Part 2 – Introduction to Microlocal Analysisyazici/ICASSPTutorial/ICASSP_Tutorial-Part2_final.pdf · Part 2 – Introduction to Microlocal Analysis BirsenYazıcı& VenkyKrishnan

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

ICASSP 2010 Dallas, TX B. Yazıcı & V. Krishnan 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


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


The Radon Transform

• Definition:

• Fourier slice theorem and

Fourier inversion formula gives

• Radon transform is an FIO.

θ

p


ψDOs and FIOs

• General integral representations:

– ψDOs:

– FIOs: Phase functionSymbols/Amplitudes/Filters

Input variableFrequency variableOutput variable


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.


Symbol of a PDE

• Linear partial differential operators:

Symbol of the PDE


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.


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.


Symbols - Examples

•

•

•

•

• where is a smooth function.

Not a symbol

Symbol

- Not a symbol


Pseudodifferential Operators

• A pseudodifferential operator of order

• satisfies the amplitude estimate

• phase term, highly oscillating.


• 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


• 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


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:


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


Fourier Integral Operators

• Fourier integral operators:

• satisfies properties of a phase function.

• satisfies estimates:


Outline






• Conclusion


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


Singular Support - Examples

• Consider the square S on the plane :

• Let be the function:

• is the boundary of the square.


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.


• 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


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


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


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


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


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


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


Method of Stationary Phase

• Expansion for certain oscillatory integrals of the form

• Conditions:

– a critical point:

– Critical point is non-degenerate:



Near the origin the function stays positive. Hence non-zero contribution to the integral.



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


Method of Stationary Phase - Application

Relation between amplitudes and symbols of PseudoDOs.

Using MSP:


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.



• Kernel representation for a ψDO:

• Use the method of stationary phase for this kernel.



• Derivative with respect to the frequency variable:

• For no stationary points.

• Kernel is smooth in the complement of the set:

• The relation:


Effect of FIOs on Singular Support

• Kernel representation for FIOs:

• Kernel of FIO:


Effect of FIOs on Singular Support

• Use the method of stationary phase: The major contribution to comes from the set of points

• The relationship:


Effect of FIOs on Singular Support-Example

• Modeling operator for SAI:

• Phase function:

• The singularities of the kernel lie in


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.


• Let be a PseudoDO.

• Let be an FIO.

Effect of ψDOs and FIOs on Wavefront Sets


SAI Example - Revisited

• Phase function in SAI:

H denotes projection on to the ground.

Hat – Unit vector Doppler


SAI Example - Revisited

This wavefront set direction is not propagated to the data


Outline






• Conclusion


Canonical Relations

Recall • Kernel for ψDOs:

• Kernel for FIOs:

Canonical Relations ≡Wavefront sets of the kernels of ψDOs and FIOs


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


Canonical Relations

• The wavefront set relation for PseudoDOsreinterpreted as

• The wavefront set relation for FIOs reinterpreted as

• Interpreted as composition of relations.


Composition of Relations

• If and

• If and


Hörmander-Sato Lemma

• Consider two FIOs: with canonical relation andwith canonical relation .

• Hörmander-Sato Lemma:

• Analyze propagation of singularities or wavefront setsusing this lemma.


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


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.


Construction of Filters

• Consider a PseudoDO

• Find another ψDO

• Choose such that



• Kernel representation for the composition:

• Choose such that

• The composition in terms of symbols:



• Using SPM:

• Set

• Let

where are homogeneous of decreasing order in



• Iteratively determine

etc…


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.

Documents

Part 2 – Introduction to Microlocal Analysisyazici/ICASSPTutorial/ICASSP_Tutorial-Part2_final.pdf · Part 2 – Introduction to Microlocal Analysis BirsenYazıcı& VenkyKrishnan