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Data Acquisition, Representation and Reconstruction of medical images. Application of Advanced Spectral Methods. Acquisition Methods for medical images. X-Rays Computer Tomography (CT or CAT) MRI (or NMR)

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Data acquisition representation and reconstruction of medical images l.jpg

Data Acquisition, Representation and Reconstruction of medical images

Application of Advanced Spectral Methods

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Acquisition Methods for medical images 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 medical images

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X-Rays - medical imagesPhysics

  • 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

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X-Rays medical images

  • X-Rays

  • similar to visible light, but higher energy!

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X-Rays - medical imagesVisibility

  • 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

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X-Rays - medical imagesImages

X-Rays can be used in computerized tomography

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Non-Intrusive Medical Diagnosis based on Computerized Tomography

  • Computer tomography CT

(From Jain’s Fig.10.1)

An X-ray CT scanning system

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Non-Intrusive Medical Diagnosis based on TomographyTransmission Tomography

Source and Detector are rotating around human’s body

(From Bovik’s Handbook Fig.10.2.1)

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Non-Intrusive Medical Diagnosis based on projections Tomography

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 structurefrom the projections

“Computerized Tomography” (CT)

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Non-Intrusive Medical Diagnosis based on TomographyEmission 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

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Radon Transform Principles Tomography

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 Jain’s Fig.10.2)

This is a transform from 2D to 2D spaces

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Tomography Tomography and Reconstruction

Lecture Overview


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 Tomography

MRI and PET showing lesions in the brain.

PET scan on the brain showing Parkinson’s Disease

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Applications & Types of Tomography – Tomographynon medical

ECT on industrial pipe flows

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CT or CAT - TomographyPrinciples

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 - TomographyMethods

measures the attenuation of X-raysfrom 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 Tomography

  • 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 Tomography


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Backpropagation Tomography

We know that objects are somewhere here in black stripes, but where?

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Example of Simple TomographyBackprojection Reconstruction

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

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Image TomographyReconstruction: ART or Algebraic Reconstruction Technique


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CT - TomographyReconstruction: ART or Algebraic Reconstruction Technique

METHOD 1: Algebraic Reconstruction Technique

iterative technique

attributed to Gordon

Initial Guess




Actual DataSlices

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CT - TomographyReconstruction: FBP Filtered Back Propagation

METHOD 2 : Filtered Back Projection

common method

uses Radon transformandFourier Slice Theorem










Spatial Domain

Frequency Domain

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COMPARISON Tomography: CT - FBP vs. ART

Computationally cheap

Clinically usually 500 projections per slice

problematic for noisy projections



Algebraic Reconstruction Technique

  • Still slow

  • better quality for fewer projections

  • better quality for non-uniform project.

  • “guided” reconstruct. (initial guess!)

Filtered Back Projection

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Fourier Slice Theorem and FFT review Tomography

Patient’s body is described by spatial distribution of attenuation coefficient

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Properties of attenuation coefficient Tomography

Our transform: f(x,y)  p(r,)

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attenuation coefficient is used in TomographyCT_number of various tissues

These numbers are represented in HU = Hounsfield Units

CT_number uses attenuation coefficients

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RADON TRANSFORM Properties Tomography

REMEMBER: f(x,y)  p(r,)

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Radon Transform is available in TomographyMatlab

  • Radon and its inverse easy to use

  • You can do your own projects with CT reconstruction

  • Data are available on internet


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Inverse Radon Transform Tomography

Inverse Radon Transform

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Matlab Tomography examples

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In TomographyMatlab there are implemented functions that use Fourier Slice Theorem

Matlab example

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Matlab Tomography example: filtering

High frequency removed

Low frequency removed

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Matlab Tomography example - convolution

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Remainder of main theorem of spectral imaging Tomography

Matlab example – filtering by convolution in spectral domain

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Example of Image Radon Transform Tomography

[Y-axis] distance, [X-axis] angle

(From Matlab Image Processing Toolbox Documentation)

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big noise Tomography

No noise

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The Lung and The CTs Tomography


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


1.Small Cell Lung Cancer (SCLC) - 20% of all lung cancers

2.Non Small Cell Lung Cancer (NSCLC) - 80% of all lung cancer


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 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 Tomography

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 Tomography


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 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:


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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Line Integrals and Projections Tomography

We review the principle

Discuss various geometries

Show the use of filtering

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Line Integrals and Projections Tomography

The function is known as the Radon transform of the function f(x,y).

  • The function P = Radon transform

  • object function f(x,y).

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Fan Beams Tomography

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 Tomography

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.

Notation for calculations in these projections

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Line Integrals and Projections Tomography

A simple diagram showing the fan beam projection

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Fourier Tomography



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Fourier Slice Theorem Tomography

  • 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 v=0

  • Define the projection at angle θ = Pθ(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 Tomography

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 Tomography Stanley and Kak

  • Full details of derivation, not for now.

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The Fourier Slice Theorem Tomography



The Fourier Slice theorem relates the Fourier transform of the object along a radial line.


Fourier transform


Space Domain

Frequency Domain

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The Fourier Slice Theorem Tomography





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


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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Ideal cylinder Tomography

Blurred edges

Crisp edges

Filtered backpropagation creates crisp edges

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CT - Tomography2D 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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CT or CAT - TomographyAdvantages

  • 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

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CT or CAT - TomographyDisadvantages

  • 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) Tomography

Magnetic Resonance Imaging (MRI)

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MRI Tomography

  • 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 Imaging TomographyMRI - 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 - TomographySignal 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 Tomography

Positron Emission TomographySingle Photon Emission Computerized Tomography

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PET/SPECT Tomography

  • 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 Tomography

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Ultrasound Tomography

  • 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) Tomography

  • 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) Tomography

  • by far least expensive

  • very safe

  • very noisy

  • 1D, 2D, 3D scanners

  • irregular sampling - reconstruction problems

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Typical Homework Tomography

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Sources of slides and information Tomography

  • Badri Roysam

  • Jian Huang,

  • Machiraju,

  • Torsten Moeller,

  • Han-Wei Shen

  • Kai Thomenius

  • Badri Roysam