The Antireflective algebra and applications M. Donatelli Universit` - PowerPoint PPT Presentation

The Antireflective algebra and applications M. Donatelli Universit` a dell’Insubria Collaborators: A. Aric` o, J. Nagy, and S. Serra-Capizzano

Outline 1 The model problem (signal deconvolution) 2 Antireflective Boundary Conditions The AR algebra The spectral decomposition 3 Regularization by filtering 4 Numerical results M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 2 / 27

The model problem (signal deconvolution) Outline 1 The model problem (signal deconvolution) 2 Antireflective Boundary Conditions The AR algebra The spectral decomposition 3 Regularization by filtering 4 Numerical results M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 3 / 27

The model problem (signal deconvolution) The model problem • Problem: to approximate f : R → R from a blurred g : I → R � g ( x ) = k ( x − y ) f ( y ) d y , x ∈ I ⊂ R , I the point spread function (PSF) k has compact support. • Discretizing the integral by a rectangular quadrature rule and imposing boundary conditions: A f = g + noise . • The structure of A depends on k and the imposed boundary conditions. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 4 / 27

The model problem (signal deconvolution) Boundary conditions Signal Zero Dirichlet Periodic Reflective Anti−Reflective M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 5 / 27

The model problem (signal deconvolution) Structure of the coefficient matrix A Type Generic PSF Symmetric PSF Zero Dirichlet Toeplitz Toeplitz Periodic Circulant Circulant Reflective Toeplitz + Hankel Cosine Antireflective Toeplitz + Hankel Sine + . . . = + rank 2 Antireflective M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 6 / 27

Antireflective Boundary Conditions Outline 1 The model problem (signal deconvolution) 2 Antireflective Boundary Conditions The AR algebra The spectral decomposition 3 Regularization by filtering 4 Numerical results M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 7 / 27

Antireflective Boundary Conditions Definition of antireflective BCs • The 1D antireflection is obtained by f 1 − j = 2 f 1 − f j +1 f n + j = 2 f n − f n − j [Serra-Capizzano, SISC. ’03] • In the multidimensional case we perform an antireflection with respect to every edge = ⇒ Tensor structure in the multidimensional case. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 8 / 27

Antireflective Boundary Conditions Approximation property The reflective BCs assure the continuity at the boundary, while the antireflective BCs assure also the continuity of the first derivative. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 9 / 27

Antireflective Boundary Conditions Structural properties • A = Toeplitz + Hankel + rank 2. • Matrix vector product in O ( n log( n )) ops. Symmetric PSF • S ∈ R ( n − 2) × ( n − 2) diagonalizable by discrete sine transforms (DST)  1  ∗ ∗    . .  . . A =   . S .     ∗ ∗   1 M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 10 / 27

Antireflective Boundary Conditions The AR algebra The AR algebra With h cosine real-valued polynomial of degree at most n-3   h (0)  , AR n ( h ) = v n − 2 ( h ) τ n − 2 ( h ) J v n − 2 ( h )  h (0) where J is the flip matrix and • τ n − 2 ( h ) = Q diag ( h ( x )) Q , with Q being the DST and x = [ j π n − 1 ] n − 2 j =1 • v n − 2 ( h ) = τ n − 2 ( φ ( h )) e 1 , with [ φ ( h )]( x ) = h ( x ) − h (0) 2 cos( x ) − 2 . AR n = { A ∈ R n × n | A = AR n ( h ) } M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 11 / 27

Antireflective Boundary Conditions The AR algebra Properties of the AR n algebra Computational properties: • α AR n ( h 1 ) + β AR n ( h 2 ) = AR n ( α h 1 + β h 2 ), • AR n ( h 1 ) AR n ( h 2 ) = AR n ( h 1 h 2 ), Diagonalization • AR n is commutative, since h = h 1 h 2 ≡ h 2 h 1 , • the elements of AR n are diagonalizable and have a common set of eigenvectors. • not all matrices in AR n are normal. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 12 / 27

