and preceding stimulus sequence, Interna- tional Journal of Psychophysiology 61(2): 244–252. Intriligator, J. & Polich, J. (1994). On the relationship between background EEG and the P300 event-related potential, Biological Psychology 37(3): 207–218. Jansen, B., Agarwal, G., Hegde, A. & Boutros, N. (2003). Phase synchronization of the ongoing EEG and auditory EP generation, Clinical Neurophysiology 114(1): 79–85. Kaipio, J. & Somersalo, E. (2005). Statistical and Computational Inverse Problems, Applied Math- ematical Sciences, Springer. Kalman, R. (1960). A new approach to linear filtering and prediction problems, Transactions of the ASME, Journal of Basic Engineering 82: 35–45. Karjalainen, P., Kaipio, J., Koistinen, A. & Vauhkonen, M. (1999). Subspace regularization method for the single trial estimation of evoked potentials, IEEE Transactions on Biomedical Engineering 46(7): 849–860. Knuth, K., Shah, A., Truccolo, W., Ding, M., Bressler, S. & Schroeder, C. (2006). Differentially variable component analysis (dVCA): Identifying multiple evoked components us- ing trial-to-trial variability, Journal of Neurophysiology 95(5): 3257–3276. Li, R., Principe, J., Bradley, M. & Ferrari, V. (2009). A spatiotemporal filtering methodology for single-trial ERP component estimation, IEEE Transactions on Biomedical Engineering 56(1): 83–92. Makeig, S., Debener, S. & Delorme, A. (2004). Mining event-related brain dynamics, Trends in Cognitive Science 8(5): 204–210. Makeig, S., Westerfield, M., Jung, T P., Enghoff, S., Townsend, J., Courchesne, E. & Sejnowski, T. (2002). Dynamic brain sources of visual evoked responses, Science 295: 690–694. Mäkinen, V., Tiitinen, H. & May, P. (2005). Auditory even-related responses are generated independently of ongoing brain activity, NeuroImage 24(4): 961–968. Malmivuo, J. & Plonsey, R. (1995). Bioelectromagnetism, Oxford university press, New York. Niedermeyer, E. & da Silva, F. L. (eds) (1999). Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, 4th edn, Williams and Wilkins. Qiu, W., Chang, C., Lie, W., Poon, P., Lam, F., Hamernik, R., Wei, G. & Chan, F. (2006). Real- time data-reusing adaptive learning of a radial basis function network for tracking evoked potentials, IEEE Transanctions on Biomedical Engineering 53(2): 226–237. Quiroga, R. Q. & Garcia, H. (2003). Single-trial evoked potentials with wavelet denoising, Clinical Neurophysiology 114: 376–390. Ranta-aho, P., Koistinen, A., Ollikainen, J., Kaipio, J., Partanen, J. & Karjalainen, P. (2003). Single-trial estimation of multichannel evoked-potential measurements, IEEE Trans- actions on Biomedical Engineering 50(2): 189–196. Rauch, H., Tung, F. & Striebel, C. (1965). Maximum likelihood estimates of linear dynamic systems, AIAA Journal 3: 1445–1450. Sorenson, H. (1980). Parameter Estimation, Principles and Problems, Vol. 9 of Control and Systems Theory, Marcel Dekker Inc., New York. Thakor, N., Vaz, C., McPherson, R. & Hanley, D. F. (1991). Adaptive Fourier series modeling of time-varying evoked potentials: Study of human somatosensory evoked response to etomidate anesthetic, Electroencephalography and Clinical Neurophysiology 80(2): 108– 118. NewDevelopmentsinBiomedicalEngineering36 Truccolo, W., Mingzhou, D., Knuth, K., Nakamura, R. & Bressler, S. (2002). Trial-to-trial vari- ability of cortical evoked responses: implications for the analysis of functional con- nectivity, Clinical Neurophysiology 113(2): 206–226. Turetsky, B., Raz, J. & Fein, G. (1989). Estimation of trial-to-trial variation in evoked potential signals by smoothing across trials, Psychophysiology 26(6): 700–712. Non-StationaryBiosignalModelling 37 Non-StationaryBiosignalModelling CarlosS.Lima,AdrianoTavares,JoséH.Correia,ManuelJ.CardosoandDanielBarbosa X Non-Stationary Biosignal Modelling Carlos S. Lima, Adriano Tavares, José H. Correia, Manuel J. Cardoso 1 and Daniel Barbosa University of Minho Portugal 1 University College of London England 1. Introduction Signals of biomedical nature are in the most cases characterized by short, impulse-like events that represent transitions between different phases of a biological cycle. As an example hearth sounds are essentially events that represent transitions between the different hemodynamic phases of the cardiac cycle. Classical techniques in general analyze the signal over long periods thus they are not adequate to model impulse-like events. High variability and the very often necessity to combine features temporally well localized with others well localized in frequency remains perhaps the most important challenges not yet completely solved for the most part of biomedical signal modeling. Wavelet