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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= density of 25

= density of 50

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ρ(x,y)

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x€

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= 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

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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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Pθ = 0(t)

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θ

= density of 25

= density of 50

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Our Second Projection€

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Pθ = 45(t)

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θ =45o

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θ

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Our Third Projection

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Pθ = 90(t)

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θ =90o

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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.