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ECE602 BME I Linear Models of Biological System

ECE602 BME I Linear Models of Biological System. Examples of linear models of biological system Gauss elimination with partial row pivoting Jacobi iterative method. Linear Models of Biosystems. Algebraic reconstruction of a biomedical image

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ECE602 BME I Linear Models of Biological System

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  1. ECE602 BME I Linear Models of Biological System

  2. Examples of linear models of biological system • Gauss elimination with partial row pivoting • Jacobi iterative method

  3. Linear Models of Biosystems • Algebraic reconstruction of a biomedical image • A square grid is imposed on the image f(x,y) • fj : the constant value in the jth cell • N: the total number of cells (n2) • A ray is a “fat” line running through the (x, y) plane • pi: ray-sum measured with the ithray, i=1,2,…M • wij: weighting factor representing the contribution of the jth cell to pi

  4. Linear Models of Biosystems • Application of spectroscopy for biological solutions • Spectroscopy: study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter

  5. Linear Models of Biosystems • Application of spectroscopy for biological solutions • Quantitative determination of the concentration of biological molecules in solution Ai: the absorbance of a solution at wavelength i,(log10(I0/I) where I0: the intensity of radiation before passing through the sample; I: the intensity after passing through the sample : the extinction coefficient of all components k (from 1 to N) obtained at wavelength i. If the radiation can pass through easily, then the coefficient is low. C(k): the concentration of the component k

  6. Linear Models of Biosystems • References • Image reconstruction • A. Rosenfeld and A.C. Kak, Digital picture Processing, 2nd ed. New York: Academic Press, 1982. • Spectroscopy: • I. Tinoco, K. Sauer, J. C. Wang, and J. D. Puglini, Physical Chemistry: Principles and Applications in Biological Sciences. Upper Saddle River, NJ: Prentice Hall, 2002.

  7. Gaussian Elimination with Row Partial Pivoting Three Steps to solve A0x=b (A0:n x n) Step 1: Initialization Step 2: Partial row pivoting and forward elimination Step 3: Back substitution

  8. Gaussian Elimination with Row Partial Pivoting Step 1: Initialization A=[A0 b]; (A: n x (n+1))

  9. Step 2: Partial row pivoting and forward elimination For k=1 : n-1 pivot=|a(k,k)| (pivot element) p=k; (pivot row) For i=k+1: n If |a(i,k)|>pivot pivot=|a(i,k)| (updating the pivot element) p=i; (updating the pivot row) End End If p>k Interchange rows k and p End For i=k+1:n (Loop on rows) For j=n+1:-1:k (Loop on columns for a given row) a(i,j)=a(i,j)-a(k,j) * (a(i,k) / a(k,k)) End End End Gaussian Elimination with Row Partial Pivoting

  10. Gaussian Elimination with Row Partial Pivoting Step 3: Back substitution x(n)=a(n,n+1)/a(n,n); For i=n-1:-1:1, (Loop of rows, working backwards) End

  11. The Jacobi Iterative method Step 1: Choose Step 2: Do Until

  12. The Jacobi Iterative method Example:

  13. The Jacobi Iterative method Example:

  14. The Jacobi Iterative method The Jacobi method is only applied to diagonally dominant matrix

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