Download Image Reconstruction

CLRS 408: Introduction to Computed Tomography
VCU Department of Radiation Sciences
Preprocessing
Reconstruction
algorithm
Image matrix of
CT numbers
For the computer to reconstruct an image in CT,
x-ray tube and detectors must rotate around the
patient at least 180°
For adequate image reconstruction, computer
must receive sufficient x-ray transmission values
or attenuation data
Algorithm that uses attenuation data measured by the
detectors to build up image for viewing / interpretation
Newer scanners generate better quality images because
they collect more attenuation data over 360°
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Linear attenuation coefficient, µ, between x–ray
tube and detectors
Attenuation coefficient is a measure of how
rapidly x-rays are absorbed
within material
2-D views = “projections” at angles all the way
around the patient
Rotate tube and detectors around patient
Sample µ at each detector for each rotation angle
Generate series of projections
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A set of rules or directions for getting a specific
output from a specific input
Necessary to solve the problems presented during the
data acquisition and image reconstruction process of CT
Rules describe simple operations that are well defined
Must terminate after a finite number of steps
For CT
To calculate attenuation coefficients
To reconstruct the data into an image
Digital image processing technique to modify
images thru filter function
Multiplication of overlapping
portions of filter function and
the detector response curve
selectively to produce a third
function, used for image
reconstruction
Image from Image Enhancement and Restoration Part 2
Mathematical technique to estimate the value of a
function from known values on either side of the
function
Helical CT sample spacing and interpolation. If data for desired slice of
thickness d (dark gray bar in figure) are interpolated between
equivalent rays from adjacent helical rotations (loops) with pitch of 1.5,
samples will be 1.5 d apart along z-axis (e.g., 10.5 mm apart for 7-mm
thickness). Larger spacing means greater chance that interpolated
estimate is in error. If 180-opposed rays are included, measurements
average half as far apart (and are more likely to actually lie within
slice). det = detector. From JNMT Principles of CT and CT Technology
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Mathematical function that converts a signal in the
spatial domain to a signal in the frequency domain
Analytical tool used to reconstruct images of a
patient’s anatomy in CT and MRI
Divides waveform (sinusoid) into series of sine and cosine
functions of different frequencies and amplitudes
These components can then be separated
In imaging, when beam
of x rays passes thru a
pt, an image profile denoted
by f(x) is obtained
Projection set of ray sums generated
as the x-ray tube and detector scan the
subject simultaneously
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• Total distribution of attenuation
coefficients in the object O is µ (x,y)
• The problem is to calculate µ (x,y)
from a set of projections specified
by angle θ and I represents beam
intensities from the source and
at the detector
• A projection is given by the line
integral of µ(x,y):
Image A: Parallel beam geometry used in the
first CT scanners
Image B: Fan beam geometry was introduced
to acquire the projection data faster than
parallel beam geometries
Image reconstruction from projections involves
several algorithms to calculate all the µ terms in
the Lambert-Beer equation from a set of
projection data
1.Back - projection
2.Iterative reconstruction
3.Analytic reconstruction
Filtered Back - Projection (convolution method)
Adaptive Statistical Iterative Projection
Fourier Reconstruction
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Summary of multiple projections to produce image
Does not produce a sharp image of the object and
therefore is not used in modern clinical CT
Characterized by star pattern artifact
air
metal
Does not produce sharp image of object
Io
Io
Io
µ1
µ2
I1
x
Io
µ3
I3
I1 = Ioe-(µ1 + µ2)x
I2 = Ioe-(µ3 + µ4)x
µ4
x
I2
I4
I3 = Ioe-(µ1 + µ3)x
I4 = Ioe-(µ2 + µ4)x
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“Starts with an assumption and compares this
assumption with measured values, makes
corrections to bring the two into agreement,
and then repeats this process over and over
until the assumed and measured values are
the same or within acceptable limits”
Initially used by Hounsfield
in 1st CT scanner
Developed to overcome the limitations of
back-projection and iterative algorithms and
are used in modern CT scanners
Types:
Filtered Back - Projection (convolution
method)
Adaptive Statistical Iterative Projection
Fourier Reconstruction
Also known as Convolution Method
Commonly used in CT today
Projection profile is filtered or convoluted to
remove the star-like blurring found in simple
back-projection techniques
Applies digital or electronic filter to projection profiles
before back projection is done
Removes blurring around edges; makes image sharper
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1. All projection profiles are obtained.
