Medical Image Analysis

1
Medical Image Analysis

• Image Reconstruction

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
2
Mathematical Preliminaries and Basic
Reconstruction Methods
An original image
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
3
Mathematical Preliminaries and Basic
Reconstruction Methods
Apply the Radon transform
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
4
Mathematical Preliminaries and Basic
Reconstruction Methods
After the inverse Radon transform
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
5
Mathematical Preliminaries and Basic
Reconstruction Methods
An test image
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
6
Mathematical Preliminaries and Basic
Reconstruction Methods
Apply the Radon transform
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
7
Mathematical Preliminaries and Basic
Reconstruction Methods
After the inverse Radon transform
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
8
Mathematical Preliminaries and Basic
Reconstruction Methods
Figure 2.8. Line integral projection P(p,q) of
the two-dimensional Radon transform.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
9
Mathematical Preliminaries and Basic
Reconstruction Methods

• The Radon transform of an object

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
10
Central Slice Theorem

• The central slice theorem
• Called the projection theorem
• A relationship between the Fourier transform of
  the object function and the Fourier transform of
  its Radon transform or projection

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
11
Central Slice Theorem
Figure comes from the Wikipedia,
www.wikipedia.org.
12
Central Slice Theorem
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
13
Figure 5.1. The frequency domain of the Fourier
transform F(u,v) with the Fourier transforms,
Sq(w) of individual projections Jq(p).
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
14
Central Slice Theorem
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
15
Central Slice Theorem

• Represents the Fourier transform of the
  projection that is taken at an angle
  in the space domain with a rotated coordinate
  system

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
16
Inverse Radon Transform
Where
17
Backprojection Method

• Modified projections
• Convolution-backprojection
• Filtered-backprojection

18
Backprojection Method

• From

19
Backprojection Method

• Ramakrishnan and Lakshiminarayanan
• In general

20
Figure 5.2. A bandlimited filter function
.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
21
Backprojection Method

• The filter kernel function
• If the projections are sampled with a time
  interval of , the projections can be
  represented as , where is an
  integer

22
Backprojection Method

• For the bandlimited projections with a sampling
  interval of
• Then

23
Backprojection Method

• The quality of the reconstructed image
• The number of projections
• The spatial interval of the acquired projection
• Limited by the detector size and the scanning
  procedure
• Suffer from poor signal-to-noise ratio if there
  is an insufficient number of photons collected by
  the detector due to its smaller size

24
Backprojection Method

• Ramakrishnan and Lakshiminarayanan filter
• Has sharp cutoffs in the frequency domain at
  and
• Cause modulated ringing artifacts in the
  reconstructed image
• Hamming window function

25
Figure 5.3. A Hamming window based filter kernel
function in the frequency domain.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
26
Figure 5.4. A comparison of the
and convolution functions in
the spatial domain.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
27
Iterative Algebraic Reconstruction Methods

• Algebraic Reconstruction Techniques (ART)
• The raw projection data from the scanner are
  distributed over a prespecified image
  reconstruction grid such that the error between
  the computed projections from the reconstructed
  image and the actual acquired projections is
  minimized

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
28
Figure 5.5. Reconstruction grid with a ray
defining the ray sum for ART.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
29
Iterative Algebraic Reconstruction Methods

• the projection data
• the pixels of the image
• weights
• Determined by geometrical consideration as the
  ratio of the area overlapping with the scanning
  ray to the total area of the pixel

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
30
Iterative Algebraic Reconstruction Methods

• the computed ray sum in the iteration

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
31
Iterative Algebraic Reconstruction Methods

• The iterative ART
• Deal with the noise and random fluctuations in
  the projection data caused by detector
  inefficiency and scattering

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
32
Estimation Methods

• Statistical estimation
• Assume a certain distribution of the measured
  photons
• Find the parameters for attenuation function (in
  the case of transmission scans such as X-ray CT)
  or emitter density (in the case of emission scans
  such as PET)

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
33
Estimation Methods

• measurement vector
  
• the random variable representing the
  number of photons collected by the detector for
  the ray
• the blank scan factor
• the attenuation coefficients

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
34
Estimation Methods

• A line integral or ray sum for ray
• The Poisson distribution model for the photon
  counts

35
Estimation Methods

• The Maximum Likelihood (ML) estimate
• The log likelihood function

36
Estimation Methods

• The Maximum Likelihood (ML) estimate
• The log likelihood function
• Find

37
Estimation Methods

• Penalty functions
• Additional constraints such as smoothness
• Find

38
Estimation Methods

• Optimization methods
• Expectation Maximization (EM)
• Complex conjugate gradient
• Gradient descent optimization
• Grouped coordinated ascent
• Fast gradient based Bayesian reconstruction
• Ordered-subsets algorithms

39
Fourier Reconstruction Methods

• Direct Fourier reconstruction
• Use the central slice theorem
• Resampling the frequency domain information from
  a polar to a Cartesian grid
• Developing sinc-based interpolation method for
  the bandlimited functions in the radial direction

40
Image Reconstruction in Medical Imaging Modalities

• Choice
• Filtered backprojection (X-ray CT)
• Statistical estimation

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
41
Figure 5.6. A 2-D divergent beam geometry.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
42
X-ray Computed Tomography

• angular step
• radial distance between the source and
  the origin
• the angle that the source makes with its
  central reference axis
• a fan projection from the divergent
  beam

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
43
X-ray Computed Tomography

• Objective convert fan projections
  into the parallel-beam projection
• Sorted

44
X-ray Computed Tomography

• Backprojected
• the total number of source positions
• the angle of the divergent beam ray
  passing through the point
• the distance between the source and the
  point for the source position

45
Nuclear Emission Computed Tomography SPECT and
PET

• X-ray CT
• Estimate the attenuation coefficient map
• SPECT or PET
• Reconstruct the source emission map within the
  object from the statistical distribution of
  photons that have gone through attenuation within
  the object but detected outside the object
• Attenuation correction

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
46
Nuclear Emission Computed Tomography SPECT and
PET

• The transmission scans in SPECT
• Computing attenuation coefficient parameter
• The iterative ML estimation-based algorithms have
  provided better results

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
47
Multi-Grid EM Algorithm

• Image reconstruction in PET

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
48
Figure 5.7. A flowchart of the MGEM algorithm for
PET image reconstruction.
49
Figure 5.8. Wavelet based interpolation method.
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
50
Figure 5.9. Shepp and Logan phantom (top left)
and reconstructed phantom images using WMREM
algorithm (top right), ML-EM algorithm (bottom
left) and filtered backprojection method (bottom
right).
51
Figure 5.10. Four reconstructed brain images of a
patient with a tumor from a PET scan. Images in
the top row are reconstructed using filtered
backprojection method, images in the middle row
are reconstructed using WMREM algorithm. Images
in the bottom row are reconstructed using a
generalized ML-EM algorithm.
52
Image Reconstruction MRI
Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.
53
Image Reconstruction Ultrasound Imaging

• Point measurements
• Line scan
• Reduction of speckle noise
• Image averaging
• Image filtering weighted median, Wiener filters

Figures come from the textbook Medical Image
Analysis, by Atam P. Dhawan, IEEE Press, 2003.