Part 2 Introduction to Microlocal Analysis
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1 Part 2 Introduction to Microlocal Analysis Birsen Yazıcı & Venky Krishnan Rensselaer Polytechnic Institute Electrical, Computer and Systems Engineering August 2 nd, 2010
2 Outline PART II Pseudodifferential (ψdos) and Fourier Integral Operators (FIOs) Definitions Motivation for the study of FIOs and ψdos Symbols/Amplitudes/Filters Propagation of singularities Singular support Wavefront Sets Method of Stationary Phase Effect of ψdos and FIOs on Singular Support and Wavefront Sets Inversion of FIOs Canonical Relations Construction of Filters Conclusion B. Yazıcı & V. Krishnan 2
3 Outline PART II ψdo and FIO Definitions Motivation for the study of FIOs and ψdos Symbols/Amplitudes/Filters Propagation of singularities Singular support Wavefront Sets Method of Stationary Phase Effect of ψdos and FIOs on Singular Support and Wavefront Sets Inversion of FIOs Canonical Relations Construction of Filters Conclusion B. Yazıcı & V. Krishnan 3
4 Definition The Radon Transform Fourier slice theorem θ Radon transform is an FIO p B. Yazıcı & V. Krishnan 4
5 ψdos and FIOs General integral representations: ψdos: FIOs: Phase function Symbols/Amplitudes/Filters Output variable Frequency variable Input variable B. Yazıcı & V. Krishnan 5
6 Motivation for study of FIOs Measured data in several imaging problems in the form of Exact determination of often not possible. Inhomogeneities in the medium carry much information. Reconstruct the inhomogeneities (modeled as singularities or sharp changes of ) FIOs propagate singularities in specific ways. Inversion of FIOs reconstructs the singularities of the medium. B. Yazıcı & V. Krishnan 6
7 Symbol of a PDE Linear partial differential operators: Symbol of the PDE B. Yazıcı & V. Krishnan 7
8 Estimates for the Symbol - Polynomial in Let and be any multi-indexes. For in a bounded subset Differentiation with respect to degree of the polynomial does not change the Differentiation with respect to the polynomial. lowers the degree of B. Yazıcı & V. Krishnan 8
9 Symbols for ψdos and FIOs Consider that satisfy the estimate Behave like polynomials or inverse of polynomials in as grows or decays in powers of. Differentiation with respect to lowers the order of growth or increases the order of decay. Symbol of order. The order need not be an integer. Needed for the method of stationary phase/high frequency approximations. B. Yazıcı & V. Krishnan 9
10 Symbols - Examples Symbol - Not a symbol where is a smooth function. Not a symbol B. Yazıcı & V. Krishnan 10
11 Pseudodifferential Operators A pseudodifferential operator of order satisfies the amplitude estimate phase term, highly oscillating B. Yazıcı & V. Krishnan 11
12 Pseudodifferential Operators - Examples Linear partial differential operators Operator Time-invariant filters in signal processing Time-invariant filters Approx. by a rational polynomial of degree m B. Yazıcı & V. Krishnan 12
13 Pseudodifferential Operators - Examples If is a time-varying convolution: In the Fourier domain Approx. by a rational polynomial with time-varying coefficients of order m This is a pseudodifferential operator. B. Yazıcı & V. Krishnan 13
14 Fourier Integral Operators Operators more general than DOs. More general phase function. Phase function retains the essential properties of that of PseudoDOs. One important difference: Dimensions of can be all different Example: Radon transform: B. Yazıcı & V. Krishnan 14
15 Phase Function of FIOs Phase function should satisfy the following properties: Real-valued and smooth in Homogeneous of order 1 in and are nonzero vectors for B. Yazıcı & V. Krishnan 15
16 Fourier Integral Operators Fourier integral operators: satisfies properties of a phase function satisfies estimates B. Yazıcı & V. Krishnan 16
17 Outline PART II ψdo and FIO Definitions Motivation for the study of FIOs and ψdos Symbols/Amplitudes/Filters Propagation of singularities Singular support Wavefront Sets Method of Stationary Phase Effect of ψdos and FIOs on Singular Support and Wavefront Sets Inversion of FIOs Canonical Relations Construction of Filters Conclusion B. Yazıcı & V. Krishnan 17
