FPGA-Based Hardware Implementation of a Stable Inverse Source Problem Algorithm in a Non-Homogeneous Circular Region

Author:

Oliveros-Oliveros José Jacobo1ORCID,Conde-Sánchez José Rubén1ORCID,Hernández-Gracidas Carlos Arturo2ORCID,Morín-Castillo María Monserrat3ORCID,Conde-Mones José Julio1ORCID

Affiliation:

1. Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 sur, Colonia San Manuel, Edificio FM1-101B, Ciudad Universitaria, Puebla C.P. 72570, Mexico

2. CONAHCYT-BUAP, Facultad de Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Edificio FM1-101B, Ciudad Universitaria, Puebla C.P. 72570, Mexico

3. Facultad de Ciencias de la Electrónica, Benemérita Universidad Autónoma de Puebla, Avenida San Claudio y 18 Sur, Colonia San Manuel, Edificio FCE1, Ciudad Universitaria, Puebla C.P. 72570, Mexico

Abstract

Objective: This work presents an implementation of a stable algorithm that recovers sources located at the boundary separating two homogeneous media in field-programmable gate arrays. Two loop unrolling architectures were developed and analyzed for this purpose. This inverse source problem is ill-posed due to numerical instability, i.e., small errors in the measurement can produce significant changes in the source location. Methodology: To handle the numerical instability when recovering these sources, the Tikhonov regularization method in combination with the Fourier series truncation method are applied in the stable algorithm. This stable algorithm is implemented in two different architectures developed in this work: The first architecture (Mode 1) allows for different operating speeds, which is an advantage depending on whether we work with fast or slow signals. The second one (Mode 2) reduces resource consumption by exploiting the characteristics of the source identification algorithm, which is an advantage for multichannel problems such as inverse electrocardiography or electroencephalography. Results: The architectures were tested on four devices of the 7 Series of Xilinx: Spartan-7 xc7s100fgga484, Virtex-7 xc7v585tffg1157, Kintex-7 xc7k70tfbg484, and Artix-7 xc7a35tcpg236. The two hardware implementations of the stable algorithm were validated using synthetic examples implemented in MATLAB, which shows the advantages of each architecture. Contributions: We developed two efficient architectures based on a loop unrolling design for source identification problems. These are effective strategies to divide and assign tasks to the configurable hardware, and they appear as an appropriate technique for implementing the algorithm. The first one is simple and allows for different operating speeds. The second one uses a control system based on multiplexors that reduce resource consumption and complexity of the design and can be used for multichannel problems. From the numerical test, we found the regularization parameters. The synthetic examples developed here can be considered for similar problems and can be extended to concentric spheres.

Funder

National Council of Science and Technology in Mexico

Publisher

MDPI AG

Reference23 articles.

1. (2023, April 10). Xilinx, 7 Series FPGAs Data Sheet: Overview, DS180 (v2.6.1), 8 September 2020. Available online: https://docs.xilinx.com/v/u/en-US/ds180_7Series_Overview.

2. Morín-Castillo, M.M., Netzahualcoyotl-Bautista, C., Conde-Mones, J.J., Oliveros-Oliveros, J.J., and Santillán-Guzmán, A. (2019). Stable identification of sources associated with epileptic focus on the cerebral cortex. Rev. Mex. Ing. Biomed., 40.

3. Morín-Castillo, M.M., Arriaga-Hernández, J., Cuevas-Otahola, B., and Oliveros-Oliveros, J.J. (2022). Analysis of Dipolar Sources in the Solution of the Electroencephalographic Inverse Problem. Mathematics, 10.

4. Gockenbach, M.S. (2016). Linear Inverse Problems and Tikhonov Regularization, The Mathematical Association of America. The Carus Mathematical Monographs, 32.

5. Kirsch, A. (2011). An Introduction to the Mathematical Theory of Inverse Problems, Springer. [2nd ed.].

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