Wavelet transform

The wavelet transform is a mathematical tool that decomposes a signal into components at different scales, using basis functions called wavelets that are localized in both time and frequency. Unlike the Fourier transform, which provides only frequency information and assumes the signal is stationary, the wavelet transform captures how frequency content evolves across the signal's duration. This makes it particularly suited to analyzing transient phenomena, edges, and multiscale structure — the properties that characterize natural images.

In image compression, the discrete wavelet transform (DWT) is the basis of JPEG 2000, replacing the discrete cosine transform used in JPEG. The DWT decomposes an image into approximation subbands and detail subbands at multiple resolutions, enabling progressive transmission and scalable quality without the block artifacts that plague DCT-based methods. The wavelet approach treats the image as a multiscale signal rather than a mosaic of independent blocks, a conceptual shift that mirrors the hierarchical organization of visual processing in the early cortex.

Wavelet transforms also appear in signal processing, denoising, and multiresolution analysis, where their ability to localize singularities makes them superior to Fourier methods for detecting edges and discontinuities. The mathematical foundation was laid by Ingrid Daubechies and others in the 1980s, though the roots reach back to Fourier analysis and the Calderón reproducing formula.

The wavelet transform is not merely a better way to decompose images. It is a different ontology of what an image is: not a grid of pixels but a multiscale field of singularities. The fact that JPEG 2000 — a wavelet-based standard — failed to displace the block-based JPEG suggests that our image infrastructure is committed to a pixel-grid ontology that wavelet methods fundamentally challenge.