Transform (WT) provides the ability to localize the information in the time-frequency plane; in particular, they are capable of trading on type of resolution for the other, which makes them especially suitable for the analysis of non-stationary signals. State of the art automatic diagnosis algorithms usually rely on pattern recognition based approaches. Hidden Markov Models (HMM’s) are statistically based pattern recognition techniques with the ability to break a signal in almost stationary segments in a framework known as quasi-stationary modeling. In this framework each segment can be modeled by classical approaches, since the signal is considered stationary in the segment, and at a whole a quasi-stationary approach is obtained. Recently Discrete Wavelet Transform (DWT) and HMM’s have been combined as an effort to increase the accuracy of pattern recognition based approaches regarding automatic diagnosis purposes. Two main motivations have been appointed to support the approach. Firstly, in each segment the signal can not be exactly stationary and in this situation the DWT is perhaps more appropriate than classical techniques that usually considers stationarity. Secondly, even if the process is exactly stationary over the entire segment the capacity given by the WT of simultaneously observing the signal at various scales (at different levels of focus), each one emphasizing different characteristics can be very beneficial regarding classification purposes. This chapter presents an overview of the various uses of the WT and HMM’s in Computer Assisted Diagnosis (CAD) in medicine. Their most important properties regarding biomedical applications are firstly described. The analogy between the WT and some of the 3 NewDevelopmentsinBiomedicalEngineering38 biological processing that occurs in the early components of the visual and auditory systems, which partially supports the WT applications in medicine is shortly described. The use of the WT in the analyses of 1-D physiological signals especially electrocardiography (ECG) and phonocardiography (PCG) are then reviewed. A survey of recent wavelet developments in medical imaging is then provided. These include biomedical image processing algorithms as noise reduction, image enhancement and detection of micro- calcifications in mammograms, image reconstruction and acquisition schemes as tomography and Magnetic Resonance Imaging (MRI), and multi-resolution methods for the registration and statistical analysis of functional images of the brain as positron emission tomography (PET) and functional MRI. The chapter provides an almost complete theoretical explanation of HMMs. Then a review of HMMs in electrocardiography and phonocardiography is given. Finally more recent approaches involving both WT and HMMs specifically in electrocardiography and phonocardiography are reviewed. 2. Wavelets and biomedical signals Biomedical applications usually require most sophisticated signal processing techniques than others fields of engineering. The information of interest is often a combination of features that are well localized in space and time. Some examples are spikes and transients in electroencephalograph signals and microcalcifications in mammograms and others more diffuse as texture, small oscillations and bursts. This universe of events at opposite extremes in the time-frequency localization can not be efficiently handled by classical signal processing techniques mostly based on the Fourier analysis. In the past few years, researchers from mathematics and signal processing have developed the concept of multiscale representation for signal analysis purposes (Vetterli & Kovacevic, 1995). These wavelet based representations have over the traditional Fourier techniques the advantage of localize the information in the time-frequency plane. They are capable of trading one type of resolution for the other, which makes them especially suitable for modelling non-stationary events. Due to these characteristics of the WT and the difficult conditions frequently encountered in biomedical signal analysis, WT based techniques proliferated in medical applications ranging from the more traditional physiological signals such as ECG to the most recent imaging modalities as PET and MRI. Theoretically wavelet analysis is a reasonably complicated mathematical discipline, at least for most biomedical engineers, and consequently a detailed analysis of this technique is out of the scope of this chapter. The interested reader can find detailed references such as (Vetterli & Kovacevic, 1995) and (Mallat, 1998). The purpose of this chapter is only to emphasize the wavelet properties more related to current biomedical applications. 