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— CH. 1 · MATHEMATICAL FOUNDATIONS AND HISTORY —

Deconvolution

~2 min read · Ch. 1 of 6
6 sections

  • Norbert Wiener published Extrapolation, Interpolation, and Smoothing of Stationary Time Series in 1949. This book laid the groundwork for deconvolution theory at the Massachusetts Institute of Technology. The research inside had been classified during World War II before its release to the public. Early applications emerged in weather forecasting and economics shortly after publication. These fields needed ways to recover original signals from distorted measurements taken by instruments.

  • Raw deconvolution collapses into simple filter reversing when measurement error remains very low. In physical measurements, noise enters the recorded signal as an epsilon term that complicates recovery. If a noisy image assumes no noise exists, statistical estimates become incorrect and amplify errors. The lower the signal-to-noise ratio, the worse the final estimate becomes. Wiener deconvolution improves results if knowledge of white noise types exists within the data set.

  • Enders Robinson worked with Norbert Wiener and Norman Levinson at MIT in 1950. They developed the convolutional model for reflection seismograms used to map Earth structure. A seismic wavelet w(t) convolves with an Earth-reflectivity function e(t) to create the recorded signal s(t). Seismologists assume reflectivity is white to simplify calculations regarding power spectra. Designing a Wiener filter shapes the estimated wavelet into a Dirac delta spike for clearer interpretation.

  • Early Hubble Space Telescope images suffered distortion due to a flawed mirror before correction. Deconvolution reverses optical distortions found in microscopes, telescopes, and electron imaging instruments. A point spread function describes how light travels through the instrument pathway theoretically. Finding the true point spread function remains impossible, so approximations based on known probes are used instead. Blind deconvolution deduces this function by systematically testing different possibilities until image quality improves.

  • Image synthesis in radio interferometry requires deconvolving the produced image with a dirty beam. The CLEAN algorithm serves as a commonly used method for this specific type of astronomy work. NMR spectroscopy records data in the time domain but analyzes it within the frequency domain. Division of time-domain data by an exponential function reduces the width of Lorentzian lines significantly. This process maps directly to division operations within the Fourier co-domain for easier application.

  • Tracer kinetics provides typical use cases where hormone concentration measurements occur in blood samples. Deconvolution estimates secretion rates from these distorted measurements taken over time intervals. Blood glucose monitoring devices estimate real blood glucose concentration from measured interstitial glucose levels. These interstitial readings represent a distorted version of actual values in both time and amplitude dimensions. Absorption spectra applications utilize the Van Cittert algorithm extensively across various scientific fields.

Common questions

When did Norbert Wiener publish Extrapolation, Interpolation, and Smoothing of Stationary Time Series?

Norbert Wiener published Extrapolation, Interpolation, and Smoothing of Stationary Time Series in 1949. This book laid the groundwork for deconvolution theory at the Massachusetts Institute of Technology.

What is the relationship between signal-to-noise ratio and deconvolution accuracy?

The lower the signal-to-noise ratio, the worse the final estimate becomes. If a noisy image assumes no noise exists, statistical estimates become incorrect and amplify errors.

Who developed the convolutional model for reflection seismograms with Norbert Wiener?

Enders Robinson worked with Norbert Wiener and Norman Levinson at MIT in 1950 to develop the convolutional model for reflection seismograms used to map Earth structure.

How does blind deconvolution determine the point spread function in imaging instruments?

Blind deconvolution deduces this function by systematically testing different possibilities until image quality improves. Finding the true point spread function remains impossible, so approximations based on known probes are used instead.

Which algorithm serves as a commonly used method for radio interferometry image synthesis?

The CLEAN algorithm serves as a commonly used method for this specific type of astronomy work involving deconvolving the produced image with a dirty beam.

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All sources

9 references cited across the entry

  1. 1webIntro to Signal Processing - DeconvolutionT. O'Haver — University of Maryland at College Park
  2. 2bookExtrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering ApplicationsNorbert Wiener — MIT Press — 1949
  3. 3bookHandbook of Biological Confocal MicroscopyP. C. Cheng — Springer — 2006
  4. 4journalRealistic modeling of the illumination point spread function in confocal scanning optical microscopyM. J. Nasse et al. — 2010
  5. 5journalDeveloping terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolutionKiarash Ahi et al. — May 26, 2016
  6. 6bookTerahertz Imaging and Remote Sensing Design for Applications in Medical ImagingShijun Sung — UCLA Electronic Theses and Dissertations — 2013
  7. 7journalReconstruction of insulin secretion rate by deconvolution: domain of validity of a monoexponential C-peptide impulse response modelGiovanni Sparacino et al. — 1996
  8. 8bookDeconvolution of Absorption SpectraW. E. Blass et al. — Academic Press — 1981
  9. 9journalAlgebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factorChengqi Wu — 1994