Data Acquisition, Representation and Reconstruction

1. Data Acquisition, Representation and Reconstruction Jian Huang, CS 594, Spring 2002 This set of slides references slides developed by Profs. Machiraju (Ohio State), Torsten Moeller (Simon Fraser) and Han-Wei Shen (Ohio State).

2. Acquisition Methods • X-Rays • Computer Tomography (CT or CAT) • MRI (or NMR) • PET / SPECT • Ultrasound • Computational

3. X-Rays • photons produced by an electron beam • similar to visible light, but higher energy!

4. X-Rays - Physics • Associated with inner shell electrons • As the electrons decelerate in the target through interaction, they emit electromagnetic radiation in the form of x-rays. • patient between an x-ray source and a film -> radiograph • cheap and relatively easy to use • potentially damaging to biological tissue

5. X-Rays - Visibility • bones contain heavy atoms -> with many electrons, which act as an absorber of x-rays • commonly used to image gross bone structure and lungs • excellent for detecting foreign metal objects • main disadvantage -> lack of anatomical structure • all other tissue has very similar absorption coefficient for x-rays

6. X-Rays - Images

7. CT or CAT - Principles • Computerized (Axial) Tomography • introduced in 1972 by Hounsfield and Cormack • natural progression from X-rays • based on the principle that a three-dimensional object can be reconstructed from its two dimensional projections • based on the Radon transform (a map from an n-dimensional space to an (n-1)-dimensional space)

8. CT or CAT - Methods • measures the attenuation of X-rays from many different angles • a computer reconstructs the organ under study in a series of cross sections or planes • combine X-ray pictures from various angles to reconstruct 3D structures

9. CT - Reconstruction: FBP • Filtered Back Projection • common method • uses Radon transform and Fourier Slice Theorem y F(u,v) f(x,y) Gf(r) x s u gf(s) f Spatial Domain Frequency Domain

10. CT - Reconstruction: ART • Algebraic Reconstruction Technique • iterative technique • attributed to Gordon Initial Guess Reconstructedmodel Back-Projection Projection Actual DataSlices

11. CT - FBP vs. ART FBP ART • Computationally cheap • Clinically usually 500 projections per slice • problematic for noisy projections • Still slow • better quality for fewer projections • better quality for non-uniform project. • “guided” reconstruct. (initial guess!)

12. CT - 2D vs. 3D • Linear advancement (slice by slice) • typical method • tumor might fall between ‘cracks’ • takes long time • helical movement • 5-8 times faster • A whole set of trade-offs

13. CT or CAT - Advantages • significantly more data is collected • superior to single X-ray scans • far easier to separate soft tissues other than bone from one another (e.g. liver, kidney) • data exist in digital form -> can be analyzed quantitatively • adds enormously to the diagnostic information • used in many large hospitals and medical centers throughout the world

14. CT or CAT - Disadvantages • significantly more data is collected • soft tissue X-ray absorption still relatively similar • still a health risk • MRI is used for a detailed imaging of anatomy

15. MRI • Nuclear Magnetic Resonance (NMR) (or Magnetic Resonance Imaging - MRI) • most detailed anatomical information • high-energy radiation is not used, i.e. “save” • based on the principle of nuclear resonance • (medicine) uses resonance properties of protons

16. MRI - polarized • all atoms (core) with an odd number of protons have a ‘spin’, which leads to a magnetic behavior • Hydrogen (H) - very common in human body + very well magnetizing • Stimulate to form a macroscopically measurable magnetic field

17. MRI - Signal to Noise Ratio • proton density pictures - measures HMRI is good for tissues, but not for bone • signal recorded in Frequency domain!! • Noise - the more protons per volume unit, the more accurate the measurements - better SNR through decreased resolution

18. PET/SPECT • Positron Emission TomographySingle Photon Emission Computerized Tomography • recent technique • involves the emission of particles of antimatter by compounds injected into the body being scanned • follow the movements of the injected compound and its metabolism • reconstruction techniques similar to CT - Filter Back Projection & iterative schemes

19. Ultrasound • the use of high-frequency sound (ultrasonic) waves to produce images of structures within the human body • above the range of sound audible to humans (typically above 1MHz) • piezoelectric crystal creates sound waves • aimed at a specific area of the body • change in tissue density reflects waves • echoes are recorded

20. Ultrasound (2) • Delay of reflected signal and amplitude determines the position of the tissue • still images or a moving picture of the inside of the body • there are no known examples of tissue damage from conventional ultrasound imaging • commonly used to examine fetuses in utero in order to ascertain size, position, or abnormalities • also for heart, liver, kidneys, gallbladder, breast, eye, and major blood vessels

21. Ultrasound (3) • by far least expensive • very safe • very noisy • 1D, 2D, 3D scanners • irregular sampling - reconstruction problems

22. Computational Methods (CM) • Computational Field Simulations • Computational Fluid Dynamics - Flow simulations • Computational Chemistry - Electron-electron interactions, Molecular surfaces • Computational Mechanics - Fracture • Computational Manufacturing - Die-casting

23. CM - Approach • (Continuous) physical model • Partial/Ordinary Differential Equation (ODE/PDE) • e.g. Navier-Stokes equation for fluid flow • e.g. Hosted Equations: • e.g. Schrödinger Equation - for waves/quantum • Continuous solution doesn’t exist (for most part) • Numerical Approximation/Solution 1. Discretize solution space - Grid generation explicit 2. Replace continuous operators with discrete ones 3. Solve for physical quantities

