Numerical Linear Algebra and. Image Restoration

Transcription

1 Numerical Linear Algebra and Image Restoration Maui High Performance Computing Center Wednesday, October 8, 2003 James G. Nagy Emory University Atlanta, GA Thanks to: AFOSR, Dave Tyler, Stuart Jefferies, Sudhaker Prasad NSF (MPS/Computational Math) Emory University Julie Kamm (Raytheon), Lisa Perrone (Emory)

2 Basic Problem Linear system of equations b = Ax + e where A, b are known A is large, structured, ill-conditioned Goal: Compute an approximation of x Example: Image restoration b = observed image A = matrix defined by point spread function (PSF) e = noise x = true image

3 Computational Considerations Ill-conditioned A implies is corrupted with noise. ˆx = A 1 b = x + A 1 e Stabilize numerical scheme using regularization. Examples of regularization: 1. Filtering Pseudo inverse, Wiener, Tikhonov, Iterative Richardson-Lucy, EM, Conjugate gradients,...

4 Computational Considerations Efficient implementation of filters depends on structure of A. We consider Circulant Toeplitz Toeplitz+Hankel Kronecker product Iterative methods can be accelerated using filters as preconditioners.

5 Course Aims Wednesday: Background, notation, tools. Thursday: Kronecker products in image restoration. Friday: Spatially variant blurs and preconditioning. Wednesday 1. Tools (matrix decompositions) for building filters. 2. Some filtering methods. 3. Structured matrices in image restoration. 4. Matrix decompositions for structured matrices.

6 Tools for Building Filters Singular value decomposition (SVD): Any matrix A can be written as A = UΣV T where U T U = I (orthogonal) V T V = I Σ =diag(σ 1, σ 2,..., σ N ), σ 1 σ 2 σ N 0 Note: For image restoration, σ 1 1, σ N 0 condition(a) = σ 1 σ N 1

7 Tools for Building Filters Singular value decomposition: In Matlab: >> [U, S, V] = svd(a); Computational cost for an n n matrix A FLOPS: O(n 3 ) Storage: O(n 2 ) Remark: Fast algorithms are possible for sparse matrices, but storage is difficult to reduce. In Matlab: >> [U, S, V] = svds(a);

8 Tools for Building Filters Spectral decomposition: Some matrices A can be written as A = V ΛV where V = complex conjugate transpose V V = I (unitary) Λ =diag(λ 1, λ 2,..., λ N ), λ i = eigenvalues, i-th column of V = v i = eigenvectors

9 Tools for Building Filters Spectral decomposition: In Matlab: >> [V, E] = eig(a); Computational cost for an n n matrix A FLOPS: O(n 3 ) Storage: O(n 2 ) Remark: Fast algorithms are possible for sparse matrices, but storage is difficult to reduce. In Matlab: >> [V, E] = eigs(a);

10 Filtering Methods Inverse solution using SVD: If A = UΣV T (or A = V ΛV ), then x = A 1 b = V Σ 1 U T b = [ v 1 v n ] 1/σ 1... u T 1. b = [ v 1 v n ] = n i=1 u T i b σ i v i u T 1 b/σ 1. u T nb/σ n 1/σ n u T n

11 Filtering Methods Inverse solution for noisy, ill-posed problems: If A = UΣV T, then ˆx = A 1 (b + e) = V Σ 1 U T (b + e) = n i=1 u T i (b + e) σ i v i = n i=1 u T i b σ i v i + n i=1 u T i e σ i v i = x + error

12 Filtering Methods Filtering Goal: Do not let error dominate computed solution. x filt = = V n i=1 φ i u T i b σ i v i φ 1 σ 1... φ n σ n U T b where the filter factors satisfy φ i { 1 if σi is large 0 if σ i is small

13 Filtering Methods Frequency space representation of filter: If U T = V T = Fourier transform, then x filt = V V T x filt = ˆx filt = φ 1 σ 1... φ 1 σ 1... φ 1 σ 1... φ n σ n φ n σ n φ n σ n U T b U T b ˆb is the frequency space representation of the filter.

14 Filtering Methods Some examples of filters: 1. Pseudo-inverse (truncated SVD) x tsvd = k i=1 u T i b σ i v i 2. Tikhonov x tik = n i=1 σi 2 u T σi 2 i b + v i α2 σ i 3. Wiener x wien = n i=1 δ i σ 2 i δ i σ 2 i + γ2 i u T i b σ i v i

15 Structured Matrices in Image Restoration Toeplitz Matrix A = a 0 a 1 a a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a a 2 a 1 a a

16 Structured Matrices in Image Restoration Circulant Matrix (Toeplitz + periodic) A = a 0 a 1 a 2 a 2 a 1 a 1 a 0 a 1 a 2 a 2 a 2 a 1 a 0 a 1 a 2 a 2 a 2 a 1 a 0 a 1 a 1 a 2 a 2 a 1 a 0 A = a 0 a 1 a a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a a 2 a 1 a a 2 a a a a 1 a

17 Structured Matrices in Image Restoration Hankel Matrix A = a 0 a 1 a 2 a 3 a 4 a 1 a 2 a 3 a 4 a 5 a 2 a 3 a 4 a 5 a 6 a 3 a 4 a 5 a 6 a 7 a 4 a 5 a 6 a 7 a 8 Toeplitz + Hankel (Toeplitz + reflection) A = a 0 a 1 a a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a a 2 a 1 a 0 + a 1 a a a a 2 a 1

