Galois Homomorphic Fractal Approach for the Recognition of Emotion

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1 Galois Hooorphic Fractal Approach for the Recognition of Eotion T. G. Grace Elizabeth Rani 1, G. Jayalalitha 1 Research Scholar, Bharathiar University, India, Associate Professor, Departent of Matheatics, Vels University, India, Abstract: A novel approach is eployed in the recognition of eotions conveyed through speech. Galois field is constructed for various segents of the signal. This is used as input for hooorphic filtering. The signal obtained by filtering is used to evaluate Higuchi's Fractal Diension. These results derived in the frequency doain are copared with results fro tie doain analysis of the signal, where the signal is initially filtered using a edian filter, and then its Higuchi's Fractal Diension is evaluated. As Galois field is capable of coputing convolution without round off error, it is eployed in our paper. The results indicate that the voice of ale is coarser in happy and angry oods than in neutral ood, which is the reverse for feales. Keywords: Galois field, Hooorphic filtering, eotion, fractals, Higuchi's fractal diension, edian filter. I. INTRODUCTION Speech can be defined as waves of air pressure created by air flow pressed out of lungs and going out through the outh and nasal cavities. The resonant frequencies of the vocal tract are called forants[1]. Forants carry inforation about eotional content [6]. Vowels and consonants characterize speech. As a result speech signal shows both periodicity and self-siilarity. The easure that is ost coonly associated with self-siilar objects is its Fractal Diension. Eotions are cognitive processes related to the architecture of the huan ind, such as decision aking, eory or attention. Eotions use coponents such as subjective, cultural, physiological and behavioral that individual's perception expresses with regard to the ental state, the body and how it interacts with the environent [5]. Classical Matheatics and signal processing techniques have played a ajor role in the developent of speech recognition systes. The kernel of fractal geoetry is the notion of self siilarity. A self-siilar object appears siilar at different scales. The work by Pickover [7] is supposed to have been the first paper on the application of fractals in speech processing. Several research is on, since then in speech and eotion analyses using fractal diension. The application of fractal geoetry to speech signals and speech recognition systes is now receiving serious attention. A very iportant characteristic of fractals, useful for their description and classification is their fractal diension D. The fractal diension provides an objective eans of quantifying the fractal property of an object and coparing objects observed in natural world. Fractals provide a siple description of any natural fors. Intuitively D easures the degree of irregularity over ultiple scales. The application of fractal geoetry depends on the accurate easureent of D, which is very uch essential in fractal based speech recognition[1]. Speech recognition can generally be defined as the process of transforing continuous speech signals into a discrete representation. The proble of speech analysis is a deconvolution process where the coponents in convolution are to be separated. This paper focuses on applying a novel approach in the recognition of eotions conveyed through speech. Two schees of analyses are adopted. The first schee uses the property of Galois field that there is no round off error in obtaining the convolution and hence Fast Fourier Transfor. Initially Galois field is constructed for various segents of the signal, which is the input to hooorphic filtering. The filtered signal is used to find Higuchi's Fractal Diension(HFD). In the second schee, the signal is filtered using edian filter and then its HFD is evaluated. The rest of the paper is organized as follows: In section, the ethodologies are presented. Section 3 presents the schees of analyses, section discusses the results and the last section gives the conclusions of the paper. II. DESCRIPTION OF METHODS The fractal diension is one of the ost coonly used easures to characterize the coplexity of a syste. Fractal Diension of natural phenoena is only easurable using statistical approaches. For non-stationary wavefors such as speech wavefors, it is intelligible to segent the signal into various fraes and then extract the inforation contained in each segent [3]

