Transcript acquisition and Radon Transform. ppt

Data Acquisition,
Representation and
Reconstruction of
medical images
Application of Advanced
Spectral Methods
Acquisition Methods for medical
images
1. X-Rays
2. Computer Tomography (CT or CAT)
3. MRI (or NMR)
4. PET / SPECT (Positron Emission Tomography,
Single Photon Emission Computerized Tomography
5. Ultrasound
6. Computational
X-Rays
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 -> radiograph
 cheap and relatively easy to use
 potentially damaging to biological tissue
X-Rays
• X-Rays
• similar to visible
light, but higher
energy!
X-Rays - Visibility
• bones contain heavy atoms -> 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 -> lack of anatomical structure
 all other tissue has very similar absorption
coefficient for x-rays
X-Rays - Images
X-Rays can be used in computerized
tomography
Computerized
(Axial)
Tomography
CT (CAT) scanners and
relevant mathematics
Non-Intrusive Medical Diagnosis based
on Computerized Tomography
– Computer
tomography CT
An X-ray CT scanning system
(From Jain’s Fig.10.1)
Non-Intrusive Medical Diagnosis based on
Transmission Tomography
Source and
Detector are
rotating
around
human’s
body
(From Bovik’s Handbook
Fig.10.2.1)
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)
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 => align to parallel or antiparallel.
– Apply RF => align to antiparallel. Remove RF => absorbed
energy is remitted and detected by Rfdetector.
f(x,y) is 2D image as before
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,)

g (s, )    f ( x, y) ( x cos  y sin   s)dxdy
(From Jain’s Fig.10.2)




f ( s cos   u sin  , s sin   u cos  )du
 s   cos 
where    
u   sin 
sin    x 
(coordinat e rotation)



cos    y 
This is a
transform
from 2D to
2D spaces
Tomography and
Reconstruction
Lecture Overview
1. Applications
2. Background/history of
tomography
3. Radon Transform
4. Fourier Slice Theorem
5. Filtered Back Projection
6. Algebraic techniques
•Measurement of Projection
data
•Example of flame
tomography
Applications & Types of Tomography
Medical Applications
Type of Tomography
Full 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)
MRI and PET showing
lesions in the brain.
PET scan on the brain
showing Parkinson’s
Disease
Applications & Types of Tomography – non medical
Non Medical
Applications
Type of Tomography
Oil Pipe Flow
Turbine Plumes
Resistive/Capacitance
Tomography
Flame Analysis
Optical Tomography
ECT on industrial pipe flows
CT or CAT Principles
• Computerized (Axial) Tomography
Radon again!
• introduced in 1972 by Hounsfield and Cormack
• natural progression from X-rays
From 2D to 3D !
• 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)
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
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.
Backpropagation
Principles
Backpropagation
We know that objects are
somewhere here in black
stripes, but where?
Example of Simple Backprojection
Reconstruction
• Given are sums, we have to reconstruct values
of pixels A, B, C and D
Image Reconstruction: ART or Algebraic
Reconstruction Technique
ART
CT - Reconstruction: ART or
Algebraic Reconstruction Technique
• METHOD 1: Algebraic Reconstruction Technique
– iterative technique
– attributed to Gordon
BackProjection
Initial Guess
Reconstructed
model
Projection
Actual Data
Slices
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)
x
f
Gf(r)
s
u
gf(s)
Spatial Domain
Frequency Domain
COMPARISON : CT -
FBP vs. ART
ART
FBP
Algebraic Reconstruction Technique
• Still slow
Filtered Back Projection
• Computationally cheap
• Clinically usually 500
projections per slice
• problematic for noisy
projections
• better quality for fewer
projections
• better quality for nonuniform project.
• “guided” reconstruct.
(initial guess!)
Fourier Slice
Theorem and
FFT review
Patient’s body is described by spatial
distribution of attenuation coefficient
Properties of attenuation
Our transform:
coefficient
f(x,y)  p(r,)
1. attenuation coefficient is used in CT_number of various tissues
2. These numbers are represented in HU = Hounsfield Units
CT_number uses attenuation
coefficients
RADON TRANSFORM
Properties
REMEMBER:
f(x,y)  p(r,)
Radon Transform is available in Matlab
1.
2.
3.
Radon and its inverse easy to use
You can do your own projects with CT reconstruction
Data are available on internet
sinogram
Inverse Radon Transform
Inverse Radon
Transform
Matlab
examples
Sinogram versus Hough
Review and notation – Fourier Transform of Image f(x,y)
In Matlab there are implemented functions that use Fourier Slice
Theorem
Matlab example
Matlab example: filtering
Low frequency
removed
High frequency
removed
Matlab example convolution
Remainder of main theorem of spectral imaging
Matlab example –
filtering by
convolution in
spectral domain
Radon Transform
and a Head
Phantom
Reconstructing
with more and
more rays
Example of Image Radon
Transform
[Y-axis] distance, [X-axis] angle
(From Matlab Image
Processing Toolbox
Documentation)
Matlab
Implementation
of Radon
Transform
No noise
big noise
The Lung Cancer
and the
reconstruction
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.
We want to reconstruct shape of the lungs
Starting Point
Border Detection
-At the moment two approaches are
available.
-Left the algorithm developed at Pisa
-Right the algorithm developed at
Lecce
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:
{Rr}( p, f ) 

