Tomography and Radon transforms

For a unit normal θ=(cosφ,sinφ), let θ perpendicular be the unit tangent obtained by a quarter-turn. The line with signed distance s from the origin consists of sθ+tθ perpendicular. Integrating f over t gives Rf(φ,s). Opposite normals describe the same unoriented lines with the sign of s reversed, so angles from zero to π and all real offsets suffice. A single projection cannot determine how mass is distributed along its integration direction, even when it is measured without noise.

Use the Fourier convention with exponent −2πi times the coordinate-frequency pairing in both one and two dimensions. Fourier transforming a projection in its offset variable and changing variables from s,t to the plane gives the two-dimensional Fourier transform of f on the radial line ωθ. The coordinate change is orthogonal with determinant of absolute value one. This proves the Fourier-slice theorem directly under the stated decay assumptions and explains which object frequencies each angle observes.

Fourier inversion integrates over the entire frequency plane. Parametrizing it by φ in [0,π) and signed radial frequency ω covers each nonzero point once, with Jacobian |ω|. Substituting the slice theorem therefore yields filtered backprojection: multiply each projection's Fourier transform by |ω|, invert in the offset variable, and integrate over normal angles. Under this convention the combined formula has no extra 2π multiplier. Changing the Fourier convention, angle interval, or backprojection normalization changes that constant.

The ramp multiplier grows with frequency, so high-frequency measurement noise is amplified. Windowing or regularizing the ramp improves stability at the cost of resolution. A discrete implementation must also address detector spacing, angular sampling, interpolation, and the field of view. Simply applying a continuous identity to finite arrays without analyzing those choices can produce artifacts and aliasing. The exact theorem explains the target operation; it does not automatically supply a stable numerical reconstruction at arbitrarily fine scales.

Full noiseless projections determine every Fourier frequency and hence uniquely determine a Schwartz object. Limited angles directly reveal only a wedge of frequency directions. Without a compact-support restriction, smooth functions with Fourier support in a missing wedge can be invisible. With compact support, analytic continuation of the Fourier transform can restore mathematical uniqueness from an open set of angles, yet the continuation is severely unstable. Thus the blanket statement that every limited-angle problem has a nontrivial nullspace is false without specifying the object class.

Directions of edges also matter: an edge normal that falls outside the measured angular range may be poorly visible even when a uniqueness theorem holds. This separates stable feature recovery from formal global identifiability. Priors or penalties can fill missing directions, but the filled structure depends on their assumptions. Reporting angular coverage, sampling, regularization, and the intended feature class makes a tomography result assessable. A smooth-looking image alone cannot establish that its unmeasured directional detail was supported by the observations.