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New Approaches Towards a Higher Resolution Biomedical Imaging
Magnetic Induction Tomography
04/06/2010
Nuno Brás PhD Thesis Presentation
Universidade Técnica de Lisboa Instituto Superior Técnico
PhD Dissertation Doctoral Program In Electrical and Computer
Engineering
António C. Serra and Raúl C. Martins (advisors)
Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of the Art Issues ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
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Outline
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! Motivation ! The Problem in Hands ! State of the Art Issues ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
Motivation
Imaging Systems
In Natural Sciences Astrophysics; Biophysics; Geophysics, Biology…
In Engineering
biomedical; industrial processes; oil and water prospecting; search and rescue instrumentation;…
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Motivation
Magnetic Induction Tomography (as a Biomedical Imaging System)
Tomography - recreates maps or images from peripheral measurements
! MIT is: ! Active Tomographic method (instead of Passive); ! Harmless (instead of Harmful) ! In-vivo or in-vitro; ! Functional or Steady Imaging
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Motivation
What does MIT try to solve? Good things about MIT: ! (very) low price comparing with other tomographic systems
(e.g. MRI machine up to 2.5 million € + installation (25%))
! biological passivity (low power radiofrequency)
! excellent penetration abilities in biological phantoms, even in bone-like tissues (unlike ultrasound tomography )
! full reconstruction is theoretically achievable
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Motivation
Not so good things about MIT: 1. The underlying image reconstruction problem is a
large, complex and non-linear problem (non-convex)
2. Its behavior is strongly dependent on the number of measurements and their intrinsic accuracy
Actual Context There are no commercial equipments, just prototypes.
First applications are being explored
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Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
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What is MIT, conceptually
It is a distributed parameter estimation (DPE) problem under Electromagnetic Partial Differential equations
(PDE)
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The Problem in Hands
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04/06/2010 Nuno B. Brás - PhD Dissertation
where is the kernel of the underlying physical process.
Consider an electromagnetic physical process ruled by the following relation:
The Problem in Hands
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Distributed Parameter Estimation Problems in electromagnetics
where is the kernel of the underlying physical process. Electric and
magnetic fields or potentials
(state variables)
Boundary Conditions
+ Sources
Set Differential Operators
+ parameters
04/06/2010 Nuno B. Brás - PhD Dissertation
The Problem in Hands
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Distributed Parameter Estimation Problems in electromagnetics
Consider an electromagnetic physical process ruled by the following relation:
Distributed Parameter Estimation Problems in electromagnetics
04/06/2010 Nuno B. Brás - PhD Dissertation
where is the kernel of the underlying physical process.
The Problem in Hands
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The inverse problem is the distributed parameter identification problem given by:
The forward problem (well-posed, and typically linear) is the PDE equation
Consider an electromagnetic physical process ruled by the following relation:
The General Parameter Estimation Model (Outline of this Thesis)
Experimental Setup (PART 1)
The Forward Problem (PART 2)
The Inverse Problem (PART 3)
04/06/2010 Nuno B. Brás - PhD Dissertation
Problem in Hands
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The MIT Setup
04/06/2010 Nuno B. Brás - PhD Dissertation
Source Currents: Harmonic , tens of kHz to some MHz;
Current Amplitude: Up to 1 A;
Setup Radius: Typically 15 cm;
Conductivity Values: complex with absolute value between 0.1 S/m to 2 S/m;
Source and sensing coils size: around 5 cm and 3 cm correspondingly
The Problem in Hands
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04/06/2010 Nuno B. Brás - PhD Dissertation
The Problem in Hands
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The MIT Setup Each d should be as large, independent and accurate as possible
Outline
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! Motivation ! The Problem in Hands ! State of the Art Issues ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
! Experimental Issues ! Typical layout generates ambiguities (gradiometers); ! Fixed system of acquisition (fixed and low number is allowed)
! Forward Problem ! Developed solvers in MIT context are accurate but typically slow;
(it is not possible to use commercial solvers)
! Inverse Problem ! Non-linear approaches are the best known reconstruction methods. ! The state of the art method is not efficient when used with a high
number of acquisitions.
