# Pottslab

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)

## Application examples

### Segmentation of vector-valued images

• 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

### Joint image reconstruction and segmentation

• 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

### Denoising of jump-sparse/piecewise-constant signals, or step detection/changepoint detection

• 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)