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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high energy photons that are detected synchronously.Positron Emission Tomography
Scale x by sx and y by sy
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
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.
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 less useful for shape.
Fourier is global, while shape is local.
Fourier requires object-specific coordinate system.Summary