A series of reflections on the physical and philosophical nature of power electronics, where the transformation of energy reveals the fundamental rules of the world.
A CT scanner never sees an organ. It sees a shadow — one thin projection of X-ray attenuation, taken from one angle, telling you only how much material the beam crossed on its way through, summed and flattened into a single number per ray. From one angle a shadow is nearly useless. From every angle, it is the whole object, hidden in plain sight.
The Radon transform is the mathematics of that flattening: an integral of the object's density along every possible line through it, sorted by angle and offset into a sinogram. Rotate the source and detector through a full half-circle and every point inside the body gets crossed by rays from every direction, contributing its density to a thousand different shadows simultaneously.
Reconstruction is the inverse problem — recovering the object from its shadows — and the direct route runs through the Fourier domain: the Projection-Slice Theorem states that the 1D Fourier transform of a single projection equals one slice through the 2D Fourier transform of the object itself, at the matching angle. Collect enough angles and you have filled the frequency plane; invert it and the image resolves.
But a plain inversion oversamples the center and starves the edges, blurring fine structure into a fog. Filtered Backprojection corrects this before smearing each projection back across the image plane: multiply each projection's spectrum by a ramp filter that boosts high frequencies in direct proportion to their distance from the center, undoing the geometric bias of polar sampling before the shadows are ever laid back down. What was flattened is unflattened, angle by angle, until the shape that cast the shadows is the only shape left standing.
Seed: Biomedical Imaging Physics & Signal Reconstruction (CT, Radon Transform, Filtered Backprojection, MRI k-space)