EIGENVALUE-BASED DIMENSIONALITY REDUCTION IN IMAGE COMPRESSION: MATHEMATICAL ANALYSIS AND EFFECTS ON IMAGE QUALITY

Abstract

Digital image storage and transmission is demanding, and lossy compression reduces this demand by removing perceptually redundant data. Principal Component Analysis is one of the most established linear tools for this purpose but the way in which the number of retained principal components influences measurable image quality is often discussed only qualitatively. In this work we present the mathematical foundations of eigenvalue-based dimensionality reduction for image compression and analyse the effects of the number of components on reconstruction fidelity.

The work formalises the compression pipeline from the image being represented as a matrix, through centring the data, constructing the covariance matrix and eigen decomposition, to component selection and reconstruction. It is shown that the squared reconstruction error is the sum of the discarded eigenvalues and the explained variance ratio is the fraction of the total variance kept by the first k components. We use a greyscale experimental setting, and resize images into a fixed 256 × 256 grid and process them row-wise, where rows are observations and columns are features. Compression performance is quantified using mean squared error, peak signal to noise ratio, structural similarity, explained variance and a theoretical storage ratio for a variety of component numbers.

The analysis shows that the reconstruction quality increases monotonically with the number of components, but with diminishing returns, so the relationship is a rate-quality curve rather than a single operating point. The result strongly depends on the shape of the eigenvalue spectrum: images with steep spectra can tolerate aggressive truncation, while images with flatter spectra cannot. Compression only happens when the number of components is below the break-even point defined by the stored scores, eigenvectors and mean vector. The study concludes that the appropriate component budget is application specific, and recommends selecting the smallest component count that meets both a cumulative explained variance target and a plateaued structural similarity score.