18 Structured Matrices in Image Restoration Hankel Matrix A = a 0 a 1 a 2 a 3 a 4 a 1 a 2 a 3 a 4 a 5 a 2 a 3 a 4 a 5 a 6 a 3 a 4 a 5 a 6 a 7 a 4 a 5 a 6 a 7 a 8 Symmetric Toeplitz + Hankel A = a 0 a 1 a a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a 1 a 2 0 a 2 a 1 a 0 a a 2 a 1 a 0 + a 1 a a a a 2 a 1 symmetric about the center

19 Structured Matrices in Image Restoration Two-dimensional analogs: e.g., block Toeplitz with Toeplitz blocks (BTTB) A = A 0 A 1 A A 1 A 0 A 1 A 2 0 A 2 A 1 A 0 A 1 A 2 0 A 2 A 1 A 0 A A 2 A 1 A 0, each A i is Toeplitz We consider: BTTB, BCCB, BHHB, BTHB, BHTB B block T Toeplitz C circulant H Hankel

20 Structured Matrices in Image Restoration Kronecker Products A = C D = c 11 D c 12 D c 1n D c 21 D. c 22 D. c 2n D. c n1 D c n2 D c nn D Note: If C and D are n n matrices, then A is n 2 n 2 C and D could be structured (e.g., Toeplitz).

21 Structured Matrices in Image Restoration Summary: Problem: Solve b = Ax + e Solution: Compute a filtered solution: x filt = n i=1 φ ut i b σ i v i Difficulty: Need to compute either Singular value decomp.: A = UΣV T Spectral decomposition: A = V ΛV How to do this efficiently? Can structure of A be exploited?

22 Decompositions for Structured Matrices Circulant and BCCB matrices: Can use FFT to get spectral decomposition: where A = F ΛF F = unitary discrete Fourier transform matrix Λ1 = nfa, a = 1st column of A Computational cost using FFTs: FLOPs = O(n log n) Storage = O(n)

23 Decompositions for Structured Matrices Circulant and BCCB matrices: In Matlab, if a is the 1st column of circulant A: To compute eigenvalues: >> Eigs = fft(a); To solve Ax = b: >> x = ifft( fft(b)./ Eigs ); If f is a vector containing filter factors, φ i : >> x_filt = ifft( fft(b).* (f./ Eigs) );

24 Decompositions for Structured Matrices Symmetric Toeplitz + Hankel Matrices: Can use discrete cosine transform (DCT): where A = C T ΛC C = orthogonal discrete cosine transform matrix Λ1 = Ca./ c, a = 1st col of A, c = 1st col of C Computational cost using fast DCTs: FLOPs = O(n log n) Storage = O(n)

25 Decompositions for Structured Matrices Symmetric Toeplitz + Hankel Matrices: In Matlab, if a is the 1st column of A: To compute eigenvalues: >> Eigs = dct(a./ dct(eye(length(a),1)); To solve Ax = b: >> x = idct( dct(b)./ Eigs ); If f is a vector containing filter factors, φ i : >> x_filt = idct( dct(b).* (f./ Eigs) );

26 Decompositions for Structured Matrices Kronecker Products: A = C D Some simple properties: (C D) T = C T D T (C D) 1 = C 1 D 1 (C 1 D 1 )(C 2 D 2 ) = C 1 C 2 D 1 D 2 A useful property: Ax = b (C D)x = b DXC T = B where x = vec(x) and b = vec(b)

27 Decompositions for Structured Matrices Kronecker Products: A = C D Using these properties, can compute SVD efficiently: If C = U c Σ c V T c and D = U d Σ d V T d, then A = C D = (U c Σ c Vc T dσ d Vd T = (U c U d )(Σ c Σ d )(V c V d ) T = UΣV T = SVD of A Computational cost for N N = n 2 n 2 matrix: FLOPs = O(n 3 ) = O(N 3/2 ) > O(N log N) Storage = O(N)

28 Decompositions for Structured Matrices Kronecker Products: A = C D In Matlab, if F c = diagonal matrix of filter factors for C F d = diagonal matrix of filter factors for D then a filtered solution can be computed as: >> [Uc, Sc, Vc] = svd(c); >> [Ud, Sd, Vd] = svd(d); >> X_filt = Vd * (Sd \ Fd) * (Ud *B*Uc) * (Sc \ Fc) * Vd ;

29 Example Gaussian PSF: e α2 (x 2 +y 2) = e α2 x 2 e α2 y 2 Discretizing this, we get: A = C D where C and D are Toeplitz matrices. Simulated data: B = D X C T + Noise

30 Summary Problem: b = Ax + e Solution: Compute a filtered solution: x filt = n i=1 φ ut i b σ i v i Efficiency: Exploiting matrix structure, can use: A = V ΛV for circulant A = V ΛV for symmetric Toeplitz + Hankel A = UΣV T for Kronecker products

31 Preview for Thursday Determine which matrix structure is most appropriate. Depends on: Properties of PSF, and boundary conditions. Exploiting Kronecker product structure for nonseparable PSFs. Implementation details for realistic data.