Deconvolution

In

Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with a filter or distortion function g, before we recorded it. Usually, h is a distorted version of f and the shape of f can't be easily recognized by the eye or simpler time-domain operations. The function g represents the impulse response of an instrument or a driving force that was applied to a physical system. If we know g, or at least know the form of g, then we can perform deterministic deconvolution. However, if we do not know g in advance, then we need to estimate it. This can be done using methods of statistical estimation or building the physical principles of the underlying system, such as the electrical circuit equations or diffusion equations.

There are several deconvolution techniques, depending on the choice of the measurement error and deconvolution parameters:

Raw deconvolution

When the measurement error is very low (ideal case), deconvolution collapses into a filter reversing. This kind of deconvolution can be performed in the Laplace domain. By computing the Fourier transform of the recorded signal h and the system response function g, you get H and G, with G as the transfer function. Using the Convolution theorem,

${\displaystyle F=H/G\,}$

where F is the estimated Fourier transform of f. Finally, the inverse Fourier transform of the function F is taken to find the estimated deconvolved signal f. Note that G is at the denominator and could amplify elements of the error model if present.

Deconvolution with noise

In physical measurements, the situation is usually closer to

${\displaystyle (f*g)+\varepsilon =h\,}$

In this case ε is

The seismologist is interested in e, which contains information about the Earth's structure. By the convolution theorem, this equation may be Fourier transformed to

${\displaystyle S(\omega )=E(\omega )W(\omega )\,}$

in the frequency domain, where ${\displaystyle \omega }$ is the frequency variable. By assuming that the reflectivity is white, we can assume that the power spectrum of the reflectivity is constant, and that the power spectrum of the seismogram is the spectrum of the wavelet multiplied by that constant. Thus,

${\displaystyle |S(\omega )|\approx k|W(\omega )|.\,}$

If we assume that the wavelet is minimum phase, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. The reflectivity may be recovered by designing and applying a Wiener filter that shapes the estimated wavelet to a Dirac delta function (i.e., a spike). The result may be seen as a series of scaled, shifted delta functions (although this is not mathematically rigorous):

${\displaystyle e(t)=\sum _{i=1}^{N}r_{i}\delta (t-\tau _{i}),}$

where N is the number of reflection events, ${\displaystyle r_{i}}$ are the reflection coefficients, ${\displaystyle t-\tau _{i}}$ are the reflection times of each event, and ${\displaystyle \delta }$ is the Dirac delta function.

In practice, since we are dealing with noisy, finite bandwidth, finite length, discretely sampled datasets, the above procedure only yields an approximation of the filter required to deconvolve the data. However, by formulating the problem as the solution of a Toeplitz matrix and using Levinson recursion, we can relatively quickly estimate a filter with the smallest mean squared error possible. We can also do deconvolution directly in the frequency domain and get similar results. The technique is closely related to linear prediction.

Optics and other imaging

In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing the

(and approximations) are the most common non-iterative algorithms.

For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically.^[6] As a result, as shown in the figure, deconvolution of the modeled PSF and the terahertz image can give a higher resolution representation of the terahertz image.

Radio astronomy

When performing image synthesis in radio interferometry, a specific kind of radio astronomy, one step consists of deconvolving the produced image with the "dirty beam", which is a different name for the point spread function. A commonly used method is the CLEAN algorithm.

Biology, physiology and medical devices

Typical use of deconvolution is in tracer kinetics. For example, when measuring a hormone concentration in the blood, its secretion rate can be estimated by deconvolution. Another example is the estimation of the blood glucose concentration from the measured interstitial glucose, which is a distorted version in time and amplitude of the real blood glucose. ^[7]

Absorption spectra

Deconvolution has been applied extensively to absorption spectra.^[8] The Van Cittert algorithm (article in German) may be used.^[9]

Fourier transform aspects

Deconvolution maps to division in the

NMR spectroscopy

where the data are recorded in the time domain, but analyzed in the frequency domain. Division of the time-domain data by an exponential function has the effect of reducing the width of Lorentzian lines in the frequency domain.

See also

• Convolution
• Bit plane
• Digital filter
• Filter (signal processing)
• Filter design
• Minimum phase
• Independent component analysis
• Wiener deconvolution
• Richardson–Lucy deconvolution
• Digital room correction
• Free deconvolution
• Point spread function
• Deblurring
• Unsharp masking

References

1. ^ O'Haver, T. "Intro to Signal Processing - Deconvolution". University of Maryland at College Park. Retrieved 2007-08-15.
2. ISBN 0-262-73005-7
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3. ^
   ISBN 0-387-25921-X
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4. PMID 20126241
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5. S2CID 114994724
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6. ^ Sung, Shijun (2013). Terahertz Imaging and Remote Sensing Design for Applications in Medical Imaging. UCLA Electronic Theses and Dissertations.
7. PMID 8773311
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8. ISBN 0121046508
   .
9. doi:10.1364/JOSAA.11.002804
   .