{"id":14095,"date":"2017-03-25T11:36:52","date_gmt":"2017-03-25T11:36:52","guid":{"rendered":"http:\/\/biomedpharmajournal.org\/?p=14095"},"modified":"2020-04-23T09:24:02","modified_gmt":"2020-04-23T09:24:02","slug":"diagnosing-sinusitis-using-fractional-b-spline-wavelet-with-near-infrared-spectroscopy","status":"publish","type":"post","link":"https:\/\/biomedpharmajournal.org\/staging\/vol10no1\/diagnosing-sinusitis-using-fractional-b-spline-wavelet-with-near-infrared-spectroscopy\/","title":{"rendered":"Diagnosing Sinusitis using Fractional B-Spline Wavelet With Near Infrared Spectroscopy"},"content":{"rendered":"

Introduction<\/strong><\/p>\n

Fractional calculus utilized in 1974 by Keith [1], which involves fractional integral and derivative.\u00a0 Many reserch works were in progress in fractional calculus concentrating in a fractional derivative.\u00a0\u00a0 This paper mainly focusses on using fractional wavelet transform to diagnose sinusitis.\u00a0 Mallat [2-3] paved a way to wavelet analysis [4] with mathematical derivation and definitions briefing about each wavelet which is quite helpful in the biomedical signal analysis [5].\u00a0 Fractional Wavelet Transform [6-7] gaining popularity in signal analysis as it combines the goodness of wavelet and SWT technique. Though Fractional Wavelet Transform [8-10] proved to be a powerful signal processing tool in 1990\u2019s itself, it is only after Unser who paved a way to the family of scaling functions, it is used for signal analysis and synthesis.\u00a0 It is used here to analyze the biomedical signal recorded from sinusitis patients and compared with healthy people with the help of FrSWT.<\/p>\n

Sinusitis [11] is a common condition due to inflammation of the lining inside the nasal cavity and the sinuses. This may be due to viral or bacterial infections and may continue for two or more weeks.\u00a0 Sinuses are cavities behind the cheeks and nasal bones, eye, and forehead. The American Academy of Otolaryngology [12] summarizes several symptoms of sinusitis with major and minor infections. As it affects 35% of the population annually, it requires proper diagnosis and treatment. Imaging techniques serve the purpose of diagnosing sinusitis at its extreme condition, but earlier diagnosis [13-14] done clinically utmost in several cases.\u00a0 This misleads sinusitis detection, as the symptoms were shared by quite a few diseases also.\u00a0 So, a compact, user-friendly, non-radiative, less expensive method is required.<\/p>\n

Near Infrared Sensor (NIR) can be used to diagnose sinusitis in a non-radiative manner. It provides high resolution spatially and temporally which may attract the user to proceed with this new technique. This paper briefs about the properties, construction, and Applications of Fractional Spline Wavelet Transform (FrSWT) and to detect sinusitis in an efficient manner using NIR sensor.<\/p>\n

Fractional Spline Wavelet Techniques<\/strong><\/p>\n

Insight to Fractional Spline Technique<\/strong><\/p>\n

A new wavelet technique based on scaling functions utilized to analyze signals.\u00a0 Wavelet techniques play a critical role in the signal analysis. Splines and wavelets are strongly related and expressed in analytical form. B-Splines stability makes it attractive for specialists.\u00a0 It requires very less degree of support, even the origin.\u00a0 The Recursive family of B-splines can be generated using factorization theory.<\/p>\n

The Polynomial splines, at first introduced by Schoenberg [15-16] in 1946 and later extended by Thierry Blu and Michael Unser in 2000 to fractional degrees \u03b1 > -1\/2. Polynomial spline is a piecewise function of order, n. Isolated discontinuities occur at the knots, joining points between polynomial segments. Fractional splines behave exactly like integer B-spline but offer extensive support for various applications. A polynomial of degree, n expressed as arithmetic relation of the scaling function.<\/p>\n

Moreover, orthogonal fractional spline wavelet [17] can be implemented as fractional differentiator for fractal signals and fractional Brownian motions. Two commonly used approaches in splines were Orthogonal, semi-orthogonal compactly supported spline wavelets and symmetric compactly supported wavelets developed by Cohen, Daubechies, and Feauveau not confining to spline space.<\/p>\n

