1. Data Acquisition, Representation and Reconstruction of medical images
Application of Advanced Spectral Methods

2. Acquisition Methods for medical images
X-Rays
Computer Tomography (CT or CAT)
MRI (or NMR)
PET / SPECT (Positron Emission Tomography, Single Photon Emission Computerized Tomography
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

4. X-Rays - Physics cheap and relatively easy to use
X-Rays are 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 is located between an x-ray source and a film -> radiograph
cheap and relatively easy to use
potentially damaging to biological tissue

5. X-Rays
X-Rays
similar to visible light, but higher energy!

6. 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

7. X-Rays - Images X-Rays can be used in computerized tomography
Add images!!!
X-Rays can be used in computerized tomography

8. Computerized (Axial) Tomography

9. CT (CAT) scanners and relevant mathematics

10. Non-Intrusive Medical Diagnosis based on Computerized Tomography
Computer tomography CT
(From Jain’s Fig.10.1)
An X-ray CT scanning system

11. Non-Intrusive Medical Diagnosis based on Transmission Tomography
Source and Detector are rotating around human’s body
(From Bovik’s Handbook Fig )

12. Non-Intrusive Medical Diagnosis based on projections
Observe a set of projections (integrations) along different angles of a cross-section
Each projection itself loses the resolution of inner structure
Types of measurements
transmission (X-ray),
emission, magnetic resonance (MRI)
Want to recover inner structure from the projections
“Computerized Tomography” (CT)

13. Non-Intrusive Medical Diagnosis based on Emission Tomography
Emission tomography: ET measure emitted gamma rays by the decay of isotopes from radioactive nuclei of certain chemical compounds affixed to body parts.
MRI: based on that protons possess a magnetic moment and spin.
In magnetic field => align to parallel or antiparallel.
Apply RF => align to antiparallel. Remove RF => absorbed energy is remitted and detected by Rfdetector.
f(x,y) is 2D image as before

14. Radon Transform Principles
A linear transform f(x,y)  g(s,)
Line integral or “ray-sum”
Along a line inclined at angle  from y-axis and s away from origin
Fix  to get a 1-D signal g(s)
We have now a set of images g(s) which represent g(s,)
(From Jain’s Fig.10.2)
This is a transform from 2D to 2D spaces

15. Tomography and Reconstruction
Lecture Overview
Applications
Background/history of tomography
Radon Transform
Fourier Slice Theorem
Filtered Back Projection
Algebraic techniques
Measurement of Projection data
Example of flame tomography

16. Applications & Types of Tomography
Medical Applications
Type of Tomography
Full body scan
X-ray
Respiratory, digestive systems, brain scanning
PET Positron Emission Tomography
Respiratory, digestive systems.
Radio-isotopes
Mammography
Ultrasound
Whole Body
Magnetic Resonance (MRI, NMR)
MRI and PET showing lesions in the brain.
PET scan on the brain showing Parkinson’s Disease

17. Applications & Types of Tomography – non medical
Non Medical Applications
Type of Tomography
Oil Pipe Flow
Turbine Plumes
Resistive/Capacitance Tomography
Flame Analysis
Optical Tomography
                        
                                            
ECT on industrial pipe flows

18. CT or CAT - Principles Computerized (Axial) Tomography Radon again!
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)
Radon again!
From 2D to 3D !

19. 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

20. The History of CAT
Johan Radon (1917) showed how a reconstruction from projections was possible.
Cormack (1963,1964) introduced Fourier transforms into the reconstruction algorithms.
Hounsfield (1972) invented the X-ray Computer scanner for medical work, (which Cormack and Hounsfield shared a Nobel prize).
EMI Ltd (1971) announced development of the EMI scanner which combined X-ray measurements and sophisticated algorithms solved by digital computers.

21. Backpropagation
Principles

22. Backpropagation
We know that objects are somewhere here in black stripes, but where?

24. Example of Simple Backprojection Reconstruction
Given are sums, we have to reconstruct values of pixels A, B, C and D

25. Image Reconstruction: ART or Algebraic Reconstruction Technique

26. CT - Reconstruction: ART or Algebraic Reconstruction Technique
METHOD 1: Algebraic Reconstruction Technique
iterative technique
attributed to Gordon
Initial Guess
Reconstructed model
Back- Projection
Projection
Actual Data Slices 26

27. CT - Reconstruction: FBP Filtered Back Propagation
METHOD 2 : 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

28. COMPARISON : CT - FBP vs. ART
Algebraic Reconstruction Technique
Still slow
better quality for fewer projections
better quality for non-uniform project.
“guided” reconstruct. (initial guess!)
Filtered Back Projection
Computationally cheap
Clinically usually 500 projections per slice
problematic for noisy projections 28

29. Fourier Slice Theorem and FFT review
Patient’s body is described by spatial distribution of attenuation coefficient

30. Properties of attenuation coefficient
Our transform: f(x,y)  p(r,)

31. attenuation coefficient is used in CT_number of various tissues
These numbers are represented in HU = Hounsfield Units
CT_number uses attenuation coefficients

32. RADON TRANSFORM Properties
REMEMBER: f(x,y)  p(r,)

33. Radon Transform is available in Matlab
Radon and its inverse easy to use
You can do your own projects with CT reconstruction
Data are available on internet
sinogram

34. Inverse Radon Transform

35. Matlab examples

36. Sinogram versus Hough

37. The object
An aside
The sinogram
AN ASIDE

38. Review and notation – Fourier Transform of Image f(x,y)

39. Matlab example
In Matlab there are implemented functions that use Fourier Slice Theorem

40. Matlab example: filtering
High frequency removed
Low frequency removed

41. Matlab example - convolution

42. Remainder of main theorem of spectral imaging
Matlab example – filtering by convolution in spectral domain

43. Radon Transform and a Head Phantom

45. Reconstructing with more and more rays

46. Example of Image Radon Transform
[Y-axis] distance, [X-axis] angle
(From Matlab Image Processing Toolbox Documentation)

