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In mathematics, deconvolution is the inverse of convolution. Both operations are used in signal processing and image processing. For example, it may be possible to recover the original signal after a filter (convolution) by using a deconvolution method with a certain degree of accuracy. [1]
Computing the inverse of the convolution operation is known as deconvolution. Definition. The convolution of and is written , denoting the operator with the symbol . ...
The Dirichlet convolution of two multiplicative functions is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring.
Here, / is the inverse of the original system, = / is the signal-to-noise ratio, and | | is the ratio of the pure filtered signal to noise spectral density. When there is zero noise (i.e. infinite signal-to-noise), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as we might ...
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
The Dirichlet inverse of a function f satisfies () = () = for all . There is a well-known recursive convolution formula for computing the Dirichlet inverse f − 1 ( n ) {\displaystyle f^{-1}(n)} of a function f by induction given in the form of [ 7 ]
where the division and multiplication are element wise, indicates a 2D convolution, and is the mirrored point spread function, or the inverse Fourier transform of the Hermitian transpose of the optical transfer function.
Most of the important attributes of the complex DFT, including the inverse transform, the convolution theorem, and most fast Fourier transform (FFT) algorithms, depend only on the property that the kernel of the transform is a principal root of unity. These properties also hold, with identical proofs, over arbitrary rings.