This package provides the functions for total variation (TV) denoising with gradient projection (GP) and fast gradient projection (FGP). GP/FGP denoises of (volume) image $\mathbf{b}$ based on the following minimization problem

$$\min_{\mathbf{x}\in{C}}\|\mathbf{x} - \mathbf{b}\|^{2}_{F} + 2\lambda\mathrm{TV}(\mathbf{x}),$$

where $C$ is a convex closed set, $\|\cdot\|_{F}$ is the Frobenius norm, $\lambda$ is the regularization parameter, and $\mathrm{TV}$ is the total variation defined by

$$\begin{align} \mathrm{TV}_{I}(\mathbf{x}) &= \begin{cases} \sum_{i}\sum_{j}\sqrt{\left(x_{i,j} - x_{i+1,j}\right)^{2} + \left(x_{i,j} - x_{i,j+1}\right)^{2}} & \text{(2D)}\\ \sum_{i}\sum_{j}\sum_{k}\sqrt{\left(x_{i,j,k} - x_{i+1,j,k}\right)^{2} + \left(x_{i,j,k} - x_{i,j+1,k}\right)^{2} + \left(x_{i,j,k} - x_{i,j,k+1}\right)^{2}} & \text{(3D)} \end{cases},\\ \mathrm{TV}_{l_{1}}(\mathbf{x}) &= \begin{cases} \sum_{i}\sum_{j}\left\{\left|x_{i,j} - x_{i+1,j}\right| + \left|x_{i,j} - x_{i,j+1}\right|\right\} & \text{(2D)}\\ \sum_{i}\sum_{j}\sum_{k}\left\{\left|x_{i,j,k} - x_{i+1,j,k}\right| + \left|x_{i,j,k} - x_{i,j+1,k}\right| + \left|x_{i,j,k} - x_{i,j,k+1}\right|\right\} & \text{(3D)} \end{cases}. \end{align}$$

$\mathrm{TV}_{I}$ and $\mathrm{TV}_{l_{1}}$ express isotropic and $l_{1}$-based anisotropic TV, respectively.

For more information on the algorithm, please refer to the following reference.

Amir Beck and Marc Teboulle, "Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems," IEEE Trans. Image Process. 18, 2419-2434 (2009)

To install this package, open the Julia REPL and run

julia> ]add FastGradientProjection

or

julia> using Pkg
julia> Pkg.add("FastGradientProjection")

Import the package first.

julia> using FastGradientProjection

For example, perform the following to denoise (volume) image $\mathbf{b}$ with FGP.

julia> x = FGP(b, 0.1, 100, lower_bound=0.0, upper_bound=1.0)

The second and third arguments are the regularization parameter $\lambda$ and the number of iterations. The values specified by the keyword arguments lower_bound and upper_bound are the upper and lower bounds of the convex closed set $C$. These default values are lower_bound=-Inf and upper_bound=Inf. If $\mathbf{b}$ is a complex array, projection to $C$ is not performed. The function FGP performs denoising based on isotropic TV by default; to perform denoising based on anisotropic TV, specify the keyword argument TV="aniso". Refer to the documentation for further information.

  
noised image            denoised image