04/06/2010 Nuno B. Brás - PhD Dissertation
State of the Art Issues
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Outline
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! Motivation ! The Problem in Hands ! State of the Art Issues ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
04/06/2010 Nuno B. Brás - PhD Dissertation
Objectives - Developed Work
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General Objective Implementation of a new Magnetic Induction Tomography prototype and numerical framework to deal with large number of acquisitions
Developed Work ! Experimental
! A new moving prototype was implemented for large number of acquisitions while attaining the state of the art sensitivity (SCR)
! Forward Problem ! A new 3D eddy current PDE solver was developed with a
competitive processing time and low relative error.
! Inverse Problem
! A new ADMM method was developed and used in 2D and 3D IP.
! Its feasibility was proved and its advantage was clearly shown for large datasets scenarios.
Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
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Experimental Work
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! Twin Coil Setup Source Current
Amplifier
Capacitive Shields
Source Coil
Motor systems
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Experimental Work
Involved Areas While Designing This Prototype
! Electromagnetic Compatibility (EMC)
! Mechanical Setup Design and Characterization
! Current Source Design
! Acquisition System
! Signal Processing
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Experimental Work
Involved Areas While Designing This Prototype
! Electromagnetic Compatibility (EMC)
! Mechanical Setup Design and Characterization
! Current Source Design
! Acquisition System
! Signal Processing Measuring System
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Symmetry Axis
Without object: V1-V2 = Residual V ≈ 0, for any angle With object: V1-V2 = ∆V
V1
V2
Experimental Work
! Twin Coil Setup
Useful V =∆V– Residual V
! Shielded Differential coils and cables
! Gain (low noise) Amplifier =100; with a low-pass filter
! ADC with 12 bits, up to 60 MSamples/s
! The Goertzel Transform (second order IIR filter ) to calculate the frequency bin amplitude and phase (others were tested)
! Average sliding window applied to avoid 50Hz modulation and reduce noise
! Principal Component Analysis (PCA) and a reference coil were used to eliminate long term trends from the source current.
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Experimental Work
Acquisition System
Signal Processing System
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Experimental Work
! Results a differential coil pair, during 300 sec time
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Experimental Work
! Results - Stability Applying an averaging sliding window with 50 values
A minimum measurable SCR of
was obtained which is the stated state of the art value in this case with a moving system of sensing coils
Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problem
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
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The Forward Solver abilities:
! Isotropic parameter model ! Constant � ! Diffusion model approach (no skin currents) ! Harmonic and stationary equations ! Multi- level grid allowing mutligrid (MG) techniques and/or
adaptive mesh refinements (AMR) in future iterations ! Analytical sources, using the Biot-Savart law
The Forward Problem
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Normal Component of = 0
The Forward Problem
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Formulation:
Imposed directly by (1)
(1)
in
Boundary Conditions Interface Conditions
where and
The Forward Problem
The Discretization : Finite Integration Technique
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The Forward Problem
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The proposed FIT formulation:
! The resultant system is non-singular - the used gauging ensures a robust regularization of the system;
! A Laplacian-type instead a curl-curl-type problem, where a larger set of numerical methods are available;
! Subgriding implementation;
Moreover…
! Several Numerical Optimization aspects were implemented;
! Solved with iterative preconditioned methods (here, iLU factorization was used);
The Forward Problem
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Results – Performance and accuracy
The Forward Problem
How this interacts with the inverse problem?
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Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problems
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
43
Inverse Problems
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! Implemented Methods :
! Two versions of the Alternating Direction Method of Multipliers (Augmented Lagrangian Method) in an elliptic 2D inverse problem with total variation regularization and Wavelets regularization
! Gauss Newton in ellipitc 2D Inverse Problem
! ADMM version adapted to solve the MIT 3D
Inverse Problems
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The ADMM – a sequence of simple problems
Closed form
Inverse Problems
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The ADMM – a sequence of simple problems
Closed form
Either implementing a fixed point iteration for a continuous approximation of
(ADMM with fixed point iteration) or
Using a second Alternate Direction split Closed Solution (DIPESAL Method)
Inverse Problems
2D Elliptical IP - The same model, a simpler problem
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Inverse Problems
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Gauss Newton using an unconstrained version of
the problem
ADMM with a fixed point iteration and TV Regularization
ADMM with a fixed point iteration and Wavelet
Regularization
Inverse Problems
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DIPESAL
ADMM with fixed point iteration
2.2% better reconstruction and 7% faster, with better image reconstruction quality
5% noise and LARGER problem
Inverse Problems
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! Some remarks for 3D MIT inverse problem
! Two things changes for MIT problem:
! Size: Each Iterations of the ADMM problem have to be solved iteratively: The Second Order Stationary method was applied.
