1. Blind Deconvolution Matlab
2. Multichannel Blind Deconvolution Software
3. Blind Image Deconvolution

In mathematics, deconvolution is an algorithm-based process used to reverse the effects of convolution on recorded data.^[1] The concept of deconvolution is widely used in the techniques of signal processing and image processing. Because these techniques are in turn widely used in many scientific and engineering disciplines, deconvolution finds many applications.

In general, the objective of deconvolution is to find the solution of a convolution equation of the form:

${\displaystyle f\ast g=h}$

This demo contains all data and batch files you need to perform blind deconvolution of a 2D image and also demonstrates the GUI capabilities of my deconvolution software to allow you to see the deconvolution as it happens (the dislay is set to udate every 5 iterations in the demo but this can be adjusted by you). Plugin for ImageJ/Fiji. Microvolution ® Software for ImageJ delivers almost instantaneous deconvolution using your computer's GPU. Key features. Runs up to 200x faster than CPU-based methods; Easy to use ImageJ plugin; Deconvolves your images with accuracy.

Usually, h is some recorded signal, and f is some signal that we wish to recover, but has been convolved with some other signal g before we recorded it. The function g might represent the transfer function 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 is most often done using methods of statisticalestimation.

In physical measurements, the situation is usually closer to

${\displaystyle (f\ast g)+\epsilon =h}$

In this case ε is noise that has entered our recorded signal. If we assume that a noisy signal or image is noiseless when we try to make a statistical estimate of g, our estimate will be incorrect. In turn, our estimate of ƒ will also be incorrect. The lower the signal-to-noise ratio, the worse our estimate of the deconvolved signal will be. That is the reason why inverse filtering the signal is usually not a good solution. However, if we have at least some knowledge of the type of noise in the data (for example, white noise), we may be able to improve the estimate of ƒ through techniques such as Wiener deconvolution.

Deconvolution is usually performed by computing the Fourier Transform of the recorded signal h and the transfer functiong, apply deconvolution in the Frequency domain, which in the case of absence of noise is merely:

${\displaystyle F=H/G}$

F, G, and H being the Fourier Transforms of f, g, and h respectively. Finally inverse Fourier TransformF to find the estimated deconvolved signal f.

The foundations for deconvolution and time-series analysis were largely laid by Norbert Wiener of the Massachusetts Institute of Technology in his book Extrapolation, Interpolation, and Smoothing of Stationary Time Series (1949).^[2] The book was based on work Wiener had done during World War II but that had been classified at the time. Some of the early attempts to apply these theories were in the fields of weather forecasting and economics.

• 1Applications

Applications[edit]

Seismology[edit]

The concept of deconvolution had an early application in reflection seismology. In 1950, Enders Robinson was a graduate student at MIT. He worked with others at MIT, such as Norbert Wiener, Norman Levinson, and economist Paul Samuelson, to develop the 'convolutional model' of a reflection seismogram. This model assumes that the recorded seismogram s(t) is the convolution of an Earth-reflectivity function e(t) and a seismicwaveletw(t) from a point source, where t represents recording time. Thus, our convolution equation is

${\displaystyle s(t)=(e\ast w)(t).}$

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. 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,

${\displaystyle S(\omega )\approx kW(\omega ).}$

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):

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

Blind Deconvolution Matlab

where N is the number of reflection events, τ_iτ_i are the reflection times of each event, and r_i are the reflection coefficients.

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[edit]

In optics and imaging, the term 'deconvolution' is specifically used to refer to the process of reversing the optical distortion that takes place in an optical microscope, electron microscope, telescope, or other imaging instrument, thus creating clearer images. It is usually done in the digital domain by a softwarealgorithm, as part of a suite of microscope image processing techniques. Deconvolution is also practical to sharpen images that suffer from fast motion or jiggles during capturing. Early Hubble Space Telescope images were distorted by a flawed mirror and were sharpened by deconvolution.

The usual method is to assume that the optical path through the instrument is optically perfect, convolved with a point spread function (PSF), that is, a mathematical function that describes the distortion in terms of the pathway a theoretical point source of light (or other waves) takes through the instrument.^[3] Usually, such a point source contributes a small area of fuzziness to the final image. If this function can be determined, it is then a matter of computing its inverse or complementary function, and convolving the acquired image with that. The result is the original, undistorted image.

In practice, finding the true PSF is impossible, and usually an approximation of it is used, theoretically calculated^[4] or based on some experimental estimation by using known probes. Real optics may also have different PSFs at different focal and spatial locations, and the PSF may be non-linear. The accuracy of the approximation of the PSF will dictate the final result. Different algorithms can be employed to give better results, at the price of being more computationally intensive. Since the original convolution discards data, some algorithms use additional data acquired at nearby focal points to make up some of the lost information. Regularization in iterative algorithms (as in expectation-maximization algorithms) can be applied to avoid unrealistic solutions.

When the PSF is unknown, it may be possible to deduce it by systematically trying different possible PSFs and assessing whether the image has improved. This procedure is called blind deconvolution.^[3] Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed exposes the PSF thus making it more feasible. It is also used in fluorescence microscopy for image restoration, and in fluorescence spectral imaging for spectral separation of multiple unknown fluorophores. The most common iterative algorithm for the purpose is the Richardson–Lucy deconvolution algorithm; the Wiener deconvolution (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[edit]

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.

Fourier transform aspects[edit]

Deconvolution maps to division in the Fourier co-domain. This allows deconvolution to be easily applied with experimental data that are subject to a Fourier transform. An example is 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 Lorenzian lines in the frequency domain.

Absorption spectra[edit]

Deconvolution has been applied extensively to absorption spectra.^[7] The Van Cittert algorithm (in German) may be used.^[8]

See also[edit]

References[edit]

1. ^O'Haver T. 'Intro to Signal Processing - Deconvolution'. University of Maryland at College Park. Retrieved 2007-08-15.
2. ^Wiener N (1964). Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Cambridge, Mass: MIT Press. ISBN0-262-73005-7.
3. ^ ^abCheng PC (2006). 'The Contrast Formation in Optical Microscopy'. Handbook of Biological Confocal Microscopy (Pawley JB, ed.) (3rd ed.). Berlin: Springer. pp. 189–90. ISBN0-387-25921-X.
4. ^Nasse M. J., Woehl J. C. (2010). 'Realistic modeling of the illumination point spread function in confocal scanning optical microscopy'. J. Opt. Soc. Am. A. 27 (2): 295–302. doi:10.1364/JOSAA.27.000295. PMID20126241.
5. ^Ahi, Kiarash; Anwar, Mehdi (May 26, 2016). 'Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution'. Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N. doi:10.1117/12.2228680.
6. ^Sung, Shijun (2013). Terahertz Imaging and Remote Sensing Design for Applications in Medical Imaging. UCLA Electronic Theses and Dissertations.
7. ^Blass, W.E.; Halsey, G.W. (1981). Deconvolution of Absorption Spectra. Academic Press. ISBN0121046508.
8. ^Wu, Chengqi; Aissaoui, Idriss; Jacquey, Serge (1994). 'Algebraic analysis of the Van Cittert iterative method of deconvolution with a general relaxation factor'. J. Opt. Soc. Am. A. 11 (11): 2804–2808. doi:10.1364/JOSAA.11.002804.

In electrical engineering and applied mathematics, blind deconvolution is deconvolution without explicit knowledge of the impulse response function used in the convolution. This is usually achieved by making appropriate assumptions of the input to estimate the impulse response by analyzing the output. Blind deconvolution is not solvable without making assumptions on input and impulse response. Most of the algorithms to solve this problem are based on assumption that both input and impulse response live in respective known subspaces. However, blind deconvolution remains a very challenging non-convex optimization problem even with this assumption.

• 2In signal processing

In image processing[edit]

In image processing, blind deconvolution is a deconvolution technique that permits recovery of the target scene from a single or set of 'blurred' images in the presence of a poorly determined or unknown point spread function (PSF).^[2] Regular linear and non-linear deconvolution techniques utilize a known PSF. For blind deconvolution, the PSF is estimated from the image or image set, allowing the deconvolution to be performed. Researchers have been studying blind deconvolution methods for several decades, and have approached the problem from different directions.

Multichannel Blind Deconvolution Software

Most of the work on blind deconvolution started in early 1970s. Blind deconvolution is used in astronomical imaging and medical imaging.

Blind deconvolution can be performed iteratively, whereby each iteration improves the estimation of the PSF and the scene, or non-iteratively, where one application of the algorithm, based on exterior information, extracts the PSF. Iterative methods include maximum a posteriori estimation and expectation-maximization algorithms. A good estimate of the PSF is helpful for quicker convergence but not necessary.

Examples of non-iterative techniques include SeDDaRA,^[3] the cepstrum transform and APEX. The cepstrum transform and APEX methods assume that the PSF has a specific shape, and one must estimate the width of the shape. For SeDDaRA, the information about the scene is provided in the form of a reference image. The algorithm estimates the PSF by comparing the spatial frequency information in the blurred image to that of the target image.

Limitation of Blind deconvolution is that both input image and blur kernel must live in fixed subspace. That means input image, represented by w, has to be written as w=Bh, where B is random matrix of size L by K (K<L) and h is of size K by 1, whereas blur kernel, if represented by x, has to be written as x=Cm, where C is random matrix of size L by N (N<L) and m is of size N by 1.Observed image, if represented by y, given by y=w*x, can only be reconstructed if L >=K +N.

Examples

Any blurred image can be given as input to blind deconvolution algorithm, it can deblur the image, but essential condition for working of this algorithm must not be violated as discussed above. In the first example (picture of shapes), recovered image was very fine, exactly similar to original image because L < K + N. In the second example (picture of a girl), L > K + N, so essential condition is violated, hence recovered image is far different from original image.

In signal processing[edit]

Seismic data[edit]

In the case of deconvolution of seismic data, the original unknown signal is made of spikes hence is possible to characterize with sparsity constraints^[4] or regularizations such as l₁ norm/l₂ norm norm ratios,^[5] suggested by W. C. Gray in 1978.^[6]

Audio deconvolution[edit]

Audio deconvolution (often referred to as dereverberation) is a reverberation reduction in audio mixtures. It is part of audio processing of recordings in ill-posed cases such as the cocktail party effect. One possibility is to use ICA.^[7]

In general[edit]

Suppose we have a signal transmitted through a channel. The channel can usually be modeled as a linear shift-invariant system, so the receptor receives a convolution of the original signal with the impulse response of the channel. If we want to reverse the effect of the channel, to obtain the original signal, we must process the received signal by a second linear system, inverting the response of the channel. This system is called an equalizer.

If we are given the original signal, we can use a supervising technique, such as finding a Wiener filter, but without it, we can still explore what we do know about it to attempt its recovery. For example, we can filter the received signal to obtain the desired spectral power density. This is what happens, for example, when the original signal is known to have no auto correlation, and we 'whiten' the received signal.

Whitening usually leaves some phase distortion in the results. Most blind deconvolution techniques use higher-order statistics of the signals, and permit the correction of such phase distortions. We can optimize the equalizer to obtain a signal with a PSF approximating what we know about the original PSF.

Blind Image Deconvolution

High-order statistics[edit]

Blind deconvolution algorithms often make use of high-order statistics, with moments higher than two. This can be implicit or explicit.^[8]

See also[edit]

External links[edit]

References[edit]

1. ^Barmby, Pauline; McLaughlin, Dean E.; Harris, William E.; Harris, Gretchen L. H.; Forbes, Duncan A. (2007). 'Structural Parameters for Globular Clusters in M31 and Generalizations for the Fundamental Plane'(PDF). The Astronomical Journal. 133 (6): 2764–2786. arXiv:0704.2057. Bibcode:2007AJ..133.2764B. doi:10.1086/516777.
2. ^Lam, Edmund Y.; Goodman, Joseph W. (2000). 'Iterative statistical approach to blind image deconvolution'. Journal of the Optical Society of America A. 17 (7): 1177–1184. Bibcode:2000JOSAA.17.1177L. doi:10.1364/JOSAA.17.001177.
3. ^Caron, James N.; Namazi, Nader M.; Rollins, Chris J. (2002). 'Noniterative blind data restoration by use of an extracted filter function'. Applied Optics. 41 (32): 6884–9. Bibcode:2002ApOpt.41.6884C. doi:10.1364/AO.41.006884. PMID12440543.
4. ^Broadhead, Michael (2010). 'Sparse seismic deconvolution by method of orthogonal matching pursuit'.Cite journal requires journal= (help)
5. ^Barmby, P.; McLaughlin, D. E.; Harris, W. E.; Harris, G. L. H.; Forbes, D. A. (2015). 'Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization'. IEEE Signal Processing Letters. 22 (5): 539–543. arXiv:1407.5465. Bibcode:2015ISPL..22.539R. doi:10.1109/LSP.2014.2362861.
6. ^Gray, W. C. (1978). 'Variable norm deconvolution'(PDF). Archived from the original(PDF) on 2015-04-09.Cite journal requires journal= (help)
7. ^Koldovsky, Zbynek; Tichavsky, Petr (2007). 'Time-domain blind audio source separation using advanced ICA methods'. The Proceedings of the 8th Annual Conference of the International Speech Communication Association (Interspeech 2007). pp. 846–849.
8. ^Cardoso, J.-F. (1991). 'Super-symmetric decomposition of the fourth-order cumulant tensor. Blind identification of more sources than sensors'. [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing. 5. pp. 3109–3112. CiteSeerX10.1.1.8.9380. doi:10.1109/ICASSP.1991.150113. ISBN978-0-7803-0003-3.