Pottslab is a toolbox for **jump-sparse reconstruction based on the Potts model. **

The **Potts model** is given by

\[ u^* = \arg\min_u \gamma \| \nabla u\|_0 + \| Au - f\|_p^p,\] where \(f \) is noisy data and \(A\) a linear operator.

- the regularizing term \(\| \nabla u\|_0 \) enforces
**jump-sparse minimizers,** - \(\gamma > 0\) is an empirical parameter which controls the balance between the regularizing term and the data term \(\| Au - f\|_p^p\), (\(p = 1\) or \(p = 2\) )
- also known as piecewise constant Mumford-Shah model (recently also called l0-gradient model)

- Supports segmentation of vector-valued images (e.g. multispectral images, feature images)
- Linear complexity in number of color channels
- Label-free: No label discretization required

Left: A natural image; Right: Result using Potts model

Texture segmentation using highdimensional curvelet-based feature vectors

- Applicable to many imaging operators, e.g. convolution, Radon transform, MRI, PET, MPI: only implementation of proximal mapping reuqired
- Supports vector-valued data
- Label-free: Labels need NOT be chosen in advance

Left: Shepp-Logan phantom; Center: Filtered backprojection from 7 angular projections; Right: Joint reconstruction and segmentation using the Potts model from 7 angular projections

Left: Blurred noisy image; Right: Joint deconvolution and segmentation using the Potts model

- L1 Potts model is robust to noise and to moderately blurred data
- Fast and exact solver for L1 Potts model
- Approximative strategies for severely blurred data

Top: Noisy signal; Bottom: Minimizer of Potts functional (ground truth in red)

- Pottslab 0.5 (Java-based with Matlab frontend)
- Icy plugin - an interactive image segmentation plugin based on Pottslab (written by Vasileios Angelopoulos)
- ImageJ plugin - an ImageJ frontend for Pottslab (written by Michael Kaul)

- M. Storath, A. Weinmann, M. Unser. Jump-penalized least absolute values estimation of scalar or circle-valued signals. Information and Inference, to appear
- A. Weinmann, M. Storath, L. Demaret. The L1-Potts functional for robust jump-sparse reconstruction. SIAM Journal on Numerical Analysis, 53(1):644-673, 2015

- M. Storath, A. Weinmann, J. Frikel, M. Unser. Joint image reconstruction and segmentation using the Potts model.
*Inverse Problems, 31(2):025003, 2015*

- A. Weinmann, M. Storath. Iterative Potts and Blake-Zisserman minimization for the recovery of functions with discontinuities from indirect measurements. Proceedings of The Royal Society A, 471(2176), 2015
- M. Storath, A. Weinmann, M. Unser. Unsupervised texture segmentation using monogenic curvelets and the Potts model. Proceedings of the IEEE International Conference on Image Processing (ICIP), 2014
- M. Storath, A. Weinmann. Fast partitioning of vector-valued images.
*SIAM Journal on Imaging Sciences, 7(3):1826-1852, 2014* - M. Storath, A. Weinmann, L. Demaret. Jump-sparse and sparse recovery using Potts functionals. IEEE Transactions on Signal Processing, 62(14):3654-3666, 2014

- Main developers: Martin Storath, Andreas Weinmann
- Contributors: Vasileios Angelopoulos, Laurent Demaret, Jürgen Frikel, Kilian Hohm, Michael Kaul