Properties of Fractional B-Spline<\/strong><\/p>\n

Some of the attractive properties of fractional B-splines commonly shared with classical B- spline, discussed in brief.\u00a0 Fractional splines have a fractional order of approximations. For non-integer \u03b1, it could not make an equivalency between the order of approximation, L and the polynomial degree, n. For integer, \u03b1 > -1\/2 fractional B-splines provide multi-resolution analyses even though it cannot produce optimal wavelet bases for integers (-1\/2< \u03b1 <0).\u00a0 It also satisfies the two-scale relation and contributes optimal Riesz bases. Fractional splines used in variational interpolation problems even. It interpolates between polynomial degrees. It interpolates polynomial splines similar to factorial interpolation using gamma function (generalization property).\u00a0 Regularity property implies that they are \u03b1- H\u04e7lder Continuous. They decay in such a way that even their causal polynomials becomes compactly supported. They are deficit in positivity and compact support alone.<\/p>\n

Experimental Works and Methods<\/strong><\/p>\n

This section enables us to learn about the basic idea and construction of B-spline and fractional B-spline technique. The proposed methodology along with the database and the method of examination were presented in detail.<\/p>\n

Fractional Differential Operators<\/strong><\/p>\n

The Fourier transform of the discrete fractional differential operator, \u2206 \u03b3 <\/sup>\u00a0of order \u03b3 \u03b5 R+<\/sup> is given by the equation [18-20];<\/p>\n

\"Formula<\/p>\n

Here \u2018+\u2019 refers to the causal function.\u00a0 The inverse FT is<\/p>\n

Computed by applying generalized binomial expansion, the fractional forward finite difference operator (convolution operator) is<\/p>\n

\"Vol10No1_Diag_Kama_for1\"<\/p>\n

Fractional B-Spline<\/strong><\/p>\n

Fractional causal B-splines constructed by considering the analogy of classical B-splines by finite fractional differences of the symmetric and piece wise power functions which is a continuation of polynomial splines for all non-integers, \u03b1 > -1.\u00a0 Fractional B-splines expressed as,<\/p>\n

\"Vol10No1_Diag_Kama_for2\"<\/p>\n

The generalized fractional binomial coefficients:<\/p>\n

\"Vol10No1_Diag_Kama_for3\"<\/p>\n

Where \u0403(u), Euler\u2019s Gamma function defined as \u0403(u) = \u00a0xu-1<\/sup>e-x<\/sup>dx<\/p>\n

Symmetric fractional B-splines were used to know the inner products of fractional B-spline with degree, \u03b1 >-1 \u0392*<\/sub>\u03b1<\/sup> = \u03b2+<\/sub>(\u03b1-1)\/2<\/sup> * \u03b2–<\/sub>(\u03b1-1)\/2<\/sup><\/p>\n

Overview of Orthogonal Spline Wavelet Design and Estimation<\/strong><\/p>\n

Orthogonal Spline wavelet transform first developed by Battle and Lemarie in 1987, later impelemented by Mallat and \u00a0Daubechies illustrated compactly supported transform for signal analysis. Wavelet Transform decomposes the signal into two basis functions.\u00a0 Scaling function, \u03d5 (t) and Wavelet function, \u03c8(t) are the basis function estimated using scaling filter coefficient, g(n) and wavelet filter coefficient, h(n) respectively.\u00a0 The Wavelet Coefficient represents the details of the signal acts a high-pass filter and the scaling function representing the approximate signal like low-pass filter.\u00a0 Decomposition and reconstruction of these signals were done using tree algorithm.<\/p>\n

Designing Orthogonal Wavelet Transform requires scaling function estimate, \u03d5 (t) and the filter coefficients, h (n) and g (n).\u00a0 Design procedure can be initiated by selecting any one of the parameters. Battle and Lamerie started designing by selecting the scaling function, but the output suffers from non-compactly supported wavelet functions.\u00a0 Later, Daubechies designed the wavelet transform considering the filter coefficients, thereby constructed a compactly supported orthonormal bases wavelet.\u00a0 Pyramid algorithm used to implement the wavelet transform using low pass and high pass filter.<\/p>\n

Orthogonal Spline wavelet can be implemented [21] by successive decomposition of approximate signals and detail signal for a maximum number of wavelet scales.\u00a0 Implementation done by deciding the impulse response coefficient of the filter coefficients and then estimate the approximate and detail signal by downsampling and upsampling respectively leading to efficient and fast computation than Fourier transform.\u00a0 This method is quite efficient for analysis and synthesis of signals, especially for biomedical signals.<\/p>\n

