Deconvolution

With deconvolution (Eng. "unfolding " ) is called the inverse of the so-called convolution operation. This is a mathematical transformation that is applied among others in the signal and image processing. A convolution can always be computed, while its inverse is not always possible, because in the folding may lose information that can not be restored. In order to still be able to calculate a possible good, inverse convolution, compute-intensive algorithms and methods have been developed.

A simple example is the sharpening of an image. The process of blurring ( blur) is represented by a convolution. The sharpening of the image as it support many image editing programs, then corresponds to a deconvolution (see picture right).

Mathematics

In mathematics, deconvolution or unfolding refers to the reversal of a folding ( symbolically: "*", to avoid confusion with the pointwise multiplication, see below) of two functions. Illustrated General, this corresponds to the attempt from the result of the convolution of two functions f and g h

To determine the unknown function g with known h and f; This problem is also referred to as an inverse convolution problem. A common solution approach results from the convolution theorem, which states that the Fourier transform of a convolution of two functions is equal to the product of the Fourier transforms of the two functions. Accordingly, the above equation can be written as

Wherein, and the Fourier transform of f, g and h, respectively. Thus, in principle, could be determined as

And from this by Fourier transform g However, this general approach is usually not applicable because, first, the function g does not have to be unique, secondly, can contain the function zeros and thirdly real data usually n with an additive noise, corresponding to an additional term, are afflicted, so that in such cases the original problem

Complicated. For this reason, various methods are used, try to determine from h and f, the most likely result for g as a unique analytical solution does not exist. It turns out that the noise n in a naive refolding with the above division method is disproportionately amplified:

The gain is because usually at higher frequencies drops to 0 (eg smoothing filter = low-pass filter), while the noise contains just there too frequencies, which are then amplified by.

Image processing

Deconvolution is used for example for sharpening images in astrophotography and microscopy. Deconvolution filter try to capture the blur and mathematically to undo. Some methods are:

• Van Cittert deconvolution
• Wiener deconvolution
• Richardson -Lucy deconvolution
• Blind deconvolution and blind deconvolution dt
• Meinel deconvolution
• ZNova algorithm or ZNova deconvolution
• Agard Sedat deconvolution

The sharpening is done via the so-called PSF matrix ( engl: point spread function, Point Spread Function ). This describes the process that generated the blur. It may be, for example, to the filter mask a blur filter to act (eg, binomial ). A PSF can be calculated for any optical imaging system, such as the lens of a camera or a microscope (for example, with the software of PSF lab for a confocal microscope ). A complete restoration of an image is often not possible, because in the " Unschärfung " lost information. However, the method presented here attempt to recover as much information from the PSF and the image. "Blind deconvolution " attempted to estimate the optimal PSF matrix from the image.