! Different discretization and equations: The new equations were referred before;
Inverse Problems
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! 3D MIT inverse problem
Inversion scenarios
Inverse Problems
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! 3D MIT inverse problem
Inversion scenarios
Inverse Problems
! Relative Error Evolution and Constrain Imposition
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Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problems
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
60
Conclusions and Original Contributions
! Experimental Work
! A new prototype was implemented for moving sensing and source coils, allowing acquire large sets of accurate data
! System sensitivity, a state of the art value, is stable during a large amount of time (>300 sec), allowing to implement a moving setup with high sensitivity.
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This work resulted in 7 congress papers during the PhD period
Conclusions and Original Contributions
! Forward Problem
! A new hybrid formulation of Finite Integration Technique
! The processing time was optimized to be included in an inverse problem attaining a low relative error
! Total relative error ~ 1.5 %
! Processing time ~19 sec. per eddy current problem
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This work was published on IEEE Transactions on Magnetics (May 2010)
Conclusions and Original Contributions
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! 2D Elliptic Inverse Problem
! A new method ADMM algorithm (DIPESAL) was implemented with closed solution for Total Variation regularization
! Lower relative error (2,2%);
! 7% faster.
! Better qualitative images in general
This work was submitted to IEEE Transactions on Image Processing (May 2010)
Conclusions and Original Contributions
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! 3D MIT Inverse Problem Solution
! The ADMM algorithm feasibility in 3D MIT problems was achieved
! A complete new approach with clear advantages in large number of measurements conditions.
! Its application to the MIT originates state of the art simulated reconstructed maps.
This work was submitted to IEEE Transactions on Medical Imaging (May 2010)
Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problems
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
65
Further Work
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! Experimental ! Use large arrays of Giant Magnetic Resistors as sensors – This can
dramatically change MIT since it can increase dramatically the number of acquisitions and their accuracy.
! Use strong permeability material cores to increase magnetic field. ! Use of transient signals to increase SNR
! Numerical ! Multigrid scheme for preconditioning and code parallelization to
accelerate even more the forward and inverse problem. ! Field Regularization – there is very experiments clearly shows
advantages in using this ! Improve ADMM to MIT – There is space to improve, namely in the
regularization process
Outline
04/06/2010 Nuno B. Brás - PhD Dissertation
! Motivation ! The Problem in Hands ! State of The Art ! Objectives and Developed Work
! Experimental Work ! Forward Problem ! Inverse Problems
! Conclusions and Original Contributions ! Further Work ! Acknowledgements
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Acknowledgements
! Supervisors: A. C. Serra And Raúl C. Martins
! Other professor and researchers ! Professor José Bioucas Dias (Elect. and Comp. Dep.) ! Professor Artur Lopes Ribeiro (Elect. and Comp. Dep.) ! Professor Paulo Martins (Mechanical Dep.) ! Professor Helena Ramos (Elect. and Comp. Dep.) ! Dr. Tomas Radil (Elect. and Comp. Dep.) ! Professor Pedro Santos (Mathematics Dep.) ! Professor Carlos Alves (Mathematics Dep.) ! Eng. Luis Soares ! Eng. Alexandre Pestana ! Eng. José Gouveia
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New Approaches Towards a Higher Resolution Biomedical Imaging
Magnetic Induction Tomography
04/06/2010
Nuno Brás PhD Thesis Presentation
Universidade Técnica de Lisboa Instituto Superior Técnico
PhD Dissertation Doctoral Program In Electrical and Computer
Engineering
António C. Serra and Raúl C. Martins (advisors)