Deconvolution is simply known as inverse filtering. First of all, when we consider digital image processing, Inverse filtering is used for image restoration to obtain a recovered image from the image data. When the image is blurred by a known lowpass filter, it is possible to recover the image by inverse filtering, such that,
DFT of blurred image = DFT of the original image × DFT of blurred kernel
Where DFT - Discrete Fourier Transform
If the DFT of the blurring kernel and that of blurred image are known, then we can recover DFT of the original image, as the division between DFT of blurred image to that of blurred kernel.
However the important assumption, we have to consider is that no noise is present in the system since inverse filtering is very sensitive to additive noise.
What is Deconvolution?
Deconvolution
Deconvolution can be defined as the process of undoing the convolution to get back to the input signal. In layman terms we can say deconvolution is the inverse of convolution. So it is clear that if you want to understand the term deconvolution, then the term convolution is unavoidable.
Types of deconvolution
In seismology, the term deconvolution can be categorized as explained here below
Spiking (whitening) deconvolution
This reduces the source wavelet to a spike by using Wiener filtering
Time-variant deconvolution
Here the deconvolution operator D changes with time to account for the different frequency content of energy that has traveled greater distances.
Predictive deconvolution
This deconvolution considers the arrival times of primary reflections which are used to predict the arrival times of multiples which are then removed.
But when we consider spiking deconvolution as the inverse filtering which can be defined as the process which is used to recover an original signal, let say x(t) from a response signal let say y(t), such as y(t) is obtained as the convolution between original signal x(t), and the impulse response P, you may refer to this post to gain insight on convolution.
y(t) = P * x(t)
Then the inverse Filter is given as P−1 as deconvolution operator, it can be used to get back the original signal x(t) by using the below expression.
x(t) = P−1y(t)
Deconvolution solves inverse problems as it finds x(t) from known y(t) and P, as shown in the expression above.
A good example to demonstrate the deconvolution is the case of seismic reflection data processing, if we want to recover the reflectivity series from a recorded Seismogram. If our seismogram, S which is recorded on seismograph is obtained by convolution between reflectivity series (R) and the seismic source wavelet (W), then Seismogram (S) is given by
S = R*W
If we let the deconvolution operator (inverse filter) represented by D, which is obtained by
D*W = δ
Where δ is the dirac (delta) function.
Then, to recover the reflectivity series can be done as by taking
D*S = D*R*W.
Then, D*W*R, but D*W = δ
Then, δ*R = R
Therefore, D*S = R
The diagrams below illustrating, on how the deconvolution is done as inverse filtering, with the deconvolution operator is flipped and moved from your left to right over the recorded signal (S)
Figure : Signal (S) is inverse filtered and the wave train is removed by deconvolution.
That is deconvolution in simple terms!
Thanks for your time!
References
John Reynolds (1997), An introduction to Applied and Environmental Geophysics, John Wiley and Sons
Kearey, P & Brooks, M. (1991), An introduction to geophysical exploration. Blackwell Scientific Publication
O'Haver, T. "Introduction to signal processing - Deconvolution". University of Maryland at College Park. Retrieved 2007-08-15.
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