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
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 high energy photons that are detected synchronously. Positron Emission Tomography
MRI (Magnetic Resonance Imaging) OCT (Optical Coherence Tomography) Other Imaging Modalities
3D Higher speed Greater resolution Measure function as well as structure Combining modalities (including direct vision) Current Trends in Imaging
Dissection: Medical School, Day 1: Meet the Cadaver. From Vesalius to the Visible Human The Gold Standard
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.
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
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 (tx , ty) Scale x by sx and y by sy
Rotation in 2D • 2 x 2 rotation portion is orthogonal (orthonormal vectors). • Therefore only 1 degree of freedom, .
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 • 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.
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
A pixel-wise intensity mapping is found that produces a uniform density of pixel intensity across the dynamic range. Histogram Equalization
Assumes bimodal distribution. Trough represents boundary points between homogenous areas. Adaptive Thresholding from Histogram
Assumes registration. Averaging multiple acquisitions for noise reduction. Subtracting sequential images for motion detection, or other changes (eg. Digital Subtractive Angiography). Masking. Algebraic Operators
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
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. “Inner scale” is the original pixel grid. Size of the kernel determines scale. Concept of Scale Space, Course-to-Fine. Neighborhood PDE Operators
Vector Direction of maximum change of scalar intensity I. Normal to the boundary. Nicely n-dimensional. Intensity Gradient
Scalar Maximum at the boundary Orientation-invariant. Intensity Gradient Magnitude
100% opaque watertight surface Fast, 28 = 256 combinations, pre-computed Isosurface, Marching Cubes (Lorensen)
Marching cubes works well with raw CT data. • Hounsfield units (attenuation). • Threshold calcium density.
Ixy = Iyx= curvature Orientation-invariant. What about in 3D? Jacobian of the Intensity Gradient
Divergence of the Gradient. Zero at the inflection point of the intensity curve. Laplacian of the Intensity I Ix Ixx
Repeated averaging of neighbors => Gaussian by Central Limit Theorem. Binomial Kernel
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)
Two regions with the same intensity but differentiated by texture are easily discriminated by the human visual system. Texture Boundaries
2D Fourier Transform analysis or synthesis
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 Combine with rotation, have arbitrary projection.
Gaussian seperable Since the Fourier Transform is also separable, the spectra of the 1D Gaussians are, themselves, separable.
Hankel Transform For radially symmetrical functions
Elliptical Fourier Series for 2D Shape 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 Fourier shape in 3D
Quaternions – 3D phasors Product is defined such that rotation by arbitrary angles from arbitrary starting points become simple multiplication.