Medical Imaging

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Medical Imaging

• Simultaneous measurements on a spatial grid.
• Many modalities mainly EM radiation and sound.

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To invent you need a good imagination and a pile
of junk.   Thomas Edison
1879
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Bremsstrahlung

• Electron rapidly decelerates at heavy metal
  target, giving off X-Rays.

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1896
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X-Ray and Fluoroscopic Images

• Projection of X-Ray silhouette onto a piece of
  film or detector array, with intervening
  fluorescent screen.

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Computerized Tomography

• From a series of projections, a tomographic image
  is reconstructed using Filtered Back Projection.

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Mass Spectrometer

• Radioactive isotope separated by difference in
  inertia while bending in magnetic field.

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Nuclear Medicine

• Gamma camera for creating image of radioactive
  target. Camera is rotated around patient in
  SPECT (Single Photon Emission Computed
  Tomography).

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Phased Array Ultrasound

• Ultrasound beam formed and steered by controlling
  the delay between the elements of the transducer
  array.

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Real Time 3D Ultrasound
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Positron Emission Tomography

• Positron-emitting organic compounds create pairs
  of
• high energy photons that are detected
  synchronously.

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Other Imaging Modalities

• MRI (Magnetic Resonance Imaging)
• OCT (Optical Coherence Tomography)

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Current Trends in Imaging

• 3D
• Higher speed
• Greater resolution
• Measure function as well as structure
• Combining modalities (including direct vision)

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The Gold Standard

• Dissection
• Medical School, Day 1 Meet the Cadaver.
• From Vesalius to the Visible Human

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Local Operators and Global Transforms
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Images are n dimensional signals.

• Some things work in n dimensions, some dont.
• It is often easier to present a concept in 2D.
• I will use the word pixel for n dimensions.

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Global Transforms in n dimensions

• Geometric (rigid body)
• n translations and rotations.
• Similarity
• Add 1 scale (isometric).
• Affine
• Add n scales (combined with rotation gt skew).
• Parallel lines remain parallel.
• Projection

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Orthographic Transform Matrix

• Capable of geometric, similarity, or affine.
• Homogeneous coordinates.
• Multiply in reverse order to combine
• SGI graphics engine 1982, now standard.

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Translation by (tx , ty)
Scale x by sx and y by sy
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Rotation in 2D

• 2 x 2 rotation portion is orthogonal (orthonormal
  vectors).
• Therefore only 1 degree of freedom, .

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Rotation in 3D

• 3 x 3 rotation portion is orthogonal (orthonormal
  vectors).
• 3 degree of freedom (dotted circled), , as
  expected.

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Non-Orthographic Projection in 3D

• For X-ray or direct vision, projects onto the
  (x,y) plane.
• Rescales x and y for perspective by changing
  the 1 in the homogeneous coordinates, as a
  function of z.

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Point Operators

• f is usually monotonic, and shift invariant.
• Inverse may not exist due to discrete values of
  intensity.
• Brightness/contrast, windowing.
• Thresholding.
• Color Maps.
• f may vary with pixel location, eg., correcting
  for inhomogeneity of RF field strength in MRI.

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Histogram Equalization

• A pixel-wise intensity mapping is found that
  produces a uniform density of pixel intensity
  across the dynamic range.

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Adaptive Thresholding from Histogram

• Assumes bimodal distribution.
• Trough represents boundary points between
  homogenous areas.

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Algebraic Operators

• Assumes registration.
• Averaging multiple acquisitions for noise
  reduction.
• Subtracting sequential images for motion
  detection, or other changes (eg. Digital
  Subtractive Angiography).
• Masking.

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Re-Sampling on a New Lattice

• Can result in denser or sparser pixels.
• Two general approaches
• Forward Mapping (Splatting)
• Backward Mapping (Interpolation)
• Nearest Neighbor
• Bilinear
• Cubic
• 2D and 3D texture mapping hardware acceleration.

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Convolution and Correlation

• Template matching uses correlation, the
  primordial form of image analysis.
• Kernels are mostly used for convolution
  although with symmetrical kernels equivalent to
  correlation.
• Convolution flips the kernel and does not
  normalize.
• Correlation subtracts the mean and generally does
  normalize.

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Neighborhood PDE Operators

• Discrete images always requires a specific scale.
• Inner scale is the original pixel grid.
• Size of the kernel determines scale.
• Concept of Scale Space, Course-to-Fine.

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Intensity Gradient

• Vector
• Direction of maximum change of scalar intensity
  I.
• Normal to the boundary.
• Nicely n-dimensional.

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Intensity Gradient Magnitude

• Scalar
• Maximum at the boundary
• Orientation-invariant.

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Classic Edge Detection Kernel (Sobel)
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Isosurface, Marching Cubes (Lorensen)

• 100 opaque watertight surface
• Fast, 28 256 combinations, pre-computed

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• Marching cubes works well with raw CT data.
• Hounsfield units (attenuation).
• Threshold calcium density.

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Jacobian of the Intensity Gradient

• Ixy Iyx curvature
• Orientation-invariant.
• What about in 3D?

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Laplacian of the Intensity

• Divergence of the Gradient.
• Zero at the inflection point of the intensity
  curve.

I
Ix
Ixx
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Binomial Kernel

• Repeated averaging of neighbors gt Gaussian by
  Central Limit Theorem.

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Binomial Difference of Offset Gaussian (DooG)

• Not the conventional concentric DOG
• Subtracting pixels displaced along the x axis
  after repeated blurring with binomial kernel
  yields Ix

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Texture Boundaries

• Two regions with the same intensity but
  differentiated by texture are easily
  discriminated by the human visual system.

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2D Fourier Transform
analysis
or
synthesis
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Properties

• Most of the usual properties, such as linearity,
  etc.
• Shift-invariant, rather than Time-invariant
• Parsevals relation becoms Rayleighs Theorem
• Also, Separability, Rotational Invariance, and
  Projection (see below)

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Separability
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Rotation Invariance
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Projection
Combine with rotation, have arbitrary projection.
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Gaussian
seperable
Since the Fourier Transform is also separable,
the spectra of the 1D Gaussians are, themselves,
separable.
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Hankel Transform
For radially symmetrical functions
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Elliptical Fourier Series for 2D Shape
Parametric function, usually with constant
velocity.
Truncate harmonics to smooth.
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Fourier shape in 3D

• Fourier surface of 3D shapes (parameterized on
  surface).
• Spherical Harmonics (parameterized in spherical
  coordinates).
• Both require coordinate system relative to the
  object. How to choose? Moments?
• Problem of poles sigularities cannot be avoided

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Quaternions 3D phasors
Product is defined such that rotation by
arbitrary angles from arbitrary starting points
become simple multiplication.
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Summary

• Fourier useful for image processing,
  convolution becomes multiplication.
• Fourier less useful for shape.
• Fourier is global, while shape is local.
• Fourier requires object-specific coordinate
  system.