2.1 The wavelet transform - An overview The wavelet transform (WT) is a signal representation in a scale-time space, where each scale represents a focus level of the signal and therefore can be seen as a result of a band- pass filtering. Given a time-varying signal x(t), WTs are a set of coefficients that are inner products of the signal with a family of wavelets basis functions obtained from a standard function known as mother wavelet. In Continuous Wavelet Transform (CWT) the wavelet corresponding to scale s and time location τ is given by (1) where ψ(t) is the mother wavelet, which can be viewed as a band-pass function. The term s ensures energy preservation. In the CWT the time-scale parameters vary continuously. The wavelet transform of a continuous time varying signal x(t) is given by (2) where the asterisk stands for complex conjugate. Equation (2) shows that the WT is the convolution between the signal and the wavelet function at scale s. For a fixed value of the scale parameter s, the WT which is now a function of the continuous shift parameter τ, can be written as a convolution equation where the filter corresponds to a rescaled and time- reversed version of the wavelet as shown by equation (1) setting t=0. From the time scaling property of the Fourier Transform the frequency response of the wavelet filter is given by (3) One important property of the wavelet filter is that for a discrete set of scales, namely the dyadic scale i s 2 a constant-Q filterbank is obtained, where the quality factor of the filter is defined as the central frequency to bandwidth ratio. Therefore WT provides a decomposition of a signal into subbands with a bandwidth that increases linearly with the frequency. Under this framework the WT can be viewed as a special kind of spectral analyser. Energy estimates in different bands or related measures can discriminate between various physiological states (Akay & al. 1994). Under this approach, the purpose is to analyse turbulent hearth sounds to detect coronary artery disease. The purpose of the approach followed by (Akay & Szeto 1994) is to characterize the states of fetal electrocortical activity. However, this type of global feature extraction assumes stationarity, therefore similar results can also be obtained using more conventional Fourier techniques. Wavelets viewed as a filterbank have motivated several approaches based on reversible wavelet decomposition such as noise reduction and image enhancement algorithms. The principle is to handle selectively the wavelet components prior to reconstruction. (Mallat & Zhong, 1992) used such a filterbank system to obtain a multiscale edge representation of a signal from its wavelets maxima. They proposed an iterative algorithm that reconstructs a very close approximation of the original from this subset of features. This approach has been adapted for noise reduction in evoked response potentials and in MR images and also in image enhancement regarding the detection of microcalcifications in mammograms.            dt s t tx s s x    * )( 1 ),(         s t s s    1 ,    sΨs s τ ψ s 1 *         Non-StationaryBiosignalModelling 39 biological processing that occurs in the early components of the visual and auditory systems, which partially supports the WT applications in medicine is shortly described. The use of the WT in the analyses of 1-D physiological signals especially electrocardiography (ECG) and phonocardiography (PCG) are then reviewed. A survey of recent wavelet developments in medical imaging is then provided. These include biomedical image processing algorithms as noise reduction, image enhancement and detection of micro- calcifications in mammograms, image reconstruction and acquisition schemes as tomography and Magnetic Resonance Imaging (MRI), and multi-resolution methods for the registration and statistical analysis of functional images of the brain as positron emission tomography (PET) and functional MRI. The chapter provides an almost complete theoretical explanation of HMMs. Then a review of HMMs in electrocardiography and phonocardiography is given. Finally more recent approaches involving both WT and HMMs specifically in electrocardiography and phonocardiography are reviewed. 