Characteristics of Paranasal Sinus and Optical Properties of A Turbid Medium<\/strong><\/p>\n

Paranasal sinuses (PNS) comprises of narrow, air-filled cavities surrounding the nasal cavity.\u00a0 Four paired sinuses [22] were Frontal, Maxillary, Sphenoidal and Ethmoidal which lies between the eyebrows, inferior to eye in maxillary bone, at the center of the head near the optic nerve and between eyes and nose separating the brain from nasal cavity respectively.\u00a0 Nasal cavities and sinus cavities were connected by Ostia to allow air and mucus flow freely.\u00a0 When infection occurs, this passage is blocked causing disturbance to the flow which creates pain in the sinus cavities.\u00a0 The Nasal blockage may be due to viral or bacterial infection which suspends tiny particles of microorganisms, dust or allergens.\u00a0 The nasal fluid becomes turbid due to such particles, which prevents easy flow of air and mucus through the cavity.<\/p>\n

Near Infrared (NIR) [23] light [750-1100) nm offers high transparency to light and greater depth of penetration in the biological tissues (translucent).\u00a0 Biophotonics field offers an ultimate diagnostic feature in exploiting the tissue-light relationship based on reflection, absorption, scattering, fluorescence and diffuse reflectance.\u00a0 As per Beer Lambert\u2019s Law, light undergoes attenuation in relation to the property of the material.\u00a0 However, turbid medium (within most of the biological tissues) highly scatters the NIR light due to various molecular interactions and so non-linear attenuation occurs.\u00a0 Multiple scattering occurs as different molecular interactions, exhibit various wavelengths leading to multiple frequency signal output.\u00a0 Scattering [24] occurs when there is a difference in the refractive index with the material or within the material.\u00a0 Air-filled cavities exhibit less absorption and reduced scattering coefficient compared to nasal cartilage [25].<\/p>\n

Optical properties describing the photon transmission inside the turbid medium were absorption coefficient, \u03bca<\/sub>, scattering coefficient, \u03bcs<\/sub>, refractive index, nref<\/sub> and scattering anisotropy factor, g [26].\u00a0 Variation in the refractive index [27] produces scattering which can be best illustrated using Maxwell\u2019s equation.\u00a0 Photon transmission inside the tissues can be analyzed analytically using Radiative transfer equations (RTE) or modeled using Monte Carlo simulations (very expensive).\u00a0 When absorption is less than the scattering coefficient, then it allows photons to propagate for large tissue volume before getting absorbed.\u00a0 Turbid media scatters more than being absorbed validating the diffusion approximation to Boltzmann\u2019s transport equation in determining \u03bca<\/sub> and \u03bcs.<\/sub>\u00a0 Photon transport [28] in highly scattering medium reduces the independent variables involved in RTE leading to diffusion theory under the assumption of directional broadening.\u00a0 Transport equation reduces further when radiance becomes a function of isotropic fluence rate and directional flux in the case of a highly scattering medium.<\/p>\n

As turbid media undergoes multiple scattering leading to total internal reflection (TIR), it necessitates the measurement of a complex refractive index.\u00a0 The diffusion approximations for determining varying refractive index were discussed in many papers [29-31] with controversies to deal with spatially varying refractive index.\u00a0 To resolve discrepancies of Fresnel\u2019s equation for TIR, few researchers [32] work with angle dependent penetration.\u00a0 Detecting in-homogeneities in the turbid medium is quite important in biomedical optics. In optics [33], photon path bends in an inhomogeneous medium calculated by Bouguer\u2019s equation.\u00a0 The photon transport equation [34] re-derived for isotropic, variable refractive index with spatial resolution in a highly turbid medium.\u00a0 Two equations derived with trajectory photon path and the standard format in case of constant refractive index, scatter less and lossless medium.<\/p>\n

Database<\/strong><\/p>\n

Patients registered to undertake the examination, included in the study after getting their consent.\u00a0 Clinical findings, objective findings, and history of symptoms identified and given preference to undertake the study and went on immediate examination through radiographic technique.\u00a0 Fifty subjects included, along with the patients with history and objective findings.\u00a0 Data recorded in Sathyabama University under the supervision of a medical practitioner.\u00a0 Signals recorded for 20 sec using sigview software using the prototype hardware [35] developed earlier.\u00a0 Chronic patients and normal people were also included for the study.\u00a0 Signals recorded using NIR sensor is further processed using signal processing tool to evaluate the findings from the hardware with the radiographic results.\u00a0 Recording of signal sample is available in the link provided https:\/\/www.youtube.com\/watch?v=pIItyK1Hp0A&feature=youtu.be.<\/p>\n

