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Discrete Tomography and Its Applications in Discrete Tomography and Its Applications in Medical ImagingMedical Imaging**

Attila Kuba

Department of Image Processing and Computer Graphics

University of Szeged

*This presentation is based on the joint paper

G.T. Herman, A. Kuba, Discrete Tomography in Medical Imaging,

Proc. of IEEE 91, 1612-1626 (2003).

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OUTLINEOUTLINE

What is Discrete Tomography (DT) ? Application in Medical Imaging

AngiographySPECTPETEM

Discussion

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DISCRETE TOMOGRAPHY (DT)

Reconstruction of functions from their projections, when the functions have known discrete range D = {d1,...dk}

Example: CT image of homogeneous object object – made of wood,projections – X-ray images,function – absorption coefficient,

D = {dair,dwood:}: absorption coefficients

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WHY DISCRETE TOMOGRAPHY ?

use the fact that the range of the function to be reconstructed is discrete and known

Consequence:

in DT we need a few (e.g., 2-10) projections,

(in CT we need a few hundred projections)

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WHY DISCRETE TOMOGRAPHY?

# projs. FBP Discretized FBP DT method

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12

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THE RECONSTRUCTION PROBLEM

y = A f ,

where

f: vector representing the object,

y: vector of measurements,

A: matrix describing the projections

f1 f2

f3 f4

y1 = 1·f1 + 1·f2 + 0·f3 + 0·f4

y2 = 2·f1 + 0·f2 + 0·f3 + 2·f4

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RECONSTRUCTION METHOD

cost function to be minimized, e.g.

C(f) = || Af - y ||2

Task: find f such that C(f) is minimal

solution by optimization (e.g., simulated annealing)

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APPLICATIONS

HRTEM (High Resolution Transmission

Emission Microscopy)

NDT(Non-destructive testing)

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DT IN MEDICAL IMAGING

but the human body is not a homogeneous object,

it cannot even be considered to contain just a few homogenous regions

DT can be applied to medical imaging in special circumstances,

e.g. contrast material is injected (angiography), then

two regions:

contrast enhanced organ and

surrounding tissues

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OBJECT TO BE RECONSTRUCTED

object – function f(x)

2D , 3D, … - two, three, … variables

range: D = {d1, d2, …, dc} known

binary object/image/function D = {0,1} c = 2

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OBJECT TO BE RECONSTRUCTED

a priori knowledge:

F : class of functions f (having discrete, known range) that

may describe an object in that application area

examples:

convex sets,

functions having constant values on closed 3D regions with triangulated boundary surfaces,

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PROJECTIONS

projections – integrals (e.g., along straight lines)

SgdxxfSfS

Pf(x)

S

g(S)

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PROJECTIONS

binary object (a (0,1)-matix),

its horizontal and vertical projections (row and column sums)

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PROJECTIONS

other kinds of projections

fan-beam

cone-beam (3D)

strips

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PROJECTIONS

if f defined on a discrete domain (e.g., digital image on pixels/voxels):

f1

fI

gj

Jjgfai

jiij ,...,1,

gAf

linear equation system

projection data y ≈ g, (an approximation to g)

available from measurements,

Af ≈ y

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RECONSTRUCTION PROBLEM

Let F be a class of functions having discrete, known range D

Given: The projections data y(S)

Task: Find a function f in F such that

[P f](S) ≈ y(S)

in the case of CT the class F is more general, e.g., D = [0,+∞)

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RECONSTRUCTION METHODS

Heuristic

Discretization of classical reconstruction methods

Optimization

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HEURISTIC RECONSTRUCTION METHODS

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DISCRETIZATION OF CLASSICAL RECONSTRUCTION METHODS

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RECONSTRUCTION METHODSBASED ON OPTIMIZATION

Af = g ≈ y

big size

under determined (not enough data)

contradiction (measurement errors, no solution)

instead of exact solution minimization of a cost function:

C = ║Af - y║2 + Φ(f)

how far is an f

from the measurements

how undesirable

is a solution f

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REGULARIZATION

C = ║Af - y║2 + γ·║f║2

different selections of C

γ regularization parameter

C = ║Af - y║2 + γ·║f – f(0)║2

f(0) a given prototype (a similar object)

C = ║Af - y║2 + γ·∑ici·fi

ci weight of cost of the position i

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OPTIMIZATION METHODS

ICM (Iterated Conditional Modes)

a local descent-based method

SA (Simulated Annealing)

initialization: let f be in Finteration: change f to reach less cost C

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COMPARISON OF DT IMAGES

%100'

'2

%100'

'2

%100'

ii

ii

ii

ii

i

ii

ff

ffVE

ff

ffSE

f

ffRrelative mean error

shape error

volume error

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ANGIOGRAPHY

digital subtraction angiography

object with two homogeneous regions: contrast medium having known absorption value (μcontrast), background (μ=0)

it can be reduced to a binary object (μ = 0,1)

a small number of projections can be taken, e.g., 2

biplane angiography

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ANGIOGRAPHY OF CARDIAC VENTRICLES

Chang, Chow 1973

clay model of a dog’s heart

two X-ray projections

cross-sections: convex, symmetric polygons

heuristic reconstruction method

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ANGIOGRAPHY OF CARDIAC VENTRICLES

Onnasch, Heintzen 1976

heart ventricles

two X-ray projections

prior information: similar neighbor slices, similar to 3D model

heuristic reconstruction method

enddiastolic RV cast

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Onnasch, Prause, 1999

ANGIOGRAPHY OF CARDIAC VENTRICLES

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Onnasch, Prause, 1999

ANGIOGRAPHY OF CARDIAC VENTRICLES

29Onnasch, Prause, 1999

ANGIOGRAPHY OF CARDIAC VENTRICLES

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ANGIOGRAPHY OF CORONARY ARTERIES

coronary arterial segments from two projectionsReiber, 1982

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Human iliac bifurcation, elliptic-based model

Pellot, 1994

ANGIOGRAPHY OF CORONARY ARTERIES

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PET

Bayesian reconstruction and use of anatomical a priori informationBowsher, 1996

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SPECT

MAP reconstructions of the bolus boundary surfaceCunningham, Hanson, Battle, 1998

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ET

Phantom FBP DT

Chan, Herman, Levitan, 1998

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SPECT

MAP reconstructions of the bolus boundary surfaceCunningham, Hanson, Battle, 1998

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SPECT

Tomography reconstruction using free-form deformation modelsFFDs reconstructions for different levels of noise

Battle, Bizais, Le Rest, Turzo, 1999

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SPECT

Tomography reconstruction using free-form deformation modelsFFDs reconstructions for different levels of noise

Battle, 1999

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PET

Edge-preserving tomography reconstruction with nonlocal regularization.Emission phantom, FBP, Huber penalty, proposed penalty

Yu, 2002

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DIscrete REConstruction Tomography

software tool for

generating/reading projections

reconstructing discrete objects

displaying discrete objects (2D/3D)

available via Internet

http://www.inf.u-szeged.hu/~direct/

it is under development

E-mail: direct@inf.u-szeged.hu

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DISCUSSIONDISCUSSION

If the object to be reconstructed consists of known materials, then DT can be applied.

DT offers new theory and techniques for reconstructing images from less number of projections.

Further experiments are necessary (including e.g. fan-beam projections, scattering)