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
Application of Advanced Spectral Methods
X-Rays can be used in computerized tomography
Computerized (Axial) Tomography
CT (CAT) scanners and relevant mathematics
(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)
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
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
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
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,)
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
[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).
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
Start by defining the 2D Fourier transform of the object function as
For simplicity θ=0 which leads to v=0
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:
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
Space Domain
Frequency Domain
Ideal cylinder
Blurred edges
Crisp edges
Filtered backpropagation creates crisp edges
Evolution of CT technology
Nuclear Magnetic Resonance (NMR)
Magnetic Resonance Imaging (MRI)
Positron Emission TomographySingle Photon Emission Computerized Tomography