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doi: 10.3934/ipi.2020048

## Reproducible kernel Hilbert space based global and local image segmentation

 1 Centre for Mathematical Imaging Techniques and Department of Mathematical Sciences, University of Liverpool, United Kingdom 2 Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve, University, Cleveland, OH 44106, USA 3 Liverpool Vascular & Endovascular Service, Royal Liverpool and Broadgreen University, Hospitals NHS Trust, Liverpool, L7 8XP, United Kingdom

Received  December 2019 Revised  May 2020 Published  August 2020

Image segmentation is the task of partitioning an image into individual objects, and has many important applications in a wide range of fields. The majority of segmentation methods rely on image intensity gradient to define edges between objects. However, intensity gradient fails to identify edges when the contrast between two objects is low. In this paper we aim to introduce methods to make such weak edges more prominent in order to improve segmentation results of objects of low contrast. This is done for two kinds of segmentation models: global and local. We use a combination of a reproducing kernel Hilbert space and approximated Heaviside functions to decompose an image and then show how this decomposition can be applied to a segmentation model. We show some results and robustness to noise, as well as demonstrating that we can combine the reconstruction and segmentation model together, allowing us to obtain both the decomposition and segmentation simultaneously.

Citation: Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, doi: 10.3934/ipi.2020048
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Comparison of Geodesic and Euclidean distance constraints
Left panel: 3D surface images of $\psi$ of nine parameter pairs $(\theta, c)$; right panel: the corresponding 2D images. In each panel, from left to right and then from top to bottom: $( \frac{4 \pi}{5} , \frac{51}{1024} ), ( \frac{4\pi}{5} , \frac{25}{64} ), ( \frac{4\pi}{5} , \frac{175}{256} ), ( \frac{6\pi}{5}, \frac{135}{1024} ), ( \frac{6\pi}{5}, \frac{1}{2} ), ( \frac{6\pi}{5}, \frac{25}{32} ), ( \frac{8\pi}{5}, \frac{5}{64} )$, $( \frac{8\pi}{5}, \frac{75}{256} ), ( \frac{8\pi}{5}, \frac{75}{128} )$
The decomposition of the input image (top left), into the smooth parts (top middle) and edge parts (top right), obtained from RKHS model (8), and edge indicator function $g(\Psi \beta)$ (bottom left). In comparison, we display the gradient of the image, $|\nabla z|$ and the edge indicator function $g(|\nabla z|)$ in the bottom middle and bottom right respectively
Segmentation of white matter from MRI brain image. We show both the contour overlaid on the original image, and the associated binary segmentation result for each example. First column: segmentation using edge weighted Chan-Vese using $|\nabla z|$ model (24). Second column: Edge weighted Chan-Vese model (24) using Canny ($g = g_{\mathcal{C}}$). Third column: Cai et al. [9]. Fourth column: Proposed two stage global model (8) and (25). Final column: The proposed combined global segmentation model (26)
Row 1: 3% noise, Row 2: 5% noise, Row 3: 7% noise, Row 4: 9% noise. We show both the contour overlaid on the original image, and the associated binary segmentation result for each example. First two columns: Using model (24). Final two columns: Combined model, algorithm (1)
Geodesic distances computed using gradient (middle) and $\Psi \beta$ (right)
Row one: Original clean image and the user input. Row two: Image and result with 20% noise. Row three: Image and result with 40% noise. Row four: Image and result with 60% noise
Segmentation of blood vessel
Segmentation of blood vessel
Result of the combined selective model
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