18 Singular Support The set of points where a function is not smooth Not smooth not infinitely differentiable The set of non-smooth points singular support Notation: Singular points Singular support of is B. Yazıcı & V. Krishnan 18
19 Singular Support - Examples Consider the square S on the plane : Let be the function: is the boundary of the square. B. Yazıcı & V. Krishnan 19
20 Singular Support - Examples Consider the function on the plane: First order derivatives exist. Higher order derivatives do not exist at points on the unit circle. B. Yazıcı & V. Krishnan 20
21 Wavefront Sets Wavefront sets Directions of singularity at the location of a singularity Smoothness of a function corresponds to decay of its Fourier transform Localize a function near the location of a singularity and consider the localized Fourier transform Microlocal information Consider directions in which the localized Fourier transform does not decay B. Yazıcı & V. Krishnan 21
22 Wavefront Sets Windowing function Function with a fold singularity Windowed function localizing a singular point Localization The two functions are multiplied Fourier transform in this direction Not a singular direction. Behavior of the directional Fourier transform Singular directions are characterized based on the behavior of the localized Fourier transform. B. Yazıcı & V. Krishnan 22
23 Windowing function Function with a fold singularity Windowed function localizing a singular point Localization Wavefront Sets The two functions are multiplied Fourier transform in this direction Singular directions. Singular directions are characterized based on the behavior of the localized Fourier transform. B. Yazıcı & V. Krishnan 23
24 Wavefront Sets a location of singularity for a function and consider a direction is microlocally smooth near if the localized Fourier transform decays rapidly The complement of the set of points near which is microlocally smooth is called the wavefront set of Denoted B. Yazıcı & V. Krishnan 24
25 Wavefront Sets-Examples Left figure Circle in red is the singular support of Right figure- Arrows indicate the wavefront set directions at points belonging to B. Yazıcı & V. Krishnan 25
26 Wavefront Sets-Examples Left figure Square in red is the singular support of Right figure- Arrows indicate the wavefront set directions at points belonging to B. Yazıcı & V. Krishnan 26
27 Propagation of Singularities Goal Understand how a PseudoDO or an FIO carries the singularities of to or. This singularity does not appear in the output space This singularity appears in three places in the output space B. Yazıcı & V. Krishnan 27
28 Method of Stationary Phase Expansion for certain oscillatory integrals of the form Conditions: a critical point: Critical point is non-degenerate: B. Yazıcı & V. Krishnan 28
29 Method of Stationary Phase Near the origin the function stays positive. Hence nonzero contribution to the integral. B. Yazıcı & V. Krishnan 29
30 Method of Stationary Phase Asymptotic expansion for the integral The first term in the asymptotic expansion n - Dimension of the integral Special case of the stationary phase method sgn Difference of the number of positive and negative eigenvalues B. Yazıcı & V. Krishnan 30
31 Method of Stationary Phase - Application Relation between amplitudes and symbols of PseudoDOs. Using MSP: B. Yazıcı & V. Krishnan 31
32 Effect of ψdos on Singular Support Goal: Understand the effect of ψdo on the singular support of a function What is the relation between and? Important tools: Amplitude estimates Method of stationary phase B. Yazıcı & V. Krishnan 32
33 Effect of ψdos on Singular Support Kernel representation for a ψdo: Use the method of stationary phase for this kernel B. Yazıcı & V. Krishnan 33
34 Effect of ψdos on Singular Support Derivative with respect to the frequency variable: For no stationary points Kernel is smooth in the complement of the set: The relation: B. Yazıcı & V. Krishnan 34
35 Effect of FIOs on Singular Support Kernel representation for FIOs Kernel of FIO B. Yazıcı & V. Krishnan 35
