rampfilter

Ramp filter, also known as the Ram-Lak filter, is a linear filter with a frequency response proportional to the absolute value of frequency, H(f) ∝ |f|. In the time domain its impulse response is proportional to 1/π t^2 (in the sense of distributions). This makes it a high-pass filter that emphasizes high-frequency components and sharp features.

In X-ray computed tomography, the ramp filter is used in the filtered back projection algorithm to counteract the blurring that occurs during the inversion of the Radon transform. Projection data collected at multiple angles are transformed to the frequency domain, multiplied by the ramp response, and then inverse transformed. This yields a back projection that more accurately reconstructs edge details. The ideal ramp filter is often modified by smoothing windows to reduce noise, producing windowed ramp filters such as Shepp-Logan, Hamming, and cosine variants. The unwindowed Ram-Lak filter tends to amplify noise.

Implementation notes: In practice, the projection data are discretely sampled, and the ramp filter is applied via a discrete Fourier transform or a fast convolution with the equivalent impulse response. Zero-padding and windowing help control high-frequency noise and ringing. The ramp filter concept also appears in other signal-processing contexts as a high-pass filter with a linear magnitude response.

See also: filtered back projection, Ram-Lak filter, Shepp-Logan filter, CT reconstruction.