2. Wavelets and biomedical signals Biomedical applications usually require most sophisticated signal processing techniques than others fields of engineering. The information of interest is often a combination of features that are well localized in space and time. Some examples are spikes and transients in electroencephalograph signals and microcalcifications in mammograms and others more diffuse as texture, small oscillations and bursts. This universe of events at opposite extremes in the time-frequency localization can not be efficiently handled by classical signal processing techniques mostly based on the Fourier analysis. In the past few years, researchers from mathematics and signal processing have developed the concept of multiscale representation for signal analysis purposes (Vetterli & Kovacevic, 1995). These wavelet based representations have over the traditional Fourier techniques the advantage of localize the information in the time-frequency plane. They are capable of trading one type of resolution for the other, which makes them especially suitable for modelling non-stationary events. Due to these characteristics of the WT and the difficult conditions frequently encountered in biomedical signal analysis, WT based techniques proliferated in medical applications ranging from the more traditional physiological signals such as ECG to the most recent imaging modalities as PET and MRI. Theoretically wavelet analysis is a reasonably complicated mathematical discipline, at least for most biomedical engineers, and consequently a detailed analysis of this technique is out of the scope of this chapter. The interested reader can find detailed references such as (Vetterli & Kovacevic, 1995) and (Mallat, 1998). The purpose of this chapter is only to emphasize the wavelet properties more related to current biomedical applications. 2.1 The wavelet transform - An overview The wavelet transform (WT) is a signal representation in a scale-time space, where each scale represents a focus level of the signal and therefore can be seen as a result of a band- pass filtering. Given a time-varying signal x(t), WTs are a set of coefficients that are inner products of the signal with a family of wavelets basis functions obtained from a standard function known as mother wavelet. In Continuous Wavelet Transform (CWT) the wavelet corresponding to scale s and time location τ is given by (1) where ψ(t) is the mother wavelet, which can be viewed as a band-pass function. The term s ensures energy preservation. In the CWT the time-scale parameters vary continuously. The wavelet transform of a continuous time varying signal x(t) is given by (2) where the asterisk stands for complex conjugate. Equation (2) shows that the WT is the convolution between the signal and the wavelet function at scale s. For a fixed value of the scale parameter s, the WT which is now a function of the continuous shift parameter τ, can be written as a convolution equation where the filter corresponds to a rescaled and time- reversed version of the wavelet as shown by equation (1) setting t=0. From the time scaling property of the Fourier Transform the frequency response of the wavelet filter is given by (3) One important property of the wavelet filter is that for a discrete set of scales, namely the dyadic scale i s 2 a constant-Q filterbank is obtained, where the quality factor of the filter is defined as the central frequency to bandwidth ratio. Therefore WT provides a decomposition of a signal into subbands with a bandwidth that increases linearly with the frequency. Under this framework the WT can be viewed as a special kind of spectral analyser. Energy estimates in different bands or related measures can discriminate between various physiological states (Akay & al. 1994). Under this approach, the purpose is to analyse turbulent hearth sounds to detect coronary artery disease. The purpose of the approach followed by (Akay & Szeto 1994) is to characterize the states of fetal electrocortical activity. However, this type of global feature extraction assumes stationarity, therefore similar results can also be obtained using more conventional Fourier techniques. Wavelets viewed as a filterbank have motivated several approaches based on reversible wavelet decomposition such as noise reduction and image enhancement algorithms. The principle is to handle selectively the wavelet components prior to reconstruction. (Mallat & Zhong, 1992) used such a filterbank system to obtain a multiscale edge representation of a signal from its wavelets maxima. They proposed an iterative algorithm that reconstructs a very close approximation of the original from this subset of features. This approach has been adapted for noise reduction in evoked response potentials and in MR images and also in image enhancement regarding the detection of microcalcifications in mammograms.            