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Tomographic Lecture

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  • 7/29/2019 Tomographic Lecture

    1/17

    School of Engineering

    Tomography and Reconstruction

    Lecture Overview

    Applications

    Background/history of

    tomography

    Radon Transform

    Fourier Slice Theorem

    Filtered Back Projection

    Algebraic techniques

    Measurement of Projection

    data

    Example of flame tomography

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    Applications & Types of Tomography

    Medical Applications Type of TomographyFull body scan X-ray

    Respiratory, digestive

    systems, brain scanning

    PET Positron Emission

    Tomography

    Respiratory, digestive

    systems.

    Radio-isotopes

    Mammography Ultrasound

    Whole Body Magnetic Resonance (MRI,

    NMR)

    PET scan on the brain

    showing Parkinsons

    Disease

    MRI and PET showing

    lesions in the brain.

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    School of Engineering

    Applications & Types of Tomography

    Non Medical Applications Type of Tomography

    Oil Pipe Flow

    Turbine Plumes

    Resistive/Capacitance

    Tomography

    Flame Analysis Optical Tomography

    ECT on industrial pipe flows

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    The History

    Johan Radon (1917) showed how a reconstruction from

    projections was possible.

    Cormack (1963,1964) introduced Fourier transforms into the

    reconstruction algorithms.

    Hounsfield (1972) invented the X-ray Computer scanner formedical work, (which Cormack and Hounsfield shared a Nobel

    prize).

    EMI Ltd (1971) announced development of the EMI scanner

    which combined X-ray measurements and sophisticated

    algorithms solved by digital computers.

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    School of Engineering

    dxdytyxyxftP )sincos(),()(

    linet

    dsyxftP),(

    ),()(

    tyx sincos

    1sincos tyx

    1)(tP

    ),( yxf

    y

    x

    Line Integrals and Projections

    The function is

    known as the Radon transform

    of the function f(x,y).

    1)(tP

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    School of Engineering

    )(1 tP

    ),( yxf

    y

    x

    )(2 tP

    A projection is formed by combining a set

    of line integrals. Here the simplestprojection, a collection of parallel ray

    integrals i.e constant , is shown.

    Line Integrals and Projections

    ),( yxf

    y

    x

    )(1 tP)(2 tP

    A simple diagram showing the fan

    beam projection

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    Fourier Slice Theorem

    The Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a

    parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform

    of the original object. It follows that given the projection data, it should then be possible to

    estimate the object by simply performing the 2D inverse Fourier transform.

    dtetPwS wtj 2)()(

    dxdyeyxfvuF vyuxj )(2),(),(

    dxdyeyxfuF uxj 2),()0,(

    dxedyyxfuF uxj 2),()0,(

    dxexPuF uxj 2

    0 )()0,(

    dyyxfxP ),()(0

    )()0,( 0 uSuF

    Start by defining the 2D Fourier transform of the

    object function as

    Define the projection at angle , P(t) and its

    transform by

    For simplicity =0 which leads to v=0

    As the phase factor is no-longer dependent on

    y, the integral can be split.

    The part in brackets is the equation for a projection

    along lines of constantx

    Substituting in

    Thus the following relationship between the

    vertical projection and the 2D transform of theobject function:

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    )(1 tP

    ),( yxf

    y

    x

    t

    v

    u

    Space Domain Frequency Domain

    Fourier transform

    The Fourier Slice theorem relates the Fourier

    transform of the object along a radial line.

    The Fourier Slice Theorem

    v

    u

    Collection of projections of an objectat a number of angles

    For the reconstruction to be made it is

    common to determine the values onto a

    square grid by linear interpolation from

    the radial points. But for high frequencies

    the points are further apart resulting in

    image degradation.

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    Filtered Back Projection

    Filtered back projection is the most commonly used algorithm for

    straight ray tomography.

    (a)The ideal Situation

    (b) Fourier Slice

    Theorem

    (c) The filter back

    projection takes the FourierSlice and applies a

    weighting so that it

    becomes an approximation

    of that in (a).

