CCD Report Radial Basis Function Modeling of CARS Data

Transcription

1 CCD Report Radial Basis Function Modeling of CARS Data March 28, 2011 Doyle Knight Center for Computational Design Dept Mechanical and Aerospace Engineering Rutgers University 98 Brett Road Piscataway, NJ

2 RADIAL BASIS FUNCTION MODELING OF CARS DATA REPORT CCD Doyle Knight Center for Computational Design Dept Mechanical and Aerospace Engineering Rutgers University, New Brunswick, NJ USA March 28, Description The Biomaterials Group of the Polymers Division of the National Institute of Standards and Technology (NIST) is generating Coherent Anti-Stokes Raman Spectroscopy (CARS) data for cells. The dataset may be organized as (1) A(x i, y j ; k l ) where A is the CARS spectral measurement, (x i, y j ) is the location of each discrete pixel (i = 1, m; j = 1, n), and k l is the discrete set of wavenumbers (l = 1, s). A Reduced Order Model (ROM) is generated for the CARS dataset using one or more Radial Basis Functions (RBFs) defined for each wavenumber k l as [ ( ) 2 ( ) ] 2 x xo (k) y yo (k) (2) R(x, y; k) = a(k) + b(k) exp x r (k) y r (k) The six parameters a(k), b(k), x o (k), y o (k), x r (k) and y r (k) are determined by minimizing the square error E defined by (3) E(k) = [R(x i, y j ; k l ) A(x i, y j ; k l )] 2 i j using a Genetic Algorithm [1]. The ROM results in a significant reduction in the size of the dataset. The BMSC Day 4 cell data provided to Rutgers on 26 February 2011 encompasses pixels for 527 wavenumbers for a total size of 3.4 M data values for each cell. The ROM encompasses six parameters for 527 wavenumbers for a total size of 3.1 K data values for each cell, a reduction in total size of more than one thousand. 2. Results The ROM was applied to the five datasets from the BMSC Day 4 cell data provided to Rutgers on 26 February In the following figures, the results of the ROM for all wavenumbers are shown adjacent to the CARS data for a single wavenumber for each of the five datasets. In each datset, two RBFs are used. For 1

3 2 CCD REPORT simplicity, the experimental data is scaled to the unit interval 1, and any negative spectral values are set to zero. The ROM presents the magnitude b(k) as a function of the pixel number in the x and y directions corresponding to x o and y o in Eq (2). The magnitude of each RBF is plotted as a sphere above the corresponding pixel location of the centroid of the RBF for each wavenumber. The height and size of the sphere is proportional to the magnitude b(k), and therefore the RBFs for those wavenumbers where b(k) is negligibly small do not appear (or appear as a small dot) F44-29-z1-CARS DetrendedMVA.txt. In Fig. 1(a), the RBFs are stacked on effectively the same location, implying that there is a single distinct feature in the CARS spectra for all wavenumbers. Fig. 1(b) displays the CARS experimental data at k = 2918 cm 1 which indicates a single physical feature typical of this dataset F44-30-z1-CARS DetrendedMVA.txt. In Fig. 2(a), the RBFs are stacked about different locations, implying more than a single physical feature in the CARS spectra. Fig. 2(b) displays the CARS experimental data at k = 2918 cm 1 which indicates two separate physical features typical of this dataset F44-31-z1-CARS DetrendedMVA.txt. In Fig. 3(a), the RBFs are stacked on effectively the same location, implying that there is a single distinct feature in the CARS spectra for all wavenumbers. Fig. 3(b) displays the CARS experimental data at k = 2918 cm 1 which indicates a single physical feature typical of this dataset F44-32-z1-CARS DetrendedMVA.txt. In Fig. 4(a), the RBFs are stacked about two distinct locations, implying more than a single physical feature in the CARS spectra. Fig. 4(b) displays the CARS experimental data at k = 2918 cm 1 which indicates two separate physical features typical of this dataset F44-33-z1-CARS DetrendedMVA.txt. In Fig. 5(a), the RBFs are stacked about two distinct locations, implying more than a single physical feature in the CARS spectra. Fig. 5(b) displays the CARS experimental data at k = 2918 cm 1 which indicates two separate physical features typical of this dataset. 3. Conclusion The Reduced Order Model (ROM) comprised of multiple Radial Basis Functions (RBFs) is capable of identifying physical features in the CARS spectra. The ROM is generated automatically using a Genetic Algorithm. 1 The scaling is performed for the entire spectrum and entire set of pixels. 2 There is a second peak near the edge of the field of view which appears to be anomalous.

4 CCD REPORT References 1. K. Rasheed and A. Gelsey, Adaptation of Genetic Algorithms for Continuous Design Space Search, Fourth International Conference on Artificial Intelligence in Design: Evolutionary Systems in Design Workshop, 1996.

5 4 CCD REPORT Figure 1. F44-29-z1-CARS DetrendedMVA.txt Figure 2. F44-30-z1-CARS DetrendedMVA.txt

6 CCD REPORT Figure 3. F44-31-z1-CARS DetrendedMVA.txt Figure 4. F44-32-z1-CARS DetrendedMVA.txt

7 6 CCD REPORT Figure 5. F44-33-z1-CARS DetrendedMVA.txt