dt s t tx s s x    * )( 1 ),(         s t s s    1 ,    sΨs s τ ψ s 1 *         NewDevelopmentsinBiomedicalEngineering40 From the filterbank point of view the shape of the mother wavelet seems to be important in order to emphasize some signal characteristics, however this topic is not explored in the ambit of the present chapter. Regarding implementation issues both s and τ must be discretized. The most usual way to sample the time-scale plane is on a so-called dyadic grid, meaning that sampled points in the time-scale plane are separated by a power of two. This procedure leads to an increase in computational efficiency for both WT and Inverse Wavelet Transform (IWT). Under this constraint the Discrete Wavelet Transform (DWT) is defined as (4) which means that DWT coefficients are sampled from CWT coefficients. As a dyadic scale is used and therefore s 0 =2 and τ 0 =1, yielding s=2 j and τ=k2 j where j and k are integers. As the scale represents the level of focus from the which the signal is viewed, which is related to the frequency range involved, the digital filter banks are appropriated to break the signal in different scales (bands). If the progression in the scale is dyadic the signal can be sequentially half-band high-pass and low-pass filtered. Fig. 1. Wavelet decomposition tree The output of the high-pass filter represents the detail of the signal. The output of the low- pass filter represents the approximation of the signal for each decomposition level, and will be decomposed in its detail and approximation components at the next decomposition level. The process proceeds iteratively in a scheme known as wavelet decomposition tree, which is     00 2 0,   ktsst j j kj    h[n] g [n] h[n] g [n] 2 2 2 2 DWT coeff. –Level 1 DWT coeff. –Level 2 … x[n] shown in figure 1. After filtering, half of the samples can be eliminated according to the Nyquist’s rule, since the signal now has only half of the frequency. This very practical filtering algorithm yields as Fast Wavelet Transform (FWT) and is known in the signal processing community as two-channel subband coder. One important property of the DWT is the relationship between the impulse responses of the high-pass (g[n]) and low-pass (h[n]) filters, which are not independent of each other and are related by (5) where L is the filter length in number of points. Since the two filters are odd index alternated reversed versions of each other they are known as Quadrature Mirror Filters (QMF). Perfect reconstruction requires, in principle, ideal half-band filtering. Although it is not possible to realize ideal filters, under certain conditions it is possible to find filters that provide perfect reconstruction. Perhaps the most famous were developed by Ingrid Daubechies and are known as Daubechies’ wavelets. This processing scheme is extended to image processing where temporal filters are changed by spatial filters and filtering is usually performed in three directions; horizontal, vertical and diagonal being the filtering in the diagonal direction obtained from high pass filters in both directions. Wavelet properties can also be viewed as other approaches than filterbanks. As a multiscale matched filter WT have been successful applied for events detection in biomedical signal processing. The matched filter is the optimum detector of a deterministic signal in the presence of additive noise. Considering a measure model       tntttf s      where     stt s /   is a known deterministic signal at scale s, Δt is an unknown location parameter and n(t) an additive white Gaussian noise component. The maximum likelihood solution based on classical detection theory states that the optimum procedure for estimating Δt is to perform the correlations with all possible shifts of the reference template (convolution) and to select the position that corresponds to the maximum output. Therefore, using a WT-like detector whenever the pattern that we are looking for appears at various scales makes some sense. Under correlated situations a pre-whitening filter can be applied and the problem can be solved as in the white noise case. In some noise conditions, specifically if the noise has a fractional Brownian motion structure then the wavelet-like structure of the detector is preserved. In this condition the noise average spectrum has the form     wwN / 2  with α=2H+1 with H as the Hurst exponent and the optimum pre-whitening matched filter at scale s as       s t CtDj ss ψψ α α  (6) where  D is the αth derivative operator which corresponds to    jw in the Fourier domain. In other words, the real valued wavelet   t  is proportional to the fractional derivative of the pattern  that must be detected. For example the optimal detector for finding a Gaussian in   2 wO noise is the second derivative of a Gaussian known as Mexican hat       nhnLg n 11  [...]