Data Acquisition, Representation and Reconstruction of medical images

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Data Acquisition, Representation and
Reconstruction of medical images
Application of Advanced Spectral Methods
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Acquisition Methods for medical images

• X-Rays
• Computer Tomography (CT or CAT)
• MRI (or NMR)
• PET / SPECT (Positron Emission Tomography,
  Single Photon Emission Computerized Tomography
• Ultrasound
• Computational

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X-Rays
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X-Rays - Physics

• X-Rays are associated with inner shell electrons
• As the electrons decelerate in the target through
  interaction, they emit electromagnetic radiation
  in the form of x-rays.
• patient is located between an x-ray source and a
  film -gt radiograph
• cheap and relatively easy to use
• potentially damaging to biological tissue

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X-Rays

• X-Rays
• similar to visible light, but higher energy!

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X-Rays - Visibility

• bones contain heavy atoms -gt with many electrons,
  which act as an absorber of x-rays
• commonly used to image gross bone structure and
  lungs
• excellent for detecting foreign metal objects
• main disadvantage -gt lack of anatomical structure
  
• all other tissue has very similar absorption
  coefficient for x-rays

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X-Rays - Images
X-Rays can be used in computerized tomography
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• Computerized (Axial) Tomography

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CT (CAT) scanners and relevant mathematics
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Non-Intrusive Medical Diagnosis based on
Computerized Tomography

• Computer tomography CT

(From Jains Fig.10.1)
An X-ray CT scanning system
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Non-Intrusive Medical Diagnosis based on
Transmission Tomography
Source and Detector are rotating around humans
body
(From Boviks Handbook Fig.10.2.1)
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Non-Intrusive Medical Diagnosis based on
projections

• Observe a set of projections (integrations) along
  different angles of a cross-section
• Each projection itself loses the resolution of
  inner structure
• Types of measurements
• transmission (X-ray),
• emission, magnetic resonance (MRI)
• Want to recover inner structure from the
  projections
• Computerized Tomography (CT)

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Non-Intrusive Medical Diagnosis based on Emission
Tomography

• Emission tomography ET measure emitted gamma
  rays by the decay of isotopes from radioactive
  nuclei of certain chemical compounds affixed to
  body parts.
• MRI based on that protons possess a magnetic
  moment and spin.
• In magnetic field gt align to parallel or
  antiparallel.
• Apply RF gt align to antiparallel. Remove RF gt
  absorbed energy is remitted and detected by
  Rfdetector.

f(x,y) is 2D image as before
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Radon Transform Principles

• A linear transform f(x,y) ? g(s,?)
• Line integral or ray-sum
• Along a line inclined at angle ? from y-axis and
  s away from origin
• Fix ? to get a 1-D signal g?(s)

We have now a set of images g?(s) which represent
g(s,?)
(From Jains Fig.10.2)
This is a transform from 2D to 2D spaces
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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
MRI and PET showing lesions in the brain.
PET scan on the brain showing Parkinsons Disease
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Applications Types of Tomography non medical
ECT on industrial pipe flows
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CT or CAT - Principles

• Computerized (Axial) Tomography
• introduced in 1972 by Hounsfield and Cormack
• natural progression from X-rays
• based on the principle that a three-dimensional
  object can be reconstructed from its two
  dimensional projections
• based on the Radon transform (a map from an
  n-dimensional space to an (n-1)-dimensional space)

Radon again!
From 2D to 3D !
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CT or CAT - Methods

• measures the attenuation of X-rays from many
  different angles
• a computer reconstructs the organ under study in
  a series of cross sections or planes
• combine X-ray pictures from various angles to
  reconstruct 3D structures

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The History of CAT

• 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 for medical 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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• Backpropagation
• Principles

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Backpropagation
We know that objects are somewhere here in black
stripes, but where?
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Example of Simple Backprojection Reconstruction

• Given are sums, we have to reconstruct values of
  pixels A, B, C and D

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Image Reconstruction ART or Algebraic
Reconstruction Technique
ART
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CT - Reconstruction ART or Algebraic
Reconstruction Technique

• METHOD 1 Algebraic Reconstruction Technique
• iterative technique
• attributed to Gordon

Initial Guess
Reconstructedmodel
Back-Projection
Projection
Actual DataSlices
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CT - Reconstruction FBP Filtered Back Propagation

• METHOD 2 Filtered Back Projection
• common method
• uses Radon transform and Fourier Slice Theorem

y
F(u,v)
f(x,y)
Gf(r)
x
s
u
gf(s)
f
Spatial Domain
Frequency Domain
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COMPARISON CT - FBP vs. ART
ART
FBP
Algebraic Reconstruction Technique

• Still slow
• better quality for fewer projections
• better quality for non-uniform project.
• guided reconstruct. (initial guess!)

Filtered Back Projection

• Computationally cheap
• Clinically usually 500 projections per slice
• problematic for noisy projections

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• Fourier Slice Theorem and FFT review

Patients body is described by spatial
distribution of attenuation coefficient
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Properties of attenuation coefficient
Our transform f(x,y) ? p(r,?)
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• attenuation coefficient is used in CT_number of
  various tissues
• These numbers are represented in HU Hounsfield
  Units

CT_number uses attenuation coefficients
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RADON TRANSFORM Properties
REMEMBER f(x,y) ? p(r,?)
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Radon Transform is available in Matlab

• Radon and its inverse easy to use
• You can do your own projects with CT
  reconstruction
• Data are available on internet

sinogram
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Inverse Radon Transform
Inverse Radon Transform
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• Matlab examples

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Sinogram versus Hough
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An aside
The object
The sinogram

• AN ASIDE

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Review and notation Fourier Transform of Image
f(x,y)
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Matlab example
In Matlab there are implemented functions that
use Fourier Slice Theorem
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Matlab example filtering
High frequency removed
Low frequency removed
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Matlab example - convolution
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Remainder of main theorem of spectral imaging
Matlab example filtering by convolution in
spectral domain
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• Radon Transform and a Head Phantom

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Reconstructing with more and more rays
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Example of Image Radon Transform
Y-axis distance, X-axis angle
(From Matlab Image Processing Toolbox
Documentation)
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• Matlab Implementation of Radon Transform

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big noise
No noise
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The Lung Cancer and the reconstruction
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The Lung and The CTs
LUNG 1.Either of the pair of organs occupying
the cavity of the thorax that effect the aeration
of the blood. 2.Balloon-like structures in the
chest that bring oxygen into the body and expel
carbon dioxide from the body
TYPES 1.Small Cell Lung Cancer (SCLC) - 20 of
all lung cancers 2.Non Small Cell Lung Cancer
(NSCLC) - 80 of all lung cancer
Risks In the United States alone, it is
estimated that 154,900 died from lung cancer in
2002. In comparison,is estimated that 126,800
people died from colon, breast and prostate
cancer combined, in 2002.
LUNG CANCER Lung Cancer happens when cells in
the lung begin to grow out of control and can
than invade nearby tissues or spread throughout
the body Large collections of this out of
control tissues are called tumors.
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Starting Point
We want to reconstruct shape of the lungs
Border Detection

• At the moment two approaches are available.
• Left the algorithm developed at Pisa
• Right the algorithm developed at Lecce

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Image Interpolation - Theory
IDEA In order to provide a richer environment
we are thinking of using interpolation methods
that will generate artificial images thus
revealing hidden information.
RADON RECONSTRUCTION Radon reconstruction is
the technique in which the object is
reconstructed from its projections. This
reconstruction method is based on approximating
the inverse Radon Transform.
RADON Transform The 2-D Radon transform is the
mathematical relationship which maps the spatial
domain (x,y) to the Radon domain (p,phi). The
Radon transform consists of taking a line
integral along a line (ray) which passes through
the object space. The radon transform is
expressed mathematically as
FILTERED BACK PROJECTION - INVERSE R.T. It is
an approximation of the Inverse Radon
Transform. The principle Several x-ray images
of a real-world volume are acquired The Data
X-ray images (projections) of known orientation,
given by data samples. The Goal Reconstruct a
numeric representation of the volume from these
samples. The Mean Obtain each voxel value from
its pooled trace on the several
projections. Resampling At this point one can
obtain the artificial slices Reslicing An
advantage of the volume reconstruction is the
capability of obtaining new perpendicular slices
on the original ones.
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Image Interpolation - Graphical Representation (I)
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Image Interpolation - Graphical Representation
(II)
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Line Integrals and Projections

• We review the principle
• Discuss various geometries
• Show the use of filtering

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Line Integrals and Projections
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Fan Beams
Parallel Beams
A fan beam projection is taken if the rays meet
in one location
Parallel beams projections are taken by measuring
a set of parallel rays for a number of different
angles
Various types of beams can be used
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Line Integrals and Projections
A projection is formed by combining a set of line
integrals. Here the simplest projection, a
collection of parallel ray integrals i.e constant
?, is shown.
Notation for calculations in these projections
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Line Integrals and Projections
A simple diagram showing the fan beam projection
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Fourier Slice Theorem
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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.

Start by defining the 2D Fourier transform of the
object function as
For simplicity ?0 which leads to v0

• Define the projection at angle ? P?(t)
• Define its transform by

As the phase factor is no-longer dependent on y,
the integral can be split.
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Fourier Slice Theorem
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 constant x
Substituting in
Thus the following relationship between the
vertical projection and the 2D transform of the
object function
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Fourier Slice Theorem Stanley and Kak

• Full details of derivation, not for now.

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The Fourier Slice Theorem
The Fourier Slice theorem relates the Fourier
transform of the object along a radial line.
t
Fourier transform
?
Space Domain
Frequency Domain
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The Fourier Slice Theorem
The Fourier Slice theorem relates the Fourier
transform of the object along a radial line.
Collection of projections of an object at a
number of angles
t
Fourier transform
?

• 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.

Space Domain
Frequency Domain
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Backprojection of Radon Transform
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Backprojection of Radon Transform
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Ideal cylinder
Blurred edges
Crisp edges
Filtered backpropagation creates crisp edges
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Computerized Tomography Equipment
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CT - 2D vs. 3D

• Linear advancement (slice by slice)
• typical method
• tumor might fall between cracks
• takes long time
• helical movement
• 5-8 times faster
• A whole set of trade-offs

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Evolution of CT technology
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CT or CAT - Advantages

• significantly more data is collected
• superior to single X-ray scans
• far easier to separate soft tissues other than
  bone from one another (e.g. liver, kidney)
• data exist in digital form -gt can be analyzed
  quantitatively
• adds enormously to the diagnostic information
• used in many large hospitals and medical centers
  throughout the world

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CT or CAT - Disadvantages

• significantly more data is collected
• soft tissue X-ray absorption still relatively
  similar
• still a health risk
• MRI is used for a detailed imaging of anatomy
  no Xrays involved.

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• Nuclear Magnetic Resonance (NMR)
• Magnetic Resonance Imaging (MRI)

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MRI

• Nuclear Magnetic Resonance (NMR) (or Magnetic
  Resonance Imaging - MRI)
• most detailed anatomical information
• high-energy radiation is not used, i.e. this is
  safe method
• based on the principle of nuclear resonance
• (medicine) uses resonance properties of protons

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Magnetic Resonance ImagingMRI - polarized

• all atoms (core) with an odd number of protons
  have a spin, which leads to a magnetic behavior
• Hydrogen (H) - very common in human body very
  well magnetizing
• Stimulate to form a macroscopically measurable
  magnetic field

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MRI - Signal to Noise Ratio

• proton density pictures - measures HMRI is good
  for tissues, but not for bone
• signal recorded in Frequency domain!!
• Noise - the more protons per volume unit, the
  more accurate the measurements - better signal to
  noise ratio (SNR) through decreased resolution

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PET/SPECT

• Positron Emission TomographySingle Photon
  Emission Computerized Tomography

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PET/SPECT

• Positron Emission TomographySingle Photon
  Emission Computerized Tomography
• recent technique
• involves the emission of particles of antimatter
  by compounds injected into the body being scanned
• follow the movements of the injected compound and
  its metabolism
• reconstruction techniques similar to CT - Filter
  Back Projection iterative schemes

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Ultrasound
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Ultrasound

• the use of high-frequency sound (ultrasonic)
  waves to produce images of structures within the
  human body
• above the range of sound audible to humans
  (typically above 1MHz)
• piezoelectric crystal creates sound waves
• aimed at a specific area of the body
• change in tissue density reflects waves
• echoes are recorded

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Ultrasound (2)

• Delay of reflected signal and amplitude
  determines the position of the tissue
• still images or a moving picture of the inside of
  the body
• there are no known examples of tissue damage from
  conventional ultrasound imaging
• commonly used to examine fetuses in utero in
  order to ascertain size, position, or
  abnormalities
• also for heart, liver, kidneys, gallbladder,
  breast, eye, and major blood vessels

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Ultrasound (3)

• by far least expensive
• very safe
• very noisy
• 1D, 2D, 3D scanners
• irregular sampling - reconstruction problems

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Typical Homework
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Sources of slides and information

• Badri Roysam
• Jian Huang,
• Machiraju,
• Torsten Moeller,
• Han-Wei Shen
• Kai Thomenius
• Badri Roysam