1. By Mary Hudachek-Buswell

2. Overview

3. Atmospheric Turbulence Blur

4. 2-D Separable Gaussian Blur This point spread function represents atmospheric turbulence blur.

5. Toeplitz Matrix Blur Toeplitz matrix structure is a matrix with constants along the descending diagonal. These type matrices are spatially invariant and have zero boundary conditions when combined together.

6. Illustration of Spatially Invariant BLur Original image Invariant BlurVariant Blur

7. Illustration of Zero Boundary Conditions Surrounds the object with a black boundary, or zeros on the outside borders of the image

8. Minimize the 2-Norm of the Residual A is the column blur, B is the row blur, X is the restored image, and C is distorted image. A & B are overdetermined and positive definite. Overdetermined systems have more equations than unknowns and positive definite matrices have a unique quality where given any nonzero vector, z, the product

9. Least Squares of an Overdetermined System

10. Least Squares cont.

11. Necessary Properties The system of recurrence formulas that generates a unique sequence of iterates with such that With the property that at step n, the norm is minimized

12. Conjugate Gradient Algorithm Initial values for approximation, residual and gradient matrices Scalar to minimize the norm and determine step length Approximate restoration matrix Residual matrix Scalar to measure residual ratio and update gradient Update conjugate gradient search direction

13. Conjugate Gradient MATLAB C = g; ro = g; p = ro; x = zeros(m, n); for i = 1:maxiter alpha = trace (ro'*ro)/trace(p'*(A*p*B)); x = x + alpha * p; rn = ro - alpha * A * p * B; beta = trace (rn'*rn)/trace(ro'*ro); p = rn + beta * p; ro = rn; End The trace function computes the sum of the diagonal elements of the matrix

14. Comparison between CG & Inverse Methods

15. Visual Comparison of Iterations

16. Concluding Thoughts Direct Methods to find the restoration of a blurred image involve inverting matrices which costs O(n 3 ) Conjugate gradient iterations require transposing matrices, then multiplying by scalars and matrices which costs O(n 2 ), considerably less than direct methods The CG computations are faster and can provide better visual results.