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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Simultaneous measurements on a spatial grid.
Many modalities: mainly EM radiation and sound.
“To invent you need a good imagination and a pile of junk.”
Electron rapidly decelerates at heavy metal target, giving off X-Rays.
Projection of X-Ray silhouette onto a piece of film or detector array, with intervening fluorescent screen.
From a series of projections, a tomographic image is reconstructed using Filtered Back Projection.
Radioactive isotope separated by difference in inertia while bending in magnetic field.
Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).
Ultrasound beam formed and steered by controlling the delay between the elements of the transducer array.
Positron-emitting organic compounds create pairs of
high energy photons that are detected synchronously.
MRI (Magnetic Resonance Imaging)
OCT (Optical Coherence Tomography)
Measure function as well as structure
Combining modalities (including direct vision)
Medical School, Day 1: Meet the Cadaver.
From Vesalius to the Visible Human
Local Operators and Global Transforms
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.
Geometric (rigid body)
n translations and rotations.
Add 1 scale (isometric).
Add n scales (combined with rotation => skew).
Parallel lines remain parallel.
Capable of geometric, similarity, or affine.
Multiply in reverse order to combine
SGI “graphics engine” 1982, now standard.
Scale x by sx and y by sy
f is usually monotonic, and shift invariant.
Inverse may not exist due to discrete values of intensity.
f may vary with pixel location, eg., correcting for inhomogeneity of RF field strength in MRI.
A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.
Assumes bimodal distribution.
Trough represents boundary points between homogenous areas.
Averaging multiple acquisitions for noise reduction.
Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography).
Can result in denser or sparser pixels.
Two general approaches:
Forward Mapping (Splatting)
Backward Mapping (Interpolation)
2D and 3D texture mapping hardware acceleration.
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.
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.
Direction of maximum change of scalar intensity I.
Normal to the boundary.
Maximum at the boundary
100% opaque watertight surface
Fast, 28 = 256 combinations, pre-computed
Ixy = Iyx= curvature
What about in 3D?
Divergence of the Gradient.
Zero at the inflection point of the intensity curve.
Repeated averaging of neighbors => Gaussian by Central Limit Theorem.
Not the conventional concentric DOG
Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields Ix
Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.
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)
Combine with rotation, have arbitrary projection.
Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.
For radially symmetrical functions
Parametric function, usually with constant velocity.
Truncate harmonics to smooth.
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
Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.
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.