24. CM - Methods • Grid Generation • non-elliptical methods:algebraic, conformal, hyperbolic, parabolic, biharmonic • elliptical methods (based on elliptical PDE’s) • Numerical Methods • Newton • Runge-Kutta • Finite Element • Finite Differences • Time Varying

25. CM - Solutions (Structured) • PDE usually constrained/given at boundary • map from computational to physical space • polar maps, elliptical, non-elliptical • structured grids computational parametric physical f x t

26. CM - Solutions (Unstructured) • usually scattered data set • Delaunay Triangulation • Element Size Optimization • start with initial tetrahedral grid • interactively insert grid points • insertion guided by curvature and distance to surface • Advancing Front Method • start with boundary • advance boundary towards inside until “filled”

27. CM - Grid Types (2) • Multiblock structured grids • multiple structured grids • connected not necessarily structured • hybrid grids • structured + unstructured • chimera grids • multiple structured grids • partially overlapping • hierarchical grids • generated by quad-tree and octree like subdivision schemes (AKA embedded or semi-structured grids)

28. CM - Grid Examples

29. CM - Structured vs. Unstructured • consider discretization points assamples, points, cells, or voxels • Structured • Addressing - Cell [i,j,k] provides location of neighbors • Boundaries of volume - Easy to determine • Unstructured • No addressing mechanism - adjacency list required • Cannot determine the boundaries easily • Cells never of same size • Cells are hexahedrons, tetrahedrons, curved patches

30. Synthetic Methods • 3D Discretization Techniques Voxelization • Scan Conversion of Geometric Objects • Planes / Triangles • Cylinders • Sphere • Cone • NURBS, Bezier patches

31. Synthetic Methods • Solid Textures • Hyper Texture - 3D Textures • Fur • Marble • Hair • Turbulent flow • 3D Regular grid has texture values

32. Grid Types Structured Grids: rectilinear curvilinear uniform regular Unstructured Grids: regular irregular hybrid curved

33. Data Representation • Regular • Contour Stacks • Raster • RLE raster Point list RLE

34. Data Representation (2) • Polygon mesh, Curvilinear, Unstructured • Vertex list • Adjacency list of cells (not for curvilinear) • No convenient structure • Compressed Grids - RLE, JPEG, Wavelets • Multi-resolution Grids

35. Data Characteristics • Large Data Sets • Modest Head 2563 = 16Mbytes !! • Visible Human - MRI (2562) + CT (5122) + photo • Male - 15GB (1mm slice dist.); Female - 40GB (0.33mm) • Noisy - Ultrasound (How about CFD, CAGD ?) • Band-limited - Textured Images (Mandrill) wc 0 Frequency FFT Noisy Data set

36. Data Characteristics • Limited Dynamic Range - values between min, max populated with unequal probability • Non-Uniform Spatial Occupancy - • Quadtree/Octree • Rectangular spatial subdivision

37. Data Objects • Data Object: dataset • (representation of information) • Structures: how the information is organized • - Topology • - Geometry • Attributes: store the information we want • to visualize. e.g. function values

38. Data Objects: structures • Topology: • Invariant under geometric transformation • (rotation, translation, scaling etc) • - Topological structures: Cells • Geometry: • The instantiation of the topology • Geometric structures: cells with • positions in 3D space

39. Cell Types for Unstructured Grid (d) polyline (e) triangle (b) Polyvertex (c) line (a) vertex (e) Quadrilateral (e) Polygon (f) Tetrahedron (f) Hexahedron And more (I am too tired drawing now …)

40. Data Attributes • The information stored at each vertex of the cell • Scalars: temperature, pressure, etc • Vector: velocity • Normal: surface directions • Texture coordinates: graphics specific • Tensors: matrices

41. Where are we now ... Remember: • Data Object: dataset • (representation of information) • Structures: how the information is organized • - Topology • - Geometry • Attributes: store the information we want • to visualize. e.g. function values

42. (d) polyline (e) triangle (b) Polyvertex (c) line (a) vertex (e) Quadrilateral (e) Polygon (f) Tetrahedron (f) Hexahedron Interpolation (1) • Visualization deals with discrete data Values defined only at cell vertices

43. Color Mapping R G B v1 v2 v3 ... Values at vertices Color lookup Result table

44. 13 10 P = ? 9 12 p Interpolation: compute data from known points Interpolation (2) • We often need information at positions other than cell vertices

45. di n-1 D = S Wi * di i=0 Interpolation (3) • Three essential information: • Cell type • Data values at cell vertices • Parametric coordinates of the point p di: cell point value Wi: weight (S wi = 1) D: interpolated result

46. r = 1 n-1 r = 0 d1 D = S Wi * di 0 <= r <=1 d0 i=0 (a) line Interpolation (4) Parametric Coordinates: Used to specify the location of a point within a cell W0 = (1-r) W1 = r r

47. Interpolation (5) (b) Triangle s W0 = 1-r-s W1 = r W2 = s p2 r+s = 1 (why?) r=0 Why? r p0 p1 s=0

48. Interpolation (6) (C) Pixel s s=1 p2 p3 W0 = (1-r)(1-s) W1 = r(1-s) W2 = (1-r)s W3 = rs (s,t) r=0 r =1 r p0 p1 s=0 Why? This is also called bi-linear interpolation

49. Interpolation (7) (D) Polygon 2 p3 2 p4 Wi = (1/ri) / S(1/ri) r3 p2 p5 Weighted distance function p0 p1

50. W0 = 1-r-s-t W1 = r W2 = s W3 = t Interpolation (8) (D) Tetrahedron t s p3 p2 p0 r p1