    The result of back projecting

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    The Array: Algebraic Reconstruction

    Technique (ART)

    ART is used in indeterminate problems

    and was first used by Gordon et alin

    the reconstruction of biological

    material.

    1 3

    1 1

    2

    3

    2 1 2 5

    5

    6

    754y

    x

    Figure a. Initial 3 by 3 grid with

    ray sums and coefficients.

    ? ?

    ? ?

    ?

    ?

    ? ? ? 5

    5

    6

    754y

    x

    Figure b. The indeterminate

    problem.

    6/3 6/3

    5/3 5/3

    6/3

    5/3

    5/3 5/3 5/3 5

    5

    6

    754y

    x

    Figure c. Step 1: All entries in

    unity, scaled by ray sum over

    number of row elements.

    6/3 6/3

    5/3 5/3

    6/3

    5/3

    5/3 5/3 5/3 5

    5

    6

    5.335.335.33y

    x

    Figure d. Step 2: Recalculated

    column sums.

    1.5 1.88

    1.25 1.57

    2.63

    2.19

    1.25 1.57 2.19 5.01

    5.01

    6.01

    7.015.024y

    x

    Figure e. Step 3. Recalculated row

    and column sums and elements.

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    School of Engineering

    Measurement of projection data

    Attenuation of X-rays

    Assume no loss of intensity of the beam due to

    divergence, however the beam does attenuate due to

    photons either being absorbed or scattered by the object.

    Photoelectric Absorption

    This consists of an x-ray photon

    imparting all of its energy to an inner

    electron of an atom. The electron uses

    this energy to overcome the binding

    energy within its shell, and the rest

    appearing as kinetic energy in this freedelectron.

    Compton Scattering

    This consists of the interaction of the

    photon with either the free electron or a

    loosely bound outer shell electron. As a

    result the x-ray is deflected from its

    original direction.

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    Measurement of projection data

    Attenuation of X-rays

    Consider N photons cross the lower

    boundary of this layer in some

    measured time interval, and N+N

    emerge from the top side. (N will be

    negative). N follows the relationship,

    xN

    N 1

    N

    N

    x

    dxN

    dN

    0 0

    dxdNN

    1 xNN 0lnln

    xeNxN 0)(=photon loss rate (per unit distance)

    of the Compton and photoelectric

    effects. In the limit x goes to zero sowe get

    Solving this across the thicknessof the slab

    Where N0 is the number of

    photons that enter the object.

    The number of photons as a

    function of the position

    within the slab is given by,

    or

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    Signal enteringflame

    Signal leaving

    flame

    Radiantintensity of

    backlight, L1 Radiance emitted

    by gas, L3

    Transmitted portion

    of back light

    radiation, L2

    Mercury

    lamp

    Burner

    Fibre optic to

    spectrograph

    Optical arrangement used to determine the optical

    thickness of a flame.

    Background Lamp, L1

    Flame, L3

    Flame+Lamp

    , L

    Counts 77.71 93.76 439.29 450.82

    Minus

    Background

    - 16.06 361.59 373.12

    33.0ln2

    1

    L

    LD

    Interpretation of Results

    Transmitted portion of backlight radiation, L2: 11.53 counts

    Radiation incident on fibre from backlight, L1: 16.06 counts

    72% transmission at 309 nm

    Optical thickness at 309 nm,

    Absorption Coefficient:

    11* 079.0ln

    mm

    x

    L

    L

    Emission Coefficient:

    1

    *

    **

    1*

    23.31exp1

    exp

    mmx

    xL

    Flame Thickness, Emission and

    Absorption

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    School of Engineering

    3.8 Fibre

    optic

    Acceptance

    cone of fibre

    Tomographic array

    The acceptance cone of the fibres fitted to the area

    The Array: Fibre Geometry

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    Array Resolution:

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    The Array: Preliminary Results

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    Comparing Results

    Single Photograph of OH modified for

    colour intensitySingle thermocouple scan

    Averaged thermocouple result Average of three photographs

    The burner has been modified

    by placing two coins on itsbase. The array result is

    shown, superimposed on a

    photograph of the modified

    burner.

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