36 Effect of FIOs on Singular Support Use the method of stationary phase: The major contribution to comes from the set of points The relationship: B. Yazıcı & V. Krishnan 36
37 Effect of FIOs on Singular Support- Example Forward model for SAI: Phase function: The singularities of the kernel lie in B. Yazıcı & V. Krishnan 37
38 Effect of FIOs on Singular Support- Example Based on the relation between singular supports: If a point, the singularity at could propagate to (some or all of) Obtain a precise description of how propagation of singularities occur. B. Yazıcı & V. Krishnan 38
39 Effect of ψdos and FIOs on Wavefront Sets Let be a PseudoDO Let be an FIO B. Yazıcı & V. Krishnan 39
40 SAI Example - Revisited Phase function in SAI: Hat Unit vector H denotes projection on to the ground. Doppler B. Yazıcı & V. Krishnan 40
41 SAI Example - Revisited This wavefront set direction is not propagated to the data B. Yazıcı & V. Krishnan 41
42 Outline PART II ψdo and FIO Definitions Motivation for the study of FIOs and ψdos Symbols/Amplitudes/Filters Propagation of singularities Singular support Wavefront Sets Method of Stationary Phase Effect of ψdos and FIOs on Singular Support and Wavefront Sets Inversion of FIOs Canonical Relations Construction of Filters Conclusion B. Yazıcı & V. Krishnan 42
43 Canonical Relations Recall Kernel for ψdos: Kernel for FIOs: Canonical Relations Wavefront sets of the kernels of ψdos and FIOs B. Yazıcı & V. Krishnan 43
44 Canonical Relations Treat kernels as functions: Wavefront set of the kernel of PseudoDO: Wavefront set of input data Wavefront set of the kernel of FIO: Wavefront set of output data Criticality condition in SPM B. Yazıcı & V. Krishnan 44
45 Canonical Relations The wavefront set relation for PseudoDOs reinterpreted as The wavefront set relation for FIOs reinterpreted as Interpreted as composition of relations B. Yazıcı & V. Krishnan 45
46 Composition of Relations If and If and B. Yazıcı & V. Krishnan 46
47 Hörmander-Sato Lemma Consider two FIOs: with canonical relation and with canonical relation. Hörmander-Sato Lemma: Analyze propagation of singularities or wavefront sets using this lemma B. Yazıcı & V. Krishnan 47
48 Inversion of FIOs Construct approximate inverse of Choose so that the location and orientation of the singularities are preserved Microlocal Analysis If is a pseudo-differential operator preserves location and orientation of singularities Analyze propagation of singularities /wavefront sets B. Yazıcı & V. Krishnan 48
49 Inversion of FIOs To choose such that is a PseudoDO: is the L 2 -adjoint (backprojection operator). is a PseudoDO. Reconstructs the singularities at the correct location and orientation. Will not reconstruct at the right strength. Design a filter to reconstruct singularities at the right strength. B. Yazıcı & V. Krishnan 49
50 Construction of Filters Consider a PseudoDO Find another ψdo Choose such that B. Yazıcı & V. Krishnan 50
51 Construction of Filters Kernel representation for the composition: Choose such that The composition in terms of symbols: B. Yazıcı & V. Krishnan 51
52 Construction of Filters Using SPM: Set Let where in are homogeneous of decreasing order B. Yazıcı & V. Krishnan 52
53 Construction of Filters Iteratively determine etc B. Yazıcı & V. Krishnan 53
54 Conclusion Several wave-based imaging problems Data models are FIOs Inhomogeneities of a medium modeled as singularities Singularities Wavefront Set (location and orientation) Pseudodifferential operators and Fourier integral operators propagate wavefront sets in specific ways Back propagation Reconstructs the singularities at the right location and orientation An appropriate choice of filter reconstructs the singularities at the right strength B. Yazıcı & V. Krishnan 54
Part 2 Introduction to Microlocal Analysis
Part 2 Introduction to Microlocal Analysis Birsen Yazıcı& Venky Krishnan Rensselaer Polytechnic Institute Electrical, Computer and Systems Engineering March 15 th, 2010 Outline PART II Pseudodifferential(ψDOs)
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