Antireflective Boundary Conditions The spectral decomposition AR n ( · ) Jordan Canonical Form Theorem Let h be a cosine real-valued polynomial of degree at most n-3. Then x )) T − 1 AR n ( h ) = T n diag ( h (ˆ n , x = [0 , x T , 0] T , x = [ j π n − 1 ] n − 2 where ˆ j =1 and 1 − ˜ x ) , ˜ x x � � T n = π , sin(˜ x ) , . . . , sin(( n − 2)˜ , π x = [0 , x T , π ] T . with ˜ M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 13 / 27

Antireflective Boundary Conditions The spectral decomposition Computational issues • Inverse antireflective transform T − 1 has a structure analogous to T n . n • The matrix vector product with T n and T − 1 can be computed in n O ( n log( n )), but they are not unitary. • The eigenvalues are mainly obtained by DST. • h (0) with multiplicity 2 • DST of the first column of τ n − 2 ( h ) M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 14 / 27

Regularization by filtering Outline 1 The model problem (signal deconvolution) 2 Antireflective Boundary Conditions The AR algebra The spectral decomposition 3 Regularization by filtering 4 Numerical results M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 15 / 27

Regularization by filtering Antireflective BCs and AR algebra If the PSF is symmetric, imposing antireflective BCs the matrix A belongs to AR . A possible problem The AR algebra is not closed with respect to transposition. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 16 / 27

Regularization by filtering Spectral properties • Large eigenvalues are associated to lower frequencies. • h (0) is the largest eigenvalue and the corresponding eigenvector is the sampling of a linear function. • Hanke et al. in [SISC ’08] firstly compute the components of the solution related to the two linear eigenvectors and then regularize the inner part that is diagonalized by DST. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 17 / 27

Regularization by filtering Regularization by filtering • A = T n D n T − 1 where d = h (ˆ x ) and n  t T  ˜ 1 . T − 1 � � . T n = t 1 · · · t n , D n = diag( d ) , =   . n   ˜ t T n • A spectral filter solution is given by n t T ˜ i g � f reg = φ i t i , (1) d i i =1 where g is the observed image and φ i are the filter factors. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 18 / 27

Regularization by filtering Filter factors • Truncated spectral value decomposition (TSVD) � 1 if d i ≥ δ φ tsvd = i 0 if d i < δ • Tikhonov regularization d 2 φ tik i = i + α , α > 0 , i d 2 • Imposing φ 1 = φ n = 1, the solution f reg is exactly that obtained by the homogeneous antireflective BCs in [Hanke et al. SISC ’08]. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 19 / 27

Regularization by filtering Reblurring Filtering with the Tikhonov filter φ tik is equivalent to solve i ( A 2 + α I ) f reg = A g • This is the reblurring approach where for a symmetric PSF A T is replace by A itself [D. and Serra-Capizzano, IP ’05]. • In the general case (nonsymmetric PSF), the reblurring replace the transposition with the correlation. • Reblurring is equivalent to regularize the continuous problem and then to discretize imposing the boundary conditions. M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 20 / 27

Numerical results Outline 1 The model problem (signal deconvolution) 2 Antireflective Boundary Conditions The AR algebra The spectral decomposition 3 Regularization by filtering 4 Numerical results M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 21 / 27

Numerical results Tikhonov regularization • Gaussian blur • 1% of white Gaussian noise True image Observed image M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 22 / 27

Numerical results Restored images. Reflective Antireflective M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 23 / 27

Numerical results Best restoration errors Relative restoration error defined as � ˆ f − f � 2 / � f � 2 , where ˆ f is the computed approximation of the true image f . noise Reflective Antireflective 10% 0.1284 0.1261 1% 0.1188 0.1034 0 . 1% 0.1186 0.0989 M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 24 / 27

Numerical results 1D Example (Tikhonov with Laplacian) True signal Observed signal (noise = 0.01) 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Restored signals Relative restoration errors 1.2 true reflective 0 10 reflective antireflective 1 antireflective 0.8 −0.2 10 0.6 −0.4 10 0.4 0.2 −0.6 10 0 −0.8 10 −0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −4 −3 −2 −1 0 1 10 10 10 10 10 10 M. Donatelli (Universit` a dell’Insubria) The Antireflective algebra 25 / 27