2-D wedgelet transform using top-down quadtree decomposition

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2-D wedgelet transform using top-down quadtree decomposition

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2-D wedgelet transform is a powerful technique in image processing that aims to represent images efficiently by partitioning them into regions with distinct geometric features. The method employs a wedgelet function—typically a piecewise constant function defined over a wedge-shaped region—to approximate local image content.

The top-down quadtree decomposition serves as the underlying framework for this transform. It starts by considering the entire image as a single block and recursively subdivides it into four quadrants. At each subdivision level, the algorithm evaluates whether a wedgelet approximation adequately represents the current block. If the approximation error exceeds a predefined threshold, the block is further divided into smaller quadrants, enabling finer geometric detail capture.

This adaptive approach ensures computational efficiency by focusing refinements only on areas with complex structures (e.g., edges or textures) while coarsely approximating homogeneous regions. The resulting hierarchical representation balances accuracy and compactness, making it useful for applications like image compression, denoising, and edge detection.

A key advantage lies in the transform's ability to preserve geometric features with fewer coefficients compared to traditional wavelets, especially for images dominated by smooth regions separated by sharp edges. The top-down strategy also simplifies implementation, as decisions are made locally at each tree node without requiring global optimization.