The Mumford-Shah Functional for Image Segmentation Applied to Landscape Planning: A Comparison of Numerical Approximation Methods

This study investigates the application of the Mumford-Shah functional, a foundational variational model in image segmentation, to the challenge of Land Use/Land Cover (LuLc) mapping within an Object-Based Image Analysis (OBIA) framework. Recognizing that no single segmentation algorithm is universally optimal, the research focuses on comparing two distinct numerical approximations to assess their suitability for processing satellite imagery. The methodological approach involved the systematic evaluation of two algo-rithms: the Ambrosio-Tortorelli method and the Active Contour Snake model. To ensure controlled conditions, a series of synthetic test images were created, featuring basic geo-metric shapes representing common landscape features. These images were designed across multiple scenarios, ranging from those with clean, high-contrast edges to more challenging cases with low contrast and introduced intra-class noise, simulating re-al-world complexities. The performance of each algorithm was rigorously measured using established statistical metrics, namely Cohen's Kappa and the Jaccard Index, to quantify segmentation accuracy against a known ground truth. The findings reveal a clear distinc-tion in algorithmic behavior. While both methods achieved high accuracy in ideal, high-contrast conditions, their performance diverged significantly under stress. The Am-brosio-Tortorelli algorithm proved notably more robust, effectively maintaining closed and coherent object boundaries even in the presence of noise and low spectral contrast. Conversely, the Snake model was highly sensitive to these conditions, often resulting in fragmented contours or complete failure to delineate objects. In conclusion, this compara-tive analysis demonstrates that the choice between these two approaches is not arbitrary but critically dependent on the nature of the input data. The study provides practical guidance, suggesting that the global, variational approach of Ambrosio-Tortorelli is better suited for the noisy and spectrally complex scenes often encountered in territorial analy-sis. Meanwhile, the Snake model may be reserved for more controlled scenarios with sharp, well-defined edges. This work thus contributes a reasoned framework for algorithm selection, aiming to enhance the precision and reliability of segmentation in sustainable landscape monitoring workflows.