... pass filtering in a specific direction, therefore encoding details in different directions Thus these parameters contain directional detail information at scale n This recursive filtering is no more than the extension of the scheme represented in figure 1 to a bi-dimensional space as shown in figure 2 Gc An-1 Dn3 Hc Gr ↓1 ,2 ↓1 ,2 Dn2 Gc ↓1 ,2 Dn1 Hc ↓1 ,2 An 2, 1 Hr 2, 1 rows Fig 2 Wavelet 2D decomposition... and to capture point discontinuities then followed by a directional filter bank to link point discontinuities into linear structures Therefore the contourlet transform provides a multiscale and directional decomposition in the frequency domain, as can be seen in figure 6, where is clear the division of the Fourier plane by scale and angle  52 New Developments in Biomedical Engineering Fig 6 The contourlet... denoising techniques, where in this case instead of suppressing the unwanted wavelet coefficients one should amplify the interesting image features Given the original data quality, redundant wavelet transforms are usually used in enhancement algorithms Examples of enhancement algorithms using wavelets are presented in (Heinlein et al 20 03, Papadopoulos et al 20 08, Przelaskowski et al 20 07) 2. 8 Breaking... finding a   Gaussian in O w 2 noise is the second derivative of a Gaussian known as Mexican hat 42 New Developments in Biomedical Engineering wavelet Several biomedical signal processing tasks have been based on the detection properties of the WT such as the detection of interictal spikes in EEG recordings of epileptic patients or cardiology based applications as the detection of the QRS complex in. .. removal of noisy artifacts 50 New Developments in Biomedical Engineering This regularization can be improved through the use of wavelet thresholding estimators (Kalifa 20 03) Jin et al (Jin 20 03) proposed the noise reduction in the reconstructed through cross-regularization of wavelet coefficients Wavelet-encoded MRI – Wavelet basis can be used in MRI encoding schemes, taking advantage from the better... n, c )( y t  ' 2, c,i  n t 1 t ,i T   ' n,c,i ) 2   ( n, c ) (45) t t 1 The reestimation formulas given by equations (45), (43), (38), (35) and ( 32) can be easily calculated using the definitions of forward sequence t(i)=f(y1,y2, yt,st=i/) and backward 62 sequence t(i)=f(yt+1,yt +2, yT,st=i/) implementation New Developments in Biomedical Engineering This procedure is standard in the HMM 4 Wavelets,... follows certain rules dependent on the sequence of events allowed These rules define a grammar with six main symbols (four main waves and two silences of different nature) and an associated language model as shown in figure 10 64 New Developments in Biomedical Engineering sil S1 sil S2 Fig 10 Heart sound Markov Model This HMM does not take into consideration the S3 and S4 heart sounds since these sounds... at the Coronary Care Unit at the Royal Infirmary of Edinburgh and a sensitivity of 99.70% and a positive predictivity of 99.68% were reported in the MIT-BIH database 48 New Developments in Biomedical Engineering Wavelet based filters have been proposed to minimize the wandering distortions (Park et al., 1998) and to remove motion artifacts in ECG’s (Park et al., 20 01) Wavelet based noise reduction methods... modeled by its own HMM and training can be done by HMM concatenation Non-Stationary Biosignal Modelling 47 according to the labeling file prepared by the physician (Lima & Barbosa 20 08) The order of occurrence of A2 and P2 can be obtained by the likelihood of both hypothesis (A2 preceding P2 and vice versa) and the split can be estimated by the backtracking procedure in the Viterbi algorithm which... modeling an isoelectric segment the happening of a strong R wave tends to force a transition to state one which helps in model/beat synchronization The adopted training strategy accommodate both the MMIE training and parameter sharing, or in other words an MMIE training procedure in only one HMM platform with capabilities to model two classes must be required This compromise was obtained by estimating . wavelet given by   2 2 0 t tjw eet    (8) D 22 D 12 D 13 D 11 D 21 D 23 Fi g . 3. Decomposition of 2D DWT in sub-bands New Developments in Biomedical Engineering4 4 has the best. Transactions on Biomedical Engineering 56(1): 83– 92. Makeig, S., Debener, S. & Delorme, A. (20 04). Mining event-related brain dynamics, Trends in Cognitive Science 8(5): 20 4 21 0. Makeig, S.,. data-reusing adaptive learning of a radial basis function network for tracking evoked potentials, IEEE Transanctions on Biomedical Engineering 53 (2) : 22 6 23 7. Quiroga, R. Q. & Garcia, H. (20 03).