Computational harmonic analysis for image compression

Digital images consist of two‑dimensional arrays of pixel values, and storing such arrays without compression quickly becomes prohibitive. Harmonic analysis offers a mathematical framework for representing images by superpositions of basic functions that capture global and local features. This paper explores the use of computational harmonic analysis in image compression. Classical Fourier representations capture global frequency content but lack spatial localization, while wavelet and related multiresolution approaches provide time–frequency localization that is ideal for images. The challenges addressed include balancing compression ratio and perceptual quality, dealing with high‑dimensional data and edge artefacts, and designing computationally efficient transforms. The methodology combines Fourier, discrete cosine and wavelet transforms, energy compaction measures and coefficient quantization to produce compact codes. A combination of transform coding and entropy coding is used to exploit sparsity in the transformed domain. Results on standard test images show that wavelet-based compression achieves higher compression ratios and peak signal‑to‑noise ratios than traditional Fourier methods. The outcomes highlight that harmonic analysis tools enable compact representation of image information and open avenues for further hybrid and adaptive schemes