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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Many modalities: mainly EM radiation and sound.Medical Imaging
1896 off X-Rays.
Projection of X-Ray silhouette onto a piece of film or detector array, with intervening fluorescent screen.X-Ray and Fluoroscopic Images
From a series of projections, a tomographic image is reconstructed using Filtered Back Projection.Computerized Tomography
Radioactive isotope separated by difference in inertia while bending in magnetic field.Mass Spectrometer
Gamma camera for creating image of radioactive target. Camera is rotated around patient in SPECT (Single Photon Emission Computed Tomography).Nuclear Medicine
Ultrasound beam formed and steered by controlling the delay between the elements of the transducer array.Phased Array Ultrasound
Positron-emitting organic compounds create pairs of between the elements of the transducer array.
high energy photons that are detected synchronously.Positron Emission Tomography
MRI (Magnetic Resonance Imaging) between the elements of the transducer array.
OCT (Optical Coherence Tomography)Other Imaging Modalities
3D between the elements of the transducer array.
Measure function as well as structure
Combining modalities (including direct vision)Current Trends in Imaging
Dissection: between the elements of the transducer array.
Medical School, Day 1: Meet the Cadaver.
From Vesalius to the Visible HumanThe Gold Standard
Some things work in between the elements of the transducer array.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.
Geometric (rigid body) between the elements of the transducer array.
n translations and rotations.
Add 1 scale (isometric).
Add n scales (combined with rotation => skew).
Parallel lines remain parallel.
ProjectionGlobal Transforms in n dimensions
Capable of geometric, similarity, or affine. between the elements of the transducer array.
Multiply in reverse order to combine
SGI “graphics engine” 1982, now standard.Orthographic Transform Matrix
Scale x by sx and y by sy
f between the elements of the transducer array. 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.Point Operators
A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range.Histogram Equalization
Assumes bimodal distribution. uniform density of pixel intensity across the dynamic range.
Trough represents boundary points between homogenous areas.Adaptive Thresholding from Histogram
Assumes registration. uniform density of pixel intensity across the dynamic range.
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. uniform density of pixel intensity across the dynamic range.
Two general approaches:
Forward Mapping (Splatting)
Backward Mapping (Interpolation)
2D and 3D texture mapping hardware acceleration.Re-Sampling on a New Lattice
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
Discrete images always requires a specific scale. image analysis.
“Inner scale” is the original pixel grid.
Size of the kernel determines scale.
Concept of Scale Space, Course-to-Fine.Neighborhood PDE Operators
Vector image analysis.
Direction of maximum change of scalar intensity I.
Normal to the boundary.
Nicely n-dimensional.Intensity Gradient
Scalar image analysis.
Maximum at the boundary
Orientation-invariant.Intensity Gradient Magnitude
100% opaque watertight surface image analysis.
Fast, 28 = 256 combinations, pre-computedIsosurface, Marching Cubes (Lorensen)
I image analysis.xy = Iyx= curvature
What about in 3D?Jacobian of the Intensity Gradient
Divergence of the Gradient. image analysis.
Zero at the inflection point of the intensity curve.Laplacian of the Intensity
Not the conventional concentric DOG Theorem.
Subtracting pixels displaced along the x axis after repeated blurring with binomial kernel yields IxBinomial Difference of Offset Gaussian (DooG)
Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system.Texture Boundaries
Most of the usual properties, such as linearity, etc. texture are easily discriminated by the human visual system.
Shift-invariant, rather than Time-invariant
Parsevals relation becoms Rayleigh’s Theorem
Also, Separability, Rotational Invariance, and Projection (see below)Properties
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). texture are easily discriminated by the human visual system.
Spherical Harmonics (parameterized in spherical coordinates).
Both require coordinate system relative to the object. How to choose? Moments?
Problem of poles: sigularities cannot be avoidedFourier shape in 3D
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.Summary