Görsel uyarılmış potansiyallerin optimal filtrelenmesi
Görsel uyarılmış potansiyallerin optimal filtrelenmesi
Dosyalar
Tarih
1992
Yazarlar
Şamlı, Seniha Özlem Türkkan
Süreli Yayın başlığı
Süreli Yayın ISSN
Cilt Başlığı
Yayınevi
Fen Bilimleri Enstitüsü
Institute of Science and Technology
Institute of Science and Technology
Özet
Bir uyaran ile uyarılmış beynin elektriksel akti vitesi, uyarılmış potansiyeller (UP) olarak bilinmektedir- uyarılmış potansiyellerin ölçülmesiyle, en önemli üç duyu olan görme, jşitme ve dokunma duyularının normallik seviyeleri değerlendirilebilmektedir. Ancak, UP kaydı sırasında, beynin spontan aktivitesi olan elektroensefalogram (EEG) da kaydedilmektedir. Bu, istenilmeyen bir durumdur. Çünkü yaşam sona ermediği sürece, beyin aktivitesi kesintisiz devam etmektedir. Gen lik ve frekans bandı yönünden EEG *nin OP 'lerden büyük olması nedeniyle, asıl işaret olan uyarılmış işaretlerin kaydı, gürültülü bir şekilde elde edilmektedir. Bu tezde, görsel OP 'lerin gürültüden temizlenmesi amacıyla, bigisayar ekranından, dama tahtası modeli yardımıyla, 100 kez uyarı verilmiş ve her defasında, 400 ms 'lik kayıtlar alınmıştır. Data, 4 ms aralıklarla örneklenmiştir. Daha sonra, datanın ortalaması alınarak, bir optimal filtreden geçirilmiştir. Bu tezde anlatılan yöntemlerin, UP fitrelenmesinde kullanılabilmesi için, bazı varsayımlar gözönüne alınmıştır. Bunlar, şöyle sıralanabilir: (i) UP * ler, stasyonerdir. (ii) Gürültü (EEG), sıfır ortalamalı random (rastgele) bir işarettir. (iiiJ UP, EEG ile ilişkili değildir. (iv) Gürültü, kendi arasında ilişki li değildir. (v) Gürültü toplamsaldır. Filtre katsayılarının hesaplanması için iki yöntem kullanılmıştır. Birinci yöntemde filtre katsayıları, işaret ile gürültü arasındaki çapraz ilişkiler ve işaretin özilişkisi cinsinden hesaplanmıştır. Ancak bu öntemde, bazı matematiksel işlemler nedeniyle hesaplama zorluğu bulunmaktadır. Bu nedenle, ikinci bir yöntem araştırılmıştır, ikinci yöntem ise, filtre katsayıla rının saptanması için LMS (Least Mean Square) metodunu kullanmaktadır. Sonuç bölümünde, çeşitli durumlar için yalnızca ortalanan ve ortalandıktan sonra her iki yöntemle de hesaplanmış filtre katsayılarını içeren optimal filtreden geçirilen görsel uyarılmış potansiyel dataları ile, Wiener filtresi sonuçları karşılaştırılmıştır
The recording of the electrical activity of the brain is known as electroencephalography (EEG) and the electrical activity of the brain evoked by a sensory stimulus is known as the evoked potetials (EP) or evoked responses (ER). It is usually measured over the sensory region of the brain corresponding to the stimulating modality. It is possible to evaluate the normality levels of the three most important senses which are sight, hearing and touch by recording and measuring the evoked potentials. The main problem with the process of evoked potentials is that several kinds of noises are re corded as super impozed on the signal while signal recording. The most effective noise is EEG, because the amplitude interval of EEG is 1-100 ^V as it is 0.1-10 /jV for evoked potentials. The widely used evoked potentials are visual EP (VEP), brainstem auditory EP (BEAP> and somatosensory EP (SEP). In this thesis, in order to get the EP the averaging technic is used firstly. To minimize the errors, the averaged data is filtered by using an optimal filter. The filter coefficients are defined by applying two different ways one of which uses the cross -correlations between signal and noise and the auto-correlations of the signal and the second applies the LMS (Least Mean Square) method. In site of validity of used methods, the fallowing assumptions is made; (i) The EP is stationary. For instance, the response to the same stimulus is always the same for the whole recording. (ii) The noise (EEG) is zero-mean random signal. (iii) EP is uncor related with the EEG. (iv) The noise (EEG) recordings are uncorrelated with one another. (ix) (v) The noise is additional. When measuring evoked potentials, The response s(t), is added to a background noise n(t). The measurement is done by collecting ensemble of N sweeps. In each sweep, x.