47. Matlab Implementation of Radon Transform

50. big noise
No noise

51. The Lung Cancer and the reconstruction51

52. The Lung and The CTs 52 [LUNG]
1.Either of the pair of organs occupying the cavity of the thorax that effect the aeration of the blood.
2.Balloon-like structures in the chest that bring oxygen into the body and expel carbon dioxide from the body
[TYPES]
1.Small Cell Lung Cancer (SCLC) - 20% of all lung cancers
2.Non Small Cell Lung Cancer (NSCLC) - 80% of all lung cancer
[Risks]
In the United States alone, it is estimated that 154,900 died from lung cancer in In comparison,is estimated that 126,800 people died from colon, breast and prostate cancer combined, in 2002.
[LUNG CANCER]
Lung Cancer happens when cells in the lung begin to grow out of control and can than invade nearby tissues or spread throughout the body; Large collections of this out of control tissues are called tumors. 52

53. We want to reconstruct shape of the lungs
Starting Point
Border Detection
At the moment two approaches are available.
Left the algorithm developed at Pisa
Right the algorithm developed at Lecce 53

54. Image Interpolation - Theory
[IDEA]
In order to provide a richer environment we are thinking of using interpolation methods that will generate “artificial images” thus revealing hidden information.
[RADON RECONSTRUCTION]
Radon reconstruction is the technique in which the object is reconstructed from its projections. This reconstruction method is based on approximating the inverse Radon Transform.
[RADON Transform]
The 2-D Radon transform is the mathematical relationship which maps the spatial domain (x,y) to the Radon domain (p,phi). The Radon transform consists of taking a line integral along a line (ray) which passes through the object space. The radon transform is expressed mathematically as:
[FILTERED BACK PROJECTION - INVERSE R.T.]
It is an approximation of the Inverse Radon Transform.
[The principle] Several x-ray images of a real-world volume are acquired
[The Data] X-ray images (projections) of known orientation, given by data samples.
[The Goal] Reconstruct a numeric representation of the volume from these samples.
[The Mean] Obtain each voxel value from its pooled trace on the several projections.
[Resampling] At this point one can obtain the “artificial slices”
[Reslicing] An advantage of the volume reconstruction is the capability of obtaining new perpendicular slices on the original ones. 54

55. Image Interpolation - Graphical Representation (I)55

56. Image Interpolation - Graphical Representation (II)56

57. Line Integrals and Projections
We review the principle
Discuss various geometries
Show the use of filtering

58. Line Integrals and Projections
The function P = Radon transform
object function f(x,y).
The function is known as the Radon transform of the function f(x,y).

59. Various types of beams can be used
Fan Beams
Parallel Beams
A fan beam projection is taken if the rays meet in one location
Parallel beams projections are taken by measuring a set of parallel rays for a number of different angles
Various types of beams can be used

60. Line Integrals and Projections
A projection is formed by combining a set of line integrals.
Here the simplest projection, a collection of parallel ray integrals i.e constant θ, is shown.
Notation for calculations in these projections

61. Line Integrals and Projections
A simple diagram showing the fan beam projection

62. Fourier
Slice
Theorem

63. Fourier Slice Theorem
The Fourier slice theorem is derived by taking the one-dimensional Fourier transform of a parallel projection and noting that it is equal to a slice of the two-dimensional Fourier transform of the original object.
It follows that given the projection data, it should then be possible to estimate the object by simply performing the 2D inverse Fourier transform.
Start by defining the 2D Fourier transform of the object function as
For simplicity θ=0 which leads to v=0
Define the projection at angle θ = Pθ(t)
Define its transform by
As the phase factor is no-longer dependent on y, the integral can be split.

64. Fourier Slice Theorem
As the phase factor is no-longer dependent on y, the integral can be split.
The part in brackets is the equation for a projection along lines of constant x
Substituting in
Thus the following relationship between the vertical projection and the 2D transform of the object function:

65. Fourier Slice Theorem Stanley and Kak
Full details of derivation, not for now.

67. The Fourier Slice Theorem
The Fourier Slice theorem relates the Fourier transform of the object along a radial line. t
Fourier transform v u θ
Space Domain
Frequency Domain

68. The Fourier Slice Theorem
The Fourier Slice theorem relates the Fourier transform of the object along a radial line.
Collection of projections of an object at a number of angles t v u
Fourier transform v u θ
For the reconstruction to be made it is common to determine the values onto a square grid by linear interpolation from the radial points.
But for high frequencies the points are further apart resulting in image degradation.
Space Domain
Frequency Domain

71. Backprojection of Radon Transform

75. Backprojection of Radon Transform

77. Filtered backpropagation creates crisp edges
Ideal cylinder
Blurred edges
Crisp edges
Filtered backpropagation creates crisp edges

79. Computerized Tomography Equipment

81. 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

84. Evolution of CT technology

85. 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

86. 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 – no Xrays involved.

87. Nuclear Magnetic Resonance (NMR) Magnetic Resonance Imaging (MRI)

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

89. Magnetic Resonance Imaging 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

90. 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 signal to noise ratio (SNR) through decreased resolution

91. PET/SPECT
Positron Emission Tomography Single Photon Emission Computerized Tomography

92. 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

93. Ultrasound

94. 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

95. 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

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

100. Typical Homework

101. Sources of slides and information
Badri Roysam
Jian Huang,
Machiraju,
Torsten Moeller,
Han-Wei Shen
Kai Thomenius