Deconvolution

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

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


 * $$f * g = h$$

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 statistical estimation.

In physical measurements, the situation is usually closer to


 * $$(f * g) + \epsilon  = h$$

In this case $$\epsilon$$ 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 $$f$$ will also be incorrect. The lower the signal-to-noise ratio, the poorer our estimate of the deconvolved signal will be. That is the reason why usually inverse filtering the signal is 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 $$f$$ through techniques such as Wiener deconvolution.

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). 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.

Seismology
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 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 seismic wavelet $$w(t)$$ from a point source, where $$t$$ represents recording time. Thus, our convolution equation is


 * $$s(t) = e(t) * w(t)$$.

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


 * $$S(\omega) = E(\omega)W(\omega)$$

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,


 * $$|S(\omega)| \approx k|W(\omega)|$$.

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):


 * $$e(t)=\sum_{i=1}^N r_i\delta(t-\tau_i)$$,

where $$N$$ is the number of reflection events, $$\tau_i$$ 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
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 software algorithm, 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 could be 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. 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 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. Blind deconvolution is a well-established image restoration technique in astronomy, where the point nature of the objects photographed makes 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.