(t) is: x.(t) = s(t) + n.(t) (1) x x The average of all the sweeps is usually used to estimate the signal: N x(t> = l/N. E ^(t) (2) i=l The average estimator can be filtered to improve its MSE (Mean Square Error). When the optimal filter, h, is applied on the average x, the estimator is given by: m s(t) = £ h(k) x(t-k) (3) k=l The filter coefficients are calculated from the estimated autocorrelation of the signal and noise. The filter h minimizes the MSE of the estimator. A necessary condition for that minimum is: 9 E [ C s(t) - s(t) >2 ] = 0 (4) dh(j) Which yields a set of linear equations: R _ = E htlc> R j = l,...,m (5) sx k=l xx Where for sweeps with length T: (x) R__(k> = l/T £ 3E(t) x(t+k) (6) ** t=l and R _(k) = l/T E s (t) x(t+k) (7) t-1 R _, which is the average of the eros s cor relation. sx between the signal and the average is also equal to R : sx R _(k) = R (k) (8) sx sx R is the average of the crosscorrelations between the signal and the sweeps. R cannot be calculated directly, because the signal is unknown. It can be estimated from the autocorrelation of the average of the sweeps (R which is defined in eq. 6) and from the xx average of the autocorrelation of the individual sweeps (R ), which is defined by: N xx ** x.x. i-i x x From equations (1), (2), (6), (9), it can be derived: R = R (k> + l/N R (k) (11) -~ sx nn xx Which yields: R (k) = T N R (k) - R (k) "I / < N - 1 ) (12) (xi) Calculating R from (6) and R from (12), substituting xx them in the linear equations (5) and solving for h, yields the filter coefficients of the optimal filter. Substituting this filter in (3), the optimal filter signal estimator can be calculated. The LMS algorithm is used in filter design since it does not need measuring the correlation functions and inverting matrices. Firstly it is necessary to describe the "method of steepest descent", which is a recursive method for finding the minimum point of the error-performance surface without knowledge of the error-performance surface itself. This method provides some heuristics for writing the expression for the LMS algorithm. The mean-squared error J(k) is: 2 +.+ ?*? _, ?»? ?+-,.* J(k) = a - h (t) r(k) - r (k) h(k> + h (k) JUk) h(k> 5 (12) 2 Where a is the variance of the desired response s(t). At the minimum point of the error-performance surface, the tap-weight vector (h) takes on the optimum value h, which is defined by the normal equation opt #.(k) h = r(k) (13) Opt The minimum mean-squared error: 2 *t * J. = a - r (k) h (14) mm 5 op t According to the steepest-descent algorithm, the updated value of the tap-weight vector: ?+ ?» ?+ h(k+l) = hCk) + 1/2 fj [- V(t)] (15) k = 0,1,... where /j is a positive real-valued constant and V(k) is the value of the gradient vector. (xii) Differentiating the mean- squared error J(k) with respect to the tap-weight vector h(k), the fallowing value for the gradient vector is got: -? 3j(k) V(k> = dn(k) = -2r(k> + 23Uk)h(k> (16) Thus, the updated value of the tap-weight vector h(t+l) may be computed by using the simple recursive relation: hCk+1) = h(k) + fj [ r(k) - ^.(k)h(k) ] (17) k = 0,1, The simplest choice of estimators for #.(k) and r(k) is: JUk> = x (t> xT (t-k) (18) ört ort r(k> = x ft) s(t-k) (19) ort Correspondingly, the instantaneous estimate of the gradient vector is as fallows: 7(k) = -2x (t) s (t-k) + 2x (t) xT (t-k) h(k) ort ort ort (20) Substituting the estimate of eq. (20) for the gradient vector V(k) in the steepest-descent algorithm as described in eg. (15), a new recursive relation for updating the tap-weight vector is got for LMS algorithm: h(k+l) = h(k> + *j x.(t) [s(t-k) - x (t-k) h(k>] ort ort (21) Equivalently, The result in the form of a pair of relations may be written as follows: (xiii) ?* e(t) = s(t-k) - h x (t-k) (22) Opt ?* * h(k+l> = h(kl + /j x (t) e(t) (23) ort Tbe iterative procedure is started with the initial guess h(Q). A convenient choise for this initial guess is the null vector, h(0) = 0. In this thesis these methods have been applied for filtering the Visual Evoked Potentials.
