Integrating Principal Component Analysis and Deep Learning Methods for Data Representation and Image Denoising

The present thesis studies dimensionality reduction and denoising with a focus on Principal Component Analysis (PCA) and its nonlinear counterparts. The main aim is to review the PCA machinery and assess its effectiveness on real data and images. Two prototype tasks and a comparative denoising study are considered. First, PCA is derived via standardization, covariance estimation, and eigenvalue, eigenvector decomposition; component selection is guided by scree plots and cumulative variance. We demonstrate the PCA using the Wine Dataset (178 samples), that provides outcomes from a chemical assessment of wines produced in the same Italian region but originating from three distinct grape varieties. The study measured the levels of 13 different components present in all three wine categories. We show that the first three components capture about two-thirds of the variance and preserve class-relevant structure, as confirmed by DBSCAN and K-Means clustering in the reduced space. To illustrate the properties of PCA, we consider a classical problem of image denoising where four denoisers, including linear PCA, kernel PCA, a shallow autoencoder, and a deep learning model based on convolutional layers (CNN) are compared on the classical USPS digits dataset. Performance is assessed through metrics such as mean squared error (MSE) and peak signal to noise ratio (PSNR). Results show that PCA provides a strong, interpretable baseline; kernel PCA improves smoothness but is sensitive to kernel choice; autoencoders offer flexible nonlinear reconstructions with higher tuning demands; and CNNs deliver the best image quality at greater computational cost. Overall, PCA emerges as a robust, efficient foundation for exploration and preprocessing, while modern CNNs are preferred when highest denoising fidelity is required.

Name thesis:degree_name

Master of Science

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2025

Author dc:creator

• Akhtar, Naveed

Advisor dc:contributor.advisor

• Labate, Demetrio

Committee members dc:contributor.committeemember

• Khurana, Seema
• Ott, William
• Quaini, Annalisa

• Convolutional Neural Networks (CNNs)
• Image Denoising
• Dimensionality Reduction
• Principal Component Analysis (PCA)
• Wine Dataset
• Eigenvalues and Eigenvectors
• Kernel PCA
• Covariance Matrix
• Autoencoders
• MSE
• K-Means
• DBSCAN
• Clustering
• PSNR

Language dc:language.iso

English

Handle dc:identifier.uri

https://hdl.handle.net/10657/20826

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/20826