Localizing Lung Sounds: Eigen Basis Decomposition for Localizing Sources Within a Circular Array of Sensors

Abstract

Lung disorders or injury can result in changes in the production of lung sounds both spectrally and regionally. Localizing these lung sounds can provide information to the extent and location of the disorder. Difference in arrival times at a set of sensors and triangulation were previously proposed for acoustic imaging of the chest. We propose two algorithms for acoustic imaging using a set of eigen basis functions of the Helmholtz wave equation. These algorithms remove the sensor location contribution from the multi sensor recordings using either an orthogonality property or a least squares based estimation after which a spatial minimum variance (MV) spectrum is applied to estimate the source locations. The use of these eigen basis functions allows possible extension to a lung sound model consisting of layered cylindrical media. Theoretical analysis of the relationship of resolution to frequency and noise power was derived and simulations verified the results obtained. Further, a Nyquist’s criteria for localizing sources within a circular array shows that the radius of region where sources can be localized is inversely proportional to the frequency of sound.The resolution analysis and modified Nyquist criteria can be used for determining the number of sensors required at a given noise level, for a required resolution, frequency range, and radius of region for which sources need to be localized.

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Notes

  1. 1.

    Here acoustic imaging refers to obtaining the location of all sound sources, and is different from x-ray imaging and CT scans which show the lungs in a visual format.

  2. 2.

    \({\boldsymbol{a}}(\theta_v,k) \triangleq \left[e^{ik{\boldsymbol{x}}_1 sin(\phi)},\ \ldots, \ e^{ik{\boldsymbol{x}}_Q sin(\phi)}\right]^T\) where ϕ is the DOA.

  3. 3.

    The discrete form of the orthogonality relationship for exponential functions applied in Eq. 19 is valid only if there is no aliasing. A discrete number of sensors sample the imping wavefront and is analogous to sampling the function e inθ at the angular positions of the sensors. For large n, a greater number of sensors spanning the circumference of a circle is required in order to avoid aliasing.

  4. 4.

    In this paper, the subscript n can refer to either noise or the mode, whenever ambiguity arises in the equations clarifications are provided in the description of these equations.

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Correspondence to S. M. A. Salehin.

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Salehin, S.M.A., Abhayapala, T.D. Localizing Lung Sounds: Eigen Basis Decomposition for Localizing Sources Within a Circular Array of Sensors. J Sign Process Syst 64, 205–221 (2011). https://doi.org/10.1007/s11265-009-0435-3

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Keywords

  • Localization
  • Lung sounds
  • Helmholtz equation
  • Basis decomposition
  • Cylindrical harmonics
  • Nyquist’s criteria
  • Resolution