Post on 09-Feb-2016
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Image Processing Project Tomographic Image Reconstruction
Introduction
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This presentation will cover a brief description of tomographic imaging, image reconstruction methods, and design challenges.
1. Tomography
3. Design Challenges 4. Your Project
2. Image Reconstruction
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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.
Genes
Proteins
Cells
Tissue
Organs
Organism
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Tomography
X-ray CT Scanner X-ray CT Images of the Human Abdomen
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Tomograms can be created using a variety of physical mechanisms X-ray Attenuation
Nuclear Magnetic Resonance
Position-Electron Annihilation
Ultrasound Interactions
Tomography means to image using sections or slices, tomo is Greek for cut, graph means form (plot) an image.
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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= density of 25
= density of 50
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ρ(x, y)€
x€
y= density of 0
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ρ(x1,y1) = 50€
x€
y
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x1
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y1 = density of 25
= density of 50
= density of 0
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€
ρ(x, y)€
x€
y
= density of 25
= density of 50
= density of 0
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€
ρ(x,y)
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x€
y
= density of 25
= density of 50
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€
ρ(x,y)= density of 25
= density of 50
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€
ρ(x,y)
Our First Projection
= density of 25
= density of 50
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€
ρ(x,y)
Our First Projection
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t
€
P(t)
= density of 25
= density of 50
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€
ρ(x,y)
Our First Projection
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t
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P(t)
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θ
= density of 25
= density of 50
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€
ρ(x,y)
Our First Projection
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t
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Pθ = 0(t)
= density of 25
= density of 50
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θ
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€
ρ(x,y)
Our First Projection
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t
€
Pθ = 0(t)
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t
€
θ
= density of 25
= density of 50
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θ
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Our Second Projection€
t
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Pθ = 45(t)
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θ =45o
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t
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θ
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Our Third Projection
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t€
Pθ = 90(t)
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θ =90o
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t
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θ
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t
Sinogram€
θ
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Sinogram€
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Sinogram€
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Sinogram€
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Sinogram€
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Sinogram€
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Sinogram€
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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
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t€
Pθ (t)
Filter
Backprojection
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Backprojection
Backprojection “smears” the data back.
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f (x,y)
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ˆ f (x,y)
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Backprojection
Backprojection “smears” the data back.
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f (x,y)
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ˆ f (x,y)
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Backprojection
Backprojection “smears” the data back.
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f (x,y)
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ˆ f (x,y)
infinite number of projections
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Filtered-Backprojection
If we filter the projections before backprojection we can recover the original object.
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f (x, y)
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ˆ 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:
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g(t) = (P∗h)(t) = P(τ )h(t − τ )−∞
∞
∫ ∂τ
filtered signal
convolution operator
filter kernel
input signal (projection)
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TPS: Think-Pair-Share Match up the following Sinograms with their objects
S1
S2
S3
O1
O2
O3
O? = S?
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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
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CNR = μ1 − μ2
σ 12 + σ 2
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Region 1
Region 2€
μ1 = 250.1σ 1 = 70.3
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μ2 =191.9σ 2 = 206.1
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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
where N is the number of projections and d is the (fixed) dose per projection.
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D(N) = N * d
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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.