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## Inhaltsbereich / Content

# Software

# Fast Finite Shearlet Transform (FFST)

The FFST package provides a fast implementation of the Finite Shearlet Transform. Following the the path via the continuous shearlet transform, its counterpart on cones and finally its discretization on the full grid we obtain the translation invariant discrete shearlet transform. Our discrete shearlet transform can be efficiently computed by the fast Fourier transform (FFT). The discrete shearlets constitute a Parseval frame of the finite Euclidean space such that the inversion of the shearlet transform can be simply done by applying the adjoint transform.

for further informations see here.

# Fast Fourier Transform at Nonequispaced Nodes (NFFT)

Excerpt from http://www-user.tu-chemnitz.de/~potts/nfft/index.php (see there for further informations)

Fast Fourier transforms (FFTs) belong to the '10 algorithms with the greatest influence on the development and practice of science and engineering in the 20th century'. The classic algorithm computes the discrete Fourier transform

\[ f_j= \sum_{k=-\frac{N}{2}}^{\frac{N}{2}-1} \hat{f}_{k} {\rm e}^{2\pi{\rm i}\frac{kj}{N}} \]

for $j=-\frac{N}{2},\ldots,\frac{N}{2}-1$ and given complex coefficients $\hat{f}_{k}\in\mathbb{C}$. Using a divide and conquer approach, the number of floating point operations is reduced from ${\cal O}(N^2)$ for a straightforward computation to only ${\cal O}(N\log N)$. In conjunction with publicly available efficient implementations the fast Fourier transform has become of great importance in scientific computing.

However, two shortcomings of traditional schemes are the need for equispaced sampling and the restriction to the system of complex exponential functions. The NFFT is a C subroutine library for computing the nonequispaced discrete Fourier transform (NDFT) and its generalisations in one or more dimensions, of arbitrary input size, and of complex data.

More precisely,we collect the possible frequencies $\mathbf{k}\in\mathbb{Z}^d$ in the multi-index set

\[ I_{\mathbf{N}} := \left\{ \mathbf{k}=\left(k_t\right)_{t=0,\ldots,d-1} \in \mathbb{Z}^d: - \frac{N_t}{2} \le k_t < \frac{N_t}{2} ,\;t=0,\ldots,d-1\right\}, \]

where $\mathbf{N}=\left(N_t\right)_{t=0,\ldots,d-1}$ is the multibandlimit, i.e., $N_t\in 2\mathbb{N}$. For a finite number of given Fourier coefficients $\hat f_{\mathbf{k}} \in \mathbb{C}$, $\mathbf{k}\in I_{\mathbf{N}}$, we consider the fast evaluation of the trigonometric polynomial

\[ f\left(\mathbf{x}\right) := \sum_{ \mathbf{k}\in I_{ N}} \hat{f}_{\mathbf{ k}} {\rm e}^{-2\pi{\rm i}\mathbf{k}\mathbf{ x}} \]

at given nonequispaced nodes $\mathbf{x}_j \in \mathbb{T}^d$, $j=0,\ldots, M-1$, from the $d$-dimensional torus as well as the adjoint problem, the fast evaluation of sums of the form

\[ \hat h_{\mathbf{k}} := \sum_{j=0}^{M-1} {f}_{j} {\rm e}^{2\pi{\rm i}\mathbf{k}\mathbf{ x}_j}. \]

# Hue and Range Preserving RGB Image Enhancement (RGB-HP-ENHANCE)

Color image enhancement is a complex and challenging task in digital imaging with abundant applications. Preserving the hue of the input image is crucial in a wide range of situations. We propose a simple image enhancement algorithm which conserves the hue and preserves the range (gamut) of the R, G, B channels of an image in an optimal way. With the Matlab toolbox RGB-HP-ENHANCE we offer a simple to use implementation of the described enhancement. A method is incorporated to handle connected image areas having nearly the same intensity value, so-called ,,large pixel areas''.

Developed by Sören Häuser, Mila Nikolova and Gabriele Steidl. For further information see here.

# Manifold-valued Image Restoration Toolbox

In many application data is nonlinear, i.e. restricted in a certain range and equipped with a different distance measure. For example measuring angles in InSAR imaging or when working on the phase of complex valued wavelets. Other applications include denoising in several color spaces like RGB, HSV and CB. These data live on the circle \(\mathbb S^1\), the sphere \(\mathbb S^2\) and vector spaces of combined real valued and phase valued data \((\mathbb S^1)^m\times\mathbb R^n\). Furthermore, in Diffusion Tensor Imaging (DTI) every data item of an image is given by an \(r\times r\) symmetric positive definite matrix, i.e. from the space \(\mathrm{Sym}(r)\). Often, all these data are obstructed by noise due to measurement or data transfer.

All mentioned spaces are (products of) manifold(s). A very common model for denoising is the well known ROF-model of TV denoising, which was recently generalised to manifolds and has several generalisations itself to overcome the well known stair casing effect.

This package provides an easy-to-use Toolbox for processing manifold valued data. Several examples illustrate and explain the usage of the Toolbox.

for further informations see here.