Measurement Method<\/strong><\/p>\n

Signals recorded from the patients using hardware, radio graphic images and results along with the signal processing output processed and compared to diagnose sinusitis. Fractional B-Spline wavelet generates a smooth output for a piece wise continuous signal.\u00a0 Wavelet analysis proves to diagnose sinusitis accurately.<\/p>\n

Simulation<\/strong><\/p>\n

Results nd Discussion<\/strong><\/p>\n

Patient\u2019s signals recorded under the supervision of a medical practitioner in Sathyabama dental hospital.\u00a0 Patients with the history of sinusitis, Clinical symptoms, and objective findings included in the study along with the patient enrolled for the study.\u00a0 Healthy subjects also included for validating the study after getting the consent of the individual subjects.\u00a0 NIR sensor used to record signal from individuals for 20sec with the help of Sigview software and stored separately for normal and diseased.<\/p>\n

Signals recorded were processed using Fractional Spline Wavelet Transform in Matlab to analyze signals in piecewise fashion, and it is represented in time-fractional\u2013domain representation simultaneously.<\/p>\n

Previously, we tried with Least Square Support Vector Machine (LS_SVM) classification algorithm [35] and compared normal, and sinusitis signal with each and every received FFT value and deviation from the normal curve confirms the condition in the chest.\u00a0 Later, the same signals were analyzed [36] using Empirical Wavelet Transform (EWT) and Multiple Decomposition Empirical Wavelet Transform (MUSIC-EWT).\u00a0 These techniques accurately generate instantaneous frequencies from a non-stationary and non-linear signal even with poor SNR.\u00a0 Thus it proves to be optimum in case of real signals where noise interference and spurious signals generate within the human body.\u00a0 However, it lags strong theoretical proof which fails to explain the non-linear behavior of the algorithm.<\/p>\n

Then, we move on to Fractional Spline Wavelet Transform (FrSWT) to analyze the signal in a piecewise manner to smoothen the output at various levels.\u00a0 We can get more detailed information as the signals were considered fractional.\u00a0 It has strong theoretical derivation which best illustrates the algorithm.<\/p>\n\n\n\n
\"Figure<\/td>\n\n

Figure 1a: Original Signal (b) Sub-band signal (c)Fractional Spline Wavelet <\/strong>Transform output (d) Re-synthesized signal (e) Scaling and Wavelet Functions\u00a0<\/strong>of Fractional Spline Wavelet Transform output of a chronic sinusitis Patient<\/strong><\/p>\n

Click here to View figure\u00a0<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Using Fractional Spline Wavelet Transform, the recorded signals processed.\u00a0 The signal decomposes at four levels with an alpha = 2.4 and \u03c4 = 0.3 of ortho type.\u00a0 Signals separated into a low pass and high pass using FIR filter.\u00a0 The high pass filter represents the impulse filter coefficient of the wavelet function which provides approximate signal, and the low pass filter represents the scaling filter coefficient of the scaling function representing the more detailed information of the signal.\u00a0 Further, it is resynthesized, scaled and wavelet transform output obtained.<\/p>\n

The output produces multiple frequencies due to multiple scattering and TIR with varying refractive indices as the medium is highly turbid for acute and chronic sinusitis.\u00a0 Thus, the patient with sinusitis generates multi-frequency output as illustrated in fig.1 for a chronic patient.\u00a0\u00a0 Fig. 1(a) represents the original signal of a sinusitis patient.\u00a0 Fig. 1(b) represents the sub-band obtained for the signal taken for study.\u00a0 Fig.1(c) represents the Fractional Wavelet transformed output decomposed into four levels with an alpha and tau predefined.\u00a0 These values can be varied, but this value gives optimum result.\u00a0 Similarly, it is analyzed for several patients with acute or chronic condition in the sinus cavity to validate the result.\u00a0 Low pass band representing the detailed signal and the high pass band representing the approximate signal.\u00a0 When NIR light is transmitted through opacified sinus cavities using NIR sensor; the light gets reflected, absorbed, scattered and diffused.\u00a0 Due to water accumulation, there is a more chance of TIR and less absorption at NIR range.\u00a0 Multiple scattering offers diffused photon path well explained by transport equation discussed earlier and thereby generates multiple frequencies owing to total internal reflection of varying refractive index.\u00a0 Fig.1 (d) illustrates the de-synthesized signal from the low pass band, combining the features of both detail and approximation signal in to approximate signal with more pr\u00e9cised details by up sampling.\u00a0 Fig. 1(e) represents the scaling and the wavelet function obtained for the sinusitis signal.\u00a0 However, for a normal person, the output consists of one or very few frequencies with less high pass band.\u00a0 Thus, the normal person\u2019s sinus cavity (fully aerated) allows greater penetration of light with no loss of signal.\u00a0 Light penetrates through the air-filled sinus cavities whereas; it undergoes reflection, absorption and scattering when passed through opacified cavities.<\/p>\n

