Efficient segmentation based on Eikonal and diffusion equations

Abstract

Segmentation of regions of interest in an image has important applications in medical image analysis, particularly in computer aided diagnosis. Segmentation can enable further quantitative analysis of anatomical structures. We present efficient image segmentation schemes based on the solution of distinct partial differential equations (PDEs). For each known image region, a PDE is solved, the solution of which locally represents the weighted distance from a region known to have a certain segmentation label. To achieve this goal, we propose the use of two separate PDEs, the Eikonal equation and a diffusion equation. In each method, the segmentation labels are obtained by a competition criterion between the solutions to the PDEs corresponding to each region. We discuss how each method applies the concept of information propagation from the labelled image regions to the unknown image regions. Experimental results are presented on magnetic resonance, computed tomography, and ultrasound images and for both two-region and multi-region segmentation problems. These results demonstrate the high level of efficiency as well as the accuracy of the proposed methods.

Keywords:

• Computer vision
• Image processing
• Pattern recognition
• Variational methods; Segmentation

Acknowledgements

We thank Dr. M. Harisinghani and Dr. R. Weissleder, Massachusetts General Hospital (MGH), Boston, and Dr. J. Barentsz, University Medical Center, Nijmegen, Netherlands, for clinical motivation, feedback and providing data, and Dr. R. Seethamraju for discussions and Dr. R. Krieg, Siemens Medical Solution, for support of this work.

Notes

^†The implementation of this technique does not require the use of limits or a specific choice of ϵ since we can implement the locally infinite travel cost present at edges by simply not allowing information to propagate with the fast marching algorithm.

^†Note that, for consistency with the previous sections, we refer to these functions as distance functions; however, they more closely resemble heat functions as is typically the case with such diffusion equations.