The recording of the electrical activity of the brain is known as electroencephalography (EEG) and the electrical activity of the brain evoked by a sensory stimulus is known as the evoked potetials (EP) or evoked responses (ER). It is usually measured over the sensory region of the brain corresponding to the stimulating modality. It is possible to evaluate the normality levels of the three most important senses which are sight, hearing and touch by recording and measuring the evoked potentials. The main problem with the process of evoked potentials is that several kinds of noises are re corded as super impozed on the signal while signal recording. The most effective noise is EEG, because the amplitude interval of EEG is 1-100 ^V as it is 0.1-10 /jV for evoked potentials. The widely used evoked potentials are visual EP (VEP), brainstem auditory EP (BEAP> and somatosensory EP (SEP). In this thesis, in order to get the EP the averaging technic is used firstly. To minimize the errors, the averaged data is filtered by using an optimal filter. The filter coefficients are defined by applying two different ways one of which uses the cross -correlations between signal and noise and the auto-correlations of the signal and the second applies the LMS (Least Mean Square) method. In site of validity of used methods, the fallowing assumptions is made; (i) The EP is stationary. For instance, the response to the same stimulus is always the same for the whole recording. (ii) The noise (EEG) is zero-mean random signal. (iii) EP is uncor related with the EEG. (iv) The noise (EEG) recordings are uncorrelated with one another. (ix) (v) The noise is additional. When measuring evoked potentials, The response s(t), is added to a background noise n(t). The measurement is done by collecting ensemble of N sweeps. In each sweep, x.(t) is: x.(t) = s(t) + n.(t) (1) x x The average of all the sweeps is usually used to estimate the signal: N x(t> = l/N. E ^(t) (2) i=l The average estimator can be filtered to improve its MSE (Mean Square Error). When the optimal filter, h, is applied on the average x, the estimator is given by: m s(t) = £ h(k) x(t-k) (3) k=l The filter coefficients are calculated from the estimated autocorrelation of the signal and noise. The filter h minimizes the MSE of the estimator. A necessary condition for that minimum is: 9 E [ C s(t) - s(t) >2 ] = 0 (4) dh(j) Which yields a set of linear equations: R _ = E htlc> R j = l,...,m (5) sx k=l xx Where for sweeps with length T: (x) R__(k> = l/T £ 3E(t) x(t+k) (6) ** t=l and R _(k) = l/T E s (t) x(t+k) (7) t-1 R _, which is the average of the eros s cor relation. sx between the signal and the average is also equal to R : sx R _(k) = R (k) (8) sx sx R is the average of the crosscorrelations between the signal and the sweeps. R cannot be calculated directly, because the signal is unknown. It can be estimated from the autocorrelation of the average of the sweeps (R which is defined in eq. 6) and from the xx average of the autocorrelation of the individual sweeps (R ), which is defined by: N xx ** x.x. i-i x x From equations (1), (2), (6), (9), it can be derived: R = R (k> + l/N R (k) (11) -~ sx nn xx Which yields: R (k) = T N R (k) - R (k) "I / < N - 1 ) (12) (xi) Calculating R from (6) and R from (12), substituting xx them in the linear equations (5) and solving for h, yields the filter coefficients of the optimal filter. Substituting this filter in (3), the optimal filter signal estimator can be calculated. The LMS algorithm is used in filter design since it does not need measuring the correlation functions and inverting matrices. Firstly it is necessary to describe the "method of steepest descent", which is a recursive method for finding the minimum point of the error-performance surface without knowledge of the error-performance surface itself. This method provides some heuristics for writing the expression for the LMS algorithm. The mean-squared error J(k) is: 2 +.+ ?*? _, ?»? ?+-,.* J(k) = a - h (t) r(k) - r (k) h(k> + h (k) JUk) h(k> 5 (12) 2 Where a is the variance of the desired response s(t). At the minimum point of the error-performance surface, the tap-weight vector (h) takes on the optimum value h, which is defined by the normal equation opt #.(k) h = r(k) (13) Opt The minimum mean-squared error: 2 *t * J. = a - r (k) h (14) mm 5 op t According to the steepest-descent algorithm, the updated value of the tap-weight vector: ?+ ?» ?+ h(k+l) = hCk) + 1/2 fj [- V(t)] (15) k = 0,1,... where /j is a positive real-valued constant and V(k) is the value of the gradient vector. (xii) Differentiating the mean- squared error J(k) with respect to the tap-weight vector h(k), the fallowing value for the gradient vector is got: -? 3j(k) V(k> = dn(k) = -2r(k> + 23Uk)h(k> (16) Thus, the updated value of the tap-weight vector h(t+l) may be computed by using the simple recursive relation: hCk+1) = h(k) + fj [ r(k) - ^.(k)h(k) ] (17) k = 0,1, The simplest choice of estimators for #.(k) and r(k) is: JUk> = x (t> xT (t-k) (18) ört ort r(k> = x ft) s(t-k) (19) ort Correspondingly, the instantaneous estimate of the gradient vector is as fallows: 7(k) = -2x (t) s (t-k) + 2x (t) xT (t-k) h(k) ort ort ort (20) Substituting the estimate of eq. (20) for the gradient vector V(k) in the steepest-descent algorithm as described in eg. (15), a new recursive relation for updating the tap-weight vector is got for LMS algorithm: h(k+l) = h(k> + *j x.(t) [s(t-k) - x (t-k) h(k>] ort ort (21) Equivalently, The result in the form of a pair of relations may be written as follows: (xiii) ?* e(t) = s(t-k) - h x (t-k) (22) Opt ?* * h(k+l> = h(kl + /j x (t) e(t) (23) ort Tbe iterative procedure is started with the initial guess h(Q). A convenient choise for this initial guess is the null vector, h(0) = 0. In this thesis these methods have been applied for filtering the Visual Evoked Potentials.
Açıklama
Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1992
Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1992
Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 1992
Anahtar kelimeler
Filtreler,
Görsel uyarılmış potansiyeller,
Uyarılmış potansiyeller,
Filters,
Visual evoked potentials,
Evoked potentials