A comparison made with the normal and diseased signal using Fractional Spline Wavelet Transform along with the radiographic images taken consecutively while recording the signal with the consent of the patients.\u00a0 This avoids further discrepancies in analyzing the data obtained at various instants.\u00a0 Fig. 2 shows the comparison of normal and diseased FrSWT output with the radiographic images.<\/p>\n\n\n\n
\"Figure<\/td>\nFigure 2: Comparison of Fractional Spline Wavelet Transform output with\u00a0<\/strong>radiographic image for a Normal and Diseased (Chronic sinusitis) patient<\/strong><\/p>\n

 <\/p>\n

Click here to View figure\u00a0<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

Likewise, the comparison made with the previous output obtained using Empirical Wavelet Transform (EWT) with the FrSWT as shown in Fig.3.\u00a0 This output represents the radiographic image with Time-Frequency representation of EWT and the Time- Fourier domain representation of Fractional Spline Wavelet Transform.\u00a0 In both EWT and FrSWT, sinusitis patients show multiple frequencies and the normal person shows very few variations in frequency.<\/p>\n\n\n\n
\"Figure<\/td>\n\n

Figure 3: Comparison of radiographic image (a &d), TFR representation of EWT\u00a0<\/strong>(b &e) and Time- Fourier domain representation of Fractional Spline Wavelet\u00a0<\/strong>Transform (c & f) of a diseased and a normal person respectively<\/strong><\/p>\n

Click here to View figure\u00a0<\/a><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

The Orthogonal Spline wavelet transforms analyze and synthesis signal at its best.\u00a0 The output shows that the diseased signal multi-scatters with multiple approximate signals in the high pass band and less low pass band to represent the detailed signal. However in normal case, the LPF represents more details about the original signal with a resolution of 22j<\/sup> whereas it is 2j <\/sup>for approximate signals. J is the no. of maximum wavelet scales.\u00a0 Thus this transforms best suits for analyzing biomedical signals.<\/p>\n

Conclusions<\/strong><\/p>\n

Near Infrared Sensor used successfully to diagnose sinusitis with the help of prototype hardware.\u00a0 Fractional Spline Wavelet Transform used to analyze the signal in fractional order to improve the resolution of the output signal. Multiple frequencies generated, as the signals scatter due to various refractive index of fluid inside the sinus cavity.\u00a0 Sinusitis diagnosed easily by the amount of fluid inside the cavity, generating multiple frequencies.\u00a0 For aerated normal person, light penetrates through the cavity without attenuation or scattering leading to single frequency output.\u00a0 Thus, an efficient algorithm has been developed to identify the turbidity of the sinus cavity and to diagnose sinusitis without undergoing radiation imaging techniques.\u00a0 This paper compares the output obtained from the hardware with the radiographic image and the Empirical and Fractional Spline Wavelet Transforms in an efficient, cost-effective and non-radiative manner.\u00a0 The diagnostic procedure of identifying sinusitis, simplified at an early stage.<\/p>\n

Acknowledgment<\/strong><\/p>\n

The authors wish to thank the management of Sathyabama University for rendering their support towards the work in facilitating testing on patients and data collection in Sathyabama Dental hospital and our sincere thanks goes to the patients who volunteered for this study.<\/p>\n

