1 Discrete Tomography and Its Applications in Medical 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, (2003).

2 OUTLINE What is Discrete Tomography (DT) ? Application in Medical Imaging  Angiography  SPECT  PET  EM Discussion

3 DISCRETE TOMOGRAPHY (DT) Reconstruction of functions from their projections, when the functions have known discrete range D = {d 1,...d k } Example: CT image of homogeneous object object – made of wood, projections – X-ray images, function – absorption coefficient, D = {d air,d wood :}: absorption coefficients

4 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)

5 WHY DISCRETE TOMOGRAPHY?

6 THE RECONSTRUCTION PROBLEM y = A f, where f: vector representing the object, y: vector of measurements, A: matrix describing the projections f1f1 f2f2 f3f3 f4f4 y 1 = 1·f 1 + 1·f 2 + 0·f 3 + 0·f 4 y 2 =  2·f 1 + 0·f 2 + 0·f 3 +  2·f 4

7 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)

8 APPLICATIONS HRTEM (High Resolution Transmission Emission Microscopy) NDT (Non-destructive testing)

9 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

10 OBJECT TO BE RECONSTRUCTED object – function f(x) 2D, 3D, … - two, three, … variables range: D = {d 1, d 2, …, d c } known binary object/image/function D = {0,1} c = 2

11 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,

12 PROJECTIONS projections – integrals (e.g., along straight lines) f(x)f(x) S g(S)g(S)

13 PROJECTIONS binary object (a (0,1)-matix), its horizontal and vertical projections (row and column sums)

14 PROJECTIONS other kinds of projections fan-beam cone-beam (3D) strips

15 PROJECTIONS if f defined on a discrete domain (e.g., digital image on pixels/voxels): f1f1 fIfI gjgj linear equation system projection data y ≈ g, (an approximation to g) available from measurements, Af ≈ y

16 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,+∞)

17 RECONSTRUCTION METHODS Heuristic Discretization of classical reconstruction methods Optimization

18 HEURISTIC RECONSTRUCTION METHODS

19 DISCRETIZATION OF CLASSICAL RECONSTRUCTION METHODS

20 RECONSTRUCTION METHODS BASED 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

21 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·∑ i c i ·f i c i weight of cost of the position i

22 OPTIMIZATION METHODS ICM (Iterated Conditional Modes) a local descent-based method SA (Simulated Annealing) initialization: let f be in F interation: change f to reach less cost C

23 COMPARISON OF DT IMAGES relative mean error shape error volume error

24 ANGIOGRAPHY digital subtraction angiography object with two homogeneous regions: contrast medium having known absorption value (μ contrast ), 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

25 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

26 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

27 Onnasch, Prause, 1999 ANGIOGRAPHY OF CARDIAC VENTRICLES

28 Onnasch, Prause, 1999 ANGIOGRAPHY OF CARDIAC VENTRICLES

29 Onnasch, Prause, 1999 ANGIOGRAPHY OF CARDIAC VENTRICLES

30 ANGIOGRAPHY OF CORONARY ARTERIES coronary arterial segments from two projections Reiber, 1982

31 Human iliac bifurcation, elliptic-based model Pellot, 1994 ANGIOGRAPHY OF CORONARY ARTERIES

32 PET Bayesian reconstruction and use of anatomical a priori information Bowsher, 1996

33 SPECT MAP reconstructions of the bolus boundary surface Cunningham, Hanson, Battle, 1998

34 ET Phantom FBP DT Chan, Herman, Levitan, 1998

35 SPECT MAP reconstructions of the bolus boundary surface Cunningham, Hanson, Battle, 1998

36 SPECT Tomography reconstruction using free-form deformation models FFDs reconstructions for different levels of noise Battle, Bizais, Le Rest, Turzo, 1999

37 SPECT Tomography reconstruction using free-form deformation models FFDs reconstructions for different levels of noise Battle, 1999

38 PET Edge-preserving tomography reconstruction with nonlocal regularization. Emission phantom, FBP, Huber penalty, proposed penalty Yu, 2002

39 DIscrete REConstruction Tomography software tool for generating/reading projections reconstructing discrete objects displaying discrete objects (2D/3D) available via Internet it is under development

40 DISCUSSION 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)