Introduction to Inverse Problems (2 lectures)


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1 Introduction to Inverse Problems (2 lectures) Summary Direct and inverse problems Examples of direct (forward) problems Deterministic and statistical points of view Illposed and illconditioned problems An illustrative example: The deconvolution problem Truncated Fourier decomposition (TFD); Tikhonov regularization Generalized Tikhonov regularization; Bayesian perspective. Iterative optimization. IP, José Bioucas Dias, 2007, IST 1
2 Direct/Inverse problems Direct (forward) problem Causes Inverse problem Effects Example: Direct problem: the computation of the trajectories of bodies from the knowledge of the forces. Inverse problem: determination of the forces from the knowledge of the trajectories Newton solved the first direct/inverse problem: the determintion of the gravitation force from the Kepler laws describing the trajectories of planets 2
3 An example: a linear time invariant (LTI) system Direct problem: Fourier domain Inverse problem: Source of difficulties: is unbounded A perturbation on leads to a perturbation on given by high frequencies of the perturbation are amplified, degrading the estimate of f 3
4 Image deblurring Observation model in (linear) image restoration/reconstruction observed image noise Linear operator original image (e.g., blur, tomography, MRI,...) Goal: estimate f from g 4
5 Image deblurring via regularization original Blurred, 9x9 uniform restored 5
6 MRI example Hydrogen density 2D frequency samples (9.4%) 6
7 Compressed Sensing (sparse representation) Sparse vector f Random matrix Observed data y Compressed Sensing N=1000 M=
8 Classes of direct problems Deterministic observation mechanism Original data (image) Operator + Observed data (image) perturbation 8
9 Classes of direct problems (deterministic) Linear spaceinvariant imaging systems Blur (motion, outoffocus, Diffractionlimited imaging atmospheric) Near field acoustic holography Channel equalization Parameter identification Linear spacevariant imaging systems (first kind Fredholm equation) Xray tomography MR imaging Radar imaging Sonar imaging Inverse diffraction Inverse source Linear regression 9
10 Classes of direct problems Statistical observation mechanism Original data (image) Observed data (image) Ex: Linear/nonlinear observations in additive Gaussian noise + 10
11 Classes of direct problems (statistic) Linear/nonlinear observation driven by nonadditive noise Parameters of a distribution Random signal/ image Rayleigh noise in coherent imaging Poisson noise in photoelectric conversion SPET (single photon emission tomography) PET (positron emission tomography) Ex: Amplitude in a coherent imaging system (radar, ultrasound) Terrain reflectance Inphase/quadrature backscattered signal 11
12 Wellposed/illposed inverse problems [Hadamard, 1923] Definition: Let be a (possible nonlinear) operator The inverse problem of solving Hadamard sense if: is wellposed in the 1) A solution exists for any in the observed data space 2) The solution is unique 3) The inverse mapping is continuous An inverse problem that is not wellposed is termed illposed The operator A of an inverse well/illposed problem is termed well/illposed 12
13 Finite/Infinite dimensional linear operators Linear Operators: The linear inverse problem equivalently, is wellposed if 1) and 2) holds or, and If is finitedimensional, the corresponding inverse problem is wellposed iif either one of the properties 1) and 2) holds Example: In infinitedimensional spaces Consider A defined on, If a solution of exists, it is unique since However, there are elements not in Thus, A is illposed (point 1 of the Hadamard conditions does not hold) Stability is also lacking: Take Then, does not converge when 13
14 Illconditioned inverse problems lllposed lllconditioned Many wellposed inverse problems are illconditioned, in the sense that For linear operators (tight bound) 14
15 Example: Discrete deconvolution Cyclic convolution Matrix notation Nperiodic funtions A is cyclic Toeplitz 15
16 Example: Discrete deconvolution 16
17 Eigendecomposition of cyclic matrices (unitary) Eigenvector (Fourier) matrix Eigenvalue matrix (diagonal) is the DFT of at frequency 17
18 Example: Discrete deconvolution (geometric viewpoint) Action of A on f Note: 18