References<\/strong><\/p>\n

    \n
  1. Keith B. Oldham Jerome Spanier, \u201cThe fractional calculus, Theory and applications of Differentiation and integration to arbitrary order\u201d Academic Press Inc. London 1974.<\/li>\n
  2. Stephane Mallat, \u201cA Theory for Multiresolution Signal Decompositon: The Wavelet representation\u201d IEEE Transact. On Pattern Analysis and Machine Intelligence, Col.11, No.7, pp.674-693, Jul. 1989.<\/li>\n
  3. Stephane Mallat, “A Wavelet tour of Signal Processing”, Academic Press,San Diego, 1998.<\/li>\n
  4. Chui, C.K. An Introduction to Wavelets. San Diego: Academic Press, 1992.<\/li>\n
  5. Akay, M. Detection and Estimation of Biomedical signals, San Diego: Academic press, 1195.<\/li>\n
  6. K. Chui and J. Wang, \u201cOn Compactly supported spline wavelets and duality priniciple, Trans. Amer. Math. Soc. 330(992), pp. 93-916.<\/li>\n
  7. J. Lai, \u201cOn Computation of Battle- Lemarie\u2019s wavelets, Math. Comp. 63 (1994), pp. 689-699.<\/li>\n
  8. Unser, Splines: a perfect (t for signal\/ Image Processing, IEEE Signal Process. Magazine 16 (6), pp. 22-38, 1999.
    \n    CrossRef<\/a><\/li>\n
  9. Unser, T. Blu, Fractional Splines and wavelets, SIAM Rev. 42(1), pp.43-67, 2000.
    \n    CrossRef<\/a><\/li>\n
  10. Unser, A.Aldroubi and M. Eden, A family of polynomial spline wavelet transforms, Signal Processing, 30, 141-162, 1993.
    \n    CrossRef<\/a><\/li>\n
  11. D. Lanza and D. D. Kennedy (1997), \u201cAdult rhinosinusitis defined\u201d, Otolaryngol.-Head Neck Surg., 117, S1\u20137.
    \n    CrossRef<\/a><\/li>\n
  12. Fokkens W, Lund V, Mullol J (2007), \u201cEuropean position paper on Rhinosinusitis and Nasal polyps, Rhinology\u201d, Vol. 45(supp.20):,1-139.<\/li>\n
  13. Hwang, A. Getz, M. Aronson et al (2010), “Acute sinusitis and rhinosinusitis in adults” (UpToDate, 2010), www.uptodate.com.<\/li>\n
  14. Jarvis D, Newson R, Lotvall J, Hastan D, et al, \u201cP: Asthma in adults and its association with chronic rhinosinusitis: the GA2LEN survey in Europe. Allergy\u201d, 67(1), pp. 91\u201398, 2012.
    \n    CrossRef<\/a><\/li>\n
  15. J. Schoenberg, Contribution to the problem of approximation of equidistant data byanalytic functions, Quart. Appl. Math. 4 (1946), 45-99, 112-141.
    \n    CrossRef<\/a><\/li>\n
  16. J. Schoenberg, On spline functions<\/em>, In \u201dInequalities\u201d, (O. Shisha ed.) (1967), 255-291.<\/li>\n
  17. Okkyung cho and Ming Jun-Lai, \u201cA Class of Compactly Supported Orthonormal B-Spline Wavelets\u201d, Splines and Wavelets: Athens 2005, pp. 123-151.<\/li>\n
  18. Blu and M. Unser, \u201c The fractional spline wavelet transform: Definition and implementation\u201d, in Proc.\u00a0 IEEE Int. Conf. Acoust., Speech, Signal process., Istanbul, Turkey, June 2000, Vol. I, pp. 512-515.<\/li>\n
  19. Blu and M. Unser. A complete family of scaling functions: The (\u03b1,\u03c4)- fractional splines.\u00a0 In Proc. Int. Conf. Acoustics, Speech, and Signal Processing, volume VI, pages 421-424, Hong Kong SAR, People\u2019s Republic of China, April 6-10, 2003.<\/li>\n
  20. Michael M. Unser, \u201c Fractional Splines Wavelet Bases and Applications, Jan 2004.<\/li>\n
  21. Metin Akay, \u201dwavelets in Biomedical Engineering\u201d, Annals of Biomedical Engineering, Vol. 23, pp.531-542, 1995.
    \n    CrossRef<\/a><\/li>\n
  22. Jing Huang, Hao Zhang, Tianqi Li, Huiying Lin, Katarina Svanberg, Sune Svanberg, \u201cAssessment of human sinus cavity air volume using tunable diode laser spectroscopy, with application to sinusitis diagnostics\u201d, J. Biophotonics 8, No.11-12, pp.985-992, 2015.
    \n    CrossRef<\/a><\/li>\n
  23. F.Jobsis (1977), \u201cNon-invasive infrared monitoring of cerebral and myocardial oxygen sufficiency and circulatory parameters\u201d, Science Vol. 198, 1264-67.
