Editing Deconvolution (section)

===Optics and other imaging===
[[File:Depth Coded Phalloidin Stained Actin Filaments Cancer Cell.png|thumb|Example of a deconvolved microscope image.|245x245px]]
In optics and imaging, the term "deconvolution" is specifically used to refer to the process of reversing the [[Aberration in optical systems#Distortion of the image|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 [[Hubble Space Telescope#Flawed mirror|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.<ref name=Pawley_2006>{{cite book |last=Cheng |first=P. C. |chapter =The Contrast Formation in Optical Microscopy |title=Handbook of Biological Confocal Microscopy |url=https://archive.org/details/handbookbiologic00pawl |url-access=limited |editor-last=Pawley |editor-first=J. B. |publisher=Springer |location=Berlin |year=2006 |pages= [https://archive.org/details/handbookbiologic00pawl/page/n214 189]&ndash;90 |edition=3rd |isbn=0-387-25921-X}}</ref> 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 function|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<ref>{{cite journal |last1=Nasse |first1=M. J. |last2=Woehl |first2=J. C. |title=Realistic modeling of the illumination point spread function in confocal scanning optical microscopy |journal=Journal of the Optical Society of America A |volume=27 |issue=2 |pages=295–302 |year=2010 |doi=10.1364/JOSAA.27.000295 |pmid=20126241|bibcode=2010JOSAA..27..295N }}</ref> 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 (mathematics)|Regularization]] in iterative algorithms (as in [[expectation-maximization algorithm]]s) 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]]''.<ref name=Pawley_2006 /> Blind deconvolution is a well-established [[Iterative reconstruction|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 [[fluorophore]]s. The most common [[Iteration|iterative]] algorithm for the purpose is the [[Richardson–Lucy deconvolution]] algorithm; the [[Wiener deconvolution]] (and approximations) are the most common non-iterative algorithms.
[[File:High Resolution THz image.png|thumb|316x316px|High Resolution THz image is achieved by deconvolution of the THz image and the mathematically modeled THz PSF. '''(a)''' THz image of an integrated circuit (IC) before enhancement; '''(b)''' Mathematically modeled THz PSF; '''(c)''' High resolution THz image which is achieved as a result of deconvolution of the THz image shown in (a) and the PSF which is shown in (b); '''(d)''' High resolution X-ray image confirms the accuracy of the measured values.<ref>{{Cite journal |last1=Ahi |first1=Kiarash |first2=Mehdi |last2=Anwar |editor3-first=Tariq |editor3-last=Manzur |editor2-first=Thomas W |editor2-last=Crowe |editor1-first=Mehdi F |editor1-last=Anwar |date=May 26, 2016 |title=Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution |url=https://www.researchgate.net/publication/303563271 |journal=Proc. SPIE 9856, Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense, 98560N |series=Terahertz Physics, Devices, and Systems X: Advanced Applications in Industry and Defense |volume=9856 |pages=98560N |doi=10.1117/12.2228680|bibcode=2016SPIE.9856E..0NA |s2cid=114994724 }}</ref>]]
For some specific imaging systems such as laser pulsed terahertz systems, PSF can be modeled mathematically.<ref>{{Cite book|title=Terahertz Imaging and Remote Sensing Design for Applications in Medical Imaging |last=Sung |first=Shijun |publisher=UCLA Electronic Theses and Dissertations |year=2013}}</ref> 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.