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Simultaneous measurements on a spatial grid. Many modalities: mainly EM radiation and sound. Medical Imaging “To invent you need a good imagination and a pile of junk.” Thomas Edison 1879 Electron rapidly decelerates at heavy metal target, giving off X-Rays. Bremsstrahlung 1896

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medical imaging
Simultaneous measurements on a spatial grid.

Many modalities: mainly EM radiation and sound.

Medical Imaging
nuclear medicine
Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).Nuclear Medicine
current trends in imaging
3D

Higher speed

Greater resolution

Measure function as well as structure

Combining modalities (including direct vision)

Current Trends in Imaging
the gold standard
Dissection:

Medical School, Day 1: Meet the Cadaver.

From Vesalius to the Visible Human

The Gold Standard
images are n dimensional signals
Some things work in n dimensions, some don’t.

It is often easier to present a concept in 2D.

I will use the word “pixel” for n dimensions.

Images are n dimensional signals.
global transforms in n dimensions
Geometric (rigid body)

n translations and rotations.

Similarity

Add 1 scale (isometric).

Affine

Add n scales (combined with rotation => skew).

Parallel lines remain parallel.

Projection

Global Transforms in n dimensions
orthographic transform matrix
Capable of geometric, similarity, or affine.

Homogeneous coordinates.

Multiply in reverse order to combine

SGI “graphics engine” 1982, now standard.

Orthographic Transform Matrix
translation by t x t y
Translation by (tx , ty)

Scale x by sx and y by sy

rotation in 2d
Rotation in 2D
  • 2 x 2 rotation portion is orthogonal (orthonormal vectors).
  • Therefore only 1 degree of freedom, .
rotation in 3d
Rotation in 3D
  • 3 x 3 rotation portion is orthogonal (orthonormal vectors).
  • 3 degree of freedom (dotted circled), , as expected.
non orthographic projection in 3d
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.
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.

Point Operators
histogram equalization
A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.Histogram Equalization
adaptive thresholding from histogram
Assumes bimodal distribution.

Trough represents boundary points between homogenous areas.

Adaptive Thresholding from Histogram
algebraic operators
Assumes registration.

Averaging multiple acquisitions for noise reduction.

Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography).

Masking.

Algebraic Operators
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.

Re-Sampling on a New Lattice
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.

Convolution and Correlation
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.

Neighborhood PDE Operators
intensity gradient
Vector

Direction of maximum change of scalar intensity I.

Normal to the boundary.

Nicely n-dimensional.

Intensity Gradient
intensity gradient magnitude
Scalar

Maximum at the boundary

Orientation-invariant.

Intensity Gradient Magnitude
slide35

Marching cubes works well with raw CT data.

  • Hounsfield units (attenuation).
  • Threshold calcium density.
jacobian of the intensity gradient
Ixy = Iyx= curvature

Orientation-invariant.

What about in 3D?

Jacobian of the Intensity Gradient
laplacian of the intensity
Divergence of the Gradient.

Zero at the inflection point of the intensity curve.

Laplacian of the Intensity

I

Ix

Ixx

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

Binomial Difference of Offset Gaussian (DooG)
texture boundaries
Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.Texture Boundaries
2d fourier transform
2D Fourier Transform

analysis

or

synthesis

properties
Most of the usual properties, such as linearity, etc.

Shift-invariant, rather than Time-invariant

Parsevals relation becoms Rayleigh’s Theorem

Also, Separability, Rotational Invariance, and Projection (see below)

Properties
projection
Projection

Combine with rotation, have arbitrary projection.

gaussian
Gaussian

seperable

Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.

hankel transform
Hankel Transform

For radially symmetrical functions

elliptical fourier series for 2d shape
Elliptical Fourier Series for 2D Shape

Parametric function, usually with constant velocity.

Truncate harmonics to smooth.

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

Fourier shape in 3D
quaternions 3d phasors
Quaternions – 3D phasors

Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.

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.

Summary