Image Processing Project
Tomographic Image Reconstruction
Introduction
This presentation will cover a brief description of tomographic
imaging, image reconstruction methods, and design challenges.
1. Tomography
2. Image Reconstruction
3. Design Challenges
4. Your Project
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Background: Importance of Medical Imaging
 There are an estimated 630,000
imaging procedures every week in
the US.
 The average radiologist has a
case load around 35% higher than
just 5 years ago.
“Molecular imaging holds great promise for early detection and
treatment of numerous diseases, for providing researchers with
detailed information about cellular physiology and function, and
for facilitating the goal of personalized medicine.”
– NIH roadmap
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Modern imaging modalities cover the EM
spectrum and all scales of resolution.
Organism
Organs
Tissue
Cells
Proteins
Genes
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Tomography
X-ray CT Scanner
X-ray CT Images of the
Human Abdomen
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Tomography means to image using sections or slices, tomo
is Greek for cut, graph means form (plot) an image.
 Tomograms can be created using a
variety of physical mechanisms

X-ray Attenuation

Nuclear Magnetic Resonance

Position-Electron Annihilation

Ultrasound Interactions
 Modalities
 X-ray Computed
Tomography
 Magnetic Resonance
Imaging
 Positron Emission
Tomography
 Ultrasound
Interactions
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To do Tomography, we need to have many projections
from different angles.
 Transforms the object we are imaging to a Sinogram
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Forming the Sinogram
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y
= density of 0
= density of 25
x

= density of 50
(x, y)
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y
= density of 0
y1



= density of 25
x1
x
= density of 50
(x
1,y1)  50
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y
= density of 0
= density of 25
x

= density of 50
(x, y)
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(x, y)
= density of 25
= density of 50
y

x


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(x, y)
= density of 25
= density of 50

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(x, y)
= density of 25
= density of 50

Our First Projection
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(x, y)
= density of 25
= density of 50
P(t)


t
Our First Projection

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(x, y)

= density of 25
= density of 50
P(t)



t
Our First Projection

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(x, y)

= density of 25
= density of 50
P  0 (t)



t
Our First Projection

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= density of 25
(x, y)

= density of 50
P  0 (t)

t


t


Our First Projection


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P  45 (t)


  45

o
t
t


Our Second Projection

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P  90 (t)
t
  90
o
t




Our Third Projection

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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
25
t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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t



Sinogram
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Given all these projections, how do we reconstruct the
tomogram?
 Filtered Backprojection
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Image Reconstruction Using Filtered
Backprojection
P (t)
Filter

t

Backprojection
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Backprojection

 Backprojection “smears” the data back.
fˆ (x, y)
f (x, y)

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Backprojection

 Backprojection “smears” the data back.
fˆ (x, y)
f (x, y)

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Backprojection
 Backprojection “smears” the data back.
infinite number of projections
fˆ (x, y)
f (x, y)

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Filtered-Backprojection
 If we filter the projections before backprojection we
can recover the original object.
fˆ (x, y)
f (x, y)

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Filtering the projection

 Filtering is a basic operation in signal processing.
 Spatial or Temporal domain filtering (convolution)
 Frequency domain filtering
 Lets consider spatial domain filtering:
g(t)  (P  h)(t) 

 P( )h(t   )
input signal
(projection)

filtered
signal
filter kernel
convolution
operator
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TPS: Think-Pair-Share
 Match up the following Sinograms with their objects
O1
S1
O? = S?
O2
S2
O3
S3
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Design Challenges
 In cases where the projection is taken using x-rays,
there is a risk associated.

x-rays absorbed by the body can cause damage to DNA
directly or through the formation of free radicals.
 The larger the number of projections and the longer
the x-rays are on, the higher the dose delivered to the
patient.
 There is a fundamental tradeoff between dose, the
number of projections, and the noise in the
projections.
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Basics of Radiation Biology
 We are constantly exposed to naturally occurring
radiation (radon, cosmic rays)


about 3 milli-Sieverts per year
there is some evidence that anti-oxidants, found in fruits and
vegetables, can protect cells from free radical formation
 A single chest x-ray is equivalent to about 10 days of
natural exposure
 A whole-body x-ray CT exam is equivalent to about 3
years of equivalent natural exposure
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Radiation Monitoring
 OSHA requires all those who work with radiation to
be badged and monitored for dose.
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Noise in X-rays
 X-ray images are formed by counting the number of
photons leaving the subject compared to those
entering.
 The count is a random variable that is Poisson
distributed.
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Dose and Image Quality
 The longer the X-rays are on the lower the noise level
but the higher the dose.
 Image Quality
 Contrast-to-Noise ratio
 improve by averaging multiple projections
 Total Dose
 Number of projections*dose per projection
 increases linearly with the number of averages
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Contrast to Noise Ratio
 CNR measures how much contrast compared to
noise there is in the reconstructed image
1  2
CNR 
2
2
1   2
Region 1

Region 2
1  250.1
1  70.3
2 191.9
 2  206.1

CNR  3.5
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Total Dose
 The dose delivered is a complex formula that
depends on the geometry and the exact
characteristics of the x-rays (energy etc).
 To compare different tomographic image acquisitions
we will use a simple formula to estimate the total
dose
D(N)  N * d
where N is the number of projections and d is the (fixed)
dose per projection.

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Your Project
 There are two parts to this project.
 Part I
 Generating projections - students will use Matlab to simulate
projection images from a phantom with varying numbers of
projections.
 Image Reconstruction - students will use Matlab to
reconstruct the images using filtered backprojection.
 You will also use Matlab to add noise to the projections to
simulate the effect of photon counting and explore the effect
on image reconstruction quality.
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Your Project
 There are two parts to this project.
 Part II
 Given two sinograms, one with a simulated “tumor”, you will
experimentally determine which sinogram contains the tumor
using different reconstruction filters.
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