Signal Processing
The language of imaging and reconstruction is Fourier analysis. This base provides the math used throughout the site: the Fourier transform and the frequency domain, sampling and the Nyquist limit, convolution and the point-spread function, filtering, and the central-slice theorem that links tilt angle to frequency coverage — the origin of the missing wedge.
Partial sum of 4 sine harmonics. More harmonics approach the square wave, but the overshoot at the jumps (the Gibbs phenomenon) never vanishes — it only narrows.
Articles in this base 7 articles
The Fourier transform & frequency domain
A decomposition of a signal into sinusoids that turns convolution into multiplication and underpins nearly every step of Cryo-ET processing
The Contrast Transfer Function (CTF)
The microscope does not image faithfully — it weights every frequency by an oscillating curve and even inverts contrast
Sampling & the Nyquist limit
Recording a continuous signal on a discrete grid sets a hard upper frequency, above which detail is lost and aliasing corrupts the spectrum
The central-slice (Fourier-slice) theorem
The Fourier transform of a projection equals a central slice through the object's spectrum, linking tomographic tilt angles to coverage of frequency space
Convolution & the point-spread function
Every linear shift-invariant imaging system blurs by convolving the object with a point-spread function, which becomes a transfer function in frequency space
Filtering: ramp, low-pass, high-pass
Reweighting the spectrum to suppress noise, sharpen detail, or compensate the geometry of backprojection
Signal-to-noise ratio & electron dose
Why Cryo-ET images are dominated by noise, how dose limits the signal, and how averaging recovers structure