Part 2 Part 2 –– Introduction to Introduction to Microlocal AnalysisMicrolocal Analysis
Birsen Yazıcı & Venky Krishnan
Rensselaer Polytechnic Institute
Electrical, Computer and Systems Engineering
March 15th , 2010
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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
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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
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The Radon Transform
• Definition:
• Fourier slice theorem and
Fourier inversion formula gives
• Radon transform is an FIO.
θ
p
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ψDOs and FIOs
• General integral representations:
– ψDOs:
– FIOs: Phase functionSymbols/Amplitudes/Filters
Input variableFrequency variableOutput variable
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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.
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Symbol of a PDE
• Linear partial differential operators:
Symbol of the PDE
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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.
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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.
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Symbols - Examples
•
•
•
•
• where is a smooth function.
Not a symbol
Symbol
- Not a symbol
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Pseudodifferential Operators
• A pseudodifferential operator of order
• satisfies the amplitude estimate
• phase term, highly oscillating.
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• Linear partial differential operators.
• Operator
• Filters in signal processing. Filters ≡ convolution operators
• Time-invariant filters:
Pseudodifferential Operators - Examples
Approx. by a rational polynomial of degree m
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• If is a time-varying convolution:
• In the Fourier domain:
• This is a pseudodifferential operator.
Pseudodifferential Operators - Examples
Approx. by a rational polynomial with time-varying coefficients of order m
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Fourier Integral Operators
• Operators more general than DOs.
• More general phase function.
• Phase function retains the essential propertiesof that of PseudoDOs.
• One important difference: Dimensions ofcan be all different.
• Example: Radon transform:
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Phase Function of FIOs
Phase function should satisfy thefollowing properties:
− Real-valued and smooth in
− Homogeneous of order 1 in
for
− and are non-zero
vectors for
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Fourier Integral Operators
• Fourier integral operators:
• satisfies properties of a phase function.
• satisfies estimates:
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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
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Singular Support
• The set of points where a function is not smooth.
• Not smooth – not infinitely differentiable.
• The set of non-smooth points – singularsupport.
• Notation:
• Singular support of is
Singular points
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Singular Support - Examples
• Consider the square S on the plane :
• Let be the function:
• is the boundary of the square.
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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.
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• Wavefront sets: Directions of singularity at the locationof a singularity.
• Smoothness of a function corresponds to decay of itsFourier 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.
Wavefront Sets
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Wavefront Sets
LocalizationFourier transformin thisdirection
Singular directions are characterized based on the behavior of the localized Fourier transform.
Not a singular direction.
Windowing function
Function with a fold singularity
Windowed function localizing a singular point
The two functions are multiplied
Behavior of the directional Fourier transform
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Wavefront Sets
LocalizationFourier transform in this direction
Singular directions are characterized based on the behavior of the localized Fourier transform.
Singular directions.
Windowing function
Function with a fold singularity
Windowed function localizing a singular point
The two functions are multiplied
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Wavefront Sets
• a location of singularity for a function andconsider 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
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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
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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
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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
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Method of Stationary Phase
• Expansion for certain oscillatory integrals of the form
• Conditions:
– a critical point:
– Critical point is non-degenerate:
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Method of Stationary Phase
Near the origin the function stays positive. Hence non-zero contribution to the integral.
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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 negativeeigenvalues
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Method of Stationary Phase - Application
Relation between amplitudes and symbols of PseudoDOs.
Using MSP:
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Effect of ψDOs on 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.
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Effect of ψDOs on Singular Support
• Kernel representation for a ψDO:
• Use the method of stationary phase for this kernel.
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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:
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Effect of FIOs on Singular Support
• Kernel representation for FIOs:
• Kernel of FIO:
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Effect of FIOs on Singular Support
• Use the method of stationary phase: The major contribution to comes from the set of points
• The relationship:
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Effect of FIOs on Singular Support-Example
• Modeling operator for SAI:
• Phase function:
• The singularities of the kernel lie in
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Effect of FIOs on Singular Support-Example
• Based on the relation between singular supports: If apoint , the singularity at could propagate to (some or all of)
Obtain a precise description of how propagation of singularities occur.
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• Let be a PseudoDO.
• Let be an FIO.
Effect of ψDOs and FIOs on Wavefront Sets
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SAI Example - Revisited
• Phase function in SAI:
H denotes projection on to the ground.
Hat – Unit vector Doppler
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SAI Example - Revisited
This wavefront set direction is not propagated to the data
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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
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Canonical Relations
Recall • Kernel for ψDOs:
• Kernel for FIOs:
Canonical Relations ≡Wavefront sets of the kernels of ψDOs and FIOs
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Canonical Relations
• Treat kernels as functions:− Wavefront set of the kernel of PseudoDO:
− Wavefront set of the kernel of FIO:
Criticality condition in SPM
Wavefront set of input data
Wavefront set of output data
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Canonical Relations
• The wavefront set relation for PseudoDOsreinterpreted as
• The wavefront set relation for FIOs reinterpreted as
• Interpreted as composition of relations.
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Composition of Relations
• If and
• If and
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Hörmander-Sato Lemma
• Consider two FIOs: with canonical relation andwith canonical relation .
• Hörmander-Sato Lemma:
• Analyze propagation of singularities or wavefront setsusing this lemma.
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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
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Inversion of FIOs
• To choose such that is a PseudoDO:
• is the L2-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.
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Construction of Filters
• Consider a PseudoDO
• Find another ψDO
• Choose such that
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Construction of Filters
• Kernel representation for the composition:
• Choose such that
• The composition in terms of symbols:
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Construction of Filters
• Using SPM:
• Set
• Let
where are homogeneous of decreasing order in
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Construction of Filters
• Iteratively determine
etc…
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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.
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