2. The logarithm of the data is obtained.
3. Logarithmic values are multiplied by a digital filter
(convolution filter) to generate a set of filtered
profiles.
4. The filtered profiles are then back-projected.
5. The filtered projections are summed and the
negative and positive components are canceled,
which produces image free of blurring.
Not filtered
Filtered
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A, Back-projection results in an
unsharp image.
B, Filtered back-projection uses a
digital filter (a convolution filter) to
remove this blurring, which
produces a sharp image.
Here is an example of streak artifacts from the
metal in bilateral hip replacements (top).
When the metal artifacts are reduced using
Metal Deletion Technique (right), enlarged
lymph nodes around the rectum are seen
much more clearly.
Builds on the FBP algorithm
Starts reconstruction after 1st-pass FBP,
and shortens reconstruction time
From Aunt Minnie’s Iterative Reconstruction Proves Its Worth
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Advantages:
Maintains much lower image noise than if the
same raw data were reconstructed with FBP
alone
Reduces quantum noise substantially with no
impact on spatial or contrast resolution
Allows lower dose
Disadvantages:
Limited by computer hardware/software power
From Medscape’s Iterative Reconstruction
Equal to Filtered Back Projection
From Clinica l applications of Cardiac CT angiography
Used in MRI but not in modern CT because it
requires more complicated mathematics
than the FBP algorithm
The image in the frequency domain can
be manipulated by changing amplitudes
of the frequency components
Computer performs manipulations
Frequency information can be used to
measure image quality
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Figure 6-8 CT image represented in the spatial domain by the function f(x,y). This can be
transformed to an image in the frequency domain F(u,v) with use of the Fourier transform. In
addition, F(u,v) can be retransformed into f(x,y) with use of the inverse Fourier transform.
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Measurement Data
Raw Data
Convolved Data
Image Data
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A.K.A. Scan Data
Collected from detectors
Preprocessed to correct errors such as
beam hardening, bad detectors, or
scattered radiation
If errors are not corrected, will cause
poor image quality and
Measurement Data
generate image artifacts
Raw Data
Convolved Data
Image Data
Result of preprocessed scan data
Subjected to filter / convolution algorithm
Can be stored and retrieved, but is huge
amount of data
Measurement Data
Raw Data
Convolved Data
Image Data
Raw data filtered using mathematical filter or
kernel
Improves image quality by removing blurring
Convolution kernels can only be applied to
the raw data.
Measurement Data
Raw Data
Convolved Data
Image Data
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Reconstructed Data
Convolved data that have been
back-projected into image matrix
Digital filters suppress noise; improve detail
Standard, Smoothing, Edge Enhancement
algorithms
Measurement Data
Raw Data
Convolved Data
Image Data
Detectors
Measurement Data
Preprocessing
Raw Data
Convolution
Convolved Data
Back Projection
Image Data
FBP based on single row of detectors
Interpolation of data
Estimates values from known values on either side
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Multiple detectors (<16)
Multi-slice algorithms
Variable slice thickness
Longitudinal interpolation (Z axis) and FBP
Y
3D Localization
X-axis – from left to right
Y-axis – from front to back
Z-axis – from head to foot
Z
X
Multiple detectors (16+)
Cone beam – requires different algorithms
As beam is opened up to cover multiple detectors, becomes more
angled at edges (cone angle)
Cone-beam algorithms to eliminate associated
artifacts
Streaking and density change artifacts caused by use of fan-beam
algorithms
Artifacts are due to the plane
of interest being projected
onto several detector rows
Surface display and volume
reconstruction software
Based on graphics/visual
perception programs
Powerful computer
hardware/software
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This completes the material for Exam 2: Ch 4, 5, 6.
Be sure to go through the objectives and complete the
quiz questions during the module to be prepared.
Please let me know if you have any questions!
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