19 Example: Discrete deconvolution (inferring f) Assume that Then is invertible and Thus, assuming the direct model We have error 19
20 Example: cyclic convolution with a Gaussian kernel What went wrong? 20
21 Example: Discrete deconvolution (estimation error) Size of the error Assume that Thus Which is a set enclosed by an ellipsoid with radii 21
22 Example: Discrete deconvolution (estimation error) The estimation error is the vector The components satisfy 22
23 Cyclic convolution with a Gaussian kernel (cont.) (unit impulse function) Noise dominates at high frequencies and is amplified by 23
24 Example: Discrete deconvolution (A is illposed) Assume now that is not invertible and it may happen that i.e, some are zero Leastsquares solution Projection error 24
25 Example: Discrete deconvolution (A is illposed) Leastsquares approach Orthogonal components 25
26 Example: Discrete deconvolution (A is illposed) Invisible objects is the minimum norm solution (related to the MoorePenrose inverse) 26
27 Example: Discrete deconvolution (Regularization) A is illconditioned A is illposed In both cases small eigenvalues are sources of instabilities Often, the smaller the eigenvalue the more oscilating the corresponding eigenvector (high frequences) Regularization by filtereing: shrink/threshold large values of i.e, multiply the eigenvalues by a regularizer filter such that as 27
28 Example: Discrete deconvolution Regularization by filtering (frequency multiplication time convolution) Such that 1) as 2 ) The larger eigenvalues are retained Truncated Fourier Decomposition (TFD) Tikhonov (Wiener) filter 28
29 Example: Discrete deconvolution (Regularization by filtering) TFD Tikhonov Tikhonov regularization Thus Solution of the variational problem 29
30 Example: Discrete deconvolution (1D example) Gaussian shaped of standard deviation = f g frequency
31 Example: Discrete deconvolution (1D example TFD) f f f f 1 f f f f 1 f f 1 f f
32 Example: Discrete deconvolution (2D examplefd) uniform 32
33 Example: Discrete deconvolution (2D exampletfd) 33
34 Curing Illposed/Illconditioned inverse problems Golden rule for solving illposed/illconditioned inverse problems Search for solutions which: are compatible with the observed data satisfy additional constraints (a priori or prior information) coming from the (physics) problem 34
35 Generalized Tikhonov regularization Tikhonov and TFD regularization are not well suited to deal with data Nonhomogeneities, such as edges Generalized Tikhonov regularization Bayesian viewpoint Data Discrepancy term Penalty/ Regularization term Negative loglikelihood Negative logprior 35
36 Dominating approaches to regularization 1) 2) 3) 4) In given circumstances 2), 3), and 4) are equivalent 36
37 Example: Discrete deconvolution (Nonquadratic regularization) penalize oscillatory solutions discontinuity preserving (robust) regularization is nonconvex hard optimization problem nondiscontinuity preserving regularization is convex treatable optimization problem 37
38 Optimization  Quadratic Linear system of equations Large systems require iterative methods  Nonquadratic and smooth Methods: Steepest descent, nonlinear conjugate gradient, Newton, trust regions,  Nonquadratic and nonsmooth Constrained optimization (Linear, quadratic, secondorder cone programs) Methods: Iterative Shrinkage/Thesholding; Coordinate Subspace Optimization; forwardbackward splitting; Primaldual Newton Majorization Minimizaton (MM) class 38
39 Majorization Minorization (MM) Framework Let Majorization Minorization algorithm:...with equality if and only if Easy to prove monotonicity: Notes: should be easy to maximize EM is an algorithm of this type. 39
40 Example: Discrete deconvolution (1D example NQ Regula.) Tikhonov f f f f f f f f f f f f
41 Example: Discrete deconvolution (2D exampletotal Variation) Total variation regularization (TV) where TV regularizer penalizes highly oscilatory solutions, while it preserves the edges 41
42 Bibliography [Ch1. RB2, Ch1. L1] Important topics Euclidian and Hilbert spaces of functions [App. A, RB2] Linear operators in function spaces [App. B, RB2] Euclidian vector spaces and matrices [App. C, RB2] Properties of the DFT and the FFT algorithm [App. B, RB2] Matlab scripts TFD_regularization_1D.m TFD_regularization_2D.m TFD_Error_1D.m TV_regulatization_1D.m 42
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