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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

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

- 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
- similar to visible light, but higher energy!

- 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 can be used in computerized tomography

Computerized (Axial) Tomography

CT (CAT) scanners and relevant mathematics

- Computer tomography CT

(From Jain’s Fig.10.1)

An X-ray CT scanning system

Source and Detector are rotating around human’s body

(From Bovik’s Handbook Fig.10.2.1)

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)

- 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

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

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

Applications & Types of Tomography

MRI and PET showing lesions in the brain.

PET scan on the brain showing Parkinson’s Disease

Applications & Types of Tomography – non medical

ECT on industrial pipe flows

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 !

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

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?

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

Image Reconstruction: ART or Algebraic Reconstruction Technique

ART

METHOD 1: Algebraic Reconstruction Technique

iterative technique

attributed to Gordon

Initial Guess

Reconstructedmodel

Back-Projection

Projection

Actual DataSlices

METHOD 2 : Filtered Back Projection

common method

uses Radon transformandFourier Slice Theorem

y

F(u,v)

f(x,y)

Gf(r)

x

s

u

gf(s)

f

Spatial Domain

Frequency Domain

Computationally cheap

Clinically usually 500 projections per slice

problematic for noisy projections

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

Fourier Slice Theorem and FFT review

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

Properties of attenuation coefficient

Our transform: f(x,y) p(r,)

attenuation coefficient is used in CT_number of various tissues

These numbers are represented in HU = Hounsfield Units

CT_number uses attenuation coefficients

RADON TRANSFORM Properties

REMEMBER: f(x,y) p(r,)

- 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

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

High frequency removed

Low 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

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

(From Matlab Image Processing Toolbox Documentation)

Matlab Implementation of Radon Transform

big noise

No noise

[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

Border Detection

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

[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.

Line Integrals and Projections

We review the principle

Discuss various geometries

Show the use of filtering

Line Integrals and Projections

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

- The function P= Radon transform
- object function f(x,y).

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.

Notation for calculations in these projections

Line Integrals and Projections

A simple diagram showing the fan beam projection

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 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.

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:

- Full details of derivation, not for now.

The Fourier Slice Theorem

v

u

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

t

Fourier transform

θ

Space Domain

Frequency Domain

The Fourier Slice Theorem

v

v

u

u

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

θ

- 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

Ideal cylinder

Blurred edges

Crisp edges

Filtered backpropagation creates crisp edges

- 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

- 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

- 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)

- 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

- 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

- 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

Positron Emission TomographySingle Photon Emission Computerized 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

- 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

- 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

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

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