روشهای ماتریکسی الجبر خطی در تصاویر دیجیتالی: مرور روایتی
DOI:
https://doi.org/10.67753/frsj.v3i2.214Keywords:
الجبر خطی، تصاویر دیجیتالی، تحلیل مؤلفههای اصلی، تجزیه مقادیر منفرد، روشهای ماتریکسیAbstract
Matrix-Based Linear Algebra Methods in Digital Images: A Narrative Review
Teaching Assistant Ahmad Wali Noori1*, Senior Teaching Assistant Farhad Hakimi2, Teaching Assistant Mohammad Yazdan Parast3
Abstract
Digital images require structured numerical representations for computational processing and analysis. This study aims to examine selected matrix-based methods of linear algebra for the representation, formulation of operations, and structural analysis of digital image data. The study was conducted as a narrative review, in which relevant scholarly sources published between 2016 and 2024 were identified and examined through a targeted literature search. The review focuses on three main themes: matrix representation of digital images, principal component analysis (PCA), and singular value decomposition (SVD), while also considering their related applications in image processing and analysis. The reviewed literature indicates that these concepts have distinct but complementary roles, and that the selection of an appropriate method depends on the nature of the problem, image structure, error criterion, and available computational resources. When the image formation or degradation process can be modeled as a linear operation, image reconstruction may be formulated as a system of linear equations or a least-squares problem, in which (SVD) can contribute to rank analysis, pseudoinverse construction, and least-squares analysis. However, low-rank approximation and image reconstruction from degraded observations are not identical problems, and minimizing numerical error under a specified criterion does not necessarily guarantee superior visual quality or practical performance. No programming implementation or empirical evaluation was conducted in the present study. Future research is recommended to evaluate the reviewed methods on well-defined image datasets in terms of reconstruction quality, computational time, and memory consumption.
Keywords: Digital images; Linear algebra; Matrix-based methods; Principal component analysis; Singular value decomposition.
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