# Deconvolution.jl

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## Introduction

This package provides a set of functions to deconvolve digital signals, like images or time series. This is written in Julia, a modern high-level, high-performance dynamic programming language designed for technical computing.

## Installation

The latest version of `Deconvolution.jl`

is available for Julia 1.0 and later
versions, and can be installed with Julia built-in package
manager. In a Julia session, after
entering the package manager mode with `]`

, run the command

`pkg> add Deconvolution`

Older versions are also available for Julia 0.4-0.7.

### Documentation

The complete manual of `Deconvolution.jl`

is available at
the documentation page. It has more detailed explanation of
the methods used and the examples are complemented with pictures.

## Usage

Currently `Deconvolution.jl`

provides only two methods, but others will come in
the future.

`wiener`

`wiener(input, signal, noise[, blurring])`

The Wiener deconvolution attempts at reducing the noise in a digital signal by suppressing frequencies with low signal-to-noise ratio. The signal is assumed to be degraded by additive noise and a shift-invariant blurring function.

The `wiener`

function can be used to apply the Wiener deconvolution method to a
digital signal. The arguments are:

`input`

: the digital signal`signal`

: the original signal (or a signal with a luckily similar power spectrum)`noise`

: the noise of the signal (or a noise with a luckily similar power spectrum)`blurring`

(optional argument): the blurring kernel

All arguments must be arrays, all with the same size, and all of them in the
time/space domain (they will be converted to the frequency domain internally
using `fft`

function). Argument `noise`

can be also a real number, in which
case a constant noise with that value will be assumed (this is a good
approximation in the case of
white noise).

`lucy`

`lucy(observed, psf[, iterations])`

The Richardson-Lucy deconvolution is an iterative method based on Bayesian inference for restoration of signal that is convolved with a point spread function.

The `lucy`

function can be used to apply the Richardson-Lucy deconvolution
method to a digital signal. The arguments are:

`observed`

: the observed blurred signal`psf`

: the point spread function (the blurring kernel)`iterations`

(optional argument): the number of iterations

First two arguments must be arrays, all with the same size, and all of them
in the time/space domain (they will be converted to the frequency domain
internally using `fft`

function). Argument `iterations`

is an integer number.
The more iterations is specified the better result should be if the solution
converges and it is going to converge if PSF is estimated well.

## Examples

### Wiener deconvolution

Here is an example of use of `wiener`

function to perform the Wiener
deconvolution of an image, degraded with a blurring function and an additive
noise.

```
using Images, TestImages, Deconvolution, FFTW, ImageView
# Open the test image
img = channelview(testimage("cameraman"))
# Create the blurring kernel in frequency domain
x = hcat(ntuple(x -> collect((1:512) .- 257), 512)...)
k = 0.001
blurring_ft = @. exp(-k*(x ^ 2 + x ^ 2)^(5//6))
# Create additive noise
noise = rand(Float64, size(img))
# Fourier transform of the blurred image, with additive noise
blurred_img_ft = fftshift(blurring_ft) .* fft(img) .+ fft(noise)
# Get the blurred image from its Fourier transform
blurred_img = real(ifft(blurred_img_ft))
# Get the blurring kernel in the space domain
blurring = ifft(fftshift(blurring_ft))
# Polish the image with Deconvolution deconvolution
polished = wiener(blurred_img, img, noise, blurring)
# Wiener deconvolution works also when you don't have the real image and noise,
# that is the most common and useful case. This happens because the Wiener
# filter only cares about the power spectrum of the signal and the noise, so you
# don't need to have the exact signal and noise but something with a similar
# power spectrum.
img2 = channelview(testimage("livingroom")) # Load another image
noise2 = rand(Float64, size(img)) # Create another additive noise
# Polish the image with Deconvolution deconvolution
polished2 = wiener(blurred_img, img2, noise2, blurring)
# # Compare...
# imshow(img) # ...the original image
# imshow(blurred_img) # ...the blurred image
# imshow(polished) # ...the polished image
# imshow(polished2) # ...the second polished image
```

### Richardson-Lucy deconvolution

Here is an example of use of `lucy`

function to perform the Richardson-Lucy
deconvolution of an image blurred by kernel that models spherical lens aberration.

```
using Images, TestImages, Deconvolution, FFTW, ZernikePolynomials, ImageView
img = channelview(testimage("cameraman"))
# model of lens aberration
blurring = evaluateZernike(LinRange(-16,16,512), [12, 4, 0], [1.0, -1.0, 2.0], index=:OSA)
blurring = fftshift(blurring)
blurring = blurring ./ sum(blurring)
blurred_img = fft(img) .* fft(blurring) |> ifft |> real
@time restored_img = lucy(blurred_img, blurring, iterations=1000)
imshow(img)
imshow(blurring)
imshow(blurred_img)
imshow(restored_img)
```

## Development

The package is developed at https://github.com/JuliaDSP/Deconvolution.jl. There you can submit bug reports, propose new deconvolution methods with pull requests, and make suggestions. If you would like to take over maintainership of the package in order to further improve it, please open an issue.

### History

The ChangeLog of the package is available in NEWS.md file in top directory.

## License

The `Deconvolution.jl`

package is licensed under the MIT "Expat" License. The
original author is Mosè Giordano.