    \n    CrossRef<\/a><\/li>\n
  24. Brian W.Pogue, et al , \u201cCharacterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycle changes\u201d, J. Biomed. Optics, Vol.9 (3), pp. 541-552, 2004.
    \n    CrossRef<\/a><\/li>\n
  25. Usama Mahmood, Albert Cerussi, et al, \u201cNear Infrared Imaging of the sinuses: Priliminaray evaluation of a new technology for diagnosing maxillary sinusitis, Jounal of biomedical optics, 15(3), 036011, June 2010.<\/li>\n
  26. Lihong V. Wang and Steven L. Jacques, \u201cSource of error in calculation of optical diffuse reflectance from turbid media using diffusion theory\u201d, computer Methods and Programs in Biomedicine,61, pp. 163-170, 2000.<\/li>\n
  27. Huafeng Ding, Jun Q Lu, William A Wooden, Peter J Kragel and Xin- Hua Hu, \u201c Refractive indices of human skin tissues at eight wavelengths and estimated dispersion relations between 300 and 1600 nm\u201d, Phys. Med. Bio. 51, pp. 1479-1489, 2006.
    \n    CrossRef<\/a><\/li>\n
  28. Haskell; et al., “Boundary conditions for the diffusion equation in radiative transfer”.Journal of the Optical Society of America A.\u00a011\u00a0(10): pp. 2727\u20132741, 1994.
    \n    CrossRef<\/a><\/li>\n
  29. Khan and H. Jiang, \u201cA new diffusion approximation to the radiative transfer equation for scattering media with spatially varying refractive indices\u201d, J. Opt. A: Pure Appl. Opt. 55, pp.137-141, 2003.<\/li>\n
  30. M. Tualle and E. Tinet, \u201c Derivation of the radiative transfer equation for scattering media with a spatially varying refractive index\u201d, Opt. Commun. 228, pp.33-38, 2003.
    \n    CrossRef<\/a><\/li>\n
  31. L. Shendeleva, \u201cRadiative transfer in a turbid medium with a varying refractive index: comment, \u201c J. Opt. Soc. Am. A 21, pp.2464-2468, 2004.
    \n    CrossRef<\/a><\/li>\n
  32. R. Calhoun, H. Maeta, A. Combs, L. M Bali, and S. Bali, \u201cMeasurement of the refractive index of highly turbid media\u201d, Optics letters, Vol.35 (8), 1224-1226, 2010.
    \n    CrossRef<\/a><\/li>\n
  33. Born and E. Wolf, Principles of Optics, <\/em>4th ed. (Pergammon, 1970).<\/li>\n
  34. Malin Premaratne, Erosha Premaratne and Arthur James Lowery, \u201cThe Photon transport equation for turbid biological media with spatially varying isotropic refractive index\u201d, Opt. epress, Vol.13(2), pp.389-399, 2005.<\/li>\n
  35. Kamatchi, Dr.M. Sundararajan, \u201cEarlier diagnose of sinusitis in Frontal and Maxillary cavities using NIR radiations\u201d, Journal of Computational and Theroetical Nanoscience, Unpublished.<\/li>\n
  36. Kamatchi, Dr.M. Sundararajan, \u201cSinusitis detection with wavelet analysis using Near Infrared Spectroscopy\u201d, Proc. Of International conference on Engineering and Technology\u201916, Vol.5, pp.97-103, 2016.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Introduction Fractional calculus utilized in 1974 by Keith [1], which […]<\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[46],"tags":[],"class_list":["post-14095","post","type-post","status-publish","format-standard","hentry","category-vol10no1"],"_links":{"self":[{"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/posts\/14095","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/comments?post=14095"}],"version-history":[{"count":6,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/posts\/14095\/revisions"}],"predecessor-version":[{"id":32299,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/posts\/14095\/revisions\/32299"}],"wp:attachment":[{"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/media?parent=14095"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/categories?post=14095"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/biomedpharmajournal.org\/staging\/wp-json\/wp\/v2\/tags?post=14095"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}