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) 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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 2
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 3
The Radon Transform Definition: Fourier slice theorem and Fourier inversion formula gives θ Radon transform is an FIO. p ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 4
ψdos and FIOs General integral representations: ψdos: FIOs: Phase function Symbols/Amplitudes/Filters Output variable Frequency variable Input variable ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 5
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 6
Symbol of a PDE Linear partial differential operators: Symbol of the PDE ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 7
Estimates for the Symbol -Polynomial in Let and be any multi-indexes. For in a bounded subset Differentiation with respect to does not change the degree of the polynomial Differentiation with respect to lowers the degree of the polynomial. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 8
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 9
Symbols - Examples Symbol -Not asymbol where is a smooth function. Not asymbol ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 10
Pseudodifferential Operators A pseudodifferentialoperator of order satisfies the amplitude estimate phase term, highly oscillating. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 11
Pseudodifferential Operators - Examples Linear partial differential operators. Operator Filters in signal processing. Filters convolution operators Time-invariant filters: Approx. by a rational polynomial of degree m ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 12
Pseudodifferential Operators - Examples If is a time-varying convolution: In the Fourier domain: This is a pseudodifferential operator. Approx. by a rational polynomial with time-varying coefficients of order m ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 13
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: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 14
Phase Function of FIOs Phase function should satisfy the following properties: Real-valued and smooth in Homogeneous of order 1 in for and are non-zero vectors for ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 15
Fourier Integral Operators Fourier integral operators: satisfies properties of a phase function. satisfies estimates: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 16
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 17
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 18
Singular Support - Examples Consider the square S on the plane : Let be the function: is the boundary of the square. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 19
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 20
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. Has more information. Microlocal information: Consider directions in which the localized Fourier transform does not decay. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 21
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 22
Windowing function Function with a fold singularity Windowed function localizing a singular point Localization Wavefront Sets The two functions are multiplied Fourier transformin this direction Singular directions. Singular directions are characterized based on the behavior of the localized Fourier transform. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 23
Wavefront Sets a location of singularity for a function and consider a direction is microlocallysmooth near if the localized Fourier transform decays rapidly. The complement of the set of points near which is microlocallysmooth is called the wavefront set of Denoted ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 24
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 25
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 26
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 27
Method of Stationary Phase Expansion for certain oscillatory integrals of the form Conditions: a critical point: Critical point is non-degenerate: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 28
Method of Stationary Phase Near the origin the function stays positive. Hence nonzero contribution to the integral. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 29
Method of Stationary Phase Asymptotic expansion for the integral: The first term in the asymptotic expansion: Special case of the stationary phase method: n-dimension of the integral sgn Difference of the number of positive and negative eigenvalues ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 30
Method of Stationary Phase -Application Relation between amplitudes and symbols of PseudoDOs. Using MSP: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 31
Effect of ψdoson Singular Support Goal: Under the effect on the singular support of a function on action by a ψdo. What is the relation between and Important tools: Amplitude estimates Method of stationary phase. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 32
Effect of ψdos on Singular Support Kernel representation for a ψdo: Use the method of stationary phase for this kernel. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 33
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: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 34
Effect of FIOs on Singular Support Kernel representation for FIOs: Kernel of FIO: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 35
Effect of FIOs on Singular Support Use the method of stationary phase: The major contribution to comes from the set of points The relationship: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 36
Effect of FIOs on Singular Support-Example Modeling operator for SAI: Phase function: The singularities of the kernel lie in ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 37
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 38
Effect of ψdos and FIOs on Wavefront Sets Let be a PseudoDO. Let be an FIO. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 39
SAI Example - Revisited Phase function in SAI: Hat Unit vector H denotes projection on to the ground. Doppler ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 40
SAI Example - Revisited This wavefront set direction is not propagated to the data ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 41
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 42
Canonical Relations Recall Kernel for ψdos: Kernel for FIOs: Canonical Relations Wavefront sets of the kernels of ψdos and FIOs ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 43
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 44
Canonical Relations The wavefront set relation for PseudoDOs reinterpreted as The wavefront set relation for FIOs reinterpreted as Interpreted as composition of relations. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 45
Composition of Relations If and If and ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 46
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 47
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 ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 48
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 49
Consider a PseudoDO Construction of Filters Find another ψdo Choose such that ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 50
Construction of Filters Kernel representation for the composition: Choose such that The composition in terms of symbols: ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 51
Construction of Filters Using SPM: Set Let where are homogeneous of decreasing order in ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 52
Construction of Filters Iteratively determine etc ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 53
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. ICASSP 2010 Dallas, TX B. Yazıcı& V. Krishnan 54