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
What about microscopic scanners!
What about electron microscopes?
What about voxelization / discretization?
What about synthetic methods?
Provide Links to the numerous online tutorials!

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
Add 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
Reconstructed model
Back- Projection
Projection
Actual Data Slices

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

12. CT - 2D vs. 3D Linear advancement (slice by slice) helical movement
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 H MRI 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 Tomography Single 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 Numerical Methods Time Varying
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
physical
parametric
computational 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 hybrid grids
multiple structured grids
connected not necessarily structured
hybrid grids
structured + unstructured
chimera grids
partially overlapping
hierarchical grids
generated by quad-tree and octree like subdivision schemes (AKA embedded or semi-structured grids)
Include figures from the Nielson book

28. CM - Grid Examples

29. CM - Structured vs. Unstructured
consider discretization points as samples, 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 raster Point list RLE Contour Stacks

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
animate

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
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
Find a histogram of an image to include here!

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

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

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

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

46. Interpolation (4) Parametric Coordinates:
Used to specify the location of a point within a cell
(a) line
r = 0
r = 1
0 <= r <=1 d0 d1
W0 = (1-r)
W1 = r
D = S Wi * di
i=0
n-1 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)
(s,t)
r=0
r =1 r p0
s=0 p1
Why?
This is also called bi-linear interpolation

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

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

51. Interpolation (9) (D) Cube (voxel) t W0 = (1-r)(1-s)(1-r)
W1 = r(1-s)(1-t)
W2 = (1-r)s(1-t)
W3 = rs(1-r)
W4 = (1-r)(1-s)t
W5 = r(1-s)t
W6 = (1-r)st
W7 = rst p6 p7 p4 p5 s p2 p3 r p0 p1

52. Interpolation (10)
The interpolation function can be used to calculate the
geometric position as well.
That is, given (r,s,t), calculate the global coordinates
Local to global coordinate transformation:
P = S Wi * Pi
n-1
i=0

53. Interpolation (11) How to get (r,s,t) ?
Line, Pixel, Cube are all trivial
Triangle, Tetrahedron can be solved analytically
Qudrilateral or Hexahedra need numerical method
P = S Wi * Pi
n-1
Known: P, Pi
Unkown: Wi (i.e. r,s,t)
i=0

54. Contours 3 7 10 4
C = 6

55. Interpolation 2D 1D
Given:
Given:
Needed:
Needed:

56. The Need for Interpolation
Interpolation is needed throughout the visualization process
Rendering?
Extracting iso-contours?
Shading?
Texture mapping? …
Anytime when resampling is needed

57. Reconstructed Function
General Process
Original function
Sampled function
Acquisition
Reconstruction
Reconstructed Function
Re-sampled function
Resampling

58. Evaluated at discrete points (sum)
How? - Convolution
Spatial Domain:
Frequency Domain:
Mathematically:f(x)*h(x) =
Multiplication:
Convolution:
Evaluated at discrete points (sum)

59. Reconstruction
Mathematically:
f(x)*h(x) = (Sf[i])*h(x)
Sf[i]
h(x)

60. General Process - Frequency Domain
Original function
Sampled function
Acquisition
Reconstruction
Reconstructed Function
Re-sampled function
Resampling

61. Pre-Filtering Original function Band-limited function Pre-Filtering
Acquisition
Sampled Function
Reconstructed function
Reconstruction

62. Ideal Reconstruction with Sinc function
Spatial Domain:
convolution is exact
Frequency Domain:
cut off freq. replica

63. Reconstructing Derivatives
Spatial Domain:
convolution is exact
Frequency Domain:
cut off freq. replica

64. Possible Errors Post-aliasing Smoothing Ringing (overshoot) Anisotropy
reconstruction filter passes frequencies beyond the Nyquist frequency (of duplicated frequency spectrum) => frequency components of the original signal appear in the reconstructed signal at different frequencies
Smoothing
frequencies below the Nyquist frequency are attenuated
Ringing (overshoot)
occurs when trying to sample/reconstruct discontinuity
Anisotropy
caused by not spherically symmetric filters

65. How Good? = Error Spatial Domain: local error asymptotic error
numerical error
Frequency Domain:
global error
visual appearance
blurring
aliasing
smoothing
Approximation
Theory/Analysis
Signal Processing

66. Sources of Aliasing Non-bandlimited signal
Low sampling rate (below Nyquist)
Non perfect reconstruction

67. Reconstruction Kernels
pass band
stop band
Ideal filter
The spatial extent of reconstruction kernels, or interpolation basis functions, depend on the cut-off frequency as well.
Smoothing error
Postaliasing error
filter

68. Reconstruction Kernels
Nearest Neighbor (Box)
Triangular func
Sinc
Gaussian
+ many others
Spatial d.
Frequency d.

69. Higher Dimensions An-isotropic Filters separable filters
(radially symmetric)
separable filters

70. Interpolation (an example)
Very important; regardless of algorithm
expensive => done very often for one image
Requirements for good reconstruction
performance
stability of the numerical algorithm
accuracy
Linear
Nearest
neighbor

71. Put Things in Perspective
In visualization, need to use continuous space functions. But can only work with discrete data
So, let’s reconstruct from discrete data to continuous space (convolution) and resample
Interpolation is doing the same thing. Computing one data point in the resulting function, say, at x1.
So, which reconstruction kernel (basis function) does linear/bilinear/tri-linear interpolations use?