mipr lecture 7 copyright oleh tretiak, 2004 1 medical imaging and pattern recognition lecture 7...

MIPR Lecture 7Copyright Oleh Tretiak, 2004

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Medical Imaging and Pattern Recognition

Lecture 7 Computed Tomography

Oleh Tretiak


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Computer Tomography:How It Works

Only one plane is illuminated. Source-subject motion provides added information.


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How it Works

• Original CT Scanner– Head only– One minute scanning time– Two sections– Ten minutes to compute images– Extremely successful!


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First CT Scanner


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Before and After CT


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Contemporary CT


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Fan-Beam Computer Tomography


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Contemporary Spiral Scanner

• Configuration:– 40 slices per rotation maximum

• Other options are 32 slices or 16 slices

– 40 mm axial distance scanned in one rotation

– 0.4 sec per rotation– 60 kW generator


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Example: Head

• Bleeding due to injury

• Can cause brain injury if not treated

• Blood between brain and dura, easy to treat


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Example: Head

• FRONTAL CONTUSION WITH SUBARACHNOID HEMORRHAGE


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Chest Study

• Pneumothorax (air between lung and chest)

• Also note the bilateral lower lobe consolidation of lungs, right being greater than left. There is a chest tube within the right hemithorax.


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Abdomen

• Appendicitis (arrow)

• Contrast agents in stomach and in blood


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Mathematics of Computed Tomography

• Model for measurements• Direct problem• Inverse problem• Algorithm for computed

tomography


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Direct Problem

• Beam with intensity I0 enters body with varying attenuation

• Each layer has thickness t

m m m mI I

m m m mI II I I

€

€

Ia = e−μ1tI0, Ib = e−μ 2tIa ,

Ic = e−μ 3tIb , I1 = e−μ 4 tIc

€

I1 = e−μ 4 te−μ 3te−μ 2te−μ1tI0 = e−(μ1 +μ 2 +μ 3 +μ 4 )tI0ln(I0 /I1) = (μ1 + μ2 + μ3 + μ4 )t


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Integral Equation

x

I (y)I

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I1(y) = exp(− μ(x,y)dx∫ )I0

ln(I0 /I1(y)) = μ(x,y)dx∫

x

I (t, q)

I

y

q

t

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ln(I0 /I1(t,θ)) = μ(t cosθ − lsinθ, t sinθ + lcosθ)dl∫


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Radon Transform

x

I (t, q)

I

y

q

t

f(x,y)

g(t, )

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g(t,θ) = f (t cosθ − lsinθ, t sinθ + lcosθ)dl∫


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Inverse Radon Transform

• Given: X-ray transmission measurements I1(t, ). Find (x, y)

• Given: g(t, ). Find f(x, y)• Method:

– (a) convolution

– (b) backprojection

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f (x,y) = g1(x cosθ + y sinθ,θ)dθ0

π

∫€

g1(t,θ) = h(t − τ )g(τ ,θ)dτ∫


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Example

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f(x, y)

Lines: g(t, ), same for all Dots: g1(t, ), after convolution


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Backprojection

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One, two, and four angles of backprojection

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More Backprojection

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8, 15, and 30 angle backprojection


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Pictures

f(x, y) g(t, Theta horizontal


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Backprojection at 4, 16, and 100 angles


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History

• 1917, Joachim Radon– Solved formal inverse problem.

Interest in theory of integration and geometry

• 1958, Simeon Tetelbaum of KPI publishes a paper about X-ray tomography. – Publishes valid inverse problem

solution.


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More History

• 1963 John Cormack publishes theoretical and experimental results.– Experiment with cylindrical objects

• 1972 Godfrey Hounsfield develops CT scanner

• 1979 Hounsfield and Cormack receive Nobel prize in Medicine


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Example of Contrast

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

12 bit image, full contrast range.

Window for lowdensities


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More Contrast Operations

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Window for highdensities


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3-D Images

• Spiral scan procedures produce sets of sectional images suitable for 3-D imaging

• Resectioning: Compute new section plane

• Projection: Compute sums along rays• Rendering: Segment image and show

surface.


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Bronchoscopy

Path View


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Colonoscopy

Path View


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Medical Practice

• In the fall of 2003 Siemens became the first CT supplier ever to receive clearance from the FDA for a computer-aided technique of identifying nodules, that is, possible tumors, in the lung. CT is also used for the diagnosis of colon cancer: A virtual flight through the human colon can detect even the smallest polyps. If these are removed in time, an outbreak of colon cancer can very probably be prevented.


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Comparison

Left: A polyp seen with optical endoscopy. Right: View in virtual endoscopy.


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Summary

• Computer tomography became successful because it showed soft tissue differences that could not be seen on X-rays.

• Evolution of high-speed (spiral scan) machines came about through improvements in X-ray detectors

• This has led to 3-D imaging methods– Surgery planning– Virtual endoscopy

mipr lecture 7 copyright oleh tretiak, 2004 1 medical imaging and pattern recognition lecture 7...

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