multiresolution

Multiresolution refers to methods and representations that analyze data at multiple levels of detail or scale. It is widely used in signal processing, image analysis, and data compression to capture both coarse structure and fine details.

In mathematics, multiresolution analysis (MRA) provides a framework for building hierarchies of function spaces that correspond to increasing levels of resolution. The standard approach uses a scaling function and a wavelet. One constructs a sequence of nested subspaces V_j such that V_{j+1} = V_j ⊕ W_j, where W_j is the detail space. Each level j represents a resolution, with coarser approximations in V_j and details in W_j.

Practical implementations rely on filter banks and pyramid schemes. In images, Gaussian or Laplacian pyramids generate a sequence of blurred, downsampled images; difference images capture details. The discrete wavelet transform provides an efficient, compact multi-resolution decomposition.

Applications include image compression (JPEG 2000), denoising, feature detection, audio processing, seismic data interpretation, and pattern recognition. Multiresolution representations are often sparse for natural signals, enabling efficient storage and processing.

Advantages: enables analysis across scales, robust to noise, reduces computational load by working with coarse representations. Challenges: selecting appropriate basis functions, boundary handling, drift and artifacts, and balancing redundancy and efficiency.

Related concepts include scale-space theory and directional multi-resolution methods such as curvelets and shearlets, which extend multiresolution to better capture edges and textures.