2 A. Higuchi's Fractal Diension When a tie series changes its structure in tie doain across a certain frequency it is difficult to deterine the power law indices and a characteristic tie scale fro the power spectru. Higuchi developed a technique which gives stable indices and tie scale corresponding to a characteristic frequency even for a sall nuber of data. Consider a finite set of tie series observations taken i is at regular intervals: 1,, 3, N. Fro the given tie series the following new tie series constructed: N i :, i, i, i, with = 1,...i (1) i Here denotes Gauss' notation and both 'i', '' are integers. They represent the interval tie and initial tie respectively. For a tie interval 'i', a total of ' i ' new sets of tie series are obtained. The series is so constructed that there are no overlaps. If ' i ' = and N =100, the four tie series obtained for Higuchi's[] procedure is: 1 3 : : : : The length of the curve i is defined as follows: L ( i) N i l1 1, 5, 9,, 97, 6, 10,, 98 3, 7, 11,, 99, 8, 1,, 100 N1 1 ( l i) ( ( l 1).) i * * N i i i (3) N 1 The ter represents the noralization factor for the curve length of subset tie series. Let N i i value over 'i ' sets of L (i). If L i i D () Li be the average () then the curve is fractal with the diension D. Using the ethod of least squares a straight line is fitted. The axiu value of 'i' is eight. B. Galois Field A finite field or Galois field ust contain p eleents, where p is prie and is a positive integer[9]. The Galois field GF( ) contains eleents. GF( ) is an extension field of GF() = {0,1},with two eleents. Arithetic operations are perfored by taking the results odulo. Non-zero eleents of GF ( ) are generated by a priitive eleent α, where α is a root of the priitive F x x 1 1x f x f irreducible polynoial 1 0 f, over GF ( ). The non-zero eleents of GF ( ) can be f. 1 1 f f represented in powers of α. i.e., GF ( ) = {0, α, α,.... α -, α -1 =1}. Since F(α) = 0, 1 0 The reason for positive sign is that the operation of subtraction is regarded as the inverse of addition. Therefore an eleent of GF ( ) can also be represented as a polynoial in α with degree less than. i.e., 1 GF a... a a a GF, 0 i i 30 30

3 C. Hooorphic Filtering A hooorphic filter is siply a hooorphic syste having the property that the desired coponent in a signal passes unaltered through the syste, while the undesired coponent is reoved. An iportant aspect in the theory of hooorphic syste is that any hooorphic syste can be represented as a cascade of three systes. The first syste takes inputs cobined by convolution and transfors the into an additive cobination of corresponding outputs. The second syste is a conventional linear syste obeying the principle of superposition[8]. The third syste is the inverse of the first syste i.e., it transfors the signals cobined by addition back into signals cobined by convolution. Hooorphic systes for convolution obey the principle of superposition, which for conventional linear syste is given by Lxn L x1 n L x n Hooorphis property can be stated as al xn Laxn Hx 1n * x n Hx n * Hx n Hxn 1 III. SCHEMES OF ANALYSES A. Galois Hooorphic Fractal Approach We present below pictorially, the existing schee for hooorphic deconvolution and the proposed schee based on Galois fields. The proposed schee is tered as Galois Hooorphic Fractal Approach(GHFA). x(n) x'(n) y'(n) y(n) C *[ ] L[ ] C * -1 [ ] (a) (5) (6) Segentation Galois field Hooorphic filtering Evaluation of HFD (b) Fig.1 Schee of analysis (a) existing (b) proposed The schee presented in Fig.1 (a), is as in [8]. C *[ ], C * -1 [ ] are called the characteristic syste for convolution and inverse characteristic syste for convolution respectively and L represents a linear syste. Figure 1(b) depicts the proposed schee, where the input signal is first segented. Construction of its Galois field is the next step, followed by hooorphic filtering. The final stage is evaluation of HFD. B. Tie Doain Analysis Using Hfd The signal is filtered using a 10 th order edian filter. Its HFD is then evaluated by segenting the signal into various windows. Noise reduction algoriths ai to reduce noise while attepting to preserve the inforation content of the signal. Median filter is used here, as it is a superior noise reducing filter especially in the reoval of isolated noise spikes [1]. The selected database consists of a ale and feale speaker both aged 31. Two eotion states, anger, happiness and a neutral state are chosen. Three texts coon to the speakers are identified and used for analysis. A collective of 1 utterances are used[]

4 IV. RESULTS AND DISCUSSION Galois field with 6 eleents viz GF( 6 ) is chosen for study. The priitive irreducible polynoial obtained for a specific segent of 6 the signal is x x x 1 F x is used to generate the Galois field eleents, corresponding to the appropriate segent. F. The result is taken as input to hooorphic filtering. The filtered result is the input to HFD. The value obtained for Text 1 in neutral ood is 1.966, for Text in angry ood is.0 and for Text 3 in angry ood is.00. As hooorphic filtering involves convolution and deconvolution, Galois Field which is capable of producing results with nil round off error is a sensible application. In the second schee of analysis, fractal diension for various segents ranges between 1 and 1.9. It is inferred fro the analyses that voice of ale is coarser in happy and angry oods than in neutral ood, which is the reverse for feales. The values obtained in the forer phase of analysis for a chosen fragent of the signal reseble the axiu values of HFD obtained in the latter schee of analysis. Thus Fractal diension dearcates ajor peaks. Higuchi's Fractal Diension of ale in happy ood is greater than in neutral ood. At the sae tie for feale the value is less than in neutral ood. In case of angry ood, the Higuchi's Fractal Diension of ale is uch greater than in neutral ood. But for feale, when copared with angry ood, Higuchi's Fractal Diension is higher for happy ood. In both ale and feale Higuchi's Fractal Diension of neutral ood is less in coparison to the other oods, naely angry and happiness. It is also interesting to note that for ale HFD is higher in angry ood and for feale it is higher in happy ood (Table 1). This deonstrates that HFD is capable of differentiating eotions, which is a special case of speech. The original speech wavefor for a text spoken by ale in happy ood and the corresponding HFD values in various segents are presented in Figure., Figure.3 respectively. Text 3 Text Text 1 Table 1 Higuchi's Fractal Diension(HFD) Values* Male Feale Happy ± ± 0.38 Neutral ± ± 0.0 Happy 1.19 ± ± Angry 1.19 ± ± 0. Neutral ± ± Happy ± ± Neutral 1.11 ± ± 0.0 *expressed as ean ± standard deviation Fig. Actual wavefor for a text spoken by ale in happy ood Fig.3 Higuchi's Fractal Diension values of various segents for a text spoken by ale in happy ood V. CONCLUSIONS A novel approach based on finite field naely Galois field is eployed in the fractal analysis of eotion recognition. Speech signals are finite by nature, which justifies the choice of Galois field. As the operations of addition and ultiplication are based on 'odulo' in finite field, the input of Galois field eleents to HFD is routed through hooorphic filtering. The process of constructing fast Fourier transfor in hooorphic filtering resolves this issue. Value of in Galois field construction ay vary between 1 and 16. The results fro GHFA suggest that it is a replica of the doinant characteristic of the signal considered. Signals of eotions carry erratic ups and downs. These are to be filtered properly for effective segentation. Median filter is used in the second schee of analysis, which is a superior noise reducing filter. The value of fractal diension varies for each segent. It

5 reflects on the coplexity of the signal and also on the ulti fractal nature of speech signal, in particular on signals of eotions. To copare the results obtained in the first phase of analysis, window size has been chosen as 6. This ay be varied, which ight yield different fractal diension value. However this would not uch affect the inferences of the work. Future work ay be aied at using different values of in the Galois field construction to excavate ore intrinsic details of eotions. REFERENCES [1] Al-Akaidi, M. (00), Fractal speech processing, Cabridge University Press [] Burkhardt, F., Paeschke, A., Rolfes, M., Sendleier, W. F., & Weiss, B. A database of Geran eotional speech, in Interspeech, 5, 005, Septeber, [3] Ezeiza, A., de Ipiña, K. L., Hernández, C., et.al., Enhancing the feature extraction process for autoatic speech recognition with fractal diensions, Cognitive Coputation, 5(), (013), [] Higuchi, T. Approach to an irregular tie series on the basis of the fractal theory, Physica D: Nonlinear Phenoena, 31(), (1988),77-83 [5] López-de-Ipiña, K., Alonso, J. B., Travieso, C. M., et.al., On the selection of non-invasive ethods based on speech analysis oriented to autoatic Alzheier disease diagnosis, Sensors, 13(5), (013), [6] V. A. Eotion recognition in speech signal: experiental study, developent, and application, Studies, 3, (000): [7] Pickover, C. A., and Khorasani, A. L. Fractal characterization of speech wavefor graphs, Coputers and graphics, 10 (1), (1986), [8] Rabiner, L. R., and Schafer R.W. (1978), Digital processing of speech signals. Prentice Hall [9] Yeh, C. S., Reed, I. S., and Truong, T. K. Systolic ultipliers for finite fields GF ( ). IEEE Transactions on Coputers, 100(), 198,