Mathematics for MRI

Cosine transform

Idea: test whether a frequency is present by correlating the signal with a cosine at that frequency. If the frequency is present, the integral is non‑zero; if absent (and orthogonal), it is zero.

\[C(\nu) = \int_{-\infty}^{\infty} f(t)\,\cos(2\pi \nu t)\,dt\]

• For a signal lacking a given component (e.g., \(24\,\mathrm{Hz}\)), \(C(24\,\mathrm{Hz}) \approx 0\).

• Orthogonality of sinusoids underlies this test.

Fourier transform

Using Euler’s identity, a complex exponential compactly represents sine and cosine:

\[e^{\,i\theta} = \cos\theta + i\sin\theta \quad\Rightarrow\quad \cos(2\pi \nu t) + i\sin(2\pi \nu t) = e^{\,i 2\pi \nu t}\]

A common Fourier transform pair (unitary conventions vary):

\[F(\nu) = \int_{-\infty}^{\infty} f(t)\,e^{-i 2\pi \nu t}\,dt, \qquad f(t) = \int_{-\infty}^{\infty} F(\nu)\,e^{+i 2\pi \nu t}\,d\nu\]

Phase matters

• The spectrum is complex: \(F(\nu) = |F(\nu)| e^{i\phi(\nu)}\). Magnitude \(|F|\) sets how much, phase \(\phi\) sets when/shape.

• Time shift \(f(t-t_0)\) adds a linear phase ramp:

  \[\mathcal{F}\{f(t-t_0)\} = F(\nu)\,e^{-i 2\pi \nu t_0}\]

• Ignoring/altering phase can distort reconstructions (ringing, shifts, asymmetric waveforms), even if the magnitude is unchanged.

Reference

Signal processing video by de Graaf.