 r(x, y)(x cos f  y sin f  p)dxdy

[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.
Image Interpolation - Graphical Representation (I)
y 0 d
 r(x, y,z )dy
Rz0 (x,90) 
0
y0
Rz0 (y,0) 
l
 r(x, y,z )dx
0
0

Image Interpolation - Graphical Representation (II)
Line Integrals
and Projections
1. We review the principle
2. Discuss various geometries
3. Show the use of filtering
Line Integrals and Projections
(t )1
•The function P P
= Radon
transform
P (t )1
•object function f(x,y).
The function P (t )1
is
known as the Radon
transform of the function
f(x,y).
P (t ) 


f ( x, y )ds
y
f ( x, y )

x
( ,t ) line
 
P (t ) 
  f ( x, y)( x cos  y sin   t )dxdy
  
x cos  y sin   t
x cos  y sin   t1
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
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 θ, is
shown.
P 1 (t )
y
P 2 (t )
f ( x, y )

x
Notation for
calculations in
these projections
Line Integrals and Projections
A simple diagram
showing the fan beam
P 1 (t )
projection
P 2 (t )
y
f ( x, y )

x
Fourier
Slice
Theorem
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 θ=0 which leads to
 j 2 ( ux vy )
F (u, v) 
f ( x, y )e
dxdy
v=0

 
  
F (u,0) 
• Define the projection at angle θ = Pθ(t)

f ( x, y )e  j 2ux dxdy
  
• Define its transform by

S ( w)   P (t )e  j 2wt dt

P (t ) 
f ( x, y )ds


( ,t ) line
 
P (t ) 
As the phase factor is no-longer
dependent on y, the integral can be
split.
  f ( x, y)( x cos  y sin   t )dxdy
  
Fourier Slice Theorem
As the phase factor is no-longer dependent on y, the integral can be split.

  j 2ux
F (u,0)     f ( x, y )dy e
dx
   


The part in brackets is the equation for a projection along lines of constant x

P 0 ( x) 
 f ( x, y)dy

Substituting
in

F (u ,0)   P 0 ( x)e  j 2ux dx

Thus the following relationship between the vertical projection and the 2D transform of
the object function:
F (u,0)  S 0 (u )
Fourier Slice Theorem Stanley and Kak
• Full details of
derivation, not for
now.
The Fourier Slice Theorem
The Fourier Slice theorem relates the
Fourier transform of the object along a
radial line.
P 1 (t )
t
Fourier transform
y
f ( x, y )
v

x
θ
u
Space
Domain
Frequency
Domain
The Fourier Slice Theorem
The Fourier Slice theorem relates the Fourier transform of the object along a radial line.
P 1 (t )
Collection of projections of
an object at a number of
angles
v
t
Fourier transform
y
f ( x, y )
v

u
x
θ
u
Space Domain
Frequency Domain
• 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.
Backprojection
of Radon
Transform
Backprojection of
Radon Transform
Ideal
cylinder
Blurred
edges
Crisp
edges
Filtered
backpropagation
creates crisp edges
Computerized
Tomography
Equipment
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
Evolution of CT technology
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 -> can be analyzed quantitatively
 adds enormously to the diagnostic information
 used in many large hospitals and medical centers throughout
the world
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.
Nuclear Magnetic
Resonance (NMR)
Magnetic Resonance
Imaging (MRI)
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
Magnetic Resonance Imaging
MRI - 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
MRI - Signal to Noise Ratio
• proton density pictures - measures H
MRI 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
PET/SPECT
Positron Emission
Tomography
Single Photon
Emission
Computerized
Tomography
PET/SPECT
• Positron Emission Tomography
Single 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
Ultrasound
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
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
Ultrasound (3)
•
•
•
•
•
by far least expensive
very safe
very noisy
1D, 2D, 3D scanners
irregular sampling reconstruction
problems
Typical Homework
Sources of slides and information
•
•
•
•
•
•
•
Badri Roysam
Jian Huang,
Machiraju,
Torsten Moeller,
Han-Wei Shen
Kai Thomenius
Badri Roysam