What Is Deconvolution?

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

Deconvolution is

Deconvolution seismology

The concept of deconvolution has long been applied in reflection seismology. In 1950, Enders Robinson was a graduate student at the Massachusetts Institute of Technology. He worked with others at MIT, such as Norbert Wiener, Norman Levinson, and economist Paul Samuelson to develop a "convolution model" of reflected seismic records. The model assumes a recorded seismogram

Is the earth's reflectivity function from a point source

And seismic wavelets

Convolution, where t represents the recording time. So our convolution equation is

Seismologist

Very interested, contains information about the structure of the earth. With Convolution Theorem, the equation can be Fourier transformed to

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 seismic map is the spectrum of the wavelet times this constant. thereby,

If we assume the wavelet is the minimum phase, we can recover it by calculating the minimum phase equivalent of the power spectrum we just found. Reflectivity can be recovered by designing and applying a Wiener filter that turns the estimated small waveform into a Dirac delta function (ie, a spike). The result may be viewed as a series of scaled, shifted delta functions (although this is not mathematically strict):

Where N is the number of reflection events, ii is the reflection time for each event, and ri is the reflection coefficient.

In fact, since we are dealing with noise, finite bandwidth, finite length, discretely sampled data sets, the above process produces only approximations of the filters required for deconvolution of the data. However, by expressing the problem as a solution to the Toeplitz matrix and using Levinson recursion, we can relatively quickly estimate the filter with the smallest mean square error. We can also perform deconvolution directly in the frequency domain and get similar results. This technique is closely related to linear prediction.

Deconvolution optics and other images

In the field of optics and imaging, the term "deconvolution" refers specifically to the process used to reverse optical distortions that occur in optical microscopes, electron microscopes, telescopes, or other imaging instruments to create sharper images. It is usually done in the digital domain through software algorithms as part of a set of microscope image processing techniques. Deconvolution is also useful for sharpening images that are affected by fast motion or jitter during the capture process. Early Hubble Space Telescope images were distorted by a defective mirror and could be sharpened by deconvolution.

The usual method is to assume that the optical path through the instrument is optically ideal, convolved with a point spread function (PSF), a mathematical function that describes a theoretical point source (or other wave) of optical path through the instrument. Generally, such point sources contribute a small range of blurriness to the final image. If this function can be determined, calculate its inverse or complementary function and convolve the acquired image with it. The result is a raw, undistorted image.

In practice, finding a true PSF is not possible, and its approximate value is usually used, theoretically calculated or based on some experimental estimates using known probes. Real optics may also have different PSFs at different focal points and spatial locations, and PSFs may be non-linear. The accuracy of the PSF approximation will determine the final result. At a more computationally intensive price, different algorithms can be used to provide better results. Because the original convolution discards the data, some algorithms use additional data obtained at nearby focal points to make up for some missing information. Regularization in iterative algorithms, such as the expectation maximization algorithm, can be used to avoid unrealistic solutions.

Figure one

When the PSF is unknown, the PSF can be inferred by systematically trying different possible PSFs and assessing whether the image has improved. This process is called blind deconvolution. Blind deconvolution is a mature image restoration technology in astronomy. The point nature of the subject exposes the PSF, making it more feasible. It is also used for image restoration in fluorescence microscopy, and fluorescence spectral imaging, for spectral separation of multiple unknown fluorophores. The most commonly used iterative algorithm for this purpose is the Richardson-Lucy deconvolution algorithm; Wiener deconvolution (and approximation) is the most common non-iterative algorithm.

For some specific imaging systems, such as laser pulsed terahertz systems, PSF can be modeled mathematically. As a result, as shown, the deconvolution of the simulated PSF and terahertz images can give a higher resolution representation of the terahertz images. ^[2]

Deconvolution radio astronomy

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

Aspects of deconvolution Fourier transform

Deconvolution maps to partitions in the Fourier common domain. This allows deconvolution to be easily applied to experimental data subjected to Fourier transforms. One example is NMR spectroscopy, where data is recorded in the time domain but analyzed in the frequency domain. Dividing the time domain data by the exponential